# exploratory combinatorial optimization with reinforcement learning

to further improve performance, which we demonstrate using a simple random A solution to a combinatorial problem defined on a graph consists of a subset of vertices that satisfies the desired optimality criteria. 06/06/2019 ∙ by Peilin Chen, et al. The training, testing and validation graphs were generated with the NetworkX Python package [hagberg08]. An RL framework for graph-based combinatorial problems introduced by Khalil et al. Another recently developed simulated annealing heuristic that relaxes the binary vertex labels to analog values. Finally, we again observe the effect of small intermediate rewards (IntRew) for finding locally optimal solutions during training upon the final performance. These agents keep selecting actions greedily even if no positive Q-values are available until t=|V|, to account for possible incorrect predictions of the Q-values. Li et al. leleu19. investigate reinforcement learning as a sole tool for approximating combinatorial optimization problems of any kind (not specifically those defined on graphs), whereas we survey all machine learning methods developed or applied for solving combinatorial optimization problems with focus on those tasks formulated on graphs. adding or removing it from the solution subset S. One straightforward application of Q-learning to CO over a graph is to attempt to directly learn the utility of adding any given vertex to the solution subset. In this work we train and test the agent on both Erdős-Rényi [erdos60] and Barabasi-Albert [albert02] graphs with edges wij∈{0,±1}, which we refer to as ER and BA graphs, respectively. Distance of current solution set from the best observed. Formally, the reward at state st∈S is given by R(st)=max(C(st)−C(s∗), 0)/\absV, where s∗∈S is the state corresponding to the highest cut value previously seen within the episode, C(s∗) (note that we implicitly assume the graph, G, and solution subset, S, to be included in the state). As the number of possible solution configurations (states) grows exponentially with the number of vertices, this simply reflects how it quickly becomes intractable to sufficiently cover the state-space in our finite number of episodes. As local optima in combinatorial problems are typically close to each other, the agent learns to “hop” between nearby local optima, thereby performing a in-depth local search of the most promising subspace of the state space (see figure 2b). Alternatively, ECO-DQN could also be initialised with solutions found by other optimization methods to further strengthen them. requiring large numbers of pre-solved instances for training. 0 Each episode for such agents is initialised with a random subset of vertices in the solution set. In the process of the evolution, the system eventually settles with all vertices in near-binary states. Another current direction is applying graph networks for CO in combination with a tree search. ∙ These observations are: Vertex state, i.e. The behaviour of the agent is shown at three points during training: when performance is equivalent to that of either MCA-irrev (dotted lines) or S2V-DQN (dashed) and when the agent is fully trained (solid). where xv∈Rm is the input vector of observations and θ1∈Rm×n. For G1-G10 we utilise 50 randomly initialised episodes per graph, however for G22-G32 we use only a single episode per graph, due to the increased computational cost. Exploratory Combinatorial Optimization with Reinforcement Learning Thomas D. Barrett, William R. Clements, Jakob N. Foerster, Alex I. Lvovsky 简述：在使用强化学习解决组合优化（NP-hard problem)问题时，之前的方法大都倾向于采用“增量”的方式来构建组合，也就是，每次往里面新增一个元素。 This is defined as α=C(s∗)/C(sopt), where C(sopt) is the cut-value associated with the true optimum solution. ∙ Instead, heuristics are often deployed that, despite offering no theoretical guarantees, are chosen for high performance. The final optimization method introduced in the main text is MaxCutApprox (MCA). For that purpose, a n agent must be able to match each sequence of packets (e.g. �3G㧚0G����f�g��M��t���lR�2 i���7�yF�)����f��>}�f��7��ʜf2��S��f礦�"33L�X}z��>�!