Many traditional algorithms for solving combinatorial optimization problems involve using hand-crafted heuristics that sequentially construct a solution. Such heuristics are designed by domain experts and may often be suboptimal due to the hard nature of the problems. Reinforcement learning (RL) proposes a good alternative to automate the search of these heuristics by training an agent in a supervised or self-supervised manner. In this survey, we explore the recent advancements of applying RL frameworks to hard combinatorial problems. Our survey provides the necessary background for operations research and machine learning communities and showcases the works that are moving the field forward. We juxtapose recently proposed RL methods, laying out the timeline of the improvements for each problem, as well as we make a comparison with traditional algorithms, indicating that RL models can become a promising direction for solving combinatorial problems.
MIT Operations Research Center（麻省理工学院运筹学中心）；Google Research（谷歌研究院）
Value-function-based methods have long played an important role in reinforcement learning. However, finding the best next action given a value function of arbitrary complexity is nontrivial when the action space is too large for enumeration. We develop a framework for value-function-based deep reinforcement learning with a combinatorial action space, in which the action selection problem is explicitly formulated as a mixed-integer optimization problem. As a motivating example, we present an application of this framework to the capacitated vehicle routing problem (CVRP), a combinatorial optimiz…
University of the Basque Country（巴斯克大学）；Lehigh University（理海大学）
This paper presents a framework to tackle constrained combinatorial optimization problems using deep Reinforcement Learning (RL). To this end, we extend the Neural Combinatorial Optimization (NCO) theory in order to deal with constraints in its formulation. Notably, we propose defining constrained combinatorial problems as fully observable Constrained Markov Decision Processes (CMDP). In that context, the solution is iteratively constructed based on interactions with the environment. The model, in addition to the reward signal, relies on penalty signals generated from constraint dissatisfactio…
University of Alberta（阿尔伯塔大学）；ETH Zurich（苏黎世联邦理工学院）；University College London（伦敦大学学院）；DeepMind（深度思维）
A practical challenge in reinforcement learning are combinatorial action spaces that make planning computationally demanding. For example, in cooperative multi-agent reinforcement learning, a potentially large number of agents jointly optimize a global reward function, which leads to a combinatorial blow-up in the action space by the number of agents. As a minimal requirement, we assume access to an argmax oracle that allows to efficiently compute the greedy policy for any Q-function in the model class. Building on recent work in planning with local access to a simulator and linear function ap…
This paper presents a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning. We focus on the traveling salesman problem (TSP) and train a recurrent network that, given a set of city coordinates, predicts a distribution over different city permutations. Using negative tour length as the reward signal, we optimize the parameters of the recurrent network using a policy gradient method. We compare learning the network parameters on a set of training graphs against learning them on individual test graphs. Despite the computational expense, without …
We develop a general method for incentive-based programming of hybrid quantum-classical computing systems using reinforcement learning, and apply this to solve combinatorial optimization problems on both simulated and real gate-based quantum computers. Relative to a set of randomly generated problem instances, agents trained through reinforcement learning techniques are capable of producing short quantum programs which generate high quality solutions on both types of quantum resources. We observe generalization to problems outside of the training set, as well as generalization from the simulat…
University of Alberta（阿尔伯塔大学）；Huawei Technologies Canada Co., Ltd.（华为技术加拿大有限公司）
Branch-and-bound is a systematic enumerative method for combinatorial optimization, where the performance highly relies on the variable selection strategy. State-of-the-art handcrafted heuristic strategies suffer from relatively slow inference time for each selection, while the current machine learning methods require a significant amount of labeled data. We propose a new approach for solving the data labeling and inference latency issues in combinatorial optimization based on the use of the reinforcement learning (RL) paradigm. We use imitation learning to bootstrap an RL agent and then use P…
In sponsored search, keyword recommendations help advertisers to achieve much better performance within limited budget. Many works have been done to mine numerous candidate keywords from search logs or landing pages. However, the strategy to select from given candidates remains to be improved. The existing relevance-based, popularity-based and regular combinatorial strategies fail to take the internal or external competitions among keywords into consideration. In this paper, we regard keyword recommendations as a combinatorial optimization problem and solve it with a modified pointer network s…
Columbia University（哥伦比亚大学）；Cornell University（康奈尔大学）
In this work, we introduce Graph Pointer Networks (GPNs) trained using reinforcement learning (RL) for tackling the traveling salesman problem (TSP). GPNs build upon Pointer Networks by introducing a graph embedding layer on the input, which captures relationships between nodes. Furthermore, to approximate solutions to constrained combinatorial optimization problems such as the TSP with time windows, we train hierarchical GPNs (HGPNs) using RL, which learns a hierarchical policy to find an optimal city permutation under constraints. Each layer of the hierarchy is designed with a separate rewar…
University of Washington（华盛顿大学）；Microsoft Research（微软研究院）
