Weak Proximal Newton Oracles for Composite Convex Optimization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915682616082432 |
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| author | Garber, Dan |
| author_facet | Garber, Dan |
| contents | Second-order methods are of great importance for composite convex optimization problems due to their local super-linear convergence rates (under appropriate assumptions). However, the presence of even a simple nonsmooth function in the model most often renders the subproblems in proximal Newton methods computationally difficult to solve in high dimensions. We introduce a novel approach based on a \textit{weak proximal Newton oracle} (WPNO), which only requires solving such subproblems to accuracy that is comparable to that of the \emph{optimal solution of the global problem}, while maintaining local super-linear convergence under standard assumptions. Mainly, unlike classical inexact proximal Newton schemes, the complexity of our WPNO is not tied to (approximately) minimizing each subproblem; instead, we establish that when the optimal solution of the global problem admits a sparse structure, the inner subproblem can be solved by specialized first-order methods whose cost scales directly with the sparsity of this solution rather than with the ambient dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01432 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak Proximal Newton Oracles for Composite Convex Optimization Garber, Dan Optimization and Control Second-order methods are of great importance for composite convex optimization problems due to their local super-linear convergence rates (under appropriate assumptions). However, the presence of even a simple nonsmooth function in the model most often renders the subproblems in proximal Newton methods computationally difficult to solve in high dimensions. We introduce a novel approach based on a \textit{weak proximal Newton oracle} (WPNO), which only requires solving such subproblems to accuracy that is comparable to that of the \emph{optimal solution of the global problem}, while maintaining local super-linear convergence under standard assumptions. Mainly, unlike classical inexact proximal Newton schemes, the complexity of our WPNO is not tied to (approximately) minimizing each subproblem; instead, we establish that when the optimal solution of the global problem admits a sparse structure, the inner subproblem can be solved by specialized first-order methods whose cost scales directly with the sparsity of this solution rather than with the ambient dimension. |
| title | Weak Proximal Newton Oracles for Composite Convex Optimization |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2503.01432 |