SILAGE: Memory-Efficient, Full-Gradient-Free Nonconvex Optimization for Nested Finite Sums

• URL: http://arxiv.org/abs/2606.15832v1
• Date: Sun, 14 Jun 2026 14:11:07 GMT
• Title: SILAGE: Memory-Efficient, Full-Gradient-Free Nonconvex Optimization for Nested Finite Sums
• Abstract summary: Empirical risk on datasets naturally exhibits similarities in $N=nm$ total samples.<n>We provide an analysis that avoids pessimistic convergence analysis.<n>Our results improve existing state-of-the-art regimes.
• Score: 51.49970814177172
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Empirical risk minimization on massive datasets naturally exhibits a nested double finite-sum structure, where $N=nm$ total samples are logically or physically partitioned into $n$ blocks of size $m$ (e.g., in pooled data silos, out-of-core learning, or deliberate stratification). While variance-reduced methods achieve optimal oracle complexities for nonconvex objectives, they suffer from severe scaling bottlenecks in this centralized regime. Recursive estimators, such as PAGE, require periodic global full-gradient refreshes over all $nm$ samples, which are computationally expensive. Conversely, single-loop methods, such as SILVER, avoid such refreshes but require an impractical $\mathcal{O}(nm)$ memory footprint to store a control variate for every sample. In this paper, we propose SILAGE, a variance-reduced algorithm that addresses this trade-off. By actively exploiting the double-sum structure, SILAGE eliminates periodic global full-gradient refreshes over all $nm$ components (evaluating at most one local group gradient per iteration) while requiring only $\mathcal{O}(n)$ memory. Furthermore, we provide a tight convergence analysis that avoids pessimistic worst-case Lipschitz constants. Instead, SILAGE's complexity natively adapts to the underlying data geometry via nested functional similarities: across-group ($δ_1$) and within-group ($δ_2$) heterogeneity. Our results improve existing state-of-the-art bounds in several practically relevant regimes.

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