{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15723",
  "author_total": 5,
  "authors": [
    {
      "name": "Wei Jiang"
    },
    {
      "name": "Sifan Yang"
    },
    {
      "name": "Yibo Wang"
    },
    {
      "name": "Lijun Zhang"
    },
    {
      "name": "Zechao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ehge5t3wc87b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15723",
  "fulltext_url": null,
  "id": "2ehge5t3wc87b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15723",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7213256219",
  "pdf_url": "https://arxiv.org/pdf/2609.15723",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ehge5t3wc87b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15723",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15723"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Solving Finite-sum Coupled Compositional Optimization via Multi-block-Single-probe Estimator",
  "title_text": "Solving Finite-sum Coupled Compositional Optimization via Multi-block-Single-probe Estimator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/2ehge5t3wc87b",
  "venue": "arXiv",
  "versions": []
}