Adaptive Matrix Sparsification and Applications to Empirical Risk Minimization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917116551102464 |
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| author | Liu, Yang P. Peng, Richard Tang, Colin Weng, Albert Yang, Junzhao |
| author_facet | Liu, Yang P. Peng, Richard Tang, Colin Weng, Albert Yang, Junzhao |
| contents | Consider the empirical risk minimization (ERM) problem, which is stated as follows. Let $K_1, \dots, K_m$ be compact convex sets with $K_i \subseteq \mathbb{R}^{n_i}$ for $i \in [m]$, $n = \sum_{i=1}^m n_i$, and $n_i\le C_K$ for some absolute constant $C_K$. Also, consider a matrix $A \in \mathbb{R}^{n \times d}$ and vectors $b \in \mathbb{R}^d$ and $c \in \mathbb{R}^n$. Then the ERM problem asks to find \[ \min_{\substack{x \in K_1 \times \dots \times K_m\\ A^\top x = b}}
c^\top x. \] We give an algorithm to solve this to high accuracy in time $\widetilde{O}(nd + d^6\sqrt{n}) \le \widetilde{O} (nd + d^{11})$, which is nearly-linear time in the input size when $A$ is dense and $n \ge d^{10}$.
Our result is achieved by implementing an $\widetilde{O}(\sqrt{n})$-iteration interior point method (IPM) efficiently using dynamic data structures. In this direction, our key technical advance is a new algorithm for maintaining leverage score overestimates of matrices undergoing row updates. Formally, given a matrix $A \in \mathbb{R}^{n \times d}$ undergoing $T$ batches of row updates of total size $n$ we give an algorithm which can maintain leverage score overestimates of the rows of $A$ summing to $\widetilde{O}(d)$ in total time $\widetilde{O}(nd + Td^6)$. This data structure is used to sample a spectral sparsifier within a robust IPM framework to establish the main result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02003 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive Matrix Sparsification and Applications to Empirical Risk Minimization Liu, Yang P. Peng, Richard Tang, Colin Weng, Albert Yang, Junzhao Data Structures and Algorithms Optimization and Control Consider the empirical risk minimization (ERM) problem, which is stated as follows. Let $K_1, \dots, K_m$ be compact convex sets with $K_i \subseteq \mathbb{R}^{n_i}$ for $i \in [m]$, $n = \sum_{i=1}^m n_i$, and $n_i\le C_K$ for some absolute constant $C_K$. Also, consider a matrix $A \in \mathbb{R}^{n \times d}$ and vectors $b \in \mathbb{R}^d$ and $c \in \mathbb{R}^n$. Then the ERM problem asks to find \[ \min_{\substack{x \in K_1 \times \dots \times K_m\\ A^\top x = b}} c^\top x. \] We give an algorithm to solve this to high accuracy in time $\widetilde{O}(nd + d^6\sqrt{n}) \le \widetilde{O} (nd + d^{11})$, which is nearly-linear time in the input size when $A$ is dense and $n \ge d^{10}$. Our result is achieved by implementing an $\widetilde{O}(\sqrt{n})$-iteration interior point method (IPM) efficiently using dynamic data structures. In this direction, our key technical advance is a new algorithm for maintaining leverage score overestimates of matrices undergoing row updates. Formally, given a matrix $A \in \mathbb{R}^{n \times d}$ undergoing $T$ batches of row updates of total size $n$ we give an algorithm which can maintain leverage score overestimates of the rows of $A$ summing to $\widetilde{O}(d)$ in total time $\widetilde{O}(nd + Td^6)$. This data structure is used to sample a spectral sparsifier within a robust IPM framework to establish the main result. |
| title | Adaptive Matrix Sparsification and Applications to Empirical Risk Minimization |
| topic | Data Structures and Algorithms Optimization and Control |
| url | https://arxiv.org/abs/2512.02003 |