VFOSA: Variance-Reduced Fast Operator Splitting Algorithms for Generalized Equations

2024-12-31 · Source: JMLR · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, quick

Summary

Quoc Tran-Dinh introduces two Variance-reduced Fast Operator Splitting Algorithms (VFOSA) designed to approximate solutions for generalized equations, encompassing minimization, minimax problems, and variational inequalities. The approach integrates accelerated operator splitting, fixed-point methods, co-hypomonotonicity, and variance reduction. The work develops a class of variance-reduced estimators, including SVRG, SAGA, SARAH, and Hybrid-SGD, and establishes their variance-reduction bounds. A novel accelerated variance-reduced forward-backward splitting (FBS) method is presented, achieving O(1/k^2) and o(1/k^2) convergence rates on the expected squared norm E[ ||G_{\lambda}x^k||^2] of the FBS residual, alongside almost sure convergence. Unlike prior stochastic methods, VFOSA accommodates co-hypomonotone operators, addressing nonmonotone problems. The paper also details a variance-reduced fast backward-forward splitting (BFS) method with similar convergence and oracle complexity, validating both through numerical experiments.

Key takeaway

For research scientists developing optimization algorithms for generalized equations, VFOSA offers a significant advancement by accommodating co-hypomonotone operators and achieving improved convergence rates. You should consider integrating these variance-reduced forward-backward or backward-forward splitting methods, especially for problems involving nonmonotone structures, to potentially achieve better performance and broader applicability compared to existing stochastic operator splitting algorithms.

Key insights

VFOSA introduces novel variance-reduced operator splitting algorithms for generalized equations, achieving superior convergence rates.

Principles

• Co-hypomonotonicity expands solvable problem classes.
• Variance reduction improves convergence in stochastic optimization.

Method

VFOSA designs accelerated variance-reduced forward-backward and backward-forward splitting methods using a new class of estimators, achieving O(1/k^2) and o(1/k^2) convergence rates for generalized equations.

In practice

• Apply VFOSA to nonmonotone optimization problems.
• Utilize SVRG, SAGA, SARAH, Hybrid-SGD as estimators.

Topics

• Operator Splitting Algorithms
• Variance Reduction
• Generalized Equations
• Convergence Rates
• Co-hypomonotone Operators

Code references

• JmlrOrg/v26

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Editorial summary, takeaway, and curation by AIssential. Original article published by JMLR.