aiqbench analysis
This paper provides rigorous theoretical proof that momentum-based optimizers like Polyak Heavy-Ball and Nesterov incur an information-theoretic penalty in nonstationary environments where the objective function shifts over time. The authors demonstrate through minimax lower bounds that vanilla SGD provably outperforms these accelerated methods when drift dominates, exposing a fundamental trade-off between gradient smoothing and systematic tracking lag. These findings offer critical guidance for practitioners deploying optimization algorithms in dynamic settings such as federated learning or online reinforcement learning.
On the Provable Suboptimality of Momentum SGD in Nonstationary Stochastic Optimization
arXiv:2601.12238v5 Announce Type: replace-cross Abstract: In this paper, we provide a comprehensive theoretical analysis of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak Heavy-Ball and Nesterov) for tracking time-varying optima under strong convexity and smoothness. Our finite-time bounds reveal a sharp decomposition of tracking error into transient, noise-induced, and drift-induced components. This decomposition exposes a fundamental trade-off: while momentum is often used as a gradient-smoothing heuristic, under distribution shift it incurs an explicit drift-amplification penalty that diverges as the momentum parameter $\beta$ approaches 1, yielding systematic tracking lag. We complement these upper bounds with minimax lower bounds under gradient-variation constraints, proving this momentum-induced tracking penalty is not an analytical artifact but an information-theoretic barrier: in drift-dominated regimes, momentum is unavoidably worse because stale-gradient averaging forces systematic lag. Our results provide theoretical grounding for the empirical instability of momentum in dynamic settings and precisely delineate regime boundaries where vanilla SGD provably outperforms its accelerated counterparts.
Abstract:In this paper, we provide a comprehensive theoretical analysis of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak Heavy-Ball and Nesterov) for tracking time-varying optima under strong convexity and smoothness. Our finite-time bounds reveal a sharp decomposition of tracking error into transient, noise-induced, and drift-induced components. This decomposition exposes a fundamental trade-off: while momentum is often used as a gradient-smoothing heuristic, under distribution shift it incurs an explicit drift-amplification penalty that diverges as the momentum parameter $\beta$ approaches 1, yielding systematic tracking lag. We complement these upper bounds with minimax lower bounds under gradient-variation constraints, proving this momentum-induced tracking penalty is not an analytical artifact but an information-theoretic barrier: in drift-dominated regimes, momentum is unavoidably worse because stale-gradient averaging forces systematic lag. Our results provide theoretical grounding for the empirical instability of momentum in dynamic settings and precisely delineate regime boundaries where vanilla SGD provably outperforms its accelerated counterparts.
Accepted to ICML 2026. 76 pages, 5 figures, 4 tables
Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2601.12238 [stat.ML]
(or arXiv:2601.12238v5 [stat.ML] for this version)
Full text under the publisher's open licence · 178 words
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