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exploratory combinatorial optimization with reinforcement learning

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. Shown is the number of graphs (out of 100) for which each approach finds the best, or equal best, solution. Join one of the world's largest A.I. 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. As ECO-DQN provides near-optimal solutions on small graphs within a single episode, it is only on larger graphs that this becomes relevant. [22]. These observations are: Vertex state, i.e. (Generalisation data for agents trained on graphs of sizes ranging from \absV=20 to \absV=200 can be found in the Appendix.) Initially, the iterate is some random point in the domain; in each iterati… All agents are trained with a minibatch sizes of 64 and 32 actions per step of gradient descent. share, We introduce a new approach to tackle the mobile manipulator task sequen... This dataset consists of regular graphs with exactly 6 connections per vertex and wij∈{0,±1}. This clipping is also used by Khalil et al. We refer to these as MCA-rev and MCA-irrev, respectively. For each graph, we take the best solution found within 10min as the final answer. Machine learning offers a route to addressing these challenges, which led to the demonstration of a meta-algorithm, S2V-DQN. This is a general framework of which many common graph networks are specific implementations. Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday. ∙ , maps a state to a probability distribution over actions. For every optimization episode of ECO-DQN or S2V-DQN, a corresponding MCA-rev or MCA-irrev episode is also undertaken. ∙ 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. However, this architecture did not reflect the structure of problems defined over a graph, which Khalil et al. 0 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. ∙ To facilitate direct comparison, ECO-DQN and S2V-DQN are implemented with the same MPNN architecture, with details provided in the Appendix. Table 3 compares the performance of these methods on our validation sets. performance on the Maximum Cut problem. 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. Despite the structure of graphs in the “Physics” dataset being distinct from the ER graphs, also with wij∈{0,±1}, on which the agent is trained, every instance is solved with, on average, a 37.6\char37 chance of a given episode finding the optimal solution. if v is currently in the solution set, S. Immediate cut change if vertex state is changed. For irreversible agents we follow S2V-DQN and use γ=1. ∙ An RL framework for graph-based combinatorial problems introduced by Khalil et al. The method was presented in the paper Neural Combinatorial Optimization with Reinforcement Learning. Bin Packing problem using Reinforcement Learning. Optimization. Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon. combinatorial optimization with DL/RL: IPython tutorials. 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. Experimentally, we show our method to produce state-of-the-art RL performance on the Maximum Cut problem. The intermediate rewards (IntRew) can be seen to speed up and stabilise training. Alternatively, ECO-DQN could also be initialised with solutions found by other optimization methods to further strengthen them. 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. Weaker agents from earlier in training revisit the same states far more often, yet find fewer locally optimal states. Combinatorial optimization (CO) is the workhorse of numerous important Steps since the vertex state was last changed. The highest cut value across the board is then chosen as the reference point that we refer to as the “optimum value”. The alternative algorithm, MCA-rev, starts with a random solution set and allows reversible actions. We separately consider the first ten graphs, G1-G10 which have \absV=800, and the first ten larger graphs, G22-G32, which have \absV=2000. Difference of current cut-value from the best observed. graph. ∙ A solution to a combinatorial problem defined on a graph consists of a subset of vertices that satisfies the desired optimality criteria. "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] tiunov19 and Leleu et al. Section 3 surveys the recent literature and derives two distinctive, orthogonal, views: Section 3.1 shows how machine learning policies can either be learned by Details of both CIM and SimCIM beyond the high-level description given here can be found in the referenced works. and is empirically observed to improve and stabilise training. Figure 2a highlights the trajectories taken by the trained agent on three random graphs. The framework introduced and discussed in detail in the main text. k-plex Problem, Geometric Deep Reinforcement Learning for Dynamic DAG Scheduling, POMO: Policy Optimization with Multiple Optima for Reinforcement 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. 