Accuracy Certificates for Convex Optimization at Accelerated Rates via Primal-Dual Averaging
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910148875780096 |
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| author | Burns, Matthew X. Liang, Jiaming |
| author_facet | Burns, Matthew X. Liang, Jiaming |
| contents | Many works in convex optimization provide rates for achieving a small primal gap. However, this quantity is typically unavailable in practice. In this work, we show that solving a regularized surrogate with algorithms based on simple primal-dual averaging provides non-asymptotic convergence guarantees for a \textit{computable} optimality certificate. We first analyze primal and dual methods based on one average, namely modified dual averaging and generalized conditional gradient, and establish $\tilde{O}(\varepsilon^{-1})$ certificate complexities. Motivated by asymmetries in the one-average case, we analyze a self-dual, two-average method that preserves symmetry while losing certificate guarantees. To recover certificate convergence, we propose a three-average method that achieves an accelerated $\tilde{O}(\varepsilon^{-1/2})$ certificate complexity. Furthermore, we prove primal-dual algorithm correspondences for the one, two, and three-average cases. In particular, the primal three-average accelerated method mirrors the well-known gradient extrapolation method in the dual. By interpreting our results through the lens of zero-sum matrix games and Fisher markets, we further connect primal-dual averaging methods to game theory and market dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18321 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Accuracy Certificates for Convex Optimization at Accelerated Rates via Primal-Dual Averaging Burns, Matthew X. Liang, Jiaming Optimization and Control Many works in convex optimization provide rates for achieving a small primal gap. However, this quantity is typically unavailable in practice. In this work, we show that solving a regularized surrogate with algorithms based on simple primal-dual averaging provides non-asymptotic convergence guarantees for a \textit{computable} optimality certificate. We first analyze primal and dual methods based on one average, namely modified dual averaging and generalized conditional gradient, and establish $\tilde{O}(\varepsilon^{-1})$ certificate complexities. Motivated by asymmetries in the one-average case, we analyze a self-dual, two-average method that preserves symmetry while losing certificate guarantees. To recover certificate convergence, we propose a three-average method that achieves an accelerated $\tilde{O}(\varepsilon^{-1/2})$ certificate complexity. Furthermore, we prove primal-dual algorithm correspondences for the one, two, and three-average cases. In particular, the primal three-average accelerated method mirrors the well-known gradient extrapolation method in the dual. By interpreting our results through the lens of zero-sum matrix games and Fisher markets, we further connect primal-dual averaging methods to game theory and market dynamics. |
| title | Accuracy Certificates for Convex Optimization at Accelerated Rates via Primal-Dual Averaging |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2604.18321 |