Cheet Sheet

CS 229 – Machine Learning https://stanford.
edu/~shervine
Super VIP Cheatsheet: Machine Learning 4 Machine Learning Tips and Tricks 10
4.1 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.1.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Afshine Amidi and Shervine Amidi 4.1.2 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
September 15, 2018 4.3 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Refreshers 12
5.1 Probabilities and Statistics . . . . . . . . . . . . . . . . . . . . . . . . 12
Contents 5.1.1 Introduction to Probability and Combinatorics . . . . . . . . . 12
5.1.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . 12
1 Supervised Learning 2 5.1.3 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1 Introduction to Supervised Learning . . . . . . . . . . . . . . . . . . . 2 5.1.4 Jointly Distributed Random Variables . . . . . . . . . . . . . . 13
1.2 Notations and general concepts . . . . . . . . . . . . . . . . . . . . . 2 5.1.5 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Linear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.2 Linear Algebra and Calculus . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 Linear regression . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.2.1 General notations . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2.2 Matrix operations . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Classification and logistic regression . . . . . . . . . . . . . . . 3
5.2.3 Matrix properties . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Generalized Linear Models . . . . . . . . . . . . . . . . . . . . 3
5.2.4 Matrix calculus . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Support Vector Machines . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Generative Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5.1 Gaussian Discriminant Analysis . . . . . . . . . . . . . . . . . 4
1.5.2 Naive Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Tree-based and ensemble methods . . . . . . . . . . . . . . . . . . . . 4
1.7 Other non-parametric approaches . . . . . . . . . . . . . . . . . . . . 4
1.8 Learning Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Unsupervised Learning 6
2.1 Introduction to Unsupervised Learning . . . . . . . . . . . . . . . . . 6
2.2 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Expectation-Maximization . . . . . . . . . . . . . . . . . . . . 6
2.2.2 k-means clustering . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.3 Hierarchical clustering . . . . . . . . . . . . . . . . . . . . . . 6
2.2.4 Clustering assessment metrics . . . . . . . . . . . . . . . . . . 6
2.3 Dimension reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 Principal component analysis . . . . . . . . . . . . . . . . . . 7
2.3.2 Independent component analysis . . . . . . . . . . . . . . . . . 7
3 Deep Learning 8
3.1 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Convolutional Neural Networks . . . . . . . . . . . . . . . . . . . . . 8
3.3 Recurrent Neural Networks . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Reinforcement Learning and Control . . . . . . . . . . . . . . . . . . . 9
Stanford University 1 Fall 2018

CS 229 – Machine Learning Shervine Amidi & Afshine Amidi
1 Supervised Learning r Cost function – The cost function J is commonly used to assess the performance of a model,
and is defined with the loss function L as follows:
1.1 Introduction to Supervised Learning m
X
J(θ) = L(hθ (x(i) ), y (i) )
Given a set of data points {x(1) , ..., x(m) } associated to a set of outcomes {y (1) , ..., y (m) }, we
want to build a classifier that learns how to predict y from x. i=1
r Type of prediction – The different types of predictive models are summed up in the table
below: r Gradient descent – By noting α ∈ R the learning rate, the update rule for gradient descent
is expressed with the learning rate and the cost function J as follows:
Regression Classifier
θ ←− θ − α∇J(θ)
Outcome Continuous Class
Examples Linear regression Logistic regression, SVM, Naive Bayes
r Type of model – The different models are summed up in the table below:
Discriminative model Generative model

Goal Directly estimate P (y|x) Estimate P (x|y) to deduce P (y|x)
What’s learned Decision boundary Probability distributions of the data
Remark: Stochastic gradient descent (SGD) is updating the parameter based on each training
example, and batch gradient descent is on a batch of training examples.
Illustration r Likelihood – The likelihood of a model L(θ) given parameters θ is used to find the optimal
parameters θ through maximizing the likelihood. In practice, we use the log-likelihood `(θ) =
log(L(θ)) which is easier to optimize. We have:
θopt = arg max L(θ)

Examples Regressions, SVMs GDA, Naive Bayes θ
1.2 Notations and general concepts r Newton’s algorithm – The Newton’s algorithm is a numerical method that finds θ such
that `0 (θ) = 0. Its update rule is as follows:
r Hypothesis – The hypothesis is noted hθ and is the model that we choose. For a given input
`0 (θ)
data x(i) , the model prediction output is hθ (x(i) ). θ←θ−
`00 (θ)
r Loss function – A loss function is a function L : (z,y) ∈ R × Y 7−→ L(z,y) ∈ R that takes as
inputs the predicted value z corresponding to the real data value y and outputs how different Remark: the multidimensional generalization, also known as the Newton-Raphson method, has
they are. The common loss functions are summed up in the table below: the following update rule:
−1
Least squared Logistic Hinge Cross-entropy θ ← θ − ∇2θ `(θ) ∇θ `(θ)
1
(y − z)2 log(1 + exp(−yz)) max(0,1 − yz) − y log(z) + (1 − y) log(1 − z)
2 1.3 Linear models
1.3.1 Linear regression
We assume here that y|x; θ ∼ N (µ,σ 2 )

r Normal equations – By noting X the matrix design, the value of θ that minimizes the cost
function is a closed-form solution such that:
Linear regression Logistic regression SVM Neural Network θ = (X T X)−1 X T y

r LMS algorithm – By noting α the learning rate, the update rule of the Least Mean Squares Distribution η T (y) a(η) b(y)
(LMS) algorithm for a training set of m data points, which is also known as the Widrow-Hoff
learning rule, is as follows: φ

Bernoulli log 1−φ
y log(1 + exp(η)) 1
m
η2 2
X (i)
exp − y2

∀j, θ j ← θj + α y (i) − hθ (x(i) ) xj Gaussian µ y 2
√1
2π
i=1
1
Poisson log(λ) y eη
Remark: the update rule is a particular case of the gradient ascent. y!
r LWR – Locally Weighted Regression, also known as LWR, is a variant of linear regression that Geometric log(1 − φ) y log eη

1
1−eη
weights each training example in its cost function by w(i) (x), which is defined with parameter
τ ∈ R as:

(x(i) − x)2 r Assumptions of GLMs – Generalized Linear Models (GLM) aim at predicting a random
w(i) (x) = exp −
2τ 2 variable y as a function fo x ∈ Rn+1 and rely on the following 3 assumptions:
(1) y|x; θ ∼ ExpFamily(η) (2) hθ (x) = E[y|x; θ] (3) η = θT x

1.3.2 Classification and logistic regression Remark: ordinary least squares and logistic regression are special cases of generalized linear
models.
r Sigmoid function – The sigmoid function g, also known as the logistic function, is defined
as follows:
1 1.4 Support Vector Machines
∀z ∈ R, g(z) = ∈]0,1[
1 + e−z The goal of support vector machines is to find the line that maximizes the minimum distance to
the line.
r Logistic regression – We assume here that y|x; θ ∼ Bernoulli(φ). We have the following r Optimal margin classifier – The optimal margin classifier h is such that:
form:
h(x) = sign(wT x − b)
1
φ = p(y = 1|x; θ) = = g(θT x)
1 + exp(−θT x) where (w, b) ∈ Rn × R is the solution of the following optimization problem:
Remark: there is no closed form solution for the case of logistic regressions. 1
min ||w||2 such that y (i) (wT x(i) − b) > 1
2
r Softmax regression – A softmax regression, also called a multiclass logistic regression, is
used to generalize logistic regression when there are more than 2 outcome classes. By convention,
we set θK = 0, which makes the Bernoulli parameter φi of each class i equal to:
exp(θiT x)
φi =
K
X
exp(θjT x)
j=1
1.3.3 Generalized Linear Models

r Exponential family – A class of distributions is said to be in the exponential family if it can
be written in terms of a natural parameter, also called the canonical parameter or link function,
η, a sufficient statistic T (y) and a log-partition function a(η) as follows:
p(y; η) = b(y) exp(ηT (y) − a(η))
Remark: the line is defined as wT x − b = 0 .
Remark: we will often have T (y) = y. Also, exp(−a(η)) can be seen as a normalization param-
eter that will make sure that the probabilities sum to one. r Hinge loss – The hinge loss is used in the setting of SVMs and is defined as follows:
Here are the most common exponential distributions summed up in the following table:
L(z,y) = [1 − yz]+ = max(0,1 − yz)

r Kernel – Given a feature mapping φ, we define the kernel K to be defined as: 1.5.2 Naive Bayes
K(x,z) = φ(x)T φ(z) r Assumption – The Naive Bayes model supposes that the features of each data point are all
independent:
2
||x−z||
In practice, the kernel K defined by K(x,z) = exp − 2σ 2
is called the Gaussian kernel
n
and is commonly used. Y
P (x|y) = P (x1 ,x2 ,...|y) = P (x1 |y)P (x2 |y)... = P (xi |y)
i=1
r Solutions – Maximizing the log-likelihood gives the following solutions, with k ∈ {0,1},
l ∈ [[1,L]]
(j)
1 #{j|y (j) = k and xi = l}
P (y = k) = × #{j|y (j) = k} and P (xi = l|y = k) =
m #{j|y (j) = k}
Remark: we say that we use the "kernel trick" to compute the cost function using the kernel
because we actually don’t need to know the explicit mapping φ, which is often very complicated.
Instead, only the values K(x,z) are needed. Remark: Naive Bayes is widely used for text classification and spam detection.
r Lagrangian – We define the Lagrangian L(w,b) as follows:

l
X 1.6 Tree-based and ensemble methods
L(w,b) = f (w) + βi hi (w)
i=1 These methods can be used for both regression and classification problems.
r CART – Classification and Regression Trees (CART), commonly known as decision trees,
Remark: the coefficients βi are called the Lagrange multipliers. can be represented as binary trees. They have the advantage to be very interpretable.
r Random forest – It is a tree-based technique that uses a high number of decision trees
1.5 Generative Learning built out of randomly selected sets of features. Contrary to the simple decision tree, it is highly
uninterpretable but its generally good performance makes it a popular algorithm.
A generative model first tries to learn how the data is generated by estimating P (x|y), which
we can then use to estimate P (y|x) by using Bayes’ rule. Remark: random forests are a type of ensemble methods.
r Boosting – The idea of boosting methods is to combine several weak learners to form a
stronger one. The main ones are summed up in the table below:
1.5.1 Gaussian Discriminant Analysis
r Setting – The Gaussian Discriminant Analysis assumes that y and x|y = 0 and x|y = 1 are
such that: Adaptive boosting Gradient boosting
y ∼ Bernoulli(φ) - High weights are put on errors to - Weak learners trained
improve at the next boosting step on remaining errors
x|y = 0 ∼ N (µ0 ,Σ) and x|y = 1 ∼ N (µ1 ,Σ) - Known as Adaboost
r Estimation – The following table sums up the estimates that we find when maximizing the
likelihood:
1.7 Other non-parametric approaches
φ
b µbj (j = 0,1) Σ
b r k-nearest neighbors – The k-nearest neighbors algorithm, commonly known as k-NN, is a
non-parametric approach where the response of a data point is determined by the nature of its
k neighbors from the training set. It can be used in both classification and regression settings.
m Pm m
1 X 1
i=1 {y (i) =j}
x(i) 1 X
1{y(i) =1} m (x(i) − µy(i) )(x(i) − µy(i) )T
1 Remark: The higher the parameter k, the higher the bias, and the lower the parameter k, the
P
m m
i=1 i=1 {y (i) =j} i=1 higher the variance.

∃h ∈ H, ∀i ∈ [[1,d]], h(x(i) ) = y (i)
r Upper bound theorem – Let H be a finite hypothesis class such that |H| = k and let δ and
the sample size m be fixed. Then, with probability of at least 1 − δ, we have:
r
1 2k

(h) 6 min (h) + 2
b log
h∈H 2m δ
r VC dimension – The Vapnik-Chervonenkis (VC) dimension of a given infinite hypothesis

class H, noted VC(H) is the size of the largest set that is shattered by H.
Remark: the VC dimension of H = {set of linear classifiers in 2 dimensions} is 3.
1.8 Learning Theory

r Union bound – Let A1 , ..., Ak be k events. We have:
P (A1 ∪ ... ∪ Ak ) 6 P (A1 ) + ... + P (Ak )

r Theorem (Vapnik) – Let H be given, with VC(H) = d and m the number of training
examples. With probability at least 1 − δ, we have:
r
d m 1 1

h) 6
(b min (h) + O log + log
h∈H m d m δ
r Hoeffding inequality – Let Z1 , .., Zm be m iid variables drawn from a Bernoulli distribution
of parameter φ. Let φ
b be their sample mean and γ > 0 fixed. We have:
P (|φ − φ
b| > γ) 6 2 exp(−2γ 2 m)
Remark: this inequality is also known as the Chernoff bound.
r Training error – For a given classifier h, we define the training error b

(h), also known as the
empirical risk or empirical error, to be as follows:
m
1 X
b(h) = 1{h(x(i) )6=y(i) }
m
i=1
r Probably Approximately Correct (PAC) – PAC is a framework under which numerous

results on learning theory were proved, and has the following set of assumptions:
• the training and testing sets follow the same distribution
• the training examples are drawn independently
r Shattering – Given a set S = {x(1) ,...,x(d) }, and a set of classifiers H, we say that H shatters
S if for any set of labels {y (1) , ..., y (d) }, we have:

2 Unsupervised Learning 2.2.2 k-means clustering
2.1 Introduction to Unsupervised Learning We note c(i) the cluster of data point i and µj the center of cluster j.
r Algorithm – After randomly initializing the cluster centroids µ1 ,µ2 ,...,µk ∈ Rn , the k-means
r Motivation – The goal of unsupervised learning is to find hidden patterns in unlabeled data algorithm repeats the following step until convergence:
{x(1) ,...,x(m) }. m
X
r Jensen’s inequality – Let f be a convex function and X a random variable. We have the 1{c(i) =j} x(i)
following inequality: i=1
c(i) = arg min||x(i) − µj ||2 and µj = m
E[f (X)] > f (E[X]) j X
1{c(i) =j}
i=1
2.2 Clustering
2.2.1 Expectation-Maximization
r Latent variables – Latent variables are hidden/unobserved variables that make estimation
problems difficult, and are often denoted z. Here are the most common settings where there are
latent variables:
Setting Latent variable z x|z Comments
Mixture of k Gaussians Multinomial(φ) N (µj ,Σj ) µj ∈ Rn , φ ∈ Rk
Factor analysis N (0,I) N (µ + Λz,ψ) µj ∈ Rn

r Distortion function – In order to see if the algorithm converges, we look at the distortion
function defined as follows:
r Algorithm – The Expectation-Maximization (EM) algorithm gives an efficient method at m
estimating the parameter θ through maximum likelihood estimation by repeatedly constructing X
a lower-bound on the likelihood (E-step) and optimizing that lower bound (M-step) as follows: J(c,µ) = ||x(i) − µc(i) ||2
i=1
• E-step: Evaluate the posterior probability Qi (z (i) ) that each data point x(i) came from
a particular cluster z (i) as follows:
2.2.3 Hierarchical clustering
Qi (z (i) ) = P (z (i) |x(i) ; θ)
r Algorithm – It is a clustering algorithm with an agglomerative hierarchical approach that
build nested clusters in a successive manner.
• M-step: Use the posterior probabilities Qi (z (i) ) as cluster specific weights on data points
r Types – There are different sorts of hierarchical clustering algorithms that aims at optimizing
x(i) to separately re-estimate each cluster model as follows: different objective functions, which is summed up in the table below:
Xˆ
P (x(i) ,z (i) ; θ)