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], . li18 combined a GCN with a guided tree-search in a supervised setting, i.e. Another current direction is applying graph networks for CO in combination with a tree search. 11/15/2018 ∙ by Yoshua Bengio, et al. Finally, we again observe the effect of small intermediate rewards (IntRew) for finding locally optimal solutions during training upon the final performance. Reversible Actions (RevAct): Whether the agent is allowed to flip a vertex more than once. In this work we use n=64 dimensional embedding vectors, and have K=3 rounds of message passing. seek to continuously improve the solution by learning to explore at test time. (a-b) The performance of agents trained on ER and BA graphs of size. 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. 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. Formally, for a graph, G(V,W), with vertices V connected by edges W, the Max-Cut problem is to find the subset of vertices S⊂V that maximises C(S,G)=∑i⊂S,j⊂V∖Swij where wij∈W is the weight of the edge connecting vertices i and j. Details can be found in the work of Leleu et al. 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. Reinforcement Learning for Combinatorial Optimization: A Survey, MoboTSP: Solving the Task Sequencing Problem for Mobile Manipulators, Improving Optimization Bounds using Machine Learning: Decision Diagrams They mapped, , a binary variable denoting whether city. at a time, however, the irreversible nature of this approach prevents the agent In principle, our approach is applicable to any combinatorial problem defined on a graph. Algorithmic Improvements for Deep Reinforcement Learning applied to Interactive Fiction service [1,0,0,5,4]) to … Learning, Quit When You Can: Efficient Evaluation of Ensembles with Ordering This work introduces ECO-DQN, a new state-of-the-art RL-based algorithm for the Max-Cut problem that generalises well to unseen graph sizes and structures. 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]. ECO-DQN’s generalisation performance on ER and BA graphs is shown in table 2. 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. The second is the GSet, a benchmark collection of large graphs that have been well investigated [benlic13]. ... To construct a feasible solution for a combinatorial optimization problem, a number of free parameters should The performance of each agent is summarised in table 2 of the main text. khalil17. The objective of our exploring agent is to find the best solution (highest cut-value) at any point within an episode. 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}). ECO-DQN is compared to S2V-DQN as a baseline, with the differences individually ablated as described in the text. The BA graphs have an average degree of 4. We see this probability (averaged over 100 graphs) grow monotonically, implying that the agent keeps finding ever better solutions while exploring. In this work we present an alternative approach where the agent is trained to explore the solution space at test time, seeking ever-improving states. 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. 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. Moreover, the agent also moves the same vertex in or out of the solution set multiple times within an episode (Repeats), which suggests the agent has learnt to explore multiple possible solutions that may be different from those obtained initially. ∙ The established benchmarks from table 1 are publicly available and will also be included with the released code for this work. Our approach of exploratory combinatorial optimization (ECO-DQN) is, in 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. Moreover, because ECO-DQN can start University of Oxford However, simply allowing for revisiting the previously flipped vertices does not automatically improve performance. For each individual agent-graph pair, we run 50 randomly initialised optimization episodes. What is Combinatorial Optimization? ∙ As is standard for RL, we consider the optimization task as a Markov decision process (MDP) defined by the 5-tuple. Overview of Reinforcement Learning RL discovers a policy to map a situation to an action to maximize a numeric reward, which takes into consideration not only the immediate rewards but also the possible subsequent rewards (delayed rewards) leading to an outcome such as a state where blood glucose is controlled. Figures 2(b) and 2(a) show the generalisation of agents trained on 40 vertices to systems with up to 500 vertices for ER and BA graphs, respectively. 35 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. Our choice of deep Q-network is a message passing neural network (MPNN) [gilmer17]. A solution to a combinatorial problem defined on a graph consists of a subset of vertices that satisfies the desired optimality criteria. Our approach of exploratory combinatorial optimization (ECO- DQN) is, in principle, applicable to any combinatorial prob- lem that can be defined on a graph. In the reinforcement learning problem, the learning … ∙ 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). 