Exploratory Combinatorial Optimization with Reinforcement Learning (1909.04063v2)

Abstract: Many real-world problems can be reduced to combinatorial optimization on a graph, where the subset or ordering of vertices that maximize some objective function must be found. With such tasks often NP-hard and analytically intractable, reinforcement learning (RL) has shown promise as a framework with which efficient heuristic methods to tackle these problems can be learned. Previous works construct the solution subset incrementally, adding one element at a time, however, the irreversible nature of this approach prevents the agent from revising its earlier decisions, which may be necessary given the complexity of the optimization task. We instead propose that the agent should seek to continuously improve the solution by learning to explore at test time. Our approach of exploratory combinatorial optimization (ECO-DQN) is, in principle, applicable to any combinatorial problem that can be defined on a graph. Experimentally, we show our method to produce state-of-the-art RL performance on the Maximum Cut problem. Moreover, because ECO-DQN can start from any arbitrary configuration, it can be combined with other search methods to further improve performance, which we demonstrate using a simple random search.

Exploratory Combinatorial Optimization with Reinforcement Learning

Introduction

The paper "Exploratory Combinatorial Optimization with Reinforcement Learning" by Barrett et al. addresses the complex issue of combinatorial optimization on graph structures, a core challenge in computational fields due to the NP-hard nature of these problems. Traditional approaches often involve incremental construction of solutions; however, this method may result in suboptimal outcomes since it lacks the flexibility to revisit past decisions. The authors propose an innovative approach where the agent employs a strategy of continuous exploration, allowing modifications and improvements to solutions during test time. They introduce the ECO-DQN (Exploratory Combinatorial Optimization Deep Q-Network) framework, specifically applied to the Maximum Cut (Max-Cut) problem, a quintessential NP-hard problem with extensive applications in various domains.

Methodology

The proposed ECO-DQN framework diverges from classic reinforcement learning (RL) approaches by reformulating the problem into one of exploration at test time. Instead of fixing decisions sequentially, the agent iteratively refines a solution by dynamically adding or removing vertices from the solution set, hence traversing the solution space actively. This enables the model to search for optimal or near-optimal solutions in a way that adapts as the episode progresses. The agents are trained using a message-passing neural network (MPNN), which is apt for capturing the relational properties of the graph vertices and edges.

The work employs a modified reward structure to facilitate exploration, providing intermediate rewards for achieving locally optimal solutions and incentivizing state transitions that lead to new, potentially higher-quality solutions. This approach leverages a deep reinforcement learning methodology with Q-learning to approximate the optimal policy for various graph configurations, allowing the agent to seek paths that maximize the expected cumulative reward.

Experimental Results

Empirical tests demonstrated that ECO-DQN effectively outperforms previous state-of-the-art approaches like S2V-DQN, especially when applied to the Max-Cut problem across Erdos-Renyi (ER) and Barabasi-Albert (BA) graphs. Notably, ECO-DQN retained its superior performance across both unseen graph sizes and structures, achieving approximation ratios close to optimal even with varied test graph distributions. The model displayed remarkable flexibility, showcasing its capability to generalize well across different types of graph configurations that were not encountered during training.

A significant insight from the paper was the potential for further performance enhancement through multiple, randomly initialized episodes, harnessing the inherent stochastic dynamics of diverse initial states to explore the solution space more comprehensively. This stochastic exploration heavily contributes toward local and global optimization, reinforcing the advantage of exploration-based strategies in solving combinatorial problems.

Implications and Future Directions

The implications of this research are manifold. The versatility in adapting to different graph distributions suggests significant real-world utility in domains such as portfolio optimization and logistical networks. By allowing dynamic revision of decisions, ECO-DQN could redefine how combinatorial optimization problems are addressed, potentially setting a precedent for hybrid models integrating RL with traditional optimization techniques.

Future research may delve into enhancing the ECO-DQN with recurrent architectures to represent episodic memory states or exploring other combinatorial tasks to evaluate model robustness. Additionally, leveraging ECO-DQN's flexibility to be fine-tuned with other heuristic methods or domain-specific knowledge can open avenues for integrating this dynamic RL approach into broader classes of optimization challenges.

Ultimately, this paper provides a compelling argument for an exploration-centric paradigm in combinatorial optimization using reinforcement learning, demonstrating substantial improvements in solution quality and generalizability over existing RL frameworks.