Ward linkage Average linkage Complete linkage
θi = argmax Qi (z (i) ) log dz (i)
θ z (i) Qi (z (i) ) Minimize within cluster Minimize average distance Minimize maximum distance
i
distance between cluster pairs of between cluster pairs
2.2.4 Clustering assessment metrics

In an unsupervised learning setting, it is often hard to assess the performance of a model since
we don’t have the ground truth labels as was the case in the supervised learning setting.
r Silhouette coefficient – By noting a and b the mean distance between a sample and all
other points in the same class, and between a sample and all other points in the next nearest
cluster, the silhouette coefficient s for a single sample is defined as follows:
b−a
s=
max(a,b)

r Calinski-Harabaz index – By noting k the number of clusters, Bk and Wk the between

and within-clustering dispersion matrices respectively defined as
k m
X X
Bk = nc(i) (µc(i) − µ)(µc(i) − µ)T , Wk = (x(i) − µc(i) )(x(i) − µc(i) )T
j=1 i=1
the Calinski-Harabaz index s(k) indicates how well a clustering model defines its clusters, such
that the higher the score, the more dense and well separated the clusters are. It is defined as
follows:
Tr(Bk ) N −k
s(k) = ×
Tr(Wk ) k−1 2.3.2 Independent component analysis
It is a technique meant to find the underlying generating sources.
2.3 Dimension reduction r Assumptions – We assume that our data x has been generated by the n-dimensional source
vector s = (s1 ,...,sn ), where si are independent random variables, via a mixing and non-singular
2.3.1 Principal component analysis matrix A as follows:
x = As
It is a dimension reduction technique that finds the variance maximizing directions onto which
to project the data. The goal is to find the unmixing matrix W = A−1 by an update rule.
r Eigenvalue, eigenvector – Given a matrix A ∈ Rn×n , λ is said to be an eigenvalue of A if
there exists a vector z ∈ Rn \{0}, called eigenvector, such that we have: r Bell and Sejnowski ICA algorithm – This algorithm finds the unmixing matrix W by
following the steps below:
Az = λz
• Write the probability of x = As = W −1 s as:
r Spectral theorem – Let A ∈ Rn×n . If A is symmetric, then A is diagonalizable by a real n
orthogonal matrix U ∈ Rn×n . By noting Λ = diag(λ1 ,...,λn ), we have:
Y
p(x) = ps (wiT x) · |W |
∃Λ diagonal, A = U ΛU T i=1
Remark: the eigenvector associated with the largest eigenvalue is called principal eigenvector of
matrix A. • Write the log likelihood given our training data {x(i) , i ∈ [[1,m]]} and by noting g the
sigmoid function as:
r Algorithm – The Principal Component Analysis (PCA) procedure is a dimension reduction
technique that projects the data on k dimensions by maximizing the variance of the data as m n
!
follows:
X X
0
l(W ) = log g (wjT x(i) ) + log |W |
• Step 1: Normalize the data to have a mean of 0 and standard deviation of 1. i=1 j=1
Therefore, the stochastic gradient ascent learning rule is such that for each training example
(i) m m
(i)
xj − µj 1 X (i) 1 X (i) x(i) , we update W as follows:
xj ← where µj = xj and σj2 = (xj − µj ) 2
σj m m
i=1 i=1
1 − 2g(w1T x(i) )
  
1 − 2g(w2 x ) (i) T
T (i)
W ←− W + α  .. x + (W T )−1 

m
1 X T .
• Step 2: Compute Σ = x(i) x(i) ∈ Rn×n , which is symmetric with real eigenvalues.
m 1 − 2g(wn T x(i) )
i=1
• Step 3: Compute u1 , ..., uk ∈ Rn the k orthogonal principal eigenvectors of Σ, i.e. the

orthogonal eigenvectors of the k largest eigenvalues.
• Step 4: Project the data on spanR (u1 ,...,uk ). This procedure maximizes the variance
among all k-dimensional spaces.

3 Deep Learning
∂L(z,y) ∂L(z,y) ∂a ∂z
= × ×
3.1 Neural Networks ∂w ∂a ∂z ∂w
Neural networks are a class of models that are built with layers. Commonly used types of neural As a result, the weight is updated as follows:
networks include convolutional and recurrent neural networks. ∂L(z,y)
w ←− w − η
r Architecture – The vocabulary around neural networks architectures is described in the ∂w
figure below:
r Updating weights – In a neural network, weights are updated as follows:
• Step 1: Take a batch of training data.
• Step 2: Perform forward propagation to obtain the corresponding loss.
• Step 3: Backpropagate the loss to get the gradients.
• Step 4: Use the gradients to update the weights of the network.

By noting i the ith layer of the network and j the j th hidden unit of the layer, we have:
[i] [i] T [i]
zj = wj x + bj r Dropout – Dropout is a technique meant at preventing overfitting the training data by
dropping out units in a neural network. In practice, neurons are either dropped with probability
p or kept with probability 1 − p.
where we note w, b, z the weight, bias and output respectively.
r Activation function – Activation functions are used at the end of a hidden unit to introduce
non-linear complexities to the model. Here are the most common ones: 3.2 Convolutional Neural Networks
r Convolutional layer requirement – By noting W the input volume size, F the size of the
Sigmoid Tanh ReLU Leaky ReLU convolutional layer neurons, P the amount of zero padding, then the number of neurons N that
fit in a given volume is such that:
1 ez − e−z
g(z) = g(z) = g(z) = max(0,z) g(z) = max(z,z) W − F + 2P
1 + e−z ez + e−z N = +1
with 1 S
r Batch normalization – It is a step of hyperparameter γ, β that normalizes the batch {xi }.

By noting µB , σB
2 the mean and variance of that we want to correct to the batch, it is done as
follows:
xi − µB
xi ←− γ p +β
2 +
σB
It is usually done after a fully connected/convolutional layer and before a non-linearity layer and
aims at allowing higher learning rates and reducing the strong dependence on initialization.
r Cross-entropy loss – In the context of neural networks, the cross-entropy loss L(z,y) is
commonly used and is defined as follows: 3.3 Recurrent Neural Networks
h i
L(z,y) = − y log(z) + (1 − y) log(1 − z) r Types of gates – Here are the different types of gates that we encounter in a typical recurrent
neural network:
Input gate Forget gate Output gate Gate

r Learning rate – The learning rate, often noted η, indicates at which pace the weights get
updated. This can be fixed or adaptively changed. The current most popular method is called Write to cell or not? Erase a cell or not? Reveal a cell or not? How much writing?
Adam, which is a method that adapts the learning rate.
r Backpropagation – Backpropagation is a method to update the weights in the neural network
by taking into account the actual output and the desired output. The derivative with respect r LSTM – A long short-term memory (LSTM) network is a type of RNN model that avoids
to weight w is computed using chain rule and is of the following form: the vanishing gradient problem by adding ’forget’ gates.

3.4 Reinforcement Learning and Control r Maximum likelihood estimate – The maximum likelihood estimates for the state transition
probabilities are as follows:
The goal of reinforcement learning is for an agent to learn how to evolve in an environment.
#times took action a in state s and got to s0
Psa (s0 ) =
r Markov decision processes – A Markov decision process (MDP) is a 5-tuple (S,A,{Psa },γ,R) #times took action a in state s
where:
• S is the set of states r Q-learning – Q-learning is a model-free estimation of Q, which is done as follows:
• A is the set of actions

h i
Q(s,a) ← Q(s,a) + α R(s,a,s0 ) + γ max Q(s0 ,a0 ) − Q(s,a)
a0
• {Psa } are the state transition probabilities for s ∈ S and a ∈ A
• γ ∈ [0,1[ is the discount factor
• R : S × A −→ R or R : S −→ R is the reward function that the algorithm wants to

maximize
r Policy – A policy π is a function π : S −→ A that maps states to actions.

Remark: we say that we execute a given policy π if given a state s we take the action a = π(s).
r Value function – For a given policy π and a given state s, we define the value function V π
as follows:
h i
V π (s) = E R(s0 ) + γR(s1 ) + γ 2 R(s2 ) + ...|s0 = s,π
∗
r Bellman equation – The optimal Bellman equations characterizes the value function V π
of the optimal policy π ∗ :
∗ ∗
X
V π (s) = R(s) + max γ Psa (s0 )V π (s0 )
a∈A
s0 ∈S
Remark: we note that the optimal policy π ∗ for a given state s is such that:
X
π ∗ (s) = argmax Psa (s0 )V ∗ (s0 )
a∈A
s0 ∈S
r Value iteration algorithm – The value iteration algorithm is in two steps:
• We initialize the value:
V0 (s) = 0
• We iterate the value based on the values before:

" #
X
0 0
Vi+1 (s) = R(s) + max γPsa (s )Vi (s )
a∈A
s0 ∈S

4 Machine Learning Tips and Tricks Metric Formula Equivalent

TP
True Positive Rate Recall, sensitivity
4.1 Metrics TP + FN
TPR
Given a set of data points {x(1) , ..., x(m) }, where each x(i) has n features, associated to a set of FP
outcomes {y (1) , ..., y (m) }, we want to assess a given classifier that learns how to predict y from False Positive Rate 1-specificity
x. TN + FP
FPR
4.1.1 Classification r AUC – The area under the receiving operating curve, also noted AUC or AUROC, is the
area below the ROC as shown in the following figure:
In a context of a binary classification, here are the main metrics that are important to track to
assess the performance of the model.
r Confusion matrix – The confusion matrix is used to have a more complete picture when
assessing the performance of a model. It is defined as follows:
Predicted class
+ –
TP FN
+ False Negatives
True Positives
Type II error
Actual class 4.1.2 Regression
FP TN
– False Positives r Basic metrics – Given a regression model f , the following metrics are commonly used to
True Negatives assess the performance of the model:
Type I error
Total sum of squares Explained sum of squares Residual sum of squares
r Main metrics – The following metrics are commonly used to assess the performance of
classification models: m
X m
X m
X
SStot = (yi − y)2 SSreg = (f (xi ) − y)2 SSres = (yi − f (xi ))2
i=1 i=1 i=1
Metric Formula Interpretation
TP + TN
Accuracy Overall performance of model r Coefficient of determination – The coefficient of determination, often noted R2 or r2 ,
TP + TN + FP + FN
provides a measure of how well the observed outcomes are replicated by the model and is defined
TP as follows:
Precision How accurate the positive predictions are
TP + FP SSres
R2 = 1 −
TP SStot
Recall Coverage of actual positive sample
TP + FN
Sensitivity r Main metrics – The following metrics are commonly used to assess the performance of
TN regression models, by taking into account the number of variables n that they take into consid-
Specificity Coverage of actual negative sample eration:
TN + FP
2TP Adjusted R2
F1 score Hybrid metric useful for unbalanced classes Mallow’s Cp AIC BIC
2TP + FP + FN
SSres + 2(n + 1)b
σ2 (1 − R2 )(m − 1)
2 (n + 2) − log(L) log(m)(n + 2) − 2 log(L) 1−
m m−n−1
r ROC – The receiver operating curve, also noted ROC, is the plot of TPR versus FPR by
varying the threshold. These metrics are are summed up in the table below: where L is the likelihood and b
σ 2 is an estimate of the variance associated with each response.

4.2 Model selection LASSO Ridge Elastic Net

- Shrinks coefficients to 0 Makes coefficients smaller Tradeoff between variable
r Vocabulary – When selecting a model, we distinguish 3 different parts of the data that we
have as follows: - Good for variable selection selection and small coefficients
Training set Validation set Testing set

- Model is trained - Model is assessed - Model gives predictions
- Usually 80% of the dataset - Usually 20% of the dataset - Unseen data
- Also called hold-out
or development set
Once the model has been chosen, it is trained on the entire dataset and tested on the unseen
test set. These are represented in the figure below: h i
... + λ||θ||1 ... + λ||θ||22 ... + λ (1 − α)||θ||1 + α||θ||22
λ∈R λ∈R λ ∈ R, α ∈ [0,1]
r Model selection – Train model on training set, then evaluate on the development set, then
r Cross-validation – Cross-validation, also noted CV, is a method that is used to select a pick best performance model on the development set, and retrain all of that model on the whole
model that does not rely too much on the initial training set. The different types are summed training set.
up in the table below:
4.3 Diagnostics
k-fold Leave-p-out
- Training on k − 1 folds and - Training on n − p observations and r Bias – The bias of a model is the difference between the expected prediction and the correct
model that we try to predict for given data points.
assessment on the remaining one assessment on the p remaining ones
- Generally k = 5 or 10 - Case p = 1 is called leave-one-out r Variance – The variance of a model is the variability of the model prediction for given data
points.
The most commonly used method is called k-fold cross-validation and splits the training data r Bias/variance tradeoff – The simpler the model, the higher the bias, and the more complex
into k folds to validate the model on one fold while training the model on the k − 1 other folds, the model, the higher the variance.
all of this k times. The error is then averaged over the k folds and is named cross-validation
error.
Underfitting Just right Overfitting
- High training error - Training error - Low training error
Symptoms - Training error close slightly lower than - Training error much
to test error test error lower than test error
- High bias - High variance
Regression
r Regularization – The regularization procedure aims at avoiding the model to overfit the
data and thus deals with high variance issues. The following table sums up the different types
of commonly used regularization techniques:

5 Refreshers
5.1 Probabilities and Statistics

Classification
5.1.1 Introduction to Probability and Combinatorics
r Sample space – The set of all possible outcomes of an experiment is known as the sample
space of the experiment and is denoted by S.
r Event – Any subset E of the sample space is known as an event. That is, an event is a set
consisting of possible outcomes of the experiment. If the outcome of the experiment is contained
in E, then we say that E has occurred.
r Axioms of probability – For each event E, we denote P (E) as the probability of event E
Deep learning occuring. By noting E1 ,...,En mutually exclusive events, we have the 3 following axioms:
n
! n
[ X
(1) 0 6 P (E) 6 1 (2) P (S) = 1 (3) P Ei = P (Ei )
i=1 i=1
- Complexify model - Regularize

