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On optimal universal first-order methods for minimizing heterogeneous sums

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Abstract

This work considers minimizing a sum of convex functions, each with potentially different structure ranging from nonsmooth to smooth, Lipschitz to non-Lipschitz. Nesterov’s universal fast gradient method  (Nesterov, Math Program 152:381–404, 2015) provides an optimal black-box first-order method for minimizing a single function that takes advantage of any continuity structure present without requiring prior knowledge. In this paper, we show that this landmark method (without modification) further adapts to heterogeneous sums. For example, it minimizes the sum of a nonsmooth M-Lipschitz function and an L-smooth function at a rate of \( O(M^2/\epsilon ^2 + \sqrt{L/\epsilon }) \) without knowledge of M, L, or even that the objective was a sum of two terms. This rate is precisely the sum of the optimal convergence rates for each term’s individual complexity class. More generally, we show sums of varied Hölder smooth functions introduce no new complexities and require at most as many iterations as is needed to minimize each summand separately. Extensions to strongly convex and growth/error bounds are also provided.

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Notes

  1. In particular, the considered methods require an oracle producing the function value and one (sub)gradient of F at the current iterate, to be called at each iteration.

  2. Namely, supposing all \((M_j,v_j) = (M/|\mathcal {J}|, v)\), the defining equation for K simplifies to

    $$\begin{aligned} |\mathcal {J}|\left( \frac{|\mathcal {J}|^{\frac{1-v}{1+v}}(M/|\mathcal {J}|)^{\frac{2}{1+v}}\xi (x_0,x^*)}{\epsilon ^{\frac{2}{1+v}}}K^{\frac{-1-3v}{1+v}}\right) = \frac{M^{\frac{2}{1+v}}\xi (x_0,x^*)}{\epsilon ^{\frac{2}{1+v}}}K^{\frac{-1-3v}{1+v}} = 1 \, \end{aligned}$$

    solved by \(K = \varTheta \left( \left( \frac{M}{\epsilon }\right) ^{\frac{2}{1+3v}}\xi (x_0,x^*)^{\frac{1+v}{1+3v}}\right) \), matching the optimal rate for (M, v)-Hölder smooth minimization.

  3. Note this comparison is somewhat superficial as we assume a more expensive oracle giving (sub)gradients of the whole objective. Given our oracle, considering all the terms as one \(L=\sum L_i\)-smooth term gives a faster \(O\left( \frac{\sqrt{\sum _i L_i}}{\sqrt{\sum _i\mu _i}} \log (1/\epsilon )\right) \) rate.

  4. Such spectrahedrons occur as the dual feasible region of MAX-CUT SDP relaxations [14].

  5. The original source for such a lower bound is difficult to identify in the literature, leaving this optimality somewhat as folklore. The theory of Grimmer [17] allows these bounds to be concluded from the classic lower bounds without growth.

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  1. Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, USA

    Benjamin Grimmer

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Grimmer, B. On optimal universal first-order methods for minimizing heterogeneous sums. Optim Lett 18, 427–445 (2024). https://doi.org/10.1007/s11590-023-02060-2

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