Solving Finite-sum Coupled Compositional Optimization via Multi-block-Single-probe Estimator

• URL: http://arxiv.org/abs/2609.15723v1
• Date: Mon, 14 Sep 2026 15:21:32 GMT
• Title: Solving Finite-sum Coupled Compositional Optimization via Multi-block-Single-probe Estimator
• Abstract summary: We show that a novel MSVR estimator efficiently trace complexities to $mathcalO(1)$ blocks.<n>We also obtain improved dependence on $m$ when the outer function gradients are gradient.<n> Empirical studies on multi-task deep AUC demonstrate superior performance of the proposed linear estimators.
• Score: 41.84305838914204
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Traditional variance reduction methods (e.g., SPIDER, SARAH, STORM) have been extensively investigated for improving the convergence rates of stochastic optimization. These techniques typically maintain a sequence of estimators for a single function (or gradient) across iterations. However, what if we need to track multiple functions, but can only access stochastic samples of $\mathcal{O}(1)$ functions at each iteration? This scenario arises in an important emerging family of finite-sum coupled compositional optimization (FCCO) problems of the form $\frac{1}{m}\sum_{i=1}^m f_i(g_i(\mathbf{w}))$, where each $g_i$ is accessible only through a stochastic oracle. The key challenge is to track $\mathbf g(\mathbf{w})=(g_1(\mathbf{w}), \ldots, g_m(\mathbf{w}))$ over time, where $\mathbf g(\mathbf{w})$ has $m$ blocks but only $\mathcal{O}(1)$ blocks can be probed for their stochastic values at each step. To address this challenge, we propose a novel Multi-block-Single-probe Variance Reduction (MSVR) estimator to efficiently trace $\mathbf g(\mathbf{w})$ under partial block sampling. Building on the MSVR estimator, we develop several algorithms for FCCO problems, achieving improved sample complexities for non-convex, convex, strongly convex, and Polyak-Łojasiewicz (PL) objectives. We further obtain an improved dependence on $m$ when the outer function gradients $\nabla f_i$ are linear. Empirical studies on multi-task deep AUC maximization further demonstrate the superior performance of the proposed estimators.

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