We introduce an online popularity prediction and tracking task as a benchmark task for reinforcement learning with a combinatorial, natural language action space. A specified number of discussion threads predicted to be popular are recommended, chosen from a fixed window of recent comments to track. Novel deep reinforcement learning architectures are studied for effective modeling of the value function associated with actions comprised of interdependent sub-actions. The proposed model, which represents dependence between sub-actions through a bi-directional LSTM, gives the best performance acr…
University of Oxford（牛津大学）；Facebook AI Research（Facebook AI研究院）；Russian Quantum Center（俄罗斯量子中心）
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 give…
Russian Quantum Center（俄罗斯量子中心）；University of Oxford（牛津大学）
Quantum hardware and quantum-inspired algorithms are becoming increasingly popular for combinatorial optimization. However, these algorithms may require careful hyperparameter tuning for each problem instance. We use a reinforcement learning agent in conjunction with a quantum-inspired algorithm to solve the Ising energy minimization problem, which is equivalent to the Maximum Cut problem. The agent controls the algorithm by tuning one of its parameters with the goal of improving recently seen solutions. We propose a new Rescaled Ranked Reward (R3) method that enables stable single-player vers…
POSTECH（浦项科技大学）；University of Guelph（圭尔夫大学）；Vector Institute（矢量研究所）；POSCO（浦项制铁公司）
Discovering a solution in a combinatorial space is prevalent in many real-world problems but it is also challenging due to diverse complex constraints and the vast number of possible combinations. To address such a problem, we introduce a novel formulation, combinatorial construction, which requires a building agent to assemble unit primitives (i.e., LEGO bricks) sequentially -- every connection between two bricks must follow a fixed rule, while no bricks mutually overlap. To construct a target object, we provide incomplete knowledge about the desired target (i.e., 2D images) instead of exact …
Adversarial self-play in two-player games has delivered impressive results when used with reinforcement learning algorithms that combine deep neural networks and tree search. Algorithms like AlphaZero and Expert Iteration learn tabula-rasa, producing highly informative training data on the fly. However, the self-play training strategy is not directly applicable to single-player games. Recently, several practically important combinatorial optimisation problems, such as the travelling salesman problem and the bin packing problem, have been reformulated as reinforcement learning problems, increas…
Combinatorial optimization problems (COPs) on the graph with real-life applications are canonical challenges in Computer Science. The difficulty of finding quality labels for problem instances holds back leveraging supervised learning across combinatorial problems. Reinforcement learning (RL) algorithms have recently been adopted to solve this challenge automatically. The underlying principle of this approach is to deploy a graph neural network (GNN) for encoding both the local information of the nodes and the graph-structured data in order to capture the current state of the environment. Then…
Combinatorial optimization problem (COP) over graphs is a fundamental challenge in optimization. Reinforcement learning (RL) has recently emerged as a new framework to tackle these problems and has demonstrated promising results. However, most RL solutions employ a greedy manner to construct the solution incrementally, thus inevitably pose unnecessary dependency on action sequences and need a lot of problem-specific designs. We propose a general RL framework that not only exhibits state-of-the-art empirical performance but also generalizes to a variety class of COPs. Specifically, we define st…
Illinois Institute of Technology（伊利诺伊理工学院）；Clemson University（克莱姆森大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Argonne National Laboratory（阿贡国家实验室）
Quantum computing exploits basic quantum phenomena such as state superposition and entanglement to perform computations. The Quantum Approximate Optimization Algorithm (QAOA) is arguably one of the leading quantum algorithms that can outperform classical state-of-the-art methods in the near term. QAOA is a hybrid quantum-classical algorithm that combines a parameterized quantum state evolution with a classical optimization routine to approximately solve combinatorial problems. The quality of the solution obtained by QAOA within a fixed budget of calls to the quantum computer depends on the per…
Symmetric Replay Training: Enhancing Sample Efficiency in Deep Reinforcement Learning for Combinatorial Optimization
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院（KAIST））；Pohang University of Science and Technology (POSTECH)（浦项科技大学（POSTECH））；OMELET
Ecole Polytechnique de Montréal（蒙特利尔综合理工学院）；Element AI（Element AI公司）；University of Toronto Scarborough（多伦多大学士嘉堡分校）
Combinatorial optimization has found applications in numerous fields, from aerospace to transportation planning and economics. The goal is to find an optimal solution among a finite set of possibilities. The well-known challenge one faces with combinatorial optimization is the state-space explosion problem: the number of possibilities grows exponentially with the problem size, which makes solving intractable for large problems. In the last years, deep reinforcement learning (DRL) has shown its promise for designing good heuristics dedicated to solve NP-hard combinatorial optimization problems.…
Aiming at a comprehensive and concise tutorial survey, recap of variational inference and reinforcement learning with Probabilistic Graphical Models are given with detailed derivations. Reviews and comparisons on recent advances in deep reinforcement learning are made from various aspects. We offer detailed derivations to a taxonomy of Probabilistic Graphical Model and Variational Inference methods in deep reinforcement learning, which serves as a complementary material on top of the original contributions.