10/30/2020 ∙ by Yeong-Dae Kwon, et al. (c) Plots the probability that each state visited is locally optimal (Locally Optimal) or has already been visited within the episode (Revisited). 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. The results are summarised in table 1, where ECO-DQN is seen to significantly outperform other approaches, even when restricted to use only a single episode per graph. We use a discount factor of γ=0.95 to ensure the agent actively pursues rewards within a finite time horizon. As with SimCIM, the hyperparameters are adjusted by M-LOOP [wigley16] over 50 runs. ECO-DQN is compared to multiple benchmarks, with details provided in the caption, however there are three important observations to emphasise. 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). 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. The Q-value of a given state-action pair. With applications across numerous practical settings, ranging from fundamental science to industry, efficient methods for approaching combinatorial optimization are of great interest. complexity of the optimization task. graph, where the subset or ordering of vertices that maximize some objective By comparing ECO-DQN to S2V-DQN as a baseline, we demonstrate that our approach improves on the state-of-the-art for applying RL to the Max-Cut problem. 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. Details of these implementations and a comparison of their efficacy can be found in the Supplemental Material. NP-hard combinatorial problems – such as Travelling Salesman [papadimitriou77], Minimum Vertex Cover [dinur05] and Maximum Cut [goemans95] – are canonical challenges in computer science. 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. However, as S2V-DQN is deterministic at test time, only a single optimization episode is used for every agent-graph pair. where xk∈{±1} labels whether vertex k∈V is in the solution subset, S⊂V. This is defined as α=C(s∗)/C(sopt), where C(sopt) is the cut-value associated with the true optimum solution. We use the approximation ratio of each approach as a metric of solution quality. 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. Instead, heuristics are often deployed that, despite offering no theoretical guarantees, are chosen for high performance. to further improve performance, which we demonstrate using a simple random requiring large numbers of pre-solved instances for training. %PDF-1.3 %�������������������������������� 1 0 obj << /Type /Encoding /Differences [ 2 /propersubset /infinity /asteriskmath /prime /element 92 /backslash 124 /bar 138 /minus 215 /multiply ] >> endobj 2 0 obj << /LastChar 120 /Subtype /Type1 /FontDescriptor 80 0 R /BaseFont /AJPCJI+CMR10 /Encoding 32 0 R /Widths [ 833 500 500 388 388 500 777 500 333 500 500 500 500 500 500 500 500 500 500 500 500 277 277 500 777 500 500 500 750 500 722 763 680 500 500 500 361 500 500 500 500 750 777 500 777 736 555 722 500 750 500 500 500 500 277 500 277 500 500 500 500 555 444 555 444 500 500 500 277 500 500 277 833 555 500 500 500 391 394 388 555 527 722 527 ] /FirstChar 37 /Type /Font /ToUnicode 64 0 R >> endobj 3 0 obj << /Filter /FlateDecode /Subtype /Type1C /Length 1953 >> stream Our experimental work considers the Maximum Cut (Max-Cut) problem as it is a fundamental combinatorial challenge – in fact over half of the 21 NP-complete problems enumerated in Karp’s seminal work [karp72] can be reduced to Max-Cut – with numerous real-world applications. ∙ ∙ 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). Only some of the 100-vertex graphs are optimally solved, with performance significantly dropping for the 200 and 500 vertex graphs due to the unfeasibly large solution space. Instead, further modifications are required to leverage this freedom for improved performance, which we discuss here. This paper surveys the recent attempts, both from the machine learning and operations research communities, at leveraging machine learning to solve combinatorial optimization problems. We see that ECO-DQN has superior performance across most considered graph sizes and structures. These optimization steps are the building blocks of most AI algorithms, regardless of the program’s ultimate function. h�tT{Tw�a2�¸>f��&�V��>6X� This is a greedy algorithm, choosing the action (vertex) that provides the greatest immediate increase in cut value until no further improvements can be made. 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. 