Optimizing Solution-Samplers for Combinatorial Problems: The Landscape of Policy-Gradient Methods

• URL: http://arxiv.org/abs/2310.05309v2
• Date: Tue, 7 Nov 2023 00:37:26 GMT
• Title: Optimizing Solution-Samplers for Combinatorial Problems: The Landscape of Policy-Gradient Methods
• Authors: Constantine Caramanis, Dimitris Fotakis, Alkis Kalavasis, Vasilis Kontonis, Christos Tzamos
• Abstract summary: We introduce a novel theoretical framework for analyzing the effectiveness of DeepMatching Networks and Reinforcement Learning methods. Our main contribution holds for a broad class of problems including Max-and Min-Cut, Max-$k$-Bipartite-Bi, Maximum-Weight-Bipartite-Bi, and Traveling Salesman Problem. As a byproduct of our analysis we introduce a novel regularization process over vanilla descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.
• Score: 52.0617030129699
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Deep Neural Networks and Reinforcement Learning methods have empirically shown great promise in tackling challenging combinatorial problems. In those methods a deep neural network is used as a solution generator which is then trained by gradient-based methods (e.g., policy gradient) to successively obtain better solution distributions. In this work we introduce a novel theoretical framework for analyzing the effectiveness of such methods. We ask whether there exist generative models that (i) are expressive enough to generate approximately optimal solutions; (ii) have a tractable, i.e, polynomial in the size of the input, number of parameters; (iii) their optimization landscape is benign in the sense that it does not contain sub-optimal stationary points. Our main contribution is a positive answer to this question. Our result holds for a broad class of combinatorial problems including Max- and Min-Cut, Max-$k$-CSP, Maximum-Weight-Bipartite-Matching, and the Traveling Salesman Problem. As a byproduct of our analysis we introduce a novel regularization process over vanilla gradient descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.

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