University of New Hampshire（新罕布什尔大学）；Department of Computer Science（计算机科学系）
A reinforcement learning agent tries to maximize its cumulative payoff by interacting in an unknown environment. It is important for the agent to explore suboptimal actions as well as to pick actions with highest known rewards. Yet, in sensitive domains, collecting more data with exploration is not always possible, but it is important to find a policy with a certain performance guaranty. In this paper, we present a brief survey of methods available in the literature for balancing exploration-exploitation trade off and computing robust solutions from fixed samples in reinforcement learning.
With the advancement of deep learning techniques, major cloud providers and niche machine learning service providers start to offer their cloud-based machine learning tools, also known as machine learning as a service (MLaaS), to the public. According to our measurement, for the same task, these MLaaSes from different providers have varying performance due to the proprietary datasets, models, etc. Federating different MLaaSes together allows us to improve the analytic performance further. However, naively aggregating results from different MLaaSes not only incurs significant momentary cost but…
KAIST（韩国科学技术院）；King’s College London（伦敦国王学院）；Naver Corporation（Naver公司）
We consider the Markov Decision Process (MDP) of selecting a subset of items at each step, termed the Select-MDP (S-MDP). The large state and action spaces of S-MDPs make them intractable to solve with typical reinforcement learning (RL) algorithms especially when the number of items is huge. In this paper, we present a deep RL algorithm to solve this issue by adopting the following key ideas. First, we convert the original S-MDP into an Iterative Select-MDP (IS-MDP), which is equivalent to the S-MDP in terms of optimal actions. IS-MDP decomposes a joint action of selecting K items simultaneou…
This manuscript surveys reinforcement learning from the perspective of optimization and control with a focus on continuous control applications. It surveys the general formulation, terminology, and typical experimental implementations of reinforcement learning and reviews competing solution paradigms. In order to compare the relative merits of various techniques, this survey presents a case study of the Linear Quadratic Regulator (LQR) with unknown dynamics, perhaps the simplest and best-studied problem in optimal control. The manuscript describes how merging techniques from learning theory an…
The formal semantics of an interpreted first-order logic (FOL) statement can be given in Tarskian Semantics or a basically equivalent Game Semantics. The latter maps the statement and the interpretation into a two-player semantic game. Many combinatorial problems can be described using interpreted FOL statements and can be mapped into a semantic game. Therefore, learning to play a semantic game perfectly leads to the solution of a specific instance of a combinatorial problem. We adapt the AlphaZero algorithm so that it becomes better at learning to play semantic games that have different chara…
Indian Institute of Technology Delhi（德里印度理工学院）；Northwestern University（西北大学）；University of California Santa Barbara（加州大学圣巴巴拉分校）
There has been an increased interest in discovering heuristics for combinatorial problems on graphs through machine learning. While existing techniques have primarily focused on obtaining high-quality solutions, scalability to billion-sized graphs has not been adequately addressed. In addition, the impact of budget-constraint, which is necessary for many practical scenarios, remains to be studied. In this paper, we propose a framework called GCOMB to bridge these gaps. GCOMB trains a Graph Convolutional Network (GCN) using a novel probabilistic greedy mechanism to predict the quality of a node…
McGill University（麦吉尔大学）；University of Amsterdam（阿姆斯特丹大学）
Exploration is an essential component of reinforcement learning algorithms, where agents need to learn how to predict and control unknown and often stochastic environments. Reinforcement learning agents depend crucially on exploration to obtain informative data for the learning process as the lack of enough information could hinder effective learning. In this article, we provide a survey of modern exploration methods in (Sequential) reinforcement learning, as well as a taxonomy of exploration methods.
