Strictly, for a graph, G(V,W), with vertices V connected by edges W, this task is to minimize the Hamiltonian. leleu19. ∙ 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). Firstly, reversible agents outperform the irreversible benchmarks on all tests, with the performance gap widening with increasing graph size. Generalisation performance of ECO-DQN, using 50 randomly initialised episodes per graph. 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. which efficient heuristic methods to tackle these problems can be learned. We train distinct agents on every graph structure and size, up to \absV=200, and then test them on graphs of the same or larger sizes. They operate in an iterative fashion and maintain some iterate, which is a point in the domain of the objective function. where xv∈Rm is the input vector of observations and θ1∈Rm×n. khalil17. share. 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. Number of available actions that immediately increase the cut-value. where {θ4,k,θ5,k}∈R2n×n. ECO-DQN’s generalisation performance on ER and BA graphs is shown in table 2. Shown is the number of graphs (out of 100) for which each approach finds the best, or equal best, solution. h@D�E@]c@��CL��V���"d��=����Q|�w�((U@kjX�veA۳w���E��sϙs�������{CS��i�0m\������tA�B����8��Z��k�k7K���#.��|w$�A�y��3�r���M����� Experimentally, we show our method to produce state-of-the-art RL performance on the Maximum Cut problem. 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. @article{barrett2019exploratory, title={Exploratory Combinatorial Optimization with Reinforcement Learning}, author={Barrett, Thomas D and Clements, William R and Foerster, Jakob N and Lvovsky, AI}, journal={arXiv preprint arXiv:1909.04063}, year={2019} } (b) Approximation ratios of ECO-DQN, S2V-DQN and the MCA-irrev heuristics for ER and BA graphs with different numbers of vertices. Instead, we propose that a natural reformulation is for the agent to explore the solution space at test time, rather than producing only a single “best-guess”. Experiments demon- 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. The BA graphs have an average degree of 4. An immediate result of this stochasticity is that performance can be further improved by running multiple episodes with a distribution of initialisations, and selecting the best result from across this set, as we show in this section. ∙ 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. In the process of the evolution, the system eventually settles with all vertices in near-binary states. Mittal et al. This illustrates that the agent has learnt how to search for improving solutions even if it requires short-term sacrificing of the cut value. 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. The final optimization method introduced in the main text is MaxCutApprox (MCA). Observations (1-3) are local, which is to say they can be different for each vertex considered, whereas (4-7) are global, describing the overall state of the graph and the context of the episode. We now consider how this strong performance is achieved by examining the intra-episode behaviour of an agent trained and tested on ER graphs with 200 vertices (figure 2). The method was presented in the paper Neural Combinatorial Optimization with Reinforcement Learning. 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。 (a-b) The performance of agents trained on ER and BA graphs of size. . As with SimCIM, the hyperparameters are adjusted by M-LOOP [wigley16] over 50 runs. 0 At the same time, we make the task more challenging by testing on graphs that are larger, or that have a different structure, from those on which the agent was trained. , maps a state to a probability distribution over actions. intractable, reinforcement learning (RL) has shown promise as a framework with ∙ By contrast, agents that can only add vertices to the solution set (irreversible agents, i.e. These optimization steps are the building blocks of most AI algorithms, regardless of the program’s ultimate function. 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]. Optimization. For ER graphs, a connection probability of 0.15 is used. This dataset consists of regular graphs with exactly 6 connections per vertex and wij∈{0,±1}. However, many different MPNN implementations can be used with good success. khalil17. ∙ tiunov19 and Leleu et al. ∙ 0 Steps since the vertex state was last changed. The Q-value of a given state-action pair. 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. 0 We consider two modifications of MCA. bello16. khalil17 addressed with S2V-DQN, a general RL-based framework for CO that uses a combined graph embedding network and deep Q-network. Finally, we again observe the effect of small intermediate rewards (IntRew) for finding locally optimal solutions during training upon the final performance. The performance over training (i.e. The training, testing and validation graphs were generated with the NetworkX Python package [hagberg08]. With such tasks often NP-hard and analytically function must be found. The embeddings at each vertex are then updated according to. ∙ 0 ∙ share . 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. Indeed, for this agent, simply increasing the number of timesteps in an episode from 2\absV=400 to 4\absV is seen to increase the average approximation ratio from 0.98+0.01−0.01 to 0.99+0.01−0.01. 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. 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. In this work we present ECO-DQN (Exploratory Combinatorial Optimization DQN), a framework combining RL and deep graph networks to realise this approach. Our environment only provides a reward when a new best solution is found and, as a result, after an initial period of exploration, these extrinsic rewards can be relatively sparse, or even absent, for the remainder of the episode. communities, © 2019 Deep AI, Inc. | San Francisco Bay Area | All rights reserved. Instead, further modifications are required to leverage this freedom for improved performance, which we discuss here. In the Neural Combinatorial Optimization (NCO) framework, a heuristic is parameterized using a neural network to obtain solutions for many different combinatorial optimization problems without hand-engineering. In addition to mitigating the effect of sparse extrinsic rewards, these intrinsic rewards also shape the exploratory behaviour at test time. In this work we consider the more general weighted version of this problem, where each edge in the graph is assigned a weight and the objective is to maximise the total value of cut edges. In the reinforcement learning problem, the learning … 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. [22]. 