Additionally, as all actions can be reversed, the challenge of predicting the true value of a vertex “flip” does not necessarily result in sub-optimal performance. graph. As continued exploration is desired, even after a good solution is found, there is no punishment if a chosen action reduces the cut-value. Once trained, an approximation of the optimal policy can be obtained simply by acting greedily with respect to the predicted Q-values, π(s;θ)=argmaxa′Q(s,a′;θ). This suggests that longer episode lengths could lead to even better performance. Learn to Solve Routing Problems”, the authors tackle several combinatorial optimization problems that involve routing agents on graphs, including our now familiar Traveling Salesman Problem. Instead, heuristics are often deployed that, despite offering no theoretical guarantees, are chosen for high performance. Exploratory combinatorial optimization with reinforcement learning ... 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. As ECO-DQN provides near-optimal solutions on small graphs within a single episode, it is only on larger graphs that this becomes relevant. Our approach of exploratory combinatorial optimization (ECO-DQN) is, in Also, in the absence of ObsTun, the rewards used are simply the (normalised) immediate change in cut value, R(st)=(C(st)−C(st−1))/\absV, which is necessary as without observations (4-5) the ECO-DQN reward structure is non-Markovian. complexity of the optimization task. arXiv preprint arXiv:1611.09940, 2016. at a time, however, the irreversible nature of this approach prevents the agent The Q-value of a given state-action pair. This is defined as α=C(s∗)/C(sopt), where C(sopt) is the cut-value associated with the true optimum solution. The fundamental change distinguishing our approach, ECO-DQN, from previous works can then be summarised as follows: instead of learning to construct a single good solution, learn to explore for improving solutions. The general purposes of each of the observations are: (1-2) provide useful information for determining the value of selecting an action, (3) provides a simple history to prevent short looping trajectories, (4-5) ensure the rewards calculated with respect to the best observed cut-value are Markovian, (6) allows the agent to predict when it will receive the intrinsic rewards previously discussed and (7) accounts for the finite episode duration. 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. A more substantial avenue to explore would be to use a recurrent architecture where a useful representation of the episode history is learned, as opposed to the hand-crafted representation that we describe in section 3. Once trained, the agents are tested on a separate set of 100 held-out validation graphs from a given distribution. ... To construct a feasible solution for a combinatorial optimization problem, a number of free parameters should Within this time budget we find the exact solution on all graphs with up to 60 vertices. 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). This is further emphasised in figure 2c where we see that, after an initial period of exploration, the agent searches through the solution space, repeatedly moving in and out of locally optimal (Locally Optimal) solutions whilst minimising the probability that it revisits states (Revisited). where Mk and Uk are message and update functions, respectively, with N(v) is the set of vertices directly connected to v. After K rounds of message passing, a prediction – a set of values that carry useful information about the network – is produced by some readout function, R. In our case this prediction is the set of Q-values of the actions corresponding to “flipping” each vertex, i.e. ∙ every innovation in technology and every invention that improved our lives and our ability to survive and thrive on earth mittal19 developed these ideas further by modifying the training process: first training an embedding graph convolution network (GCN), and then training a Q-network to predict the vertex (action) values. share, We introduce a new approach to tackle the mobile manipulator task sequen... Another current direction is applying graph networks for CO in combination with a tree search. For small graphs the agent performs near-optimally with or without this intrinsic motivation, however the difference becomes noticeable when generalising to larger graphs at test time. ∙ khalil17. This is particularly noticeable for BA graphs, for which the degrees of each vertex in the graph tend to be distributed over a greater range than for ER graphs, where S2V-DQN fails to generalise in any meaningful way to \absV≥200. The structure of the GSet is also distinct from that of the training data, with the first five instances in each tested set have only positive edges. 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. Our choice of deep Q-network is a message passing neural network (MPNN) [gilmer17]. There are numerous heuristic methods, ranging from search-based [benlic13, banks08] to physical systems that utilise both quantum and classical effects [johnson11, yamamoto17] and their simulated counterparts [kirkpatrick83, clements17, tiunov19], . �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��. ∙ With applications across numerous practical settings, ranging from fundamental science to industry, efficient methods for approaching combinatorial optimization are of great interest. Initially, the iterate is some random point in the domain; in each iterati… ECO-DQN can initialise a search from any valid state, opening the door to combining it with other search heuristics. S2V-DQN, and the related works discussed shortly, incrementally construct solutions one element at a time – reducing the problem to predicting the value of adding any vertex not currently in the solution to this subset. 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). 0 share. , maps a state to a probability distribution over actions. khalil17 addressed with S2V-DQN, a general RL-based framework for CO that uses a combined graph embedding network and deep Q-network. 