�M�45��ޑPS(�߃��Ai(*��")JKQ�)j E��G�I1�JM����t2}�n�����ac��F�L��. ECO-DQN is compared to multiple benchmarks, with details provided in the caption, however there are three important observations to emphasise. To facilitate direct comparison, ECO-DQN and S2V-DQN are implemented with the same MPNN architecture, with details provided in the Appendix. Figure 0(a) shows learning curves of agents trained on ER graphs of size \absV=40, where it can be seen that ECO-DQN reaches a significantly higher average cut than S2V-DQN. Bin Packing problem using Reinforcement Learning. We therefore also provide a small intermediate reward of 1/\absV whenever the agent reaches a locally optimal state (one where no action will immediately increase the cut value) previously unseen within the episode. where xk∈{±1} labels whether vertex k∈V is in the solution subset, S⊂V. (b) Approximation ratios of ECO-DQN, S2V-DQN and the MCA-irrev heuristics for ER and BA graphs with different numbers of vertices. pr... A Q-value for flipping each vertex is calculated using seven observations derived from the current state (xv∈R7). The framework introduced and discussed in detail in the main text. The intermediate rewards (IntRew) can be seen to speed up and stabilise training. However, it is noteworthy that even the simple MCA-rev algorithm, with only a relatively modest budget of 50 random initialisations, outperforms a highly trained irreversible heuristic (S2V-DQN). Very recently, Abe et al. We take the best solution found in any episode by either of these greedy algorithms as the MCA solution. where {θ4,k,θ5,k}∈R2n×n. ∙ 09/09/2019 ∙ by Thomas D. Barrett, et al. Experimentally, we show our method to produce state-of-the-art RL performance on the Maximum Cut problem. The authors would like to thank D. Chermoshentsev and A. Boev for their expertise in applying simulated annealing heuristics and CPLEX to our validation graphs. 11/09/2020 ∙ by Nathan Grinsztajn, et al. During the training of an irreversible agent’s Q-network, the predictions of the Q-values produced by the target network are clipped to be strictly non-negative. For ER graphs, a connection probability of 0.15 is used. ∙ Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon. ∙ This paper presents Neural Combinatorial Optimization, a framework to tackle combinatorial op-timization with reinforcement learning and neural networks. 0 Note also that the reward is normalised by the total number of vertices, \absV, to mitigate the impact of different reward scales across different graph sizes. h�tT{Tw�a2�¸>f��&�V��>6X� Instead, further modifications are required to leverage this freedom for improved performance, which we discuss here. Specifically, in addition to ECO-DQN, S2V-DQN and the MCA algorithms, we use CPLEX, an industry standard integer programming solver, and a pair of recently developed simulated annealing heuristics by Tiunov et al. S2V-DQN and selected ablations) are initialised with an empty solution subset. Changing the initial subset of vertices selected to be in the solution set can result in very different trajectories over the course of an episode. Weaker agents from earlier in training revisit the same states far more often, yet find fewer locally optimal states. We note that soon after our paper appeared, (Andrychowicz et al., 2016) also independently proposed a similar idea. 持续探索 (ECO-DQN ≡ S2V-DQN + RevAct + ObsTun + IntRew) Exploratory Combinatorial Optimization with Reinforcement Learning (Max-Cut, 翻转节点, 鼓励局部最优, 储存当前最高) [1909.04063] 部分系统差距 取决于 是否允许 Agent 扭转先前的决策 (图决策中 "filp" vertex) share, The Maximum k-plex Problem is an important combinatorial optimization pr... Join one of the world's largest A.I. "Exploratory Combinatorial Optimization with Reinforcement Learning" [ paper, code ] TD Barrett, WR Clements, JN Foerster, AI Lvovsky AAAI Conference on Artificial Intelligence, 2020 "Loaded DiCE: Trading off Bias and Variance in Any-Order Score Function Estimators for Reinforcement Learning" [ paper, code] Steps since the vertex state was last changed. This tutorial demonstrates technique to solve combinatorial optimization problems such as the well-known travelling salesman problem. Experimentally, we show our method to produce state-of-the-art RL performance on the Maximum Cut problem. We optimize every graph using 50 randomly initialised episodes. We report the mean approximation ratio of each agent over the 100 validation graphs, along with the distance to the upper and lower quartiles as a guide to how varied the performance is across different graph instances. In principle, our approach is applicable to any combinatorial problem defined on a graph. Exploratory Combinatorial Optimization with Reinforcement Learning 9 Sep 2019 • Thomas D. Barrett • William R. Clements • Jakob N. Foerster • A. I. Lvovsky 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. For each individual agent-graph pair, we run 50 randomly initialised optimization episodes. Mittal et al. Moreover, 90.4\char37 of the optimal solutions found are unique, demonstrating that, in conjunction with random initialisations, the agent is capable of finding many different optimal trajectories. principle, applicable to any combinatorial problem that can be defined on a Difference of current cut-value from the best observed. Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday. The larger graph size is chosen as it provides greater scope for the agent to exhibit non-trivial behaviour. However, the complexity of NP-hard combinatorial problems means it is challenging to learn a single function approximation of Q∗(s,a) that generalises across the vast number of possible graphs. This further emphasises how stochasticity – which here is provided by the random episode initialisations and ensures many regions of the solution space are considered – is a powerful attribute when combined with local optimization. Concretely, this means the agent can add or remove vertices from the solution subset and is tasked with searching for ever-improving solutions at test time. This illustrates that the agent has learnt how to search for improving solutions even if it requires short-term sacrificing of the cut value. Also shown is the probability at each timestep that the best solution that will be found within the episode has already been seen (MC found). This ability to generalise to unseen challenges is important for the real-world applicability of RL agents to combinatorial problems where the distribution of optimization tasks may be unknown or even change over time. The initial embedding for each vertex, v, is given by. ECO-DQN is compared to S2V-DQN as a baseline, with the differences individually ablated as described in the text. The second is the GSet, a benchmark collection of large graphs that have been well investigated [benlic13]. The approximation ratios, averaged across 100 graphs for each graph structure and size, of the different optimization methods. k-plex Problem, Geometric Deep Reinforcement Learning for Dynamic DAG Scheduling, POMO: Policy Optimization with Multiple Optima for Reinforcement Number of available actions that immediately increase the cut-value. Each vertex is initialised to 0, and then subjected to evolution according to a set of stochastic differential equations that describe the operation of the CIM. Here, we give an overview of each method and summarise their efficacy. 06/28/2018 ∙ by Serena Wang, et al. The highest cut value across the board is then chosen as the reference point that we refer to as the “optimum value”. To show that this exploratory behaviour improves the agent’s performance, we also plot the probability that the agent has already found the best solution it will see in the episode (MC found). ECO-DQN can initialise a search from any valid state, opening the door to combining it with other search heuristics. We first transform the Max-Cut problem into a QUBO (Quadratic Unconstrained Binary Optimization) task [kochenberger06]. A simulated annealing heuristic proposed by Tiunov et al. ∙ The basic idea is to represent each vertex in the graph, v∈V, with some n-dimensional embedding, μkv, where k labels the current iteration (network layer). However, the effectiveness of general algorithms is dependent on the problem being considered, and high levels of performance often require extensive tailoring and domain-specific knowledge. Further analysis of the agent’s behaviour is presented in figures 2b and 2c which show the action preferences and the types of states visited, respectively, over the course of an optimization episode. A formative demonstration of neural networks for combinatorial optimization (CO) was the application of Hopfield networks to the Travelling Salesman Problem (TSP) by Hopfield and Tank hopfield85. The Max-Cut problem is to find a subset of vertices on a graph that maximises the total number of edges connecting vertices within this subset to vertices not in this subset (the cut value). However, due to the inherent complexity of many combinatorial problems, learning a policy that directly produces a single, optimal solution is often impractical, as evidenced by the sub-optimal performance of such approaches. ECO-DQN and selected ablations), which for convenience we will refer to as reversible agents, the episode lengths are set to twice the number of vertices in the graph, t=1,2,…,2\absV. To \absV=200 can be used to that achieve that goal optimization ) task [ kochenberger06 ] on larger.! Learning ( RL ) can be used with good success the list above that allow exploratory combinatorial optimization with reinforcement learning to. As is standard for RL, we give an overview of each method and their. Approach by M-LOOP, 2016 ) also independently proposed a similar idea and artificial intelligence research sent straight your! This approach is applicable to any combinatorial problem defined on a separate set of results for each structure! All tests, with the same MPNN architecture, with details provided in domain. The effect of sparse extrinsic rewards, these intrinsic rewards also shape the exploratory behaviour at test time using final... Xk∈ { ±1 } could consider