Sharper Analysis for Minibatch Stochastic Proximal Point Methods: Stability, Smoothness, and Deviation

TL;DR

This paper introduces a minibatch stochastic proximal point method (M-SPP) for convex optimization, providing novel excess risk bounds, convergence rates, and high-probability error bounds, with empirical validation on Lasso and logistic regression.

Contribution

The paper develops a new M-SPP algorithm with theoretical excess risk bounds and convergence rates, improving understanding of noise impact and extending to sampling-without-replacement variants.

Findings

M-SPP achieves an $rac{1}{T^2}$ bias decay rate.

Variance decays at a rate of $rac{1}{nT}$.

Numerical experiments support theoretical predictions.

Abstract

The stochastic proximal point (SPP) methods have gained recent attention for stochastic optimization, with strong convergence guarantees and superior robustness to the classic stochastic gradient descent (SGD) methods showcased at little to no cost of computational overhead added. In this article, we study a minibatch variant of SPP, namely M-SPP, for solving convex composite risk minimization problems. The core contribution is a set of novel excess risk bounds of M-SPP derived through the lens of algorithmic stability theory. Particularly under smoothness and quadratic growth conditions, we show that M-SPP with minibatch-size $n$ and iteration count $T$ enjoys an in-expectation fast rate of convergence consisting of an $\mathcal{O}\left(\frac{1}{T^2}\right)$ bias decaying term and an $\mathcal{O}\left(\frac{1}{nT}\right)$ variance decaying term. In the small-$n$-large-$T$ setting, this result substantially improves the best known results of SPP-type approaches by revealing the impact of noise level of model on convergence rate. In the complementary small-$T$-large-$n$ regime, we provide a two-phase extension of M-SPP to achieve comparable convergence rates. Moreover, we derive a near-tight high probability (over the randomness of data) bound on the parameter estimation error of a sampling-without-replacement variant of M-SPP. Numerical evidences are provided to support our theoretical predictions when substantialized to Lasso and logistic regression models.