RanSOM: Second-Order Momentum with Randomized Scaling for Constrained and Unconstrained Optimization

• URL: http://arxiv.org/abs/2602.06824v1
• Date: Fri, 06 Feb 2026 16:09:36 GMT
• Title: RanSOM: Second-Order Momentum with Randomized Scaling for Constrained and Unconstrained Optimization
• Authors: El Mahdi Chayti,
• Abstract summary: Momentum methods, such as Polyak's Heavy Ball, are the standard for training deep networks but suffer from curvature-induced bias in settings.<n>We propose textbfRanSOM, a unified framework that eliminates this bias by replacing deterministic step sizes with randomized steps drawn from distributions with mean $_t$.<n>We instantiate this framework in two algorithms: textbfRanSOM-E for unconstrained optimization and textbfRanSOM-B for constrained optimization.
• Score: 1.3537117504260623
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 ($p \in (1, 2]$) without requiring gradient clipping.

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