Solving Stochastic Fixed-Point Equations with High Probability

2026-07-13 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

The paper introduces VR-GHAL, a Variance-Reduced Gradual Halpern method, designed to solve stochastic fixed-point equations T(x)=x in quadratically smoothable Banach spaces. This method addresses scenarios where the operator T is nonexpansive or contractive and is only accessible through noisy, unbiased stochastic evaluations with bounded second central moments. VR-GHAL's core innovation is a recursive stochastic estimator that clips differences of oracle evaluations at the Lipschitz scale gamma||x-y||, ensuring pathwise Lipschitz behavior and enabling martingale concentration in the native norm. The main theorem provides an anytime high-probability residual bound, demonstrating nearly geometric residual decrease across epochs. The resulting oracle complexity is min{epsilon^-5, (1-gamma)^-3 epsilon^-2} under bounded variance, improving to epsilon^-3 for nonexpansive operators with a Lipschitz-in-expectation oracle, and epsilon^-2 under samplewise nonexpansiveness. This framework offers high-probability guarantees, uses native norm measurements, and is parameter-free.

Key takeaway

For AI Scientists and Machine Learning Engineers developing robust algorithms for stochastic fixed-point problems in non-Euclidean spaces, VR-GHAL offers a significant advancement. This method provides high-probability convergence guarantees and measures variance in the native norm, overcoming limitations of prior expectation-based or l2-norm approaches. You should consider VR-GHAL for applications like reinforcement learning Bellman equations or deep equilibrium models, especially when dealing with noisy oracles and requiring parameter-free, anytime performance.

Key insights

VR-GHAL solves stochastic fixed-point equations using a novel Lipschitz-clipped difference estimator for high-probability convergence in native norms.

Principles

• Clipping stochastic differences at the Lipschitz scale maintains pathwise Lipschitzness.
• Martingale concentration in native norms yields high-probability bounds with finite second moments.
• Gradual Halpern iteration can converge robustly without explicit iterate set diameter.

Method

VR-GHAL integrates the Gradual Halpern Algorithm with a recursive, clipped-difference variance-reduced estimator, alternating step size reduction and modified Banach-Picard iterations.

In practice

• Solve Bellman equations in reinforcement learning.
• Implement deep equilibrium models.
• Perform self-consistent field calculations.

Topics

• Stochastic Fixed-Point Equations
• Variance Reduction
• Halpern Iteration
• Banach Spaces
• High-Probability Guarantees
• Clipped Difference Estimators

Code references

• freegs-plasma/freegs

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.