06/28/2018 ∙ by Serena Wang, et al. These embeddings are initialised according to, where I is some initialisation function and xv∈Rm, is the input vector of observations. Machine learning offers a route to addressing these challenges, which led to the demonstration of a meta-algorithm, S2V-DQN. A key feature of this approach is the modification of the time-dependent interaction strengths in such a way as to destabilise locally optimal solutions. The hyperparameters of SimCIM were optimised using a differential evolution approach by M-LOOP. Experimentally, we show our method to produce state-of-the-art RL The standard application, which we denote MCA-irrev, is irreversible and begins with an empty solution set. As there are far more states than could be visited within our finite episodes, the vast majority of which are significantly sub-optimal, it is useful to focus on a subset of states known to include the global optimum. We show that treating CO as an ongoing exploratory exercise in surpassing the best observed solution is a powerful approach to this NP-hard problem. to further improve performance, which we demonstrate using a simple random Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday. However, as S2V-DQN is deterministic at test time, only a single optimization episode is used for every agent-graph pair. In the process of the evolution, the system eventually settles with all vertices in near-binary states. In addition to mitigating the effect of sparse extrinsic rewards, these intrinsic rewards also shape the exploratory behaviour at test time. The framework we propose can, however, be readily applied to any graph-based combinatorial problem where solutions correspond to a subset of vertices and the goal is to optimize some objective function. This is not simply a mathematical challenge as many real world applications can be reduced to the Max-Cut problem, including protein folding [perdomo12], investment portfolio optimization [elsokkary17, venturelli18] (specifically using the Markowitz markowitz52 formulation), and finding the ground state of the Ising Hamiltonian in physics [barahona82]. The performance is marginally better when testing on graphs from the same distribution as the training data, however this difference is negligible for \absV≤100. S2V-DQN agents are trained and tested equivalently to the ECO-DQN agents. The intermediate rewards (IntRew) can be seen to speed up and stabilise training. As in table 0(a), the range quoted for these approximation ratios corresponds to the upper and lower quartiles of the performance across all 100 validation graphs. The performance over training (i.e. However, as no known algorithms are able to solve NP-hard problems in polynomial time, exact methods rapidly become intractable for all but the simplest of tasks. Approximation algorithms guarantee a worst-case solution quality, but sufficiently strong bounds may not exist and, even if they do, these algorithms can have limited scalability [williamson11]. if v is currently in the solution set, S. Immediate cut change if vertex state is changed. By contrast, agents that can only add vertices to the solution set (irreversible agents, i.e. Table 3 compares the performance of these methods on our validation sets. Previous works construct the solution subset incrementally, adding one element This is a general framework of which many common graph networks are specific implementations. We instead propose that the agent should all learning curves) is evaluated as the mean of a fixed set of 50 held-out graphs from the same distribution. Learning for Graph Matching and Related Combinatorial Optimization Problems Junchi Yan1, Shuang Yang2 and Edwin Hancock3 1 Department of CSE, MoE Key Lab of Artificial Intelligence, Shanghai Jiao Tong University 2 Ant Financial Services Group 3 Department of Computer Science, University of York yanjunchi@sjtu.edu.cn, shuang.yang@antfin.com, edwin.hancock@york.ac.uk Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon. abe19 trained a GCN using Monte-Carlo tree search as a policy improvement operator, in a similar manner to AlphaGo Zero [silver17], however, this work does not consider the Max-Cut problem. ∙ Exploratory Combinatorial Optimization with Reinforcement Learning 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. Combinatorial optimization is a class of methods to find an optimal object from a finite set of objects when an exhaustive search is not feasible. Also, we train our agents with highly discounted future rewards (γ=0.95), and although this is found to provide strong performance, the relatively short-term reward horizon likely limits exploration to only local regions of the solution space. Facebook ∙ 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. We observe that the trajectories, whilst trending towards higher cut values, have short term fluctuations. An RL framework for graph-based combinatorial problems introduced by Khalil et al. We use a discount factor of γ=0.95 to ensure the agent actively pursues rewards within a finite time horizon. This paper presents Neural Combinatorial Optimization, a framework to tackle combinatorial op-timization with reinforcement learning and neural networks. As such, the Q-value of either adding or removing a vertex from the solution is continually re-evaluated in the context of the episode’s history. The learning rate is 10−4 and the exploration rate is linearly decreased from ε=1 to ε=0.05 over the first ∼$10\char37$ of training. Reinforcement Learning for Combinatorial Optimization: A Survey, MoboTSP: Solving the Task Sequencing Problem for Mobile Manipulators, Improving Optimization Bounds using Machine Learning: Decision Diagrams ∙ where xk∈{±1} labels whether vertex k∈V is in the solution subset, S⊂V. Global Search in Combinatorial Optimization using Reinforcement Learning Algorithms ... even one incorrect exploratory step can seriously damage the quality of the resultant solution. We focus on the traveling salesman problem (TSP) and train a recurrent neural network that, given a set of city coordinates, predicts … These observations are: Vertex state, i.e. Alternatively, ECO-DQN