Complexity reduction in online stochastic Newton methods with potential O(N d) total cost
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910012779003904 |
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| author | Godichon-Baggioni, Antoine Portier, Bruno Sallé, Guillaume |
| author_facet | Godichon-Baggioni, Antoine Portier, Bruno Sallé, Guillaume |
| contents | Optimizing smooth convex functions in stochastic settings, where only noisy estimates of gradients and Hessians are available, is a fundamental problem in optimization. While first-order methods possess a low per-iteration cost, their convergence is slow for ill-conditioned problems. Stochastic Newton methods utilize second-order information to correct for local curvature, but the O(d 3 ) per-iteration cost of computing and inverting a full Hessian, where d is the problem dimension, is prohibitive in high dimensions. This paper introduces an online mini-batch stochastic Newton algorithm. The method employs a random masking strategy that selects a subset of Hessian columns at each iteration, substantially reducing the per-step computational cost. This approach allows the algorithm, in the mini-batch setting, to achieve a total computational cost for a single pass over N data points of O(N d), which is comparable to first-order methods while retaining the advantages of second-order information. We establish the almost sure convergence and asymptotic efficiency of the resulting estimator. This property is obtained without requiring iterate averaging, which distinguishes this work from prior analyses. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05460 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complexity reduction in online stochastic Newton methods with potential O(N d) total cost Godichon-Baggioni, Antoine Portier, Bruno Sallé, Guillaume Statistics Theory Optimizing smooth convex functions in stochastic settings, where only noisy estimates of gradients and Hessians are available, is a fundamental problem in optimization. While first-order methods possess a low per-iteration cost, their convergence is slow for ill-conditioned problems. Stochastic Newton methods utilize second-order information to correct for local curvature, but the O(d 3 ) per-iteration cost of computing and inverting a full Hessian, where d is the problem dimension, is prohibitive in high dimensions. This paper introduces an online mini-batch stochastic Newton algorithm. The method employs a random masking strategy that selects a subset of Hessian columns at each iteration, substantially reducing the per-step computational cost. This approach allows the algorithm, in the mini-batch setting, to achieve a total computational cost for a single pass over N data points of O(N d), which is comparable to first-order methods while retaining the advantages of second-order information. We establish the almost sure convergence and asymptotic efficiency of the resulting estimator. This property is obtained without requiring iterate averaging, which distinguishes this work from prior analyses. |
| title | Complexity reduction in online stochastic Newton methods with potential O(N d) total cost |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2602.05460 |