Combinatorial Multivariant Multi-Armed Bandits with Applications to Episodic Reinforcement Learning and Beyond
The Chinese University of Hong Kong（香港中文大学）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Microsoft Research（微软研究院）；Shanghai Jiao Tong University（上海交通大学）
Deep reinforcement learning is poised to revolutionise the field of AI and represents a step towards building autonomous systems with a higher level understanding of the visual world. Currently, deep learning is enabling reinforcement learning to scale to problems that were previously intractable, such as learning to play video games directly from pixels. Deep reinforcement learning algorithms are also applied to robotics, allowing control policies for robots to be learned directly from camera inputs in the real world. In this survey, we begin with an introduction to the general field of reinf…
Inverse reinforcement learning (IRL) is the problem of inferring the reward function of an agent, given its policy or observed behavior. Analogous to RL, IRL is perceived both as a problem and as a class of methods. By categorically surveying the current literature in IRL, this article serves as a reference for researchers and practitioners of machine learning and beyond to understand the challenges of IRL and select the approaches best suited for the problem on hand. The survey formally introduces the IRL problem along with its central challenges such as the difficulty in performing accurate …
Reinforcement learning has recently experienced increased prominence in the machine learning community. There are many approaches to solving reinforcement learning problems with new techniques developed constantly. When solving problems using reinforcement learning, there are various difficult challenges to overcome. To ensure progress in the field, benchmarks are important for testing new algorithms and comparing with other approaches. The reproducibility of results for fair comparison is therefore vital in ensuring that improvements are accurately judged. This paper provides an overview of d…
Explainable reinforcement learning (XRL) is an emerging subfield of explainable machine learning that has attracted considerable attention in recent years. The goal of XRL is to elucidate the decision-making process of learning agents in sequential decision-making settings. In this survey, we propose a novel taxonomy for organizing the XRL literature that prioritizes the RL setting. We overview techniques according to this taxonomy. We point out gaps in the literature, which we use to motivate and outline a roadmap for future work.
National University of Singapore（新加坡国立大学）；National University Hospital（国立大学医院）
Owe to the recent advancements in Artificial Intelligence especially deep learning, many data-driven decision support systems have been implemented to facilitate medical doctors in delivering personalized care. We focus on the deep reinforcement learning (DRL) models in this paper. DRL models have demonstrated human-level or even superior performance in the tasks of computer vision and game playings, such as Go and Atari game. However, the adoption of deep reinforcement learning techniques in clinical decision optimization is still rare. We present the first survey that summarizes reinforcemen…
Georgia Institute of Technology（佐治亚理工学院）；Microsoft Research（微软研究院）
Interactive Fiction games are text-based simulations in which an agent interacts with the world purely through natural language. They are ideal environments for studying how to extend reinforcement learning agents to meet the challenges of natural language understanding, partial observability, and action generation in combinatorially-large text-based action spaces. We present KG-A2C, an agent that builds a dynamic knowledge graph while exploring and generates actions using a template-based action space. We contend that the dual uses of the knowledge graph to reason about game state and to cons…
Graph problems such as traveling salesman problem, or finding minimal Steiner trees are widely studied and used in data engineering and computer science. Typically, in real-world applications, the features of the graph tend to change over time, thus, finding a solution to the problem becomes challenging. The dynamic version of many graph problems are the key for a plethora of real-world problems in transportation, telecommunication, and social networks. In recent years, using deep learning techniques to find heuristic solutions for NP-hard graph combinatorial problems has gained much interest …
In this paper we discuss policy iteration methods for approximate solution of a finite-state discounted Markov decision problem, with a focus on feature-based aggregation methods and their connection with deep reinforcement learning schemes. We introduce features of the states of the original problem, and we formulate a smaller "aggregate" Markov decision problem, whose states relate to the features. We discuss properties and possible implementations of this type of aggregation, including a new approach to approximate policy iteration. In this approach the policy improvement operation combines…
Sequential decision making, commonly formalized as Markov Decision Process (MDP) optimization, is a important challenge in artificial intelligence. Two key approaches to this problem are reinforcement learning (RL) and planning. This paper presents a survey of the integration of both fields, better known as model-based reinforcement learning. Model-based RL has two main steps. First, we systematically cover approaches to dynamics model learning, including challenges like dealing with stochasticity, uncertainty, partial observability, and temporal abstraction. Second, we present a systematic ca…
School of Management, Technical University of Munich, Germany（慕尼黑技术大学管理学院）；Munich Data Science Institute, Technical University of Munich, Germany（慕尼黑数据科学研究所）；Inria, Université de Lille, France（法国里尔大学Inria）；CERMICS, Ecole Nationale des Ponts et Chaussées, Institut Polytechnique Paris, France（巴黎高等理工学院CERMICS）
