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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2602.06824 |
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| _version_ | 1866918501913985024 |
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| author | Chayti, El Mahdi |
| author_facet | Chayti, El Mahdi |
| contents | Momentum methods, such as Polyak's Heavy Ball, are the standard for training deep networks but suffer from curvature-induced bias in stochastic settings, limiting convergence to suboptimal $\mathcal{O}(ε^{-4})$ rates. Existing corrections typically require expensive auxiliary sampling or restrictive smoothness assumptions. We propose \textbf{RanSOM}, a unified framework that eliminates this bias by replacing deterministic step sizes with randomized steps drawn from distributions with mean $η_t$. This modification allows us to leverage Stein-type identities to compute an exact, unbiased estimate of the momentum bias using a single Hessian-vector product computed jointly with the gradient, avoiding auxiliary queries. We instantiate this framework in two algorithms: \textbf{RanSOM-E} for unconstrained optimization (using exponentially distributed steps) and \textbf{RanSOM-B} for constrained optimization (using beta-distributed steps to strictly preserve feasibility). Theoretical analysis confirms that RanSOM recovers the optimal $\mathcal{O}(ε^{-3})$ convergence rate under standard bounded noise, and achieves optimal rates for heavy-tailed noise settings ($p \in (1, 2]$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06824 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | RanSOM: Second-Order Momentum with Randomized Scaling for Constrained and Unconstrained Optimization Chayti, El Mahdi Optimization and Control Machine Learning Momentum methods, such as Polyak's Heavy Ball, are the standard for training deep networks but suffer from curvature-induced bias in stochastic settings, limiting convergence to suboptimal $\mathcal{O}(ε^{-4})$ rates. Existing corrections typically require expensive auxiliary sampling or restrictive smoothness assumptions. We propose \textbf{RanSOM}, a unified framework that eliminates this bias by replacing deterministic step sizes with randomized steps drawn from distributions with mean $η_t$. This modification allows us to leverage Stein-type identities to compute an exact, unbiased estimate of the momentum bias using a single Hessian-vector product computed jointly with the gradient, avoiding auxiliary queries. We instantiate this framework in two algorithms: \textbf{RanSOM-E} for unconstrained optimization (using exponentially distributed steps) and \textbf{RanSOM-B} for constrained optimization (using beta-distributed steps to strictly preserve feasibility). Theoretical analysis confirms that RanSOM recovers the optimal $\mathcal{O}(ε^{-3})$ convergence rate under standard bounded noise, and achieves optimal rates for heavy-tailed noise settings ($p \in (1, 2]$). |
| title | RanSOM: Second-Order Momentum with Randomized Scaling for Constrained and Unconstrained Optimization |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2602.06824 |