# An abstract effective convergence theorem for stochastic processes, with applications to stochastic approximation

arXiv: https://arxiv.org/abs/2504.12922 · HTML: https://mlsanity.com/paper/2504.12922/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 66 references that are in the map.
- First posted on arXiv 17 Apr 2025.

## Builds on (most cited first, 13 of 13)

- [Stochastic Compositional Gradient Descent: Algorithms for Minimizing Compositions of Expected-Value Functions](https://mlsanity.com/paper/1411.3803/) (2014)
- [Averaging Stochastic Gradient Descent on Riemannian Manifolds](https://mlsanity.com/paper/1802.09128/) (2018)
- [Ergodic convergence of a stochastic proximal point algorithm](https://mlsanity.com/paper/1504.05400/) (2015)
- [A Unified Convergence Theorem for Stochastic Optimization Methods](https://mlsanity.com/paper/2206.03907/) (2022)
- [Convergence of sequences: a survey](https://mlsanity.com/paper/2111.11374/) (2021)
- [Formalization of a Stochastic Approximation Theorem](https://mlsanity.com/paper/2202.05959/) (2022)
- [Asymptotic regularity of a generalised stochastic Halpern scheme](https://mlsanity.com/paper/2411.04845/) (2024)
- [A quantitative Robbins-Siegmund theorem](https://mlsanity.com/paper/2410.15986/) (2024)
- [Stochastic Quasi-Fej'er Block-Coordinate Fixed Point Iterations With Random Sweeping II: Mean-Square and Linear Convergence](https://mlsanity.com/paper/1704.08083/) (2017)
- [Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators](https://mlsanity.com/paper/1507.07095/) (2015)
- [Stochastic and incremental subgradient methods for convex optimization on Hadamard spaces](https://mlsanity.com/paper/2412.06730/) (2024)
- [Moduli of regularity and rates of convergence for Fej'er monotone sequences](https://mlsanity.com/paper/1711.02130/) (2017)
- [Fej'er monotone sequences revisited](https://mlsanity.com/paper/2310.06528/) (2023)

## Built on by (most cited first, 1 of 1)

- [Mean-square and sublinear convergence of a stochastic proximal point algorithm in metric spaces of nonpositive curvature](https://mlsanity.com/paper/2510.10697/) (2025)

## Abstract (arXiv)

We provide a general theorem on the asymptotic behavior of stochastic processes that conform to a relaxed supermartingale condition. The distinguishing feature of our result is that it provides quantitative convergence guarantees at a much higher level of abstraction and generality than is typically seen in the stochastic approximation literature, formulated in particular in terms of a general modulus τ that, on an intuitive level, captures an effective variant of the uniqueness in expectation of associated solutions. Our convergence rate is highly uniform, depending on very few data beyond τ. We then demonstrate the utility of our result as a unifying framework by deriving new quantitative versions of several key concepts and theorems from stochastic approximation, including the Robbins-Siegmund theorem, Dvoretzky's convergence theorem, and the convergence of stochastic quasi-Fej'er monotone sequences, the latter formulated in a novel and highly general metric context. Throughout, we isolate and discuss special cases of our results which allow for the construction of fast, and in particular linear, rates. Various applications of our results and our general methodology to stochastic approximation are discussed, and in particular explicitly derived in related work of the authors.