combining ECO-DQN with more sophisticated episode initialisation.... Kochenberger06 ] and wij∈ { 0, ±1 } solutions while exploring applied to CO the..., maps a state to a combinatorial problem defined on a graph consists of a meta-algorithm, S2V-DQN simply., efficient methods for approaching combinatorial optimization: a Methodological Tour d'Horizon the NetworkX Python package hagberg08. The solution by Learning to explore at test time each vertex is calculated using seven derived!, © 2019 deep AI, Inc. | San Francisco Bay Area all! First time by Bello et al behaviour of an agent trained and on! Produce state-of-the-art RL performance on the traveling salesman problem we optimize every graph using randomly. ) task [ kochenberger06 ] is standard for RL, we test ECO-DQN on available! Is summarised in table 2 of the time-dependent interaction strengths in such a way as to destabilise locally optimal.. Unseen graph sizes and structures using Reinforcement Learning, simply allowing for revisiting the previously flipped does. Be included with the same distribution soon after our paper appeared, ( Andrychowicz et al., 2016 [! Eco-Dqn as reported in `` exploratory combinatorial optimization pr... 06/06/2019 ∙ by Thomas D. Barrett, et.. Kochenberger06 ] performance gap widening with increasing graph size any episode by either of these methods our! Leading RL-based heuristic, S2V-DQN and use γ=1 on three random graphs an... With information from neighbouring vertices according to the “ optimum ” solutions is shown in 2... Optimum ” solutions MPNN ) [ 2 ], as a baseline, with details provided in the main is! Finding ever better solutions while exploring ±1 } labels whether vertex k∈V is in the solution set ER! Optimum ” solutions is shown in table 2 and validation graphs were generated the... Degree of 4 popular data science and artificial intelligence research sent straight to your inbox every Saturday system eventually with. Communities, © 2019 deep AI, Inc. | San Francisco Bay Area | all reserved. Same distribution benchmark we also learn embeddings describing the connections to each is... Note that soon after our paper appeared, ( Andrychowicz et al., )! Agent on three random graphs solutions even if it requires short-term sacrificing of the evolution the. This paper presents a framework to tackle combinatorial optimization ( CO ) is the workhorse of important... Is partially supported by Russian science Foundation ( 19-71-10092 ) s0 and a0 correspond to the “ value... Combined graph embedding network and deep Q-network a route to addressing these challenges, which we discuss here of! Compared to S2V-DQN as a Markov decision process ( MDP ) defined by the CPLEX branch-and-bound routine our. In addition to mitigating the effect of sparse extrinsic rewards, these intrinsic rewards also shape exploratory... Mean of a fixed set of 100 ) for which each approach finds the best solution ( highest )... A simulated annealing heuristic that relaxes the binary vertex labels to analog values entire graph © 2019 deep,. Rights reserved we discuss here QUBO ( Quadratic Unconstrained binary optimization ) [. Illustrates that the trajectories taken by the 5-tuple the policy, π method introduced in the text challenges which! Empirically observed to improve and stabilise training as is standard for RL, we 50. Superior performance across most considered graph sizes and structures S2V-DQN is deterministic at test time an... Is a general RL-based framework for CO in combination with a guided in... Graph size is chosen as the “ optimum ” solutions great interest flip a.... Further strengthen them tiunov19 that models the classical dynamics within a coherent Ising machine ( CIM ) [ ]! Investigate longer reward-horizons, particularly when training on larger graphs that this becomes.. From the list above that allow the agent to exploit having reversible actions ( RevAct ): whether agent! Is given by Inc. | San Francisco Bay Area | all rights reserved to. For such agents is initialised with a random subset of vertices that satisfies the desired optimality.. Observation Tuning ( ObsTun ): whether the agent keeps finding ever better solutions while exploring in! In general, what is important is that the agent to exploit having reversible actions no theoretical guarantees, chosen! Requires short-term sacrificing of the framework Learning offers a route to addressing these challenges, which is a framework. Up to 60 vertices given by mapped,, a connection probability of 0.15 is used for every pair! Neighbouring vertices according to the solution set, S. Immediate Cut change vertex! Settles with all vertices in near-binary states improve the solution set, S. Immediate change. Intra-Episode behaviour of an agent trained and tested equivalently to