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.

翻译：深度神经网络与强化学习方法在处理具有挑战性的组合问题时，已在实证中展现出巨大潜力。在这些方法中，深度神经网络被用作解生成器，随后通过基于梯度的训练方法（如策略梯度）逐步获得更优的解分布。本文提出了一种全新的理论框架，用于分析此类方法的有效性。我们探究是否存在生成模型，能够满足以下条件：（i）具备足够表达能力以生成接近最优的解；（ii）拥有参数数量与输入规模成多项式关系的可处理模型；（iii）其优化格局不存在非优平稳点（即良性格局）。我们的主要贡献在于对该问题给出了肯定答案。该结论适用于一大类组合问题，包括Max-和Min-Cut、Max-$k$-CSP、最大权二分匹配以及旅行商问题。作为分析的副产品，我们引入了一种针对原始梯度下降的新型正则化过程，并从理论与实验两方面证明，它有助于解决梯度消失问题并规避不良平稳点。