Remedies - Add more features - Get more data r Permutation – A permutation is an arrangement of r objects from a pool of n objects, in a
- Train longer given order. The number of such arrangements is given by P (n, r), defined as:
n!
P (n, r) =
r Error analysis – Error analysis is analyzing the root cause of the difference in performance (n − r)!
between the current and the perfect models.
r Ablative analysis – Ablative analysis is analyzing the root cause of the difference in perfor- r Combination – A combination is an arrangement of r objects from a pool of n objects, where
mance between the current and the baseline models. the order does not matter. The number of such arrangements is given by C(n, r), defined as:
P (n, r) n!
C(n, r) = =
r! r!(n − r)!
Remark: we note that for 0 6 r 6 n, we have P (n,r) > C(n,r).
5.1.2 Conditional Probability

r Bayes’ rule – For events A and B such that P (B) > 0, we have:
P (B|A)P (A)
P (A|B) =
P (B)
Remark: we have P (A ∩ B) = P (A)P (B|A) = P (A|B)P (B).
r Partition – Let {Ai , i ∈ [[1,n]]} be such that for all i, Ai 6= ∅. We say that {Ai } is a partition
if we have:
n
[
∀i 6= j, Ai ∩ Aj = ∅ and Ai = S
i=1
n
X
Remark: for any event B in the sample space, we have P (B) = P (B|Ai )P (Ai ).
i=1

r Extended form of Bayes’ rule – Let {Ai , i ∈ [[1,n]]} be a partition of the sample space. r Expectation and Moments of the Distribution – Here are the expressions of the expected
We have: value E[X], generalized expected value E[g(X)], kth moment E[X k ] and characteristic function
ψ(ω) for the discrete and continuous cases:
P (B|Ak )P (Ak )
P (Ak |B) = n
X
P (B|Ai )P (Ai ) Case E[X] E[g(X)] E[X k ] ψ(ω)
i=1 n n n n
X X X X
(D) xi f (xi ) g(xi )f (xi ) xki f (xi ) f (xi )eiωxi
r Independence – Two events A and B are independent if and only if we have: i=1 i=1 i=1 i=1
ˆ +∞ ˆ +∞ ˆ +∞ ˆ +∞
P (A ∩ B) = P (A)P (B) (C) xf (x)dx g(x)f (x)dx xk f (x)dx f (x)eiωx dx
−∞ −∞ −∞ −∞
5.1.3 Random Variables

Remark: we have eiωx = cos(ωx) + i sin(ωx).
r Random variable – A random variable, often noted X, is a function that maps every element
in a sample space to a real line. r Revisiting the kth moment – The kth moment can also be computed with the characteristic
function as follows:
r Cumulative distribution function (CDF) – The cumulative distribution function F ,
1 ∂k ψ
which is monotonically non-decreasing and is such that lim F (x) = 0 and lim F (x) = 1, is E[X k ] =
x→−∞ x→+∞ ik ∂ω k
defined as: ω=0
F (x) = P (X 6 x)
r Transformation of random variables – Let the variables X and Y be linked by some
function. By noting fX and fY the distribution function of X and Y respectively, we have:
Remark: we have P (a < X 6 B) = F (b) − F (a).
fY (y) = fX (x)
dx
r Probability density function (PDF) – The probability density function f is the probability dy

that X takes on values between two adjacent realizations of the random variable.
r Relationships involving the PDF and CDF – Here are the important properties to know r Leibniz integral rule – Let g be a function of x and potentially c, and a, b boundaries that
in the discrete (D) and the continuous (C) cases. may depend on c. We have:
ˆ ˆ b
∂ b ∂b ∂a ∂g
g(x)dx = · g(b) − · g(a) + (x)dx
Case CDF F PDF f Properties of PDF ∂c a ∂c ∂c a ∂c
X X
(D) F (x) = P (X = xi ) f (xj ) = P (X = xj ) 0 6 f (xj ) 6 1 and f (xj ) = 1
r Chebyshev’s inequality – Let X be a random variable with expected value µ and standard
xi 6x j deviation σ. For k, σ > 0, we have the following inequality:
ˆ x ˆ +∞ 1
dF
(C) F (x) = f (y)dy f (x) = f (x) > 0 and f (x)dx = 1 P (|X − µ| > kσ) 6
−∞ dx −∞ k2
5.1.4 Jointly Distributed Random Variables

r Variance – The variance of a random variable, often noted Var(X) or σ 2 , is a measure of the
spread of its distribution function. It is determined as follows:
r Conditional density – The conditional density of X with respect to Y , often noted fX|Y ,
Var(X) = E[(X − E[X])2 ] = E[X 2 ] − E[X]2 is defined as follows:
fXY (x,y)
fX|Y (x) =
r Standard deviation – The standard deviation of a random variable, often noted σ, is a fY (y)
measure of the spread of its distribution function which is compatible with the units of the
actual random variable. It is determined as follows:
r Independence – Two random variables X and Y are said to be independent if we have:
p
σ= Var(X) fXY (x,y) = fX (x)fY (y)

r Marginal density and cumulative distribution – From the joint density probability 5.1.5 Parameter estimation
function fXY , we have:
r Random sample – A random sample is a collection of n random variables X1 , ..., Xn that
Case Marginal density Cumulative function are independent and identically distributed with X.
r Estimator – An estimator θ̂ is a function of the data that is used to infer the value of an
unknown parameter θ in a statistical model.
X XX
(D) fX (xi ) = fXY (xi ,yj ) FXY (x,y) = fXY (xi ,yj )
j xi 6x yj 6y r Bias – The bias of an estimator θ̂ is defined as being the difference between the expected
ˆ ˆ ˆ value of the distribution of θ̂ and the true value, i.e.:
+∞ x y
(C) fX (x) = fXY (x,y)dy FXY (x,y) = fXY (x0 ,y 0 )dx0 dy 0 Bias(θ̂) = E[θ̂] − θ
−∞ −∞ −∞
Remark: an estimator is said to be unbiased when we have E[θ̂] = θ.

r Distribution of a sum of independent random variables – Let Y = X1 + ... + Xn with
X1 , ..., Xn independent. We have: r Sample mean and variance – The sample mean and the sample variance of a random
sample are used to estimate the true mean µ and the true variance σ 2 of a distribution, are
n
Y noted X and s2 respectively, and are such that:
ψY (ω) = ψXk (ω)
n n
k=1 1 X 1 X
X= Xi and s2 = σ̂ 2 = (Xi − X)2
n n−1
i=1 i=1
r Covariance – We define the covariance of two random variables X and Y , that we note σXY
2
or more commonly Cov(X,Y ), as follows:

r Central Limit Theorem – Let us have a random sample X1 , ..., Xn following a given
Cov(X,Y ) , σXY
2
= E[(X − µX )(Y − µY )] = E[XY ] − µX µY distribution with mean µ and variance σ 2 , then we have:
σ

r Correlation – By noting σX , σY the standard deviations of X and Y , we define the correlation X ∼ N µ, √
n→+∞ n
between the random variables X and Y , noted ρXY , as follows:
2
σXY
ρXY =
σX σY 5.2 Linear Algebra and Calculus
Remarks: For any X, Y , we have ρXY ∈ [−1,1]. If X and Y are independent, then ρXY = 0. 5.2.1 General notations
r Main distributions – Here are the main distributions to have in mind:
r Vector – We note x ∈ Rn a vector with n entries, where xi ∈ R is the ith entry:
x1 !
x2
Type Distribution PDF ψ(ω) E[X] Var(X) ..
x= ∈ Rn
n .
xn
X ∼ B(n, p) P (X = x) = px q n−x (peiω + q)n np npq
x
Binomial x ∈ [[0,n]]
(D) r Matrix – We note A ∈ Rm×n a matrix with m rows and n columns, where Ai,j ∈ R is the
µx iω
entry located in the ith row and j th column:
X ∼ Po(µ) P (X = x) = e−µ eµ(e −1) µ µ A1,1 · · · A1,n
!
x!
Poisson x∈N A= . . ∈ Rm×n
.. ..
1 eiωb − eiωa a+b (b − a)2 Am,1 · · · Am,n
X ∼ U (a, b) f (x) =
b−a (b − a)iω 2 12 Remark: the vector x defined above can be viewed as a n × 1 matrix and is more particularly
Uniform x ∈ [a,b] called a column-vector.
2
1 −1
x−µ
1 2
σ2 r Identity matrix – The identity matrix I ∈ Rn×n is a square matrix with ones in its diagonal
(C) X ∼ N (µ, σ) f (x) = √ e2 σ
eiωµ− 2 ω µ σ2 and zero everywhere else:
2πσ
Gaussian x∈R  1 0 ··· 0 
.. .. .
1 1 1 . . .. 
X ∼ Exp(λ) f (x) = λe−λx I= 0

1− iω λ λ2 .. . . ..

Exponential x ∈ R+
λ
. . . 0
0 ··· 0 1

Remark: for all matrices A ∈ Rn×n , we have A × I = I × A = A.
r Diagonal matrix – A diagonal matrix D ∈ Rn×n is a square matrix with nonzero values in ∀i,j, i,j = Aj,i
AT
its diagonal and zero everywhere else:
 d1 0 · · · 0 Remark: for matrices A,B, we have (AB)T = B T AT .

.. .. ..
. .
D= 0 .
r Inverse – The inverse of an invertible square matrix A is noted A−1 and is the only matrix
 
.. .. ..

. . . 0 such that:
0 ··· 0 dn
AA−1 = A−1 A = I
Remark: we also note D as diag(d1 ,...,dn ).
Remark: not all square matrices are invertible. Also, for matrices A,B, we have (AB)−1 =
5.2.2 Matrix operations B −1 A−1
r Trace – The trace of a square matrix A, noted tr(A), is the sum of its diagonal entries:
r Vector-vector multiplication – There are two types of vector-vector products:
n
• inner product: for x,y ∈ Rn , we have:
X
tr(A) = Ai,i
i=1
n
X
xT y = xi yi ∈ R
Remark: for matrices A,B, we have tr(AT ) = tr(A) and tr(AB) = tr(BA)
i=1
r Determinant – The determinant of a square matrix A ∈ Rn×n , noted |A| or det(A) is
expressed recursively in terms of A\i,\j , which is the matrix A without its ith row and j th
• outer product: for x ∈ Rm , y ∈ Rn , we have:
column, as follows:
x1 y1 ··· x1 yn n
.. ..
X
xy T = ∈ Rm×n det(A) = |A| = (−1)i+j Ai,j |A\i,\j |
. .
xm y1 ··· xm yn j=1
Remark: A is invertible if and only if |A| 6= 0. Also, |AB| = |A||B| and |AT | = |A|.
r Matrix-vector multiplication – The product of matrix A ∈ Rm×n and vector x ∈ Rn is a
vector of size Rn , such that:

aT
 5.2.3 Matrix properties
r,1 x n
..
X
Ax =  = ac,i xi ∈ R n
r Symmetric decomposition – A given matrix A can be expressed in terms of its symmetric
.
T
ar,m x i=1 and antisymmetric parts as follows:
A + AT A − AT
where aT
r,i are the vector rows and ac,j are the vector columns of A, and xi are the entries
A= +
2 2
of x. | {z } | {z }
Symmetric Antisymmetric
r Matrix-matrix multiplication – The product of matrices A ∈ Rm×n and B ∈ Rn×p is a
matrix of size Rn×p , such that:

aT ··· aT
 r Norm – A norm is a function N : V −→ [0, + ∞[ where V is a vector space, and such that
r,1 bc,1 r,1 bc,p n
for all x,y ∈ V , we have:
.. ..
X
AB =  = ac,i bT n×p
∈R
. . r,i
T
ar,m bc,1 ··· T
ar,m bc,p i=1 • N (x + y) 6 N (x) + N (y)
where aT • N (ax) = |a|N (x) for a scalar

r,i , br,i are the vector rows and ac,j , bc,j are the vector columns of A and B respec-
T
tively.
• if N (x) = 0, then x = 0
r Transpose – The transpose of a matrix A ∈ Rm×n , noted AT , is such that its entries are
flipped: For x ∈ V , the most commonly used norms are summed up in the table below:

Norm Notation Definition Use case

∂f (A)

n
X ∇A f (A) =
Manhattan, L1 ||x||1 |xi | LASSO regularization i,j ∂Ai,j
i=1
Remark: the gradient of f is only defined when f is a function that returns a scalar.
v
r Hessian – Let f : Rn → R be a function and x ∈ Rn be a vector. The hessian of f with
u n
uX
Euclidean, L2 ||x||2 t x2i Ridge regularization respect to x is a n × n symmetric matrix, noted ∇2x f (x), such that:
i=1 ∂ 2 f (x)

∇2x f (x) =
! 1 i,j ∂xi ∂xj
n p
X
p-norm, Lp ||x||p xpi Hölder inequality Remark: the hessian of f is only defined when f is a function that returns a scalar.
i=1
r Gradient operations – For matrices A,B,C, the following gradient properties are worth
Infinity, L∞ ||x||∞ max |xi | Uniform convergence having in mind:
i
∇A tr(AB) = B T ∇AT f (A) = (∇A f (A))T
r Linearly dependence – A set of vectors is said to be linearly dependent if one of the vectors ∇A tr(ABAT C) = CAB + C T AB T ∇A |A| = |A|(A−1 )T
in the set can be defined as a linear combination of the others.
Remark: if no vector can be written this way, then the vectors are said to be linearly independent.
r Matrix rank – The rank of a given matrix A is noted rank(A) and is the dimension of the
vector space generated by its columns. This is equivalent to the maximum number of linearly
independent columns of A.
r Positive semi-definite matrix – A matrix A ∈ Rn×n is positive semi-definite (PSD) and

is noted A 0 if we have:
A = AT and ∀x ∈ Rn , xT Ax > 0
Remark: similarly, a matrix A is said to be positive definite, and is noted A 0, if it is a PSD

matrix which satisfies for all non-zero vector x, xT Ax > 0.

there exists a vector z ∈ Rn \{0}, called eigenvector, such that we have:
Az = λz
r Spectral theorem – Let A ∈ Rn×n . If A is symmetric, then A is diagonalizable by a real

∃Λ diagonal, A = U ΛU T
r Singular-value decomposition – For a given matrix A of dimensions m × n, the singular-

value decomposition (SVD) is a factorization technique that guarantees the existence of U m×m
unitary, Σ m × n diagonal and V n × n unitary matrices, such that:
A = U ΣV T
5.2.4 Matrix calculus
r Gradient – Let f : Rm×n → R be a function and A ∈ Rm×n be a matrix. The gradient of f

with respect to A is a m × n matrix, noted ∇A f (A), such that:

CS 230 – Deep Learning Shervine Amidi & Afshine Amidi
Super VIP Cheatsheet: Deep Learning 1 Convolutional Neural Networks
Afshine Amidi and Shervine Amidi 1.1 Overview
November 25, 2018

r Architecture of a traditional CNN – Convolutional neural networks, also known as CNNs,
are a specific type of neural networks that are generally composed of the following layers:
Contents
1 Convolutional Neural Networks 2

1.1 Overview . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 2
1.2 Types of layer . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 2
1.3 Filter hyperparameters . . . . . . . . . . . . .. . . . . . . . . . . . . 2
1.4 Tuning hyperparameters . . . . . . . . . . . .. . . . . . . . . . . . . 3
1.5 Commonly used activation functions . . . . . .. . . . . . . . . . . . . 3
1.6 Object detection . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 4 The convolution layer and the pooling layer can be fine-tuned with respect to hyperparameters
1.6.1 Face verification and recognition . . . .. . . . . . . . . . . . . 5 that are described in the next sections.
1.6.2 Neural style transfer . . . . . . . . . .. . . . . . . . . . . . . 5
1.6.3 Architectures using computational tricks . . . . . . . . . . . . 6
2 Recurrent Neural Networks 7

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Handling long term dependencies . . . . . . . . . . . . . . . . . . . . 8
1.2 Types of layer
2.3 Learning word representation . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Motivation and notations . . . . . . . . . . . . . . . . . . . 9
2.3.2 Word embeddings . . . . . . . . . . . . . . . . . . . . . . . 9
r Convolutional layer (CONV) – The convolution layer (CONV) uses filters that perform
2.4 Comparing words . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 convolution operations as it is scanning the input I with respect to its dimensions. Its hyperpa-
2.5 Language model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 rameters include the filter size F and stride S. The resulting output O is called feature map or
activation map.
2.6 Machine translation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.7 Attention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Deep Learning Tips and Tricks 11