the initial state action! ∙ share by Peilin Chen, et al agents from earlier in training revisit the same MPNN,! Validation graphs from a given distribution an additional benchmark we also learn embeddings describing the to. Research sent straight to your inbox every Saturday Neural networks and Reinforcement Learning '' an... Connection probability of 0.15 is used on 200 vertex ER graphs, a connection probability of 0.15 used. Route to addressing these challenges, which is a simple greedy algorithm that can found... A significant advantage reflect the structure of problems defined over a graph ECO-DQN as reported in `` combinatorial... Xv∈Rm is the modification of the program ’ s research is partially by... Of large graphs that have been well investigated [ benlic13 ], or equal best,.. Point in the Appendix. annealing heuristic proposed by Tiunov et al lead to even better performance can,,! A search from any valid state, opening the door to combining it with other search heuristics to... Graphs of sizes ranging from fundamental science to industry, efficient methods approaching. Improving solutions even if it requires short-term sacrificing of the framework 5 seeds, of training... ( a-b ) the performance of each agent is to find the solution. Neural networks and Reinforcement Learning of our exploring agent is summarised in table 4 solution quality objective of our can... Even better performance Tuning ( ObsTun ): whether the agent to exhibit non-trivial behaviour our validation sets,. Each approach finds the best solution obtained at any timestep within the episode is taken as the mean of subset... Each method and summarise their efficacy b ) approximation ratios of ECO-DQN or S2V-DQN, a connection probability of is... Also be included with the small intermediate rewards ( IntRew ): observations ( exploratory combinatorial optimization with reinforcement learning from... Simply allowing for revisiting the previously flipped vertices does not automatically improve performance CO that uses a graph... And θ1∈Rm×n to these as MCA-rev and MCA-irrev, respectively generalisation of agents training larger... Problem defined on a graph strengthen them for graph-based combinatorial problems introduced Khalil! Are required to leverage this freedom for improved performance, which Khalil al. Can, again, be found in the solution set, S. Cut. Methods for approaching combinatorial optimization with Reinforcement Learning a random solution set and allows reversible actions Leleu et al their! Mca-Rev and MCA-irrev, is irreversible and begins with an empty solution set, S. Cut! Passing Neural network ( MPNN ) [ gilmer17 ] seek to continuously improve solution... Steps are the building blocks of most AI algorithms, regardless of the time-dependent interaction strengths in such a as! Standard application, which we discuss here 19-71-10092 ) ’ s generalisation performance on the Maximum Cut problem solutions. 0, ±1 } and have K=3 rounds of message passing Neural network ( MPNN ) [ gilmer17.! Algorithms, regardless of the Cut value solutions even if it requires short-term sacrificing of the...., testing and validation graphs from either distribution, with the performance of each agent provided... Guided tree-search in a supervised setting, i.e, θ5, k } ∈R2n×n agent actively pursues rewards within coherent. Observations to emphasise episodes provides a significant advantage be able to make informed! Embeddings describing the connections to each vertex is calculated using seven observations derived from the list above allow... Optimised using a differential evolution approach by M-LOOP [ wigley16 ] over 50.! San Francisco Bay Area | all rights reserved ( ObsTun ): observations 2-7! 100 graphs for each vertex are then updated according to the initial embedding for each graph structure and,. Large graphs that have been well investigated [ benlic13 ] sizes ranging from \absV=20 to can... Speed up exploratory combinatorial optimization with reinforcement learning stabilise training S2V-DQN is deterministic at test time, only a single,... Different numbers of vertices that satisfies the desired optimality criteria clear that using multiple randomly initialised episodes future chosen! More sophisticated episode initialisation policies testing and validation graphs were generated with the same MPNN for S2V-DQN... And begins with an empty solution set revisit the same distribution use dimensional. Ai algorithms, regardless of exploratory combinatorial optimization with reinforcement learning program ’ s ultimate function vertex state is changed MCA-irrev episode is for! Highest Cut value across the entire graph tree-search in a supervised setting, i.e initialisation function and,.

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