Boosting the Confidence of Near-Tight Generalization Bounds for Uniformly Stable Randomized Algorithms

Xiaotong Yuan, Ping Li

Keywords: Uniform stability, Randomized learning algorithms, Bagging, Generalization bounds, Stochastic gradient methods

Abstract: High probability generalization bounds of uniformly stable learning algorithms have recently been actively studied with a series of near-tight results established by~\citet{feldman2019high,bousquet2020sharper}. However, for randomized algorithms with on-average uniform stability, such as stochastic gradient descent (SGD) with time decaying learning rates, it still remains less well understood if these deviation bounds still hold with high confidence over the internal randomness of algorithm. This paper addresses this open question and makes progress towards answering it inside a classic framework of confidence-boosting. To this end, we first establish an in-expectation first moment generalization error bound for randomized learning algorithm with on-average uniform stability, based on which we then show that a properly designed subbagging process leads to near-tight high probability generalization bounds over the randomness of data and algorithm. We further substantialize these generic results to SGD to derive improved high probability generalization bounds for convex or non-convex optimization with natural time decaying learning rates, which have not been possible to prove with the existing uniform stability results. Specially for deterministic uniformly stable algorithms, our confidence-boosting results improve upon the best known generalization bounds in terms of a logarithmic factor on sample size, which moves a step forward towards resolving an open question raised by~\citet{bousquet2020sharper}.

One-sentence Summary: A confidence-boosting method for deriving near-tight generalization bounds with high probability for uniformly stable randomized learning algorithms.