3.1 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Training a neural network . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.2 Finding optimal weights . . . . . . . . . . . . . . . . . . . . . 12
3.3 Parameter tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Remark: the convolution step can be generalized to the 1D and 3D cases as well.
3.3.1 Weights initialization . . . . . . . . . . . . . . . . . . . . . . 12
3.3.2 Optimizing convergence . . . . . . . . . . . . . . . . . . . . . 12
r Pooling (POOL) – The pooling layer (POOL) is a downsampling operation, typically applied
3.4 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 after a convolution layer, which does some spatial invariance. In particular, max and average
3.5 Good practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 pooling are special kinds of pooling where the maximum and average value is taken, respectively.
Stanford University 1 Winter 2019

Max pooling Average pooling

Each pooling operation selects the Each pooling operation averages
Purpose
maximum value of the current view the values of the current view
r Zero-padding – Zero-padding denotes the process of adding P zeroes to each side of the
boundaries of the input. This value can either be manually specified or automatically set through
one of the three modes detailed below:
Valid Same Full

Illustration j I e−I+F −S
k
Sd S
Pstart = 2 Pstart ∈ [[0,F − 1]]
Value P =0 l I e−I+F −S
m
Sd S
Pend = Pend = F − 1
2
- Preserves detected features - Downsamples feature map
Comments
- Most commonly used - Used in LeNet
Illustration
r Fully Connected (FC) – The fully connected layer (FC) operates on a flattened input where
each input is connected to all neurons. If present, FC layers are usually found towards the end
of CNN architectures and can be used to optimize objectives such as class scores.
- Maximum padding
- No padding - Padding such that feature
l m such that end
map size has size I
convolutions are
- Drops last S
Purpose applied on the limits
convolution if - Output size is
of the input
dimensions do not mathematically convenient
match - Filter ’sees’ the input
- Also called ’half’ padding
end-to-end
1.4 Tuning hyperparameters

r Parameter compatibility in convolution layer – By noting I the length of the input
1.3 Filter hyperparameters volume size, F the length of the filter, P the amount of zero padding, S the stride, then the
output size O of the feature map along that dimension is given by:
The convolution layer contains filters for which it is important to know the meaning behind its
hyperparameters. I − F + Pstart + Pend
O= +1
r Dimensions of a filter – A filter of size F × F applied to an input containing C channels is S
a F × F × C volume that performs convolutions on an input of size I × I × C and produces an
output feature map (also called activation map) of size O × O × 1.
Remark: the application of K filters of size F × F results in an output feature map of size
O × O × K.
r Stride – For a convolutional or a pooling operation, the stride S denotes the number of pixels Remark: often times, Pstart = Pend , P , in which case we can replace Pstart + Pend by 2P in
by which the window moves after each operation. the formula above.

r Understanding the complexity of the model – In order to assess the complexity of a ReLU Leaky ReLU ELU
model, it is often useful to determine the number of parameters that its architecture will have.
In a given layer of a convolutional neural network, it is done as follows: g(z) = max(z,z) g(z) = max(α(ez − 1),z)
g(z) = max(0,z)
with 1 with α 1
CONV POOL FC
Illustration
Input size I ×I ×C I ×I ×C Nin

Non-linearity complexities Addresses dying ReLU
Differentiable everywhere
Output size O×O×K O×O×C Nout biologically interpretable issue for negative values
Number of
(F × F × C + 1) · K 0 (Nin + 1) × Nout
parameters r Softmax – The softmax step can be seen as a generalized logistic function that takes as input
a vector of scores x ∈ Rn and outputs a vector of output probability p ∈ Rn through a softmax
- Input is flattened function at the end of the architecture. It is defined as follows:
- One bias parameter
- Pooling operation - One bias parameter
per filter
done channel-wise per neuron p1
Remarks - In most cases, S < F .. e xi
- The number of FC p= where pi =
- A common choice - In most cases, S = F . n
neurons is free of pn X
for K is 2C e xj
structural constraints
j=1
r Receptive field – The receptive field at layer k is the area denoted Rk × Rk of the input
that each pixel of the k-th activation map can ’see’. By calling Fj the filter size of layer j and 1.6 Object detection
Si the stride value of layer i and with the convention S0 = 1, the receptive field at layer k can
be computed with the formula: r Types of models – There are 3 main types of object recognition algorithms, for which the
k j−1 nature of what is predicted is different. They are described in the table below:
X Y
Rk = 1 + (Fj − 1) Si
Classification
j=1 i=0 Image classification Detection
w. localization
In the example below, we have F1 = F2 = 3 and S1 = S2 = 1, which gives R2 = 1+2 · 1+2 · 1 =
5.
- Classifies a picture - Detects object in a picture - Detects up to several objects

- Predicts probability of in a picture
- Predicts probability object and where it is - Predicts probabilities of objects
of object located and where they are located
Traditional CNN Simplified YOLO, R-CNN YOLO, R-CNN
1.5 Commonly used activation functions
r Rectified Linear Unit – The rectified linear unit layer (ReLU) is an activation function g r Detection – In the context of object detection, different methods are used depending on
that is used on all elements of the volume. It aims at introducing non-linearities to the network. whether we just want to locate the object or detect a more complex shape in the image. The
Its variants are summarized in the table below: two main ones are summed up in the table below:

Bounding box detection Landmark detection

- Detects a shape or characteristics of
Detects the part of the image where
an object (e.g. eyes)
the object is located
- More granular
r YOLO – You Only Look Once (YOLO) is an object detection algorithm that performs the
following steps:
• Step 1: Divide the input image into a G × G grid.
• Step 2: For each grid cell, run a CNN that predicts y of the following form:
Box of center (bx ,by ), height bh
Reference points (l1x ,l1y ), ...,(lnx ,lny ) T
and width bw y = pc ,bx ,by ,bh ,bw ,c1 ,c2 ,...,cp ,... ∈ RG×G×k×(5+p)
| {z }
repeated k times
r Intersection over Union – Intersection over Union, also known as IoU, is a function that
quantifies how correctly positioned a predicted bounding box Bp is over the actual bounding where pc is the probability of detecting an object, bx ,by ,bh ,bw are the properties of the
box Ba . It is defined as: detected bouding box, c1 ,...,cp is a one-hot representation of which of the p classes were
detected, and k is the number of anchor boxes.
Bp ∩ Ba
IoU(Bp ,Ba ) = • Step 3: Run the non-max suppression algorithm to remove any potential duplicate over-
Bp ∪ Ba
lapping bounding boxes.
Remark: when pc = 0, then the network does not detect any object. In that case, the corre-
sponding predictions bx , ..., cp have to be ignored.
Remark: we always have IoU ∈ [0,1]. By convention, a predicted bounding box Bp is considered
as being reasonably good if IoU(Bp ,Ba ) > 0.5. r R-CNN – Region with Convolutional Neural Networks (R-CNN) is an object detection algo-
rithm that first segments the image to find potential relevant bounding boxes and then run the
r Anchor boxes – Anchor boxing is a technique used to predict overlapping bounding boxes. detection algorithm to find most probable objects in those bounding boxes.
In practice, the network is allowed to predict more than one box simultaneously, where each box
prediction is constrained to have a given set of geometrical properties. For instance, the first
prediction can potentially be a rectangular box of a given form, while the second will be another
rectangular box of a different geometrical form.
r Non-max suppression – The non-max suppression technique aims at removing duplicate
overlapping bounding boxes of a same object by selecting the most representative ones. After
having removed all boxes having a probability prediction lower than 0.6, the following steps are
repeated while there are boxes remaining:
• Step 1: Pick the box with the largest prediction probability.

Remark: although the original algorithm is computationally expensive and slow, newer archi-
• Step 2: Discard any box having an IoU > 0.5 with the previous box. tectures enabled the algorithm to run faster, such as Fast R-CNN and Faster R-CNN.

1.6.1 Face verification and recognition
r Types of models – Two main types of model are summed up in table below:
Face verification Face recognition

- Is this the correct person? - Is this one of the K persons in the database?
- One-to-one lookup - One-to-many lookup
r Activation – In a given layer l, the activation is noted a[l] and is of dimensions nH × nw × nc
r Content cost function – The content cost function Jcontent (C,G) is used to determine how
the generated image G differs from the original content image C. It is defined as follows:
1 [l](C)
Jcontent (C,G) = ||a − a[l](G) ||2
2
r Style matrix – The style matrix G[l] of a given layer l is a Gram matrix where each of its
[l]
elements Gkk0 quantifies how correlated the channels k and k0 are. It is defined with respect to
r One Shot Learning – One Shot Learning is a face verification algorithm that uses a limited activations a[l] as follows:
training set to learn a similarity function that quantifies how different two given images are. The [l] [l]
similarity function applied to two images is often noted d(image 1, image 2). n nw
H X
X
[l] [l] [l]
Gkk0 = aijk aijk0
r Siamese Network – Siamese Networks aim at learning how to encode images to then quantify
i=1 j=1
how different two images are. For a given input image x(i) , the encoded output is often noted
as f (x(i) ).
Remark: the style matrix for the style image and the generated image are noted G[l](S) and
r Triplet loss – The triplet loss ` is a loss function computed on the embedding representation G[l](G) respectively.
of a triplet of images A (anchor), P (positive) and N (negative). The anchor and the positive
example belong to a same class, while the negative example to another one. By calling α ∈ R+ r Style cost function – The style cost function Jstyle (S,G) is used to determine how the
the margin parameter, this loss is defined as follows: generated image G differs from the style S. It is defined as follows:
`(A,P,N ) = max (d(A,P ) − d(A,N ) + α,0) nc

1 1 X 2
[l] [l](S) [l](G)
Jstyle (S,G) = ||G[l](S) − G[l](G) ||2F = Gkk0 − Gkk0
(2nH nw nc )2 (2nH nw nc )2
k,k0 =1
r Overall cost function – The overall cost function is defined as being a combination of the
content and style cost functions, weighted by parameters α,β, as follows:
J(G) = αJcontent (C,G) + βJstyle (S,G)
Remark: a higher value of α will make the model care more about the content while a higher
value of β will make it care more about the style.
1.6.3 Architectures using computational tricks

1.6.2 Neural style transfer r Generative Adversarial Network – Generative adversarial networks, also known as GANs,
are composed of a generative and a discriminative model, where the generative model aims at
r Motivation – The goal of neural style transfer is to generate an image G based on a given generating the most truthful output that will be fed into the discriminative which aims at
content C and a given style S. differentiating the generated and true image.

2 Recurrent Neural Networks
2.1 Overview
r Architecture of a traditional RNN – Recurrent neural networks, also known as RNNs,
are a class of neural networks that allow previous outputs to be used as inputs while having
hidden states. They are typically as follows:
Remark: use cases using variants of GANs include text to image, music generation and syn-
thesis.
r ResNet – The Residual Network architecture (also called ResNet) uses residual blocks with a
high number of layers meant to decrease the training error. The residual block has the following
characterizing equation:
a[l+2] = g(a[l] + z [l+2] ) For each timestep t, the activation a<t> and the output y <t> are expressed as follows:
r Inception Network – This architecture uses inception modules and aims at giving a try
at different convolutions in order to increase its performance. In particular, it uses the 1 × 1 a<t> = g1 (Waa a<t−1> + Wax x<t> + ba ) and y <t> = g2 (Wya a<t> + by )
convolution trick to lower the burden of computation.
where Wax , Waa , Wya , ba , by are coefficients that are shared temporally and g1 , g2 activation
functions
? ? ?
The pros and cons of a typical RNN architecture are summed up in the table below:
Advantages Drawbacks
- Possibility of processing input of any length - Computation being slow
- Model size not increasing with size of input - Difficulty of accessing information
- Computation takes into account from a long time ago
historical information - Cannot consider any future input
- Weights are shared across time for the current state
r Applications of RNNs – RNN models are mostly used in the fields of natural language
processing and speech recognition. The different applications are summed up in the table below:

Type of RNN Illustration Example

T
∂L(T ) X ∂L(T )
=
One-to-one ∂W ∂W
(t)
t=1
Traditional neural network
Tx = Ty = 1
2.2 Handling long term dependencies

r Commonly used activation functions – The most common activation functions used in
One-to-many RNN modules are described below:
Music generation
Tx = 1, Ty > 1
Sigmoid Tanh RELU
1 ez − e−z
g(z) = g(z) = g(z) = max(0,z)
1 + e−z ez + e−z
Many-to-one
Sentiment classification
Tx > 1, Ty = 1
Many-to-many
Name entity recognition
Tx = Ty r Vanishing/exploding gradient – The vanishing and exploding gradient phenomena are
often encountered in the context of RNNs. The reason why they happen is that it is difficult
to capture long term dependencies because of multiplicative gradient that can be exponentially
decreasing/increasing with respect to the number of layers.
Many-to-many r Gradient clipping – It is a technique used to cope with the exploding gradient problem
sometimes encountered when performing backpropagation. By capping the maximum value for
Machine translation the gradient, this phenomenon is controlled in practice.
Tx 6= Ty
r Loss function – In the case of a recurrent neural network, the loss function L of all time
steps is defined based on the loss at every time step as follows:
Ty
X r Types of gates – In order to remedy the vanishing gradient problem, specific gates are used
y ,y) =
L(b y <t> ,y <t> )
L(b in some types of RNNs and usually have a well-defined purpose. They are usually noted Γ and
t=1 are equal to:
Γ = σ(W x<t> + U a<t−1> + b)
r Backpropagation through time – Backpropagation is done at each point in time. At where W, U, b are coefficients specific to the gate and σ is the sigmoid function. The main ones
timestep T , the derivative of the loss L with respect to weight matrix W is expressed as follows: are summed up in the table below:

Type of gate Role Used in 2.3 Learning word representation

Update gate Γu How much past should matter now? GRU, LSTM In this section, we note V the vocabulary and |V | its size.
Relevance gate Γr Drop previous information? GRU, LSTM
Forget gate Γf Erase a cell or not? LSTM 2.3.1 Motivation and notations
Output gate Γo How much to reveal of a cell? LSTM r Representation techniques – The two main ways of representing words are summed up in
the table below:
r GRU/LSTM – Gated Recurrent Unit (GRU) and Long Short-Term Memory units (LSTM)
deal with the vanishing gradient problem encountered by traditional RNNs, with LSTM being 1-hot representation Word embedding
a generalization of GRU. Below is a table summing up the characterizing equations of each
architecture:
Gated Recurrent Unit Long Short-Term Memory
(GRU) (LSTM)
c̃<t> tanh(Wc [Γr ? a<t−1> ,x<t> ] + bc ) tanh(Wc [Γr ? a<t−1> ,x<t> ] + bc )
c<t> Γu ? c̃<t> + (1 − Γu ) ? c<t−1> Γu ? c̃<t> + Γf ? c<t−1>
a<t> c<t> Γo ? c<t>
- Noted ow - Noted ew
- Naive approach, no similarity information - Takes into account words similarity
Dependencies r Embedding matrix – For a given word w, the embedding matrix E is a matrix that maps
its 1-hot representation ow to its embedding ew as follows:
ew = Eow
Remark: learning the embedding matrix can be done using target/context likelihood models.
Remark: the sign ? denotes the element-wise multiplication between two vectors.
r Variants of RNNs – The table below sums up the other commonly used RNN architectures: 2.3.2 Word embeddings
Bidirectional Deep r Word2vec – Word2vec is a framework aimed at learning word embeddings by estimating the
(BRNN) likelihood that a given word is surrounded by other words. Popular models include skip-gram,
(DRNN) negative sampling and CBOW.
r Skip-gram – The skip-gram word2vec model is a supervised learning task that learns word
embeddings by assessing the likelihood of any given target word t happening with a context
word c. By noting θt a parameter associated with t, the probability P (t|c) is given by:

exp(θtT ec )
P (t|c) =
|V |
X
exp(θjT ec )
j=1
Remark: summing over the whole vocabulary in the denominator of the softmax part makes
this model computationally expensive. CBOW is another word2vec model using the surrounding
words to predict a given word.
r Negative sampling – It is a set of binary classifiers using logistic regressions that aim at 2.5 Language model
assessing how a given context and a given target words are likely to appear simultaneously, with
the models being trained on sets of k negative examples and 1 positive example. Given a context r Overview – A language model aims at estimating the probability of a sentence P (y).
word c and a target word t, the prediction is expressed by:
r n-gram model – This model is a naive approach aiming at quantifying the probability that
P (y = 1|c,t) = σ(θtT ec ) an expression appears in a corpus by counting its number of appearance in the training data.
Remark: this method is less computationally expensive than the skip-gram model. r Perplexity – Language models are commonly assessed using the perplexity metric, also
known as PP, which can be interpreted as the inverse probability of the dataset normalized by
r GloVe – The GloVe model, short for global vectors for word representation, is a word em- the number of words T . The perplexity is such that the lower, the better and is defined as
bedding technique that uses a co-occurence matrix X where each Xi,j denotes the number of follows:
times that a target i occurred with a context j. Its cost function J is as follows: ! T1
T
|V |
Y 1
1 X PP =
J(θ) = f (Xij )(θiT ej + bi + b0j − log(Xij ))2
P|V | (t) (t)
yj · b
yj
2 t=1 j=1
i,j=1
Remark: PP is commonly used in t-SNE.
here f is a weighting function such that Xi,j = 0 =⇒ f (Xi,j ) = 0.
(final)
Given the symmetry that e and θ play in this model, the final word embedding ew is given
by: 2.6 Machine translation
(final) e w + θw r Overview – A machine translation model is similar to a language model except it has an
ew =
2 encoder network placed before. For this reason, it is sometimes referred as a conditional language
model. The goal is to find a sentence y such that:
Remark: the individual components of the learned word embeddings are not necessarily inter-
pretable. y= arg max P (y <1> ,...,y <Ty > |x)
y <1> ,...,y <Ty >
2.4 Comparing words r Beam search – It is a heuristic search algorithm used in machine translation and speech
recognition to find the likeliest sentence y given an input x.
r Cosine similarity – The cosine similarity between words w1 and w2 is expressed as follows:
• Step 1: Find top B likely words y <1>
w1 · w2
similarity = = cos(θ)
||w1 || ||w2 || • Step 2: Compute conditional probabilities y <k> |x,y <1> ,...,y <k−1>
• Step 3: Keep top B combinations x,y <1> ,...,y <k>

Remark: θ is the angle between words w1 and w2 .
r t-SNE – t-SNE (t-distributed Stochastic Neighbor Embedding) is a technique aimed at re-

ducing high-dimensional embeddings into a lower dimensional space. In practice, it is commonly
used to visualize word vectors in the 2D space.

Remark: if the beam width is set to 1, then this is equivalent to a naive greedy search. Remark: the attention scores are commonly used in image captioning and machine translation.
r Beam width – The beam width B is a parameter for beam search. Large values of B yield
to better result but with slower performance and increased memory. Small values of B lead to
worse results but is less computationally intensive. A standard value for B is around 10.
r Length normalization – In order to improve numerical stability, beam search is usually ap-
plied on the following normalized objective, often called the normalized log-likelihood objective,
defined as:
Ty
1 X h i
Objective = log p(y <t> |x,y <1> , ..., y <t−1> )
Tyα
t=1
Remark: the parameter α can be seen as a softener, and its value is usually between 0.5 and 1.
r Error analysis – When obtaining a predicted translation b y that is bad, one can wonder why r Attention weight – The amount of attention that the output y <t> should pay to the
0 0
we did not get a good translation y ∗ by performing the following error analysis: activation a<t > is given by α<t,t > computed as follows:
0
0 exp(e<t,t >)
Case P (y ∗ |x) > P (b
y |x) P (y ∗ |x) 6 P (b
y |x) α<t,t >
=
Tx
00
X
Root cause Beam search faulty RNN faulty exp(e<t,t >
)
- Try different architecture t00 =1
Remedies Increase beam width - Regularize Remark: computation complexity is quadratic with respect to Tx .
- Get more data
? ? ?
r Bleu score – The bilingual evaluation understudy (bleu) score quantifies how good a machine
translation is by computing a similarity score based on n-gram precision. It is defined as follows:
n
!
1 X
bleu score = exp pk
n
k=1
where pn is the bleu score on n-gram only defined as follows:

X
countclip (n-gram)
n-gram∈y
pn =
b
X
count(n-gram)
n-gram∈y b
Remark: a brevity penalty may be applied to short predicted translations to prevent an artificially
inflated bleu score.
2.7 Attention
r Attention model – This model allows an RNN to pay attention to specific parts of the input
that is considered as being important, which improves the performance of the resulting model
0
in practice. By noting α<t,t > the amount of attention that the output y <t> should pay to the
0
activation a <t > and c <t> the context at time t, we have:
0
> <t0 > 0
X X
c<t> = α<t,t a with α<t,t >
=1
t0 t0

3 Deep Learning Tips and Tricks 3.2 Training a neural network
3.1 Data processing 3.2.1 Definitions
r Data augmentation – Deep learning models usually need a lot of data to be properly trained. r Epoch – In the context of training a model, epoch is a term used to refer to one iteration
It is often useful to get more data from the existing ones using data augmentation techniques. where the model sees the whole training set to update its weights.
The main ones are summed up in the table below. More precisely, given the following input
r Mini-batch gradient descent – During the training phase, updating weights is usually not
image, here are the techniques that we can apply:
based on the whole training set at once due to computation complexities or one data point due
to noise issues. Instead, the update step is done on mini-batches, where the number of data
Original Flip Rotation Random crop points in a batch is a hyperparameter that we can tune.
r Loss function – In order to quantify how a given model performs, the loss function L is
usually used to evaluate to what extent the actual outputs y are correctly predicted by the
model outputs z.
r Cross-entropy loss – In the context of binary classification in neural networks, the cross-
entropy loss L(z,y) is commonly used and is defined as follows:
h i
L(z,y) = − y log(z) + (1 − y) log(1 − z)
- Random focus
- Flipped with respect - Rotation with on one part of
- Image without to an axis for which a slight angle the image
3.2.2 Finding optimal weights
any modification the meaning of the - Simulates incorrect - Several random
image is preserved horizon calibration crops can be r Backpropagation – Backpropagation is a method to update the weights in the neural network
done in a row by taking into account the actual output and the desired output. The derivative with respect
to each weight w is computed using the chain rule.
Color shift Noise addition Information loss Contrast change
Using this method, each weight is updated with the rule:

∂L(z,y)
w ←− w − α
- Nuances of RGB ∂w
- Addition of noise - Parts of image - Luminosity changes
is slightly changed
- More tolerance to ignored - Controls difference
- Captures noise r Updating weights – In a neural network, weights are updated as follows:
quality variation of - Mimics potential in exposition due
that can occur
inputs loss of parts of image to time of day • Step 1: Take a batch of training data and perform forward propagation to compute the
with light exposure
loss.
• Step 2: Backpropagate the loss to get the gradient of the loss with respect to each weight.
r Batch normalization – It is a step of hyperparameter γ, β that normalizes the batch {xi }.
• Step 3: Use the gradients to update the weights of the network.
By noting µB , σB
2 the mean and variance of that we want to correct to the batch, it is done as
follows:
xi − µB
xi ←− γ p +β
2 +
σB
It is usually done after a fully connected/convolutional layer and before a non-linearity layer and
aims at allowing higher learning rates and reducing the strong dependence on initialization.

3.3 Parameter tuning Method Explanation Update of w Update of b

- Dampens oscillations
3.3.1 Weights initialization Momentum - Improvement to SGD w − αvdw b − αvdb
- 2 parameters to tune
r Xavier initialization – Instead of initializing the weights in a purely random manner, Xavier - Root Mean Square propagation
dw db
initialization enables to have initial weights that take into account characteristics that are unique RMSprop - Speeds up learning algorithm w − α√ b ←− b − α √
to the architecture. by controlling oscillations
sdw sdb
r Transfer learning – Training a deep learning model requires a lot of data and more impor- - Adaptive Moment estimation
tantly a lot of time. It is often useful to take advantage of pre-trained weights on huge datasets vdw vdb
Adam - Most popular method w − α√ b ←− b − α √
that took days/weeks to train, and leverage it towards our use case. Depending on how much sdw + sdb +
data we have at hand, here are the different ways to leverage this: - 4 parameters to tune
Remark: other methods include Adadelta, Adagrad and SGD.
Training size Illustration Explanation

3.4 Regularization
r Dropout – Dropout is a technique used in neural networks to prevent overfitting the training
Freezes all layers, data by dropping out neurons with probability p > 0. It forces the model to avoid relying too
Small
trains weights on softmax much on particular sets of features.
Freezes most layers,

Medium trains weights on last
layers and softmax
Remark: most deep learning frameworks parametrize dropout through the ’keep’ parameter 1−p.
Trains weights on layers r Weight regularization – In order to make sure that the weights are not too large and that
Large and softmax by initializing the model is not overfitting the training set, regularization techniques are usually performed on
weights on pre-trained ones the model weights. The main ones are summed up in the table below:
LASSO Ridge Elastic Net

- Shrinks coefficients to 0 Tradeoff between variable
Makes coefficients smaller
- Good for variable selection selection and small coefficients
3.3.2 Optimizing convergence
r Learning rate – The learning rate, often noted α or sometimes η, indicates at which pace the
weights get updated. It can be fixed or adaptively changed. The current most popular method
is called Adam, which is a method that adapts the learning rate.
r Adaptive learning rates – Letting the learning rate vary when training a model can reduce
h i
the training time and improve the numerical optimal solution. While Adam optimizer is the ... + λ||θ||1 ... + λ||θ||22 ... + λ (1 − α)||θ||1 + α||θ||22
most commonly used technique, others can also be useful. They are summed up in the table λ∈R λ∈R
below: λ ∈ R,α ∈ [0,1]

r Early stopping – This regularization technique stops the training process as soon as the
validation loss reaches a plateau or starts to increase.
3.5 Good practices

r Overfitting small batch – When debugging a model, it is often useful to make quick tests
to see if there is any major issue with the architecture of the model itself. In particular, in order
to make sure that the model can be properly trained, a mini-batch is passed inside the network
to see if it can overfit on it. If it cannot, it means that the model is either too complex or not
complex enough to even overfit on a small batch, let alone a normal-sized training set.
r Gradient checking – Gradient checking is a method used during the implementation of
the backward pass of a neural network. It compares the value of the analytical gradient to the
numerical gradient at given points and plays the role of a sanity-check for correctness.
Numerical gradient Analytical gradient

df f (x + h) − f (x − h) df
Formula (x) ≈ (x) = f 0 (x)
dx 2h dx
- Expensive; loss has to be
computed two times per dimension
- ’Exact’ result
- Used to verify correctness
Comments of analytical implementation - Direct computation
-Trade-off in choosing h
not too small (numerical instability) - Used in the final implementation
nor too large (poor gradient approx.)
? ? ?

CS 221 – Artificial Intelligence Afshine Amidi & Shervine Amidi
3 Variables-based models 12
Super VIP Cheatsheet: Artificial Intelligence 3.1 Constraint satisfaction problems . . . . . . . . . . . . . . . . . . . . . 12
3.1.1 Factor graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.2 Dynamic ordering . . . . . . . . . . . . . . . . . . . . . . . . 12
Afshine Amidi and Shervine Amidi 3.1.3 Approximate methods . . . . . . . . . . . . . . . . . . . . . . 13
3.1.4 Factor graph transformations . . . . . . . . . . . . . . . . . . 13
May 26, 2019 3.2 Bayesian networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.2 Probabilistic programs . . . . . . . . . . . . . . . . . . . . . . 15
3.2.3 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Contents
4 Logic-based models 16
4.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1 Reflex-based models 2 4.2 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1 Linear predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.3 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Loss minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Non-linear predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Stochastic gradient descent . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Fine-tuning models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Unsupervised Learning . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6.1 k-means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6.2 Principal Component Analysis . . . . . . . . . . . . . . . . 4
2 States-based models 5
2.1 Search optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Tree search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Graph search . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Learning costs . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.4 A? search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.5 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Markov decision processes . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 When unknown transitions and rewards . . . . . . . . . . . . . 9
2.3 Game playing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Speeding up minimax . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Simultaneous games . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3 Non-zero-sum games . . . . . . . . . . . . . . . . . . . . . . . 12
Stanford University 1 Spring 2019

1 Reflex-based models 1.1.2 Regression
r Linear regression – Given a weight vector w ∈ Rd and a feature vector φ(x) ∈ Rd , the
1.1 Linear predictors output of a linear regression of weights w denoted as fw is given by:
In this section, we will go through reflex-based models that can improve with experience, by fw (x) = s(x,w)
going through samples that have input-output pairs.
r Feature vector – The feature vector of an input x is noted φ(x) and is such that: r Residual – The residual res(x,y,w) ∈ R is defined as being the amount by which the prediction
fw (x) overshoots the target y:
" φ (x) #
1 res(x,y,w) = fw (x) − y
φ(x) = .. ∈ Rd
.
φd (x)
1.2 Loss minimization

r Score – The score s(x,w) of an example (φ(x),y) ∈ Rd × R associated to a linear model of
r Loss function – A loss function Loss(x,y,w) quantifies how unhappy we are with the weights
weights w ∈ Rd is given by the inner product:
w of the model in the prediction task of output y from input x. It is a quantity we want to
minimize during the training process.
s(x,w) = w · φ(x)
r Classification case – The classification of a sample x of true label y ∈ {−1,+1} with a linear
model of weights w can be done with the predictor fw (x) , sign(s(x,w)). In this situation, a
metric of interest quantifying the quality of the classification is given by the margin m(x,y,w),
and can be used with the following loss functions:
1.1.1 Classification
Name Zero-one loss Hinge loss Logistic loss
r Linear classifier – Given a weight vector w ∈ Rd and a feature vector φ(x) ∈ Rd , the binary
linear classifier fw is given by: Loss(x,y,w) 1{m(x,y,w)60} max(1 − m(x,y,w), 0) log(1 + e−m(x,y,w) )
+1 if w · φ(x) > 0

fw (x) = sign(s(x,w)) = −1 if w · φ(x) < 0
? if w · φ(x) = 0
Illustration
r Regression case – The prediction of a sample x of true label y ∈ R with a linear model of
weights w can be done with the predictor fw (x) , s(x,w). In this situation, a metric of interest
quantifying the quality of the regression is given by the margin res(x,y,w) and can be used with
the following loss functions:
Name Squared loss Absolute deviation loss
Loss(x,y,w) (res(x,y,w))2 |res(x,y,w)|
r Margin – The margin m(x,y,w) ∈ R of an example (φ(x),y) ∈ Rd × {−1, + 1} associated to

a linear model of weights w ∈ Rd quantifies the confidence of the prediction: larger values are
better. It is given by:
Illustration
m(x,y,w) = s(x,w) × y

r Loss minimization framework – In order to train a model, we want to minimize the

training loss is defined as follows:
w ←− w − η∇w Loss(x,y,w)
1 X
TrainLoss(w) = Loss(x,y,w)
|Dtrain |
(x,y)∈Dtrain
1.3 Non-linear predictors

r k-nearest neighbors – The k-nearest neighbors algorithm, commonly known as k-NN, is a
non-parametric approach where the response of a data point is determined by the nature of its
k neighbors from the training set. It can be used in both classification and regression settings.
r Stochastic updates – Stochastic gradient descent (SGD) updates the parameters of the
model one training example (φ(x),y) ∈ Dtrain at a time. This method leads to sometimes noisy,
but fast updates.
r Batch updates – Batch gradient descent (BGD) updates the parameters of the model one
batch of examples (e.g. the entire training set) at a time. This method computes stable update
directions, at a greater computational cost.
1.5 Fine-tuning models

