A Unified Zeroth-Order Optimization Framework via Oblivious Randomized Sketching
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| Format: | Preprint |
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2025
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| _version_ | 1866909840839802880 |
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| author | Ye, Haishan Chang, Xiangyu Chen, Xi |
| author_facet | Ye, Haishan Chang, Xiangyu Chen, Xi |
| contents | We propose a new framework for analyzing zeroth-order optimization (ZOO) from the perspective of \emph{oblivious randomized sketching}.In this framework, commonly used gradient estimators in ZOO-such as finite difference (FD) and random finite difference (RFD)-are unified through a general sketch-based formulation. By introducing the concept of oblivious randomized sketching, we show that properly chosen sketch matrices can significantly reduce the high variance of RFD estimates and enable \emph{high-probability} convergence guarantees of ZOO, which are rarely available in existing RFD analyses.
\noindent We instantiate the framework on convex quadratic objectives and derive a query complexity of $\tilde{\mathcal{O}}(\mathrm{tr}(A)/L \cdot L/μ\log\frac{1}ε)$ to achieve a $ε$-suboptimal solution, where $A$ is the Hessian, $L$ is the largest eigenvalue of $A$, and $μ$ denotes the strong convexity parameter. This complexity can be substantially smaller than the standard query complexity of ${\cO}(d\cdot L/μ\log\frac{1}ε)$ that is linearly dependent on problem dimensionality, especially when $A$ has rapidly decaying eigenvalues. These advantages naturally extend to more general settings, including strongly convex and Hessian-aware optimization.
\noindent Overall, this work offers a novel sketch-based perspective on ZOO that explains why and when RFD-type methods can achieve \emph{weakly dimension-independent} convergence in general smooth problems, providing both theoretical foundations and practical implications for ZOO. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10945 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Unified Zeroth-Order Optimization Framework via Oblivious Randomized Sketching Ye, Haishan Chang, Xiangyu Chen, Xi Optimization and Control We propose a new framework for analyzing zeroth-order optimization (ZOO) from the perspective of \emph{oblivious randomized sketching}.In this framework, commonly used gradient estimators in ZOO-such as finite difference (FD) and random finite difference (RFD)-are unified through a general sketch-based formulation. By introducing the concept of oblivious randomized sketching, we show that properly chosen sketch matrices can significantly reduce the high variance of RFD estimates and enable \emph{high-probability} convergence guarantees of ZOO, which are rarely available in existing RFD analyses. \noindent We instantiate the framework on convex quadratic objectives and derive a query complexity of $\tilde{\mathcal{O}}(\mathrm{tr}(A)/L \cdot L/μ\log\frac{1}ε)$ to achieve a $ε$-suboptimal solution, where $A$ is the Hessian, $L$ is the largest eigenvalue of $A$, and $μ$ denotes the strong convexity parameter. This complexity can be substantially smaller than the standard query complexity of ${\cO}(d\cdot L/μ\log\frac{1}ε)$ that is linearly dependent on problem dimensionality, especially when $A$ has rapidly decaying eigenvalues. These advantages naturally extend to more general settings, including strongly convex and Hessian-aware optimization. \noindent Overall, this work offers a novel sketch-based perspective on ZOO that explains why and when RFD-type methods can achieve \emph{weakly dimension-independent} convergence in general smooth problems, providing both theoretical foundations and practical implications for ZOO. |
| title | A Unified Zeroth-Order Optimization Framework via Oblivious Randomized Sketching |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2510.10945 |