Mini-batch stochastic subgradient for functional constrained optimization
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917852614754304 |
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| author | Singh, Nitesh Kumar Necoara, Ion Kungurtsev, Vyacheslav |
| author_facet | Singh, Nitesh Kumar Necoara, Ion Kungurtsev, Vyacheslav |
| contents | In this paper we consider finite sum composite convex optimization problems with many functional constraints. The objective function is expressed as a finite sum of two terms, one of which admits easy computation of (sub)gradients while the other is amenable to proximal evaluations. We assume a generalized bounded gradient condition on the objective which allows us to simultaneously tackle both smooth and nonsmooth problems. We also consider the cases of both with and without a strong convexity property. Further, we assume that each constraint set is given as the level set of a convex but not necessarily differentiable function. We reformulate the constrained finite sum problem into a stochastic optimization problem for which the stochastic subgradient projection method from [17] specializes to a collection of mini-batch variants, with different mini-batch sizes for the objective function and functional constraints, respectively. More specifically, at each iteration, our algorithm takes a mini-batch stochastic proximal subgradient step aimed at minimizing the objective function and then a subsequent mini-batch subgradient projection step minimizing the feasibility violation. By specializing different mini-batching strategies, we derive exact expressions for the stepsizes as a function of the mini-batch size and in some cases we also derive insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime. We also prove sublinear convergence rates for the mini-batch subgradient projection algorithm which depend explicitly on the mini-batch sizes and on the properties of the objective function. Numerical results also show a better performance of our mini-batch scheme over its single-batch counterpart. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_10616 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mini-batch stochastic subgradient for functional constrained optimization Singh, Nitesh Kumar Necoara, Ion Kungurtsev, Vyacheslav Optimization and Control In this paper we consider finite sum composite convex optimization problems with many functional constraints. The objective function is expressed as a finite sum of two terms, one of which admits easy computation of (sub)gradients while the other is amenable to proximal evaluations. We assume a generalized bounded gradient condition on the objective which allows us to simultaneously tackle both smooth and nonsmooth problems. We also consider the cases of both with and without a strong convexity property. Further, we assume that each constraint set is given as the level set of a convex but not necessarily differentiable function. We reformulate the constrained finite sum problem into a stochastic optimization problem for which the stochastic subgradient projection method from [17] specializes to a collection of mini-batch variants, with different mini-batch sizes for the objective function and functional constraints, respectively. More specifically, at each iteration, our algorithm takes a mini-batch stochastic proximal subgradient step aimed at minimizing the objective function and then a subsequent mini-batch subgradient projection step minimizing the feasibility violation. By specializing different mini-batching strategies, we derive exact expressions for the stepsizes as a function of the mini-batch size and in some cases we also derive insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime. We also prove sublinear convergence rates for the mini-batch subgradient projection algorithm which depend explicitly on the mini-batch sizes and on the properties of the objective function. Numerical results also show a better performance of our mini-batch scheme over its single-batch counterpart. |
| title | Mini-batch stochastic subgradient for functional constrained optimization |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2401.10616 |