aiqbench analysis
This paper establishes the information-theoretic optimality of Matrix Shampoo for nonstationary stochastic optimization by deriving tight minimax lower bounds. It provides a rigorous theoretical framework that disentangles tradeoffs between noise suppression, adaptation speed, and preconditioning stability in time-varying environments. The work advances fundamental understanding of adaptive optimizers beyond standard convex or stationary assumptions.
Minimax Optimality of Shampoo for Nonstationary Stochastic Optimization
arXiv:2610.16923v1 Announce Type: cross Abstract: We develop a comprehensive theoretical analysis of Matrix Shampoo for tracking time-varying optima in nonstationary stochastic optimization. We derive finite-horizon high-probability bounds for average tracking error that disentangle initialization, optimizer drift, momentum-induced drift penalty, stochastic noise, and adaptive preconditioning, with explicit dependence on momentum parameter \(\beta_1\) and stabilization parameter \(\varepsilon\). These bounds reveal an explicit drift--noise and stability--geometry tradeoffs: noise-dominated regimes favor larger $\beta_1$, which provides stronger first-moment averaging and greater suppression of gradient noise, whereas drift-dominated regimes favor smaller $\beta_1$, allowing faster adaptation to moving optimizer. Larger $\varepsilon$ reduces the preconditioner-perturbation term while weakening Shampoo's ability to exploit matrix geometry, thereby reducing adaptivity. We establish a matching minimax lower bound under squared path-variation constraint, proving the information-theoretic optimality of Shampoo over the class of causal optimization procedures. Under a constant stepsize, we establish high-probability bounds for the average projected stationarity gap along general nonstationary nonconvex objectives. We validate our theoretical findings on both synthetic and multiple real-world problems.
Abstract:We develop a comprehensive theoretical analysis of Matrix Shampoo for tracking time-varying optima in nonstationary stochastic optimization. We derive finite-horizon high-probability bounds for average tracking error that disentangle initialization, optimizer drift, momentum-induced drift penalty, stochastic noise, and adaptive preconditioning, with explicit dependence on momentum parameter \(\beta_1\) and stabilization parameter \(\varepsilon\). These bounds reveal an explicit drift--noise and stability--geometry tradeoffs: noise-dominated regimes favor larger $\beta_1$, which provides stronger first-moment averaging and greater suppression of gradient noise, whereas drift-dominated regimes favor smaller $\beta_1$, allowing faster adaptation to moving optimizer. Larger $\varepsilon$ reduces the preconditioner-perturbation term while weakening Shampoo's ability to exploit matrix geometry, thereby reducing adaptivity. We establish a matching minimax lower bound under squared path-variation constraint, proving the information-theoretic optimality of Shampoo over the class of causal optimization procedures. Under a constant stepsize, we establish high-probability bounds for the average projected stationarity gap along general nonstationary nonconvex objectives. We validate our theoretical findings on both synthetic and multiple real-world problems.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2610.16923 [stat.ML]
(or arXiv:2610.16923v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.16923 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 256 words
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