Remark: the higher the parameter k, the higher the bias, and the lower the parameter k, the
higher the variance.
r Hypothesis class – A hypothesis class F is the set of possible predictors with a fixed φ(x)
r Neural networks – Neural networks are a class of models that are built with layers. Com- and varying w:
monly used types of neural networks include convolutional and recurrent neural networks. The
vocabulary around neural networks architectures is described in the figure below:

F= fw : w ∈ Rd
r Logistic function – The logistic function σ, also called the sigmoid function, is defined as:
1
∀z ∈] − ∞, + ∞[, σ(z) =
1 + e−z
Remark: we have σ 0 (z) = σ(z)(1 − σ(z)).

By noting i the ith layer of the network and j the j th hidden unit of the layer, we have:
r Backpropagation – The forward pass is done through fi , which is the value for the subex-
[i] [i] T [i] pression rooted at i, while the backward pass is done through gi = ∂out and represents how fi
zj = wj x + bj ∂fi
influences the output.
where we note w, b, x, z the weight, bias, input and non-activated output of the neuron respec-
tively.
1.4 Stochastic gradient descent

r Approximation and estimation error – The approximation error approx represents how
r Gradient descent – By noting η ∈ R the learning rate (also called step size), the update far the entire hypothesis class F is from the target predictor g ∗ , while the estimation error est
rule for gradient descent is expressed with the learning rate and the loss function Loss(x,y,w) as quantifies how good the predictor fˆ is with respect to the best predictor f ∗ of the hypothesis
follows: class F .

1.6 Unsupervised Learning

The class of unsupervised learning methods aims at discovering the structure of the data, which
may have of rich latent structures.
r Regularization – The regularization procedure aims at avoiding the model to overfit the
data and thus deals with high variance issues. The following table sums up the different types
of commonly used regularization techniques: 1.6.1 k-means
r Clustering – Given a training set of input points Dtrain , the goal of a clustering algorithm
LASSO Ridge Elastic Net is to assign each point φ(xi ) to a cluster zi ∈ {1,...,k}.
- Shrinks coefficients to 0 Tradeoff between variable r Objective function – The loss function for one of the main clustering algorithms, k-means,
Makes coefficients smaller is given by:
- Good for variable selection selection and small coefficients
n
X
Lossk-means (x,µ) = ||φ(xi ) − µzi ||2
i=1
r Algorithm – After randomly initializing the cluster centroids µ1 ,µ2 ,...,µk ∈ Rn , the k-means
algorithm repeats the following step until convergence:
m
X
1{zi =j} φ(xi )
i=1
zi = arg min||φ(xi ) − µj || 2
and µj = m
j X
h i 1{zi =j}
... + λ||θ||1 ... + λ||θ||22 ... + λ (1 − α)||θ||1 + α||θ||22 i=1
λ∈R λ∈R λ ∈ R, α ∈ [0,1]
r Hyperparameters – Hyperparameters are the properties of the learning algorithm, and

include features, regularization parameter λ, number of iterations T , step size η, etc.
r Sets vocabulary – When selecting a model, we distinguish 3 different parts of the data that
we have as follows:
Training set Validation set Testing set

- Model is assessed - Model gives predictions
- Model is trained
- Usually 20 of the dataset - Unseen data
- Usually 80 of the dataset
- Also called hold-out or development set 1.6.2 Principal Component Analysis
there exists a vector z ∈ Rn \{0}, called eigenvector, such that we have:
Once the model has been chosen, it is trained on the entire dataset and tested on the unseen
test set. These are represented in the figure below: Az = λz

r Spectral theorem – Let A ∈ Rn×n . If A is symmetric, then A is diagonalizable by a real 2 States-based models
∃Λ diagonal, A = U ΛU T 2.1 Search optimization
In this section, we assume that by accomplishing action a from state s, we deterministically
Remark: the eigenvector associated with the largest eigenvalue is called principal eigenvector of arrive in state Succ(s,a). The goal here is to determine a sequence of actions (a1 ,a2 ,a3 ,a4 ,...)
matrix A. that starts from an initial state and leads to an end state. In order to solve this kind of problem,
our objective will be to find the minimum cost path by using states-based models.
r Algorithm – The Principal Component Analysis (PCA) procedure is a dimension reduction
technique that projects the data on k dimensions by maximizing the variance of the data as
follows:
2.1.1 Tree search
• Step 1: Normalize the data to have a mean of 0 and standard deviation of 1.
This category of states-based algorithms explores all possible states and actions. It is quite
memory efficient, and is suitable for huge state spaces but the runtime can become exponential
in the worst cases.
(i) m m
(i)
xj − µj 1 X (i) 1 X (i)
xj ← where µj = xj and σj2 = (xj − µj )2
σj m m
i=1 i=1
m
1 X T
• Step 2: Compute Σ = x(i) x(i) ∈ Rn×n , which is symmetric with real eigenvalues.
m
i=1
• Step 3: Compute u1 , ..., uk ∈ Rn the k orthogonal principal eigenvectors of Σ, i.e. the

orthogonal eigenvectors of the k largest eigenvalues.
• Step 4: Project the data on spanR (u1 ,...,uk ). This procedure maximizes the variance
r Search problem – A search problem is defined with:
among all k-dimensional spaces.
• a starting state sstart
• possible actions Actions(s) from state s
• action cost Cost(s,a) from state s with action a
• successor Succ(s,a) of state s after action a
• whether an end state was reached IsEnd(s)
The objective is to find a path that minimizes the cost.

r Backtracking search – Backtracking search is a naive recursive algorithm that tries all
possibilities to find the minimum cost path. Here, action costs can be either positive or negative.
r Breadth-first search (BFS) – Breadth-first search is a graph search algorithm that does a
level-by-level traversal. We can implement it iteratively with the help of a queue that stores at

each step future nodes to be visited. For this algorithm, we can assume action costs to be equal r Graph – A graph is comprised of a set of vertices V (also called nodes) as well as a set of
to a constant c > 0. edges E (also called links).
r Depth-first search (DFS) – Depth-first search is a search algorithm that traverses a graph
by following each path as deep as it can. We can implement it recursively, or iteratively with
the help of a stack that stores at each step future nodes to be visited. For this algorithm, action Remark: a graph is said to be acylic when there is no cycle.
costs are assumed to be equal to 0.
r State – A state is a summary of all past actions sufficient to choose future actions optimally.
r Dynamic programming – Dynamic programming (DP) is a backtracking search algorithm

with memoization (i.e. partial results are saved) whose goal is to find a minimum cost path from
state s to an end state send . It can potentially have exponential savings compared to traditional
graph search algorithms, and has the property to only work for acyclic graphs. For any given
state s, the future cost is computed as follows:
0 if IsEnd(s)

FutureCost(s) =

min Cost(s,a) + FutureCost(Succ(s,a)) otherwise
a∈Actions(s)
r Iterative deepening – The iterative deepening trick is a modification of the depth-first

search algorithm so that it stops after reaching a certain depth, which guarantees optimality
when all action costs are equal. Here, we assume that action costs are equal to a constant c > 0.
r Tree search algorithms summary – By noting b the number of actions per state, d the
solution depth, and D the maximum depth, we have:
Algorithm Action costs Space Time
Backtracking search any O(D) O(bD )
Breadth-first search c>0 O(bd ) O(bd )
Depth-first search 0 O(D) O(bD )
DFS-Iterative deepening c>0 O(d) O(bd )
Remark: the figure above illustrates a bottom-to-top approach whereas the formula provides the
2.1.2 Graph search intuition of a top-to-bottom problem resolution.
This category of states-based algorithms aims at constructing optimal paths, enabling exponen- r Types of states – The table below presents the terminology when it comes to states in the
tial savings. In this section, we will focus on dynamic programming and uniform cost search. context of uniform cost search:

State Explanation • decreases the estimated cost of each state-action of the true minimizing path y given by
the training data,
States for which the optimal path has
Explored E • increases the estimated cost of each state-action of the current predicted path y 0 inferred
already been found
from the learned weights.
States seen for which we are still figuring out
Frontier F
how to get there with the cheapest cost Remark: there are several versions of the algorithm, one of which simplifies the problem to only
learning the cost of each action a, and the other parametrizes Cost(s,a) to a feature vector of
Unexplored U States not seen yet learnable weights.
r Uniform cost search – Uniform cost search (UCS) is a search algorithm that aims at finding 2.1.4 A? search
the shortest path from a state sstart to an end state send . It explores states s in increasing order
of PastCost(s) and relies on the fact that all action costs are non-negative. r Heuristic function – A heuristic is a function h over states s, where each h(s) aims at
estimating FutureCost(s), the cost of the path from s to send .
r Algorithm – A∗ is a search algorithm that aims at finding the shortest path from a state s to
an end state send . It explores states s in increasing order of PastCost(s) + h(s). It is equivalent
to a uniform cost search with edge costs Cost0 (s,a) given by:
Cost0 (s,a) = Cost(s,a) + h(Succ(s,a)) − h(s)
Remark: this algorithm can be seen as a biased version of UCS exploring states estimated to be
closer to the end state.
Remark 1: the UCS algorithm is logically equivalent to Djikstra’s algorithm.
Remark 2: the algorithm would not work for a problem with negative action costs, and adding a r Consistency – A heuristic h is said to be consistent if it satisfies the two following properties:
positive constant to make them non-negative would not solve the problem since this would end
up being a different problem. • For all states s and actions a,
r Correctness theorem – When a state s is popped from the frontier F and moved to explored h(s) 6 Cost(s,a) + h(Succ(s,a))
set E, its priority is equal to PastCost(s) which is the minimum cost path from sstart to s.
r Graph search algorithms summary – By noting N the number of total states, n of which
are explored before the end state send , we have:
Algorithm Acyclicity Costs Time/space

Dynamic programming yes any O(N )
Uniform cost search no c>0 O(n log(n))
Remark: the complexity countdown supposes the number of possible actions per state to be
constant. • The end state verifies the following:
h(send ) = 0
2.1.3 Learning costs
Suppose we are not given the values of Cost(s,a), we want to estimate these quantities from a
training set of minimizing-cost-path sequence of actions (a1 , a2 , ..., ak ).
r Structured perceptron – The structured perceptron is an algorithm aiming at iteratively

learning the cost of each state-action pair. At each step, it:

r Correctness – If h is consistent, then A∗ returns the minimum cost path. r Max heuristic – Let h1 (s), h2 (s) be two heuristics. We have the following property:
r Admissibility – A heuristic h is said to be admissible if we have: h1 (s), h2 (s) consistent =⇒ h(s) = max{h1 (s), h2 (s)} consistent
h(s) 6 FutureCost(s)
2.2 Markov decision processes
r Theorem – Let h(s) be a given heuristic. We have:
In this section, we assume that performing action a from state s can lead to several states s01 ,s02 ,...
h(s) consistent =⇒ h(s) admissible in a probabilistic manner. In order to find our way between an initial state and an end state,
our objective will be to find the maximum value policy by using Markov decision processes that
help us cope with randomness and uncertainty.
r Efficiency – A∗ explores all states s satisfying the following equation:
PastCost(s) 6 PastCost(send ) − h(s) 2.2.1 Notations
r Definition – The objective of a Markov decision process is to maximize rewards. It is defined
with:
• a starting state sstart
• transition probabilities T (s,a,s0 ) from s to s0 with action a
• rewards Reward(s,a,s0 ) from s to s0 with action a
Remark: larger values of h(s) is better as this equation shows it will restrict the set of states s • whether an end state was reached IsEnd(s)
going to be explored.
• a discount factor 0 6 γ 6 1
2.1.5 Relaxation
It is a framework for producing consistent heuristics. The idea is to find closed-form reduced
costs by removing constraints and use them as heuristics.
r Relaxed search problem – The relaxation of search problem P with costs Cost is noted
Prel with costs Costrel , and satisfies the identity:
Costrel (s,a) 6 Cost(s,a)
r Relaxed heuristic – Given a relaxed search problem Prel , we define the relaxed heuristic
h(s) = FutureCostrel (s) as the minimum cost path from s to an end state in the graph of costs
Costrel (s,a). r Transition probabilities – The transition probability T (s,a,s0 ) specifies the probability
of going to state s0 after action a is taken in state s. Each s0 7→ T (s,a,s0 ) is a probability
r Consistency of relaxed heuristics – Let Prel be a given relaxed problem. By theorem, we distribution, which means that:
have: X
h(s) = FutureCostrel (s) =⇒ h(s) consistent ∀s,a, T (s,a,s0 ) = 1
s0 ∈ States
r Tradeoff when choosing heuristic – We have to balance two aspects in choosing a heuristic:
r Policy – A policy π is a function that maps each state s to an action a, i.e.
• Computational efficiency: h(s) = FutureCostrel (s) must be easy to compute. It has to π : s 7→ a
produce a closed form, easier search and independent subproblems.
• Good enough approximation: the heuristic h(s) should be close to FutureCost(s) and we r Utility – The utility of a path (s0 , ..., sk ) is the discounted sum of the rewards on that path.
have thus to not remove too many constraints. In other words,

k
X
X Qopt (s,a) = T (s,a,s0 ) Reward(s,a,s0 ) + γVopt (s0 )
u(s0 ,...,sk ) = ri γ i−1
s0 ∈ States
i=1
r Optimal value – The optimal value Vopt (s) of state s is defined as being the maximum value
attained by any policy. It is computed as follows:
Vopt (s) = max Qopt (s,a)
a∈ Actions(s)
Remark: the figure above is an illustration of the case k = 4. r Optimal policy – The optimal policy πopt is defined as being the policy that leads to the
optimal values. It is defined by:
r Q-value – The Q-value of a policy π by taking action a from state s, also noted Qπ (s,a), is
the expected utility of taking action a from state s and then following policy π. It is defined as ∀s, πopt (s) = argmax Qopt (s,a)
follows: a∈ Actions(s)
X
Qπ (s,a) = T (s,a,s0 ) Reward(s,a,s0 ) + γVπ (s0 )
r Value iteration – Value iteration is an algorithm that finds the optimal value Vopt as well
s0 ∈ States
as the optimal policy πopt . It is done as follows:
r Value of a policy – The value of a policy π from state s, also noted Vπ (s), is the expected • Initialization: for all states s, we have
utility by following policy π from state s over random paths. It is defined as follows:
(0)
Vopt (s) ←− 0
Vπ (s) = Qπ (s,π(s))
• Iteration: for t from 1 to TVI , we have