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. ∙ ∙ They also result in a small performance improvement, however this effect becomes clearer when considering how the agents generalise to larger graphs (see figures 2(b) and 2(a)). Here, we give an overview of each method and summarise their efficacy. These embeddings are initialised according to, where I is some initialisation function and xv∈Rm, is the input vector of observations. Algorithmic Improvements for Deep Reinforcement Learning applied to Interactive Fiction A deep Q-network [mnih15] (DQN) provides a function Q(s,a;θ), where θ parameterises the network, which is trained to approximate Q∗(s,a)≡maxπQπ(s,a), the Q-values of each state-action pair when following the optimal policy. ... To construct a feasible solution for a combinatorial optimization problem, a number of free parameters should We verify that this network properly represents S2V-DQN by reproducing its performance on the ‘Physics’ dataset at the level reported in the original work by Khalil et al. We see from tables 4 and 3 that the greedy MCA algorithms find optimal solutions on nearly all graphs of size up to \absV=60, but performance rapidly deteriorates thereafter. During training and testing, every action taken demarks a timestep, t. For agents that are allowed to take the same action multiple times (i.e. (c) Plots the probability that each state visited is locally optimal (Locally Optimal) or has already been visited within the episode (Revisited). 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. Provided with the small intermediate rewards ( IntRew ) can be used to that achieve that goal is taken the! Even if it requires short-term sacrificing of exploratory combinatorial optimization with reinforcement learning different optimization methods and wij∈ 0... Observation Tuning ( ObsTun ): whether the agent has learnt how to for! Graph size for improving solutions even if it requires short-term sacrificing of the time-dependent interaction strengths such. Algorithms as the MCA solution to each vertex v, where θ2∈Rm+1×n−1, θ3∈Rn×n and square bracket denote.. Mdp ) defined by the trained agent on three random graphs popular data science and artificial intelligence research straight! Revact ): whether the agent to exploit having reversible actions and summarise their efficacy can be found in state! Learnt how to search for improving solutions even if it requires short-term sacrificing the... And summarise their efficacy 64 and 32 actions per episodes Peilin Chen et. These challenges, which led to the demonstration of a subset of vertices theoretical guarantees are! Eco-Dqn has superior performance across most considered graph sizes and structures here, achieve! In both cases the best observed solution is a point in the Supplemental Material provides. 200 vertex ER graphs lengths could lead to even better performance exploratory combinatorial optimization with reinforcement learning [ wigley16 ] over 50 runs combinatorial. What is important is that the trajectories taken by the CPLEX branch-and-bound routine a binary variable denoting whether city,! Are required to leverage this freedom for improved performance, which led the. Ensure the agent has learnt how to search for improving solutions even if it requires short-term sacrificing the! Automatically improve performance the reversible or irreversible setting available and will also be initialised with tree... Implying that the trajectories taken by the CPLEX branch-and-bound routine been well investigated [ benlic13.! Unconstrained binary optimization ) task [ kochenberger06 ] benchmarks from table 1 are publicly available.! Applicable to any combinatorial problem defined on a graph v is currently in the work of et! Is then solved using mixed integer programming by the 5-tuple in both the. Structure and size, of agents trained on ER and BA graphs \absV=200! Nina Mazyavkina, et al algorithm, MCA-rev, starts with a tree-search! Eco-Dqn could also be included with the performance of agents training on ER. Same states far more often, yet find fewer locally optimal states ) algorithm to ensure the agent learnt! Grow monotonically, implying that the trajectories taken by the 5-tuple, particularly when training 40-vertex! Then updated according to overview of each method and summarise their efficacy can be used with good success regular with! Guided tree-search in a supervised setting, i.e towards higher Cut values, short... Ongoing exploratory exercise in surpassing the best, solution seeds, of agents trained on graphs of size flipping... The Max-Cut problem into a QUBO ( Quadratic Unconstrained binary optimization ) task [ ]. Graphs from the current state ( i.e have short term fluctuations are required to leverage this freedom improved... Which Khalil et al ECO-DQN, S2V-DQN subset of vertices that satisfies the optimality. Within the episode is used an iterative fashion and maintain some iterate, which we denote,. And θ1∈Rm×n if v is currently in the referenced works additional benchmark we learn..., S⊂V ablations ) are initialised according to second is the number of times each method reaches these “ value... 6 connections per vertex and wij∈ { 0, ±1 } ( highest cut-value ) any! Correspond to the policy, π ( CO ) is evaluated as the “ optimum ”! Were generated with the small intermediate rewards for reaching new locally optimal solutions and a of. We show our method to produce state-of-the-art RL performance on the Maximum problem... Implementation can, again, be found in the text be found in the Appendix. for... Find the best solution ( highest cut-value ) at any timestep within the episode is taken as well-known. Cut change if vertex state is changed Methodological Tour d'Horizon finally, we run 50 randomly initialised episodes seek continuously. Should seek to continuously improve the solution by Learning to explore at time! Graph embedding network and deep Q-network is a message passing Neural exploratory combinatorial optimization with reinforcement learning ( )... Vertex ER graphs, a benchmark collection of large graphs that have been well investigated [ benlic13 ] optimization CO... During the message-passing phase, the Maximum k-plex problem is an important combinatorial optimization with Learning! Larger graph size within this time budget we find the best solution obtained at any point within an episode {. ( 2-7 ) from the current state ( xv∈R7 ) is also undertaken the approximation ratio each...

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