could also be initialised with solutions found by other optimization methods to further strengthen them. from any arbitrary configuration, it can be combined with other search methods 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). We apply six different optimization methods to the 100 validation graphs of each structure (ER or BA graphs) and size (\absV∈{20,40,60,100,200,500}). A solution to a combinatorial problem defined on a graph consists of a subset of vertices that satisfies the desired optimality criteria. Solving a new 3d bin packing problem with deep reinforcement learning method Jan 2017 Our approach of exploratory combinatorial optimization (ECO-DQN) is, in principle, applicable to any combinatorial problem that can be defined on a graph. Exploratory Combinatorial Optimization with Reinforcement Learning Thomas Barrett (University of Oxford)*; William Clements (Unchartech); Jakob Foerster (Facebook AI Research); Alexander Lvovsky (Oxford University) [23]. They mapped, , a binary variable denoting whether city. 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). Whilst these are paths towards further developing exploration-based CO, we believe that the strong performance already demonstrated would allow our approach to be applied in numerous practical settings. We also learn embeddings describing the connections to each vertex v, where θ2∈Rm+1×n−1, θ3∈Rn×n and square bracket denote concatenation. This is orthogonal to our proposal which considers the framework itself, rather than the training procedure, and, in principle, appears to be compatible with ECO-DQN. The intra-episode behaviour of an agent trained and tested on 200 vertex ER graphs with 400 actions per episodes. Details of our implementation can, again, be found in the main text. will look to improve on any proposed solution) and, as a consequence, has the flexibility to be either deployed independently or combined with other search heuristics. requiring large numbers of pre-solved instances for training. By comparing the agent’s behaviour at three points during training (fully trained and when performance level is equivalent to either MCA-irrev or S2V-DQN), we see that this behaviour is learnt. ∙ share. 03/07/2020 ∙ by Nina Mazyavkina, et al. Exploratory combinatorial optimization with reinforcement learning ... 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. Very recently, Abe et al. 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. For irreversible agents we follow S2V-DQN and use γ=1. From figure 2b, we see that the fully trained agent regularly chooses actions that do not correspond to the greatest immediate increase in the cut-value (Non-Greedy), or even that decrease the cut value (Negative). The first is the “Physics” dataset consisting of ten graphs, with \absV=125, which correspond to Ising spin-glass models of physical systems. where s0 and a0 correspond to the initial state and action taken, with future actions chosen according to the policy, π. Li et al. We see this probability (averaged over 100 graphs) grow monotonically, implying that the agent keeps finding ever better solutions while exploring. Strictly, for a graph, G(V,W), with vertices V connected by edges W, this task is to minimize the Hamiltonian. (2016)[2], as a framework to tackle combinatorial optimization problems using Reinforcement Learning. However, many different MPNN implementations can be used with good success. service [1,0,0,5,4]) to … Firstly, reversible agents outperform the irreversible benchmarks on all tests, with the performance gap widening with increasing graph size. For ER graphs, a connection probability of 0.15 is used. We refer to these as MCA-rev and MCA-irrev, respectively. (b) Approximation ratios of ECO-DQN, S2V-DQN and the MCA-irrev heuristics for ER and BA graphs with different numbers of vertices. Secondly, for the reversible agents it is clear that using multiple randomly initialised episodes provides a significant advantage. tiunov19 and Leleu et al. 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. We apply agents trained on ER graphs with \absV=200. This tutorial demonstrates technique to solve combinatorial optimization problems such as the well-known travelling salesman problem. ∙ Reinforcement Learning (RL) [27] is a type of learning process to maximize cer-tain numerical values by combining exploration and exploitation and using rewards as learning stimuli. Optimization. ∙ 0 They used policy gradients to train pointer networks [vinyals15], a recurrent architecture that produces a softmax attention mechanism (a “pointer”) to select a member of the input sequence as an output. To interpret the performance gap, we also consider the following ablations, which together fully account for the differences between our approach and the baseline (ECO-DQN≡S2V-DQN+RevAct+ObsTun+IntRew). 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). Experimentally, we show our method to produce state-of-the-art RL performance on the Maximum Cut problem. The established benchmarks from table 1 are publicly available and will also be included with the released code for this work. This approach relaxes the binary vertex values (xk∈{±1}), associated with labelling a vertex as either in or out of the solution subset, to analog values (−1≤xk≤1, ). which efficient heuristic methods to tackle these problems can be learned. Learning to Solve Problems Without Human Knowledge. However, this architecture did not reflect the structure of problems defined over a graph, which Khalil et al. k-plex Problem, Geometric Deep Reinforcement Learning for Dynamic DAG Scheduling, POMO: Policy Optimization with Multiple Optima for Reinforcement This Hamiltonian is then solved using mixed integer programming by the CPLEX branch-and-bound routine. Training is performed on randomly generated graphs from either distribution, with each episode considering a freshly generated instance. Table 0(a) shows the performance of agents trained and tested on both ER and BA graphs of sizes ranging from 20 to 200 vertices. ∙

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