Nowadays, the development of most leading web services is controlled by online experiments that qualify and quantify the steady stream of their updates achieving more than a thousand concurrent experiments per day. Despite the increasing need for running more experiments, these services are limited in their user traffic. This situation leads to the problem of finding a new or improving existing key performance metric with a higher sensitivity and lower variance. We focus on the problem of variance reduction for engagement metrics of user loyalty that are widely used in A/B testing of web services. We develop a general framework that is based on evaluation of the mean difference between the actual and the approximated values of the key performance metric (instead of the mean of this metric). On the one hand, it allows us to incorporate the state-of-the-art techniques widely used in randomized experiments of clinical and social research, but limitedly used in online evaluation. On the other hand, we propose a new class of methods based on advanced machine learning algorithms, including ensembles of decision trees, that, to the best of our knowledge, have not been applied earlier to the problem of variance reduction. We validate the variance reduction approaches on a very large set of real large-scale A/B experiments run at Yandex for different engagement metrics of user loyalty. Our best approach demonstrates average variance reduction (which is equivalent to 63% saved user traffic) and detects the treatment effect in times more A/B experiments.