Remark: Vπ (s) is equal to 0 if s is an end state.
(t) (t−1)
∀s, Vopt (s) ←− max Qopt (s,a)
a∈ Actions(s)
2.2.2 Applications
with
r Policy evaluation – Given a policy π, policy evaluation is an iterative algorithm that com-
putes Vπ . It is done as follows: X h i
(t−1) (t−1)
Qopt (s,a) = T (s,a,s0 ) Reward(s,a,s0 ) + γVopt (s0 )
• Initialization: for all states s, we have s0 ∈ States
(0)
Vπ (s) ←− 0
Remark: if we have either γ < 1 or the MDP graph being acyclic, then the value iteration
algorithm is guaranteed to converge to the correct answer.
• Iteration: for t from 1 to TPE , we have
∀s,
(t)
Vπ (s) ←− Qπ
(t−1)
(s,π(s)) 2.2.3 When unknown transitions and rewards
Now, let’s assume that the transition probabilities and the rewards are unknown.
with
r Model-based Monte Carlo – The model-based Monte Carlo method aims at estimating
T (s,a,s0 ) and Reward(s,a,s0 ) using Monte Carlo simulation with:
X h i
(t−1) 0 0 (t−1) 0
Qπ (s,π(s)) = T (s,π(s),s ) Reward(s,π(s),s ) + γVπ (s )
# times (s,a,s0 ) occurs
s0 ∈ States Tb(s,a,s0 ) =
# times (s,a) occurs
Remark: by noting S the number of states, A the number of actions per state, S 0 the number and
of successors and T the number of iterations, then the time complexity is of O(TPE SS 0 ).
0
Reward(s,a,s
\ ) = r in (s,a,r,s0 )
r Optimal Q-value – The optimal Q-value Qopt (s,a) of state s with action a is defined to be
the maximum Q-value attained by any policy starting. It is computed as follows: These estimations will be then used to deduce Q-values, including Qπ and Qopt .

Remark: model-based Monte Carlo is said to be off-policy, because the estimation does not • a starting state sstart
depend on the exact policy.
r Model-free Monte Carlo – The model-free Monte Carlo method aims at directly estimating
Qπ , as follows: • successors Succ(s,a) from states s with actions a
bπ (s,a) = average of ut where st−1 = s, at = a
Q • whether an end state was reached IsEnd(s)
where ut denotes the utility starting at step t of a given episode.
• the agent’s utility Utility(s) at end state s
Remark: model-free Monte Carlo is said to be on-policy, because the estimated value is dependent
on the policy π used to generate the data. • the player Player(s) who controls state s
r Equivalent formulation – By introducing the constant η = 1
and for
1+(#updates to (s,a))
Remark: we will assume that the utility of the agent has the opposite sign of the one of the
each (s,a,u) of the training set, the update rule of model-free Monte Carlo has a convex combi- opponent.
nation formulation:
r Types of policies – There are two types of policies:
bπ (s,a) ← (1 − η)Q
Q bπ (s,a) + ηu
as well as a stochastic gradient formulation: • Deterministic policies, noted πp (s), which are actions that player p takes in state s.
bπ (s,a) ← Q
Q bπ (s,a) − η(Q
bπ (s,a) − u) • Stochastic policies, noted πp (s,a) ∈ [0,1], which are probabilities that player p takes action
a in state s.
r SARSA – State-action-reward-state-action (SARSA) is a boostrapping method estimating

Qπ by using both raw data and estimates as part of the update rule. For each (s,a,r,s0 ,a0 ), we r Expectimax – For a given state s, the expectimax value Vexptmax (s) is the maximum expected
have: utility of any agent policy when playing with respect to a fixed and known opponent policy πopp .
h i It is computed as follows:
bπ (s,a) ←− (1 − η)Q
Q bπ (s0 ,a0 )
bπ (s,a) + η r + γ Q
Utility(s) IsEnd(s)

max Vexptmax (Succ(s,a)) Player(s) = agent



Remark: the SARSA estimate is updated on the fly as opposed to the model-free Monte Carlo a∈Actions(s)
one where the estimate can only be updated at the end of the episode. Vexptmax (s) = X

 πopp (s,a)Vexptmax (Succ(s,a)) Player(s) = opp
r Q-learning – Q-learning is an off-policy algorithm that produces an estimate for Qopt . On

a∈Actions(s)
each (s,a,r,s0 ,a0 ), we have:
h i
bopt (s,a) ← (1 − η)Q
Q bopt (s,a) + η r + γ max bopt (s0 ,a0 )
Q Remark: expectimax is the analog of value iteration for MDPs.
a0 ∈ Actions(s0 )
r Epsilon-greedy – The epsilon-greedy policy is an algorithm that balances exploration with

probability and exploitation with probability 1 − . For a given state s, the policy πact is
computed as follows:

argmax Q
bopt (s,a) with proba 1 −
πact (s) = a∈ Actions
random from Actions(s) with proba
2.3 Game playing

In games (e.g. chess, backgammon, Go), other agents are present and need to be taken into r Minimax – The goal of minimax policies is to find an optimal policy against an adversary
account when constructing our policy. by assuming the worst case, i.e. that the opponent is doing everything to minimize the agent’s
utility. It is done as follows:
r Game tree – A game tree is a tree that describes the possibilities of a game. In particular,
each node is a decision point for a player and each root-to-leaf path is a possible outcome of the  Utility(s) IsEnd(s)

game. max Vminimax (Succ(s,a)) Player(s) = agent
Vminimax (s) = a∈Actions(s)
r Two-player zero-sum game – It is a game where each state is fully observed and such that  min Vminimax (Succ(s,a)) Player(s) = opp
players take turns. It is defined with: a∈Actions(s)

Remark: we can extract πmax and πmin from the minimax value Vminimax .
r TD learning – Temporal difference (TD) learning is used when we don’t know the transi-
tions/rewards. The value is based on exploration policy. To be able to use it, we need to know
r Minimax properties – By noting V the value function, there are 3 properties around rules of the game Succ(s,a). For each (s,a,r,s0 ), the update is done as follows:
minimax to have in mind:

w ←− w − η V (s,w) − (r + γV (s0 ,w)) ∇w V (s,w)
• Property 1 : if the agent were to change its policy to any πagent , then the agent would be
no better off.
∀πagent , V (πmax ,πmin ) > V (πagent ,πmin ) 2.3.2 Simultaneous games
This is the contrary of turn-based games, where there is no ordering on the player’s moves.
• Property 2 : if the opponent changes its policy from πmin to πopp , then he will be no
better off. r Single-move simultaneous game – Let there be two players A and B, with given possible
actions. We note V (a,b) to be A’s utility if A chooses action a, B chooses action b. V is called
∀πopp , V (πmax ,πmin ) 6 V (πmax ,πopp ) the payoff matrix.
r Strategies – There are two main types of strategies:

• Property 3 : if the opponent is known to be not playing the adversarial policy, then the
minimax policy might not be optimal for the agent. • A pure strategy is a single action:
∀π, V (πmax ,π) 6 V (πexptmax ,π) a ∈ Actions
In the end, we have the following relationship: • A mixed strategy is a probability distribution over actions:
V (πexptmax ,πmin ) 6 V (πmax ,πmin ) 6 V (πmax ,π) 6 V (πexptmax ,π) ∀a ∈ Actions, 0 6 π(a) 6 1
2.3.1 Speeding up minimax r Game evaluation – The value of the game V (πA ,πB ) when player A follows πA and player
B follows πB is such that:
r Evaluation function – An evaluation function is a domain-specific and approximate estimate X
of the value Vminimax (s). It is noted Eval(s). V (πA ,πB ) = πA (a)πB (b)V (a,b)
Remark: FutureCost(s) is an analogy for search problems. a,b
r Alpha-beta pruning – Alpha-beta pruning is a domain-general exact method optimizing

the minimax algorithm by avoiding the unnecessary exploration of parts of the game tree. To do r Minimax theorem – By noting πA ,πB ranging over mixed strategies, for every simultaneous
so, each player keeps track of the best value they can hope for (stored in α for the maximizing two-player zero-sum game with a finite number of actions, we have:
player and in β for the minimizing player). At a given step, the condition β < α means that the
optimal path is not going to be in the current branch as the earlier player had a better option max min V (πA ,πB ) = min max V (πA ,πB )
at their disposal. πA πB πB πA

2.3.3 Non-zero-sum games 3 Variables-based models

r Payoff matrix – We define Vp (πA ,πB ) to be the utility for player p.
3.1 Constraint satisfaction problems
r Nash equilibrium – A Nash equilibrium is (πA
∗ ,π ∗ ) such that no player has an incentive to
change its strategy. We have:

B In this section, our objective is to find maximum weight assignments of variable-based models.
One advantage compared to states-based models is that these algorithms are more convenient
∗
∀πA , VA (πA ∗
,πB ∗
) > VA (πA ,πB ) and ∗
∀πB , VB (πA ∗
,πB ∗
) > VB (πA ,πB ) to encode problem-specific constraints.
Remark: in any finite-player game with finite number of actions, there exists at least one Nash 3.1.1 Factor graphs
equilibrium.
r Definition – A factor graph, also referred to as a Markov random field, is a set of variables
X = (X1 ,...,Xn ) where Xi ∈ Domaini and m factors f1 ,...,fm with each fj (X) > 0.
r Scope and arity – The scope of a factor fj is the set of variables it depends on. The size of
this set is called the arity.
Remark: factors of arity 1 and 2 are called unary and binary respectively.
r Assignment weight – Each assignment x = (x1 ,...,xn ) yields a weight Weight(x) defined as
being the product of all factors fj applied to that assignment. Its expression is given by:
m
Y
Weight(x) = fj (x)
j=1
r Constraint satisfaction problem – A constraint satisfaction problem (CSP) is a factor

graph where all factors are binary; we call them to be constraints:
∀j ∈ [[1,m]], fj (x) ∈ {0,1}
Here, the constraint j with assignment x is said to be satisfied if and only if fj (x) = 1.
r Consistent assignment – An assignment x of a CSP is said to be consistent if and only if

Weight(x) = 1, i.e. all constraints are satisfied.
3.1.2 Dynamic ordering

r Dependent factors – The set of dependent factors of variable Xi with partial assignment x
is called D(x,Xi ), and denotes the set of factors that link Xi to already assigned variables.

r Backtracking search – Backtracking search is an algorithm used to find maximum weight 3.1.3 Approximate methods
assignments of a factor graph. At each step, it chooses an unassigned variable and explores
its values by recursion. Dynamic ordering (i.e. choice of variables and values) and lookahead r Beam search – Beam search is an approximate algorithm that extends partial assignments
(i.e. early elimination of inconsistent options) can be used to explore the graph more efficiently, of n variables of branching factor b = |Domain| by exploring the K top paths at each step. The
although the worst-case runtime stays exponential: O(|Domain|n ). beam size K ∈ {1,...,bn } controls the tradeoff between efficiency and accuracy. This algorithm
has a time complexity of O(n · Kb log(Kb)).
r Forward checking – It is a one-step lookahead heuristic that preemptively removes incon- The example below illustrates a possible beam search of parameters K = 2, b = 3 and n = 5.
sistent values from the domains of neighboring variables. It has the following characteristics:
• After assigning a variable Xi , it eliminates inconsistent values from the domains of all its
neighbors.
• If any of these domains becomes empty, we stop the local backtracking search.
• If we un-assign a variable Xi , we have to restore the domain of its neighbors.
r Most constrained variable – It is a variable-level ordering heuristic that selects the next
unassigned variable that has the fewest consistent values. This has the effect of making incon-
sistent assignments to fail earlier in the search, which enables more efficient pruning.
r Least constrained value – It is a value-level ordering heuristic that assigns the next value
that yields the highest number of consistent values of neighboring variables. Intuitively, this
procedure chooses first the values that are most likely to work.
Remark: in practice, this heuristic is useful when all factors are constraints.
Remark: K = 1 corresponds to greedy search whereas K → +∞ is equivalent to BFS tree search.
r Iterated conditional modes – Iterated conditional modes (ICM) is an iterative approximate

algorithm that modifies the assignment of a factor graph one variable at a time until convergence.
At step i, we assign to Xi the value v that maximizes the product of all factors connected to
that variable.
Remark: ICM may get stuck in local minima.
r Gibbs sampling – Gibbs sampling is an iterative approximate method that modifies the
assignment of a factor graph one variable at a time until convergence. At step i:
• we assign to each element u ∈ Domaini a weight w(u) that is the product of all factors
connected to that variable,
• we sample v from the probability distribution induced by w and assign it to Xi .
The example above is an illustration of the 3-color problem with backtracking search coupled Remark: Gibbs sampling can be seen as the probabilistic counterpart of ICM. It has the advan-
with most constrained variable exploration and least constrained value heuristic, as well as tage to be able to escape local minima in most cases.
forward checking at each step.
r Arc consistency – We say that arc consistency of variable Xl with respect to Xk is enforced 3.1.4 Factor graph transformations
when for each xl ∈ Domainl :
• unary factors of Xl are non-zero, r Independence – Let A,B be a partitioning of the variables X. We say that A and B are
independent if there are no edges between A and B and we write:
• there exists at least one xk ∈ Domaink such that any factor between Xl and Xk is
non-zero. A,B independent ⇐⇒ A ⊥
⊥B
r AC-3 – The AC-3 algorithm is a multi-step lookahead heuristic that applies forward checking
to all relevant variables. After a given assignment, it performs forward checking and then Remark: independence is the key property that allows us to solve subproblems in parallel.
successively enforces arc consistency with respect to the neighbors of variables for which the
r Conditional independence – We say that A and B are conditionally independent given C
domain change during the process.
if conditioning on C produces a graph in which A and B are independent. In this case, it is
Remark: AC-3 can be implemented both iteratively and recursively. written:

