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Convergence Rate of the Last Iterate of Stochastic Proximal Algorithms

Kevin Kurian Thomas Vaidyan, Michael P. Friedlander, Ahmet Alacaoglu

TL;DR

This work addresses the lack of theoretical guarantees for last-iterate convergence in proximal stochastic optimization with unbounded gradient variance. By leveraging co-coercivity and a novel last-iterate reduction, it proves a $\widetilde{O}(1/\sqrt{T})$ last-iterate rate for proximal SGD under componentwise smoothness and convexity, and extends the result to randomized incremental proximal methods with a Lipschitz assumption on the component regularizers. It also provides corollaries for projected SGD and stochastic proximal point, and demonstrates the practical viability via BlockProx extensions and numerical experiments showing the last iterate often outperforms averaging. The results apply to graph-guided regularizers common in multi-task and federated learning, offering nonasymptotic, data-independent rates without requiring bounded variance. Overall, the paper advances the understanding of last-iterate behavior in regularized stochastic optimization and broadens the toolkit for large-scale, graph-structured learning tasks.

Abstract

We analyze two classical algorithms for solving additively composite convex optimization problems where the objective is the sum of a smooth term and a nonsmooth regularizer: proximal stochastic gradient method for a single regularizer; and the randomized incremental proximal method, which uses the proximal operator of a randomly selected function when the regularizer is given as the sum of many nonsmooth functions. We focus on relaxing the bounded variance assumption that is common, yet stringent, for getting last iterate convergence rates. We prove the $\widetilde{O}(1/\sqrt{T})$ rate of convergence for the last iterate of both algorithms under componentwise convexity and smoothness, which is optimal up to log terms. Our results apply directly to graph-guided regularizers that arise in multi-task and federated learning, where the regularizer decomposes as a sum over edges of a collaboration graph.

Convergence Rate of the Last Iterate of Stochastic Proximal Algorithms

TL;DR

This work addresses the lack of theoretical guarantees for last-iterate convergence in proximal stochastic optimization with unbounded gradient variance. By leveraging co-coercivity and a novel last-iterate reduction, it proves a last-iterate rate for proximal SGD under componentwise smoothness and convexity, and extends the result to randomized incremental proximal methods with a Lipschitz assumption on the component regularizers. It also provides corollaries for projected SGD and stochastic proximal point, and demonstrates the practical viability via BlockProx extensions and numerical experiments showing the last iterate often outperforms averaging. The results apply to graph-guided regularizers common in multi-task and federated learning, offering nonasymptotic, data-independent rates without requiring bounded variance. Overall, the paper advances the understanding of last-iterate behavior in regularized stochastic optimization and broadens the toolkit for large-scale, graph-structured learning tasks.

Abstract

We analyze two classical algorithms for solving additively composite convex optimization problems where the objective is the sum of a smooth term and a nonsmooth regularizer: proximal stochastic gradient method for a single regularizer; and the randomized incremental proximal method, which uses the proximal operator of a randomly selected function when the regularizer is given as the sum of many nonsmooth functions. We focus on relaxing the bounded variance assumption that is common, yet stringent, for getting last iterate convergence rates. We prove the rate of convergence for the last iterate of both algorithms under componentwise convexity and smoothness, which is optimal up to log terms. Our results apply directly to graph-guided regularizers that arise in multi-task and federated learning, where the regularizer decomposes as a sum over edges of a collaboration graph.
Paper Structure (32 sections, 20 theorems, 160 equations, 1 figure, 1 table)

This paper contains 32 sections, 20 theorems, 160 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem setting and main assumptions
  3. Notation.
  4. Variance control via co-coercivity
  5. Related Work
  6. SGD world
  7. Proximal point world
  8. Proximal SGD
  9. Statement of the result
  10. Proof setup
  11. One-iteration analysis
  12. Last-iterate reduction
  13. Corollary for Projected SGD
  14. Randomized Incremental Proximal Method
  15. Statement of the result
  16. ...and 17 more sections

Key Result

Theorem 3.1

Let assumptions hold. In eq:intro-spgd, let $\tau = 1/(3 L \sqrt{T})$. Then

Figures (1)

  • Figure 1: Comparison of last and averaged iterates.

Theorems & Definitions (31)

  • Theorem 3.1: Last-iterate convergence
  • Lemma 3.2: Per-iteration descent
  • Lemma 3.3: Last iterate reduction
  • Corollary 3.4: Projected SGD
  • Theorem 4.1: Last-iterate convergence, incremental proximal
  • Lemma 4.2: Per-iteration descent, incremental proximal
  • Corollary 4.3: Stochastic proximal point
  • Theorem 4.4: Last-iterate convergence, BlockProx
  • Theorem A.1: Last-iterate convergence, polynomial step sizes
  • Corollary A.2: Best polynomial step size
  • ...and 21 more