Stochastic Krasnoselskii-Mann Iterations: Convergence without Uniformly Bounded Variance
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911671498309632 |
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| author | Cortild, Daniel Cartis, Coralia |
| author_facet | Cortild, Daniel Cartis, Coralia |
| contents | We investigate the Stochastic Krasnoselskii-Mann iterations for expected nonexpansive fixed-point problems in a real Hilbert space. We establish convergence guarantees under significantly weaker assumptions on the variance than those typically used in the literature. In particular, instead of a uniform bound on the variance of the stochastic oracle, we only assume finite variance at a single fixed point. Under this assumption, we prove almost sure weak convergence of the iterates, derive convergence rates for the expected residual, and obtain almost sure convergence rates for the running minimum residual. Notably, we recover the best-known stochastic oracle complexity without imposing uniformly bounded variance. We illustrate the applicability of our results to Stochastic Gradient Descent, where we recover known guarantees, and to Stochastic Three-Operator Splitting and Stochastic Backward-Forward Splitting, for which we obtain the first results that avoid uniform variance bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22581 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stochastic Krasnoselskii-Mann Iterations: Convergence without Uniformly Bounded Variance Cortild, Daniel Cartis, Coralia Optimization and Control 46N10, 47J26, 65K10, 90C25, 90C15 We investigate the Stochastic Krasnoselskii-Mann iterations for expected nonexpansive fixed-point problems in a real Hilbert space. We establish convergence guarantees under significantly weaker assumptions on the variance than those typically used in the literature. In particular, instead of a uniform bound on the variance of the stochastic oracle, we only assume finite variance at a single fixed point. Under this assumption, we prove almost sure weak convergence of the iterates, derive convergence rates for the expected residual, and obtain almost sure convergence rates for the running minimum residual. Notably, we recover the best-known stochastic oracle complexity without imposing uniformly bounded variance. We illustrate the applicability of our results to Stochastic Gradient Descent, where we recover known guarantees, and to Stochastic Three-Operator Splitting and Stochastic Backward-Forward Splitting, for which we obtain the first results that avoid uniform variance bounds. |
| title | Stochastic Krasnoselskii-Mann Iterations: Convergence without Uniformly Bounded Variance |
| topic | Optimization and Control 46N10, 47J26, 65K10, 90C25, 90C15 |
| url | https://arxiv.org/abs/2604.22581 |