A and B cond. indep. given C ⇐⇒ A ⊥

⊥ B|C
r Conditioning – Conditioning is a transformation aiming at making variables independent

that breaks up a factor graph into smaller pieces that can be solved in parallel and can use
backtracking. In order to condition on a variable Xi = v, we do as follows:
• Consider all factors f1 ,...,fk that depend on Xi
• Remove Xi and f1 ,...,fk
• Add gj (x) for j ∈ {1,...,k} defined as:
gj (x) = fj (x ∪ {Xi : v}) Remark: finding the best variable ordering is a NP-hard problem.
r Markov blanket – Let A ⊆ X be a subset of variables. We define MarkovBlanket(A) to be 3.2 Bayesian networks
the neighbors of A that are not in A.
In this section, our goal will be to compute conditional probabilities. What is the probability of
r Proposition – Let C = MarkovBlanket(A) and B = X\(A ∪ C). Then we have: a query given evidence?
A⊥
⊥ B|C
3.2.1 Introduction
r Explaining away – Suppose causes C1 and C2 influence an effect E. Conditioning on the
effect E and on one of the causes (say C1 ) changes the probability of the other cause (say C2 ).
In this case, we say that C1 has explained away C2 .
r Directed acyclic graph – A directed acyclic graph (DAG) is a finite directed graph with
no directed cycles.
r Bayesian network – A Bayesian network is a directed acyclic graph (DAG) that specifies
a joint distribution over random variables X = (X1 ,...,Xn ) as a product of local conditional
distributions, one for each node:
r Elimination – Elimination is a factor graph transformation that removes Xi from the graph
and solves a small subproblem conditioned on its Markov blanket as follows: n
Y
P (X1 = x1 ,...,Xn = xn ) , p(xi |xParents(i) )
• Consider all factors fi,1 ,...,fi,k that depend on Xi
i=1
• Remove Xi and fi,1 ,...,fi,k
• Add fnew,i (x) defined as: Remark: Bayesian networks are factor graphs imbued with the language of probability.
k
Y
fnew,i (x) = max fi,l (x)
xi
l=1
r Treewidth – The treewidth of a factor graph is the maximum arity of any factor created by
variable elimination with the best variable ordering. In other words,
Treewidth = min max arity(fnew,i )
orderings i∈{1,...,n}
r Locally normalized – For each xParents(i) , all factors are local conditional distributions.
The example below illustrates the case of a factor graph of treewidth 3. Hence they have to satisfy:

Hto ∼ o )
p(Hto |Ht−1
o∈{a,b} Multiple object
Factorial HMM
tracking
X
p(xi |xParents(i) ) = 1 Et ∼ p(Et |Hta ,Htb )
xi
Y ∼ p(Y ) Document
As a result, sub-Bayesian networks and conditional distributions are consistent. Naive Bayes
Wi ∼ p(Wi |Y ) classification
Remark: local conditional distributions are the true conditional distributions.
r Marginalization – The marginalization of a leaf node yields a Bayesian network without

that node.
α ∈ RK distribution
Latent Dirichlet
Zi ∼ p(Zi |α) Topic modeling
Allocation (LDA)
Wi ∼ p(Wi |Zi )
3.2.2 Probabilistic programs
r Concept – A probabilistic program randomizes variables assignment. That way, we can write 3.2.3 Inference
down complex Bayesian networks that generate assignments without us having to explicitly
specify associated probabilities. r General probabilistic inference strategy – The strategy to compute the probability
P (Q|E = e) of query Q given evidence E = e is as follows:
Remark: examples of probabilistic programs include Hidden Markov model (HMM), factorial
HMM, naive Bayes, latent Dirichlet allocation, diseases and symptoms and stochastic block • Step 1: Remove variables that are not ancestors of the query Q or the evidence E by
models.
marginalization
r Summary – The table below summarizes the common probabilistic programs as well as their • Step 2: Convert Bayesian network to factor graph
applications:
• Step 3: Condition on the evidence E = e
• Step 4: Remove nodes disconnected from the query Q by marginalization
Program Algorithm Illustration Example • Step 5: Run probabilistic inference algorithm (manual, variable elimination, Gibbs sam-
pling, particle filtering)
Language r Forward-backward algorithm – This algorithm computes the exact value of P (H = hk |E =

Markov Model Xi ∼ p(Xi |Xi−1 )
modeling e) (smoothing query) for any k ∈ {1, ..., L} in the case of an HMM of size L. To do so, we proceed
in 3 steps:
• Step 1: for i ∈ {1,..., L}, compute Fi (hi ) = Fi−1 (hi−1 )p(hi |hi−1 )p(ei |hi )
P
hi−1
Hidden Markov Ht ∼ p(Ht |Ht−1 ) • Step 2: for i ∈ {L,..., 1}, compute Bi (hi ) = Bi+1 (hi+1 )p(hi+1 |hi )p(ei+1 |hi+1 )
P
Object tracking hi+1
Model (HMM) Et ∼ p(Et |Ht )
• Step 3: for i ∈ {1,...,L}, compute Si (hi ) = PFi (hi )Bi (hi )
Fi (hi )Bi (hi )
hi

with the convention F0 = BL+1 = 1. From this procedure and these notations, we get that 4 Logic-based models
P (H = hk |E = e) = Sk (hk )
4.1 Concepts
Remark: this algorithm interprets each assignment to be a path where each edge hi−1 → hi is
In this section, we will go through logic-based models that use logical formulas and inference
of weight p(hi |hi−1 )p(ei |hi ). rules. The idea here is to balance expressivity and computational efficiency.
r Gibbs sampling – This algorithm is an iterative approximate method that uses a small set of r Syntax of propositional logic – By noting f,g formulas, and ¬, ∧, ∨, →, ↔ connectives,
assignments (particles) to represent a large probability distribution. From a random assignment here are the following possible logical expressions that we can write:
x, Gibbs sampling performs the following steps for i ∈ {1,...,n} until convergence:
• For all u ∈ Domaini , compute the weight w(u) of assignment x where Xi = u Name Symbol Translation
• Sample v from the probability distribution induced by w: v ∼ P (Xi = v|X−i = x−i ) Negation ¬f not f
Conjunction f ∧g f and g
• Set Xi = v
Disjunction f ∨g f or g
Remark: X−i denotes X\{Xi } and x−i represents the corresponding assignment.
Implication f →g if f then g
r Particle filtering – This algorithm approximates the posterior density of state variables
given the evidence of observation variables by keeping track of K particles at a time. Starting Biconditional f ↔g f , that is to say g
from a set of particles C of size K, we run the following 3 steps iteratively:
• Step 1: proposal - For each old particle xt−1 ∈ C, sample x from the transition probability Remark: formulas can be built up recursively.
distribution p(x|xt−1 ) and add x to a set C 0 . r Model – A model w is an assignment of truth values to propositional symbols.
• Step 2: weighting - Weigh each x of the set C 0 by w(x) = p(et |x), where et is the evidence r Interpretation function – Let f be a formula, w be a model, then the interpretation function
observed at time t. I(f,w) is such that:
• Step 3: resampling - Sample K elements from the set C 0 using the probability distribution I(f,w) ∈ {0,1}
induced by w and store them in C: these are the current particles xt .
Remark: a more expensive version of this algorithm also keeps track of past particles in the r Set of models – We note M(f ) the set of models w for which we have:
proposal step. ∀w ∈ M(f ), I(f,w) = 1
r Maximum likelihood – If we don’t know the local conditional distributions, we can learn
them using maximum likelihood.
r Knowledge base – A knowledge base KB is defined to be a set of formulas representing their
Y conjunction, as follows:
max p(X = x; θ)
θ \
x∈Dtrain M(KB) = M(f )
f ∈KB
r Laplace smoothing – For each distribution d and partial assignment (xParents(i) ,xi ), add λ
to countd (xParents(i) ,xi ), then normalize to get probability estimates.
r Entailment – A knowledge base KB that is said to entail f is noted KB |= f . We have:
r Algorithm – The Expectation-Maximization (EM) algorithm gives an efficient method at KB |= f ⇐⇒ M(KB) ⊆ M(f )
estimating the parameter θ through maximum likelihood estimation by repeatedly constructing
a lower-bound on the likelihood (E-step) and optimizing that lower bound (M-step) as follows:
r Contradiction – A knowledge base KB contradicts f if and only if we have the following:
• E-step: Evaluate the posterior probability q(h) that each data point e came from a
particular cluster h as follows: M(KB) ∩ M(f ) = ∅
q(h) = P (H = h|E = e; θ) Remark: KB contradicts f if and only if KB entails ¬f .

r Contingency – f is said to be contingent when there is a non-trivial overlap between the
• M-step: Use the posterior probabilities q(h) as cluster specific weights on data points e models of KB and f , i.e. when we have:
to determine θ through maximum likelihood.
∅ ( M(KB) ∩ M(f ) ( M(KB)

Remark: we can quantify the uncertainty of the overlap of the two by computing the following r Resolution inference rule – The resolution inference rule is a generalized inference rule
quantity: and is written as follows:
X
P (W = w) f1 ∨ ... ∨ fn ∨ p, ¬p ∨ g1 ∨ ... ∨ gm
w∈M(KB∪{f }) f1 ∨ ... ∨ fn ∨ g1 ∨ ... ∨ gm
P (f |KB) = X
P (W = w) Remark: this can take exponential time.
w∈M(KB)
r Conjunctive normal form – A conjunctive normal form (CNF) formula is a conjunction of
clauses.
r Satisfiability – A knowledge base KB is said to be satisfiable if we have:
r Conversion to CNF – Every formula f in propositional logic can be converted into an
M(KB) 6= ∅ equivalent CNF formula f 0 :
M(f ) = M(f 0 )
r Model checking – Model checking is an algorithm that takes as input a knowledge base KB
and checks whether we have M(KB) 6= ∅.
r Resolution-based inference – The resolution-based inference algorithm follows the follow-
Remark: popular model checking algorithms include DPLL and WalkSat. ing steps:
r Modus ponens inference rule – For any propositional symbols p1 ,...,pk ,q, we have: • Step 1: Convert all formulas into CNF
p1 ,...,pk , (p1 ∧ ... ∧ pk ) −→ q • Step 2: Repeatedly apply resolution rule
q
• Step 3: Return unsatisfiable if and only if derive false
Remark: this can take linear time.
r Inference rule – If f1 ,...,fk ,g are formulas, then the following is an inference rule: 4.3 First-order logic
f1 ,...,fk The idea here is that variables yield compact knowledge representations.
g
r Model – A model w in first-order logic maps:
r Derivation – We say that KB derives f , and we note KB ` f , if and only if f eventually gets • constant symbols to objects
added to KB. • predicate symbols to tuple of objects
r Soundness/completeness – A set of inference rules can have the following properties:
r Definite clause – By noting x1 ,...,xn variables and a1 ,...,ak ,b atomic formulas, a definite
Property Meaning clause has the following form:
Soundness {f : KB ` f } ⊆ {f : KB |= f } ∀x1 ,...,∀xn , (a1 ∧ ... ∧ ak ) → b
Completeness {f : KB ` f } ⊇ {f : KB |= f }
r Modus ponens – By noting x1 ,...,xn variables and a1 ,...,ak ,b atomic formulas, a modus
ponens has the following form:
a1 ,...,ak ∀x1 ,...,∀xn (a1 ∧ ... ∧ ak ) → b
b
4.2 Propositional logic
r Definite clause – By noting p1 ,...,pk ,q propositional symbols, we define a definite clause as r Substitution – A substitution θ is a mapping from variables to terms. For instance,
having the following form: Subst(θ,f ) returns the result of performing substitution θ on f .
(p1 ∧ ... ∧ pk ) −→ q r Unification – Unification takes two formulas f and g and returns a substitution θ which is
the most general unifier:
Remark: the case when q = false is called a goal clause. Unify[f,g] = θ s.t. Subst[θ,f ] = Subst[θ,g]
r Horn clause – A Horn clause is defined to be either a definite clause or a goal clause. or fail if no such θ exists.
r Modus ponens on Horn clauses – Modus ponens is complete with respect to Horn clauses r Completeness – Modus ponens is complete for first-order logic with only Horn clauses.
if we suppose that KB contains only Horn clauses and p is an entailed propositional symbol.
Applying modus ponens will then derive p. r Semi-decidability – First-order logic, even restricted to only Horn clauses, is semi-decidable.

• if KB |= f , forward inference on complete inference rules will prove f in finite time
• if KB 6|= f , no algorithm can show this in finite time
r Resolution rule – By noting θ = Unify(p,q), we have:

f1 ∨ ... ∨ fn ∨ p, ¬q ∨ g1 ∨ ... ∨ gm
Subst[θ,f1 ∨ ... ∨ fn ∨ g1 ∨ ... ∨ gm ]

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CS 229 – Machine Learning https://stanford.

Stanford University 1 Fall 2018

Discriminative model Generative model

θopt = arg max L(θ)

1.3.1 Linear regression

We assume here that y|x; θ ∼ N (µ,σ 2 )

Linear regression Logistic regression SVM Neural Network θ = (X T X)−1 X T y

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(1) y|x; θ ∼ ExpFamily(η) (2) hθ (x) = E[y|x; θ] (3) η = θT x

1.3.3 Generalized Linear Models

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r Lagrangian – We define the Lagrangian L(w,b) as follows:

Stanford University 4 Fall 2018

∃h ∈ H, ∀i ∈ [[1,d]], h(x(i) ) = y (i)

r VC dimension – The Vapnik-Chervonenkis (VC) dimension of a given infinite hypothesis

1.8 Learning Theory

P (A1 ∪ ... ∪ Ak ) 6 P (A1 ) + ... + P (Ak )

Remark: this inequality is also known as the Chernoff bound.

r Training error – For a given classifier h, we define the training error b

r Probably Approximately Correct (PAC) – PAC is a framework under which numerous

• the training and testing sets follow the same distribution

• the training examples are drawn independently

Stanford University 5 Fall 2018

2 Unsupervised Learning 2.2.2 k-means clustering

Setting Latent variable z x|z Comments

Mixture of k Gaussians Multinomial(φ) N (µj ,Σj ) µj ∈ Rn , φ ∈ Rk

Factor analysis N (0,I) N (µ + Λz,ψ) µj ∈ Rn

2.2.4 Clustering assessment metrics

Stanford University 6 Fall 2018

r Calinski-Harabaz index – By noting k the number of clusters, Bk and Wk the between

• Step 3: Compute u1 , ..., uk ∈ Rn the k orthogonal principal eigenvectors of Σ, i.e. the

Stanford University 7 Fall 2018

• Step 2: Perform forward propagation to obtain the corresponding loss.

• Step 3: Backpropagate the loss to get the gradients.

• Step 4: Use the gradients to update the weights of the network.

r Batch normalization – It is a step of hyperparameter γ, β that normalizes the batch {xi }.

Input gate Forget gate Output gate Gate

Stanford University 8 Fall 2018

• A is the set of actions

• γ ∈ [0,1[ is the discount factor

• R : S × A −→ R or R : S −→ R is the reward function that the algorithm wants to

r Policy – A policy π is a function π : S −→ A that maps states to actions.

r Value iteration algorithm – The value iteration algorithm is in two steps:

• We initialize the value:

• We iterate the value based on the values before:

Stanford University 9 Fall 2018

4 Machine Learning Tips and Tricks Metric Formula Equivalent

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4.2 Model selection LASSO Ridge Elastic Net

Training set Validation set Testing set

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5.1 Probabilities and Statistics

- Complexify model - Regularize

Remark: we note that for 0 6 r 6 n, we have P (n,r) > C(n,r).

5.1.2 Conditional Probability

Remark: we have P (A ∩ B) = P (A)P (B|A) = P (A|B)P (B).

Stanford University 12 Fall 2018

5.1.3 Random Variables

Remark: similarly, a matrix A is said to be positive definite, and is noted A 0, if it is a PSD

`(A,P,N ) = max (d(A,P ) − d(A,N ) + α,0) nc