# Solving Stochastic Fixed-Point Equations with High Probability

arXiv: https://arxiv.org/abs/2607.09097 · HTML: https://mlsanity.com/paper/2607.09097/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 72 references that are in the map.
- First posted on arXiv 10 Jul 2026.

## Builds on (most cited first, 30 of 33)

- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient](https://mlsanity.com/paper/1703.00102/) (2017)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Stochastic Optimization with Heavy-Tailed Noise via Accelerated Gradient Clipping](https://mlsanity.com/paper/2005.10785/) (2020)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)
- [High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails](https://mlsanity.com/paper/2106.14343/) (2021)
- [Stochastic approximation with cone-contractive operators: Sharp ℓ_∞-bounds for Q-learning](https://mlsanity.com/paper/1905.06265/) (2019)
- [Accelerated Algorithms for Smooth Convex-Concave Minimax Problems with O(1/k2) Rate on Squared Gradient Norm](https://mlsanity.com/paper/2102.07922/) (2021)
- [A First Order Method for Solving Convex Bi-Level Optimization Problems](https://mlsanity.com/paper/1702.03999/) (2017)
- [Fast Extra Gradient Methods for Smooth Structured Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2106.02326/) (2021)
- [Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities](https://mlsanity.com/paper/2002.08872/) (2020)
- [High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance](https://mlsanity.com/paper/2302.00999/) (2023)
- [Loss minimization and parameter estimation with heavy tails](https://mlsanity.com/paper/1307.1827/) (2013)
- [Concentration of Contractive Stochastic Approximation: Additive and Multiplicative Noise](https://mlsanity.com/paper/2303.15740/) (2023)
- [Stochastic fixed-point iterations for nonexpansive maps: Convergence and error bounds](https://mlsanity.com/paper/2208.04193/) (2022)
- [High-Probability Convergence for Composite and Distributed Stochastic Minimization and Variational Inequalities with Heavy-Tailed Noise](https://mlsanity.com/paper/2310.01860/) (2023)
- [Concentration of Contractive Stochastic Approximation and Reinforcement Learning](https://mlsanity.com/paper/2106.14308/) (2021)
- [Halpern-Type Accelerated and Splitting Algorithms For Monotone Inclusions](https://mlsanity.com/paper/2110.08150/) (2021)
- [Near-Optimal Algorithms for Making the Gradient Small in Stochastic Minimax Optimization](https://mlsanity.com/paper/2208.05925/) (2022)
- [Stochastic Halpern Iteration with Variance Reduction for Stochastic Monotone Inclusions](https://mlsanity.com/paper/2203.09436/) (2022)
- [Optimal variance-reduced stochastic approximation in Banach spaces](https://mlsanity.com/paper/2201.08518/) (2022)
- [Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noise](https://mlsanity.com/paper/2206.01095/) (2022)
- [Near-Optimal Sample Complexity for MDPs via Anchoring](https://mlsanity.com/paper/2502.04477/) (2025)
- [Optimal error bounds for nonexpansive fixed-point iterations in normed spaces](https://mlsanity.com/paper/2108.10969/) (2021)
- [Stochastic Halpern iteration in normed spaces and applications to reinforcement learning](https://mlsanity.com/paper/2403.12338/) (2024)
- [Mean Estimation in Banach Spaces Under Infinite Variance and Martingale Dependence](https://mlsanity.com/paper/2411.11271/) (2024)
- [Optimal Non-Asymptotic Rates of Value Iteration for Average-Reward Markov Decision Processes](https://mlsanity.com/paper/2504.09913/) (2025)
- [Pushing the Complexity Boundaries of Fixed-Point Equations: Adaptation to Contraction and Controlled Expansion](https://mlsanity.com/paper/2506.17698/) (2025)

## Built on by (most cited first, 3 of 3)

- [Oracle Complexity of Stochastic Fixed-Point Equations with Nonexpansive Maps](https://mlsanity.com/paper/2609.09524/) (2026)
- [Stochastic Inertial Krasnosel'skii-Mann Iteration Achieves Near-Optimal Sample Complexity](https://mlsanity.com/paper/2609.28543/) (2026)
- [Dual-Anchor Acceleration Is Near-Optimal for Stochastic Monotone Root-Finding](https://mlsanity.com/paper/2609.36033/) (2026)

## Abstract (arXiv)

We study stochastic fixed-point equations T(x) = x over normed spaces (E, |·|), where the operator T is nonexpansive or contractive and is accessed only through unbiased stochastic evaluations with bounded second central moment. Given ε > 0, δ ∈ (0, 1), the goal is to output x ∈ E such that |T(x) - x| ≤ ε with probability at least 1-δ. We introduce VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. The key algorithmic ingredient is a recursive stochastic estimator based on clipped differences of oracle evaluations: instead of clipping τ(x; xi) itself, we clip stochastic differences at the Lipschitz scale γ|x - y|. This makes the estimator pathwise Lipschitz along the algorithmic trajectory while permitting martingale concentration under finite second moments in the native norm. Our main theorem gives an anytime high-probability residual bound: on a single event of probability at least 1 - δ, the residual decreases nearly geometrically across epochs, up to lower-order logarithmic factors. Under only bounded variance, displaying only the dependence on the target error ε and Lipschitz constant γ ∈ (0, 1] of T, the resulting oracle complexity is minε-5, (1-γ)-3ε-2. Under a Lipschitz-in-expectation oracle, the dependence improves to the corresponding ε-3 nonexpansive rate (i.e., for γ = 1), and under samplewise nonexpansiveness to ε-2.