Stochastic Optimization under Hidden Convexity

In this work, we consider constrained stochastic optimization problems under hidden convexity, i.e., those that admit a convex reformulation via non-linear (but invertible) map \(c(\cdot)\). A number of non-convex problems ranging from optimal control, revenue and inventory management, to convex rei...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Authors: Ilyas Fatkhullin, He, Niao, Hu, Yifan
Format: Article
Language:English
Subjects:
Convergence
Convexity
Inventory management
Optimal control
Optimization
Online Access:Get full text
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Summary:In this work, we consider constrained stochastic optimization problems under hidden convexity, i.e., those that admit a convex reformulation via non-linear (but invertible) map \(c(\cdot)\). A number of non-convex problems ranging from optimal control, revenue and inventory management, to convex reinforcement learning all admit such a hidden convex structure. Unfortunately, in the majority of applications considered, the map \(c(\cdot)\) is unavailable or implicit; therefore, directly solving the convex reformulation is not possible. On the other hand, the stochastic gradients with respect to the original variable are often easy to obtain. Motivated by these observations, we examine the basic projected stochastic (sub-) gradient methods for solving such problems under hidden convexity. We provide the first sample complexity guarantees for global convergence in smooth and non-smooth settings. Additionally, in the smooth setting, we improve our results to the last iterate convergence in terms of function value gap using the momentum variant of projected stochastic gradient descent.
ISSN:2331-8422