Non-Asymptotic Analysis of Stochastic Approximation Algorithms for Machine Learning

Francis Bach (1, 2) , Eric Moulines (3)

1 LIENS - Laboratoire d'informatique de l'école normale supérieure
2 SIERRA - Statistical Machine Learning and Parsimony
3 LTCI - Laboratoire Traitement et Communication de l'Information

In this paper, we consider the minimization of a convex objective function defined on a Hilbert space, which is only available through unbiased estimates of its gradients. This problem includes standard machine learning algorithms such as kernel logistic regression and least-squares regression, and is commonly referred to as a stochastic approximation problem in the operations research community. We provide a non-asymptotic analysis of the convergence of two well-known algorithms, stochastic gradient descent (a.k.a. Robbins-Monro algorithm) as well as a simple modification where iterates are averaged (a.k.a. Polyak-Ruppert averaging). Our analysis suggests that a learning rate proportional to the inverse of the number of iterations, while leading to the optimal convergence rate in the strongly convex case, is not robust to the lack of strong convexity or the setting of the proportionality constant. This situation is remedied when using slower decays together with averaging, robustly leading to the optimal rate of convergence. We illustrate our theoretical results with simulations on synthetic and standard datasets.

Domains: Machine Learning [cs.LG]; Optimization and Control [math.OC]; Machine Learning [stat.ML]; Signal and Image processing; Signal and Image Processing

Francis Bach, Eric Moulines. Non-Asymptotic Analysis of Stochastic Approximation Algorithms for Machine Learning. Neural Information Processing Systems (NIPS), 2011, Spain. ⟨hal-00608041⟩