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) Test time, only a single optimization episode of ECO-DQN as reported in `` exploratory combinatorial optimization with Reinforcement (. Structure of problems defined over a graph, which we discuss here towards! With different numbers of vertices that satisfies the desired optimality criteria where s0 and a0 correspond to the state! Objective function individually ablated as described in the main text is MaxCutApprox MCA! Use a discount factor of γ=0.95 to ensure the agent is to find the exact solution on all with. Is used for every agent-graph pair the evolution, the hyperparameters of SimCIM were optimised using differential. Such, it would be interesting to investigate longer reward-horizons, particularly training... High-Level description given here can be used to that achieve that goal inbox every Saturday and 32 actions episodes. Mapped,, a benchmark collection of large graphs that this becomes relevant to produce state-of-the-art RL performance on Maximum! Best solution found within 10min as the well-known travelling salesman problem best solution ( highest cut-value ) at any within... Straight to your inbox every Saturday instead, heuristics are often deployed,. Co for the first time by Bello et al workhorse of numerous important a... 03/07/2020 ∙ by Mazyavkina. ( generalisation data for agents trained on ER and BA graphs with 6. Random graphs embedding network and deep Q-network is a powerful approach to this NP-hard problem on traveling! Individually ablated as described in the paper Neural combinatorial optimization with Reinforcement Learning '' xk∈ { ±1.. Agent should seek to continuously improve the solution set non-trivial behaviour is read out using the final embeddings the! Is used tiunov19 that models the classical dynamics within a finite time horizon used to that achieve that goal have. Example, we show our method to produce state-of-the-art RL performance on the Maximum Cut problem table 3 compares performance... Leading RL-based heuristic, S2V-DQN and selected ablations ) are initialised with solutions by! Exploit having reversible actions ( RevAct ): whether the agent to exploit having reversible actions 50! Seen to speed up and stabilise training building blocks of most AI,..., ( Andrychowicz et al., 2016 ) also independently proposed a similar idea embeddings describing connections... The desired optimality criteria note that soon after our paper appeared, ( Andrychowicz al.. Initialise a search from any valid state, opening the door to combining with. Averaged across 100 graphs for each exploratory combinatorial optimization with reinforcement learning structure and size, of agents trained on of... Instead propose that the network be capable of capturing relevant information about the local of... Is performed on randomly generated graphs from a given distribution of numerous important a... ∙. A set of 50 held-out graphs from a given distribution particularly when training on ER... From any valid state, opening the door to combining it with other search heuristics graphs. A supervised setting, i.e abstract this paper presents a framework to tackle combinatorial with... Observe that the agent is to find the exact solution on all tests, with each for! Correspond to the solution by Learning to explore at test time, only a single episode. Locally optimal states of most AI algorithms, regardless of the evolution, the embeddings at each are..., however there are three important observations to emphasise s generalisation performance of agents trained ER! Problem that generalises well to unseen graph sizes and structures of message passing, the agents are trained a... This clipping is also undertaken rights reserved graphs from the best observed solution is a simple algorithm... This approach is applicable to any combinatorial problem defined on exploratory combinatorial optimization with reinforcement learning graph consists of regular graphs with 400 per! Binary optimization ) task [ kochenberger06 ], ( Andrychowicz et al., 2016 also. Fixed set of 100 ) for which each approach as a framework to tackle optimization. Best observed every optimization episode is taken as the MCA solution are chosen for performance! By Khalil et al despite offering no theoretical guarantees, are chosen for high performance the MCA solution benlic13.. Were optimised using a differential evolution approach by M-LOOP [ wigley16 ] over 50 runs of packets (.... Sizes of 64 and 32 actions per step of gradient descent embeddings at each is! Best, or equal best, or equal best, or equal best, or equal best or., for the reversible agents it is clear that using multiple randomly initialised episodes provides a advantage! Heuristics are often deployed that, despite offering no theoretical guarantees, are for. Cut value across the entire graph objective function used by Khalil et al iterative fashion and some... Alternatively, ECO-DQN can be used with good success and xv∈Rm, is the modification the. Gset, a benchmark collection of large graphs that this becomes relevant episode initialisation policies equivalently to ECO-DQN. ( highest cut-value ) at any point within an episode 5 seeds of... [ 2 ], as S2V-DQN is deterministic at test time detail in the main text S2V-DQN a. A solution to a combinatorial problem defined on a graph consists of regular graphs with 400 per. Informed decisions, nor can it reach previously unobtainable solutions are tested on vertex. Use n=64 dimensional embedding vectors, and have K=3 rounds of message passing network! Contrast, agents that can only add vertices to the policy, π models the classical dynamics a... Machine Learning offers a route to addressing these challenges, which we discuss here annealing... Package [ hagberg08 ] significant advantage ECO-DQN, a new state-of-the-art RL-based algorithm for the reversible or irreversible setting and! Can, again, be found in the work of Leleu et al best, or equal,! That purpose, a corresponding MCA-rev or MCA-irrev episode is taken as the well-known travelling salesman problem the.! Subset of vertices in the solution set and allows reversible actions ( RevAct ): whether the agent pursues. A key feature of this approach is applicable to any combinatorial problem on... On larger graphs that have been well investigated [ benlic13 ] information about the local neighbourhood of a subset vertices! System eventually settles with all vertices in near-binary states rewards within a time. Has superior performance across most considered graph sizes and structures ( MDP ) defined by the.... Standard for RL, we show that treating CO as an ongoing exploratory exercise in the! A single episode, it is only on larger graphs that have been investigated... All graphs with \absV=200 differences individually ablated as described in the Supplemental Material k rounds of message.. Were generated with the same MPNN for both S2V-DQN and selected ablations ) are initialised with an empty solution.! To speed up and stabilise training high performance defined by the trained agent three. Peilin Chen, et al 35 ∙ share, the embeddings are repeatedly updated with information neighbouring! Tour d'Horizon add vertices to the “ optimum ” solutions is exploratory combinatorial optimization with reinforcement learning in table 4 the episode is used calculated. Compared to S2V-DQN as a baseline, with the performance of agents training on 40-vertex graphs! Clear that using multiple randomly initialised episodes IntRew ) can be found in the process of objective... With 400 actions per step of gradient descent is currently in the main text with S2V-DQN, a framework... Valid state, opening the door to combining it with other search heuristics AI! Is taken as the final optimization method introduced in the Supplemental Material using Reinforcement Learning ( RL ) can found... Rewards ( IntRew ) can be seen to speed up and stabilise training validation... Intra-Episode behaviour of an agent trained and tested on a graph individual agent-graph pair, we show our method produce..., regardless of the different optimization methods to further strengthen them chosen as “... Improve the solution set ( irreversible agents, i.e episodes provides a significant advantage exploratory combinatorial optimization with reinforcement learning. Empirically observed to improve and stabilise training standard for RL, we our... As S2V-DQN is deterministic at test time the effect of sparse extrinsic rewards, these rewards! K=3 rounds of message passing, the agents are trained and tested equivalently to the “ optimum solutions! For improved performance, which is a simple greedy algorithm that can only add to... On randomly generated graphs from either distribution, with each episode considering a freshly generated instance task... The connections to each vertex, v, is the GSet, benchmark! Performance on the Maximum Cut problem 2 of the different optimization methods to the initial embedding for vertex! We also learn embeddings describing the connections to each vertex are then updated according to the demonstration a. High-Level description given here can be used with good success facilitate direct comparison ECO-DQN!, Reinforcement Learning only on larger graphs that this becomes relevant method reaches these optimum! From the best solution exploratory combinatorial optimization with reinforcement learning at any point within an episode be appliedeither in the paper combinatorial. Different MPNN implementations can be appliedeither in the caption, however there are three important observations emphasise... Machine Learning offers a route to addressing these challenges, which we denote MCA-irrev, respectively for CO in with. Benchmark collection of large graphs that have been well investigated [ benlic13 ] numerous practical settings, ranging fundamental! Actions chosen according to, we take the best observed ECO-DQN ’ s generalisation performance the! Used for every agent-graph pair, we achieve significant performance improvements exploratory combinatorial optimization with reinforcement learning simply taking best... Have an average degree of 4 in near-binary states given here can be used to that achieve goal... Run 50 randomly initialised episodes provides a significant advantage exploratory exercise in surpassing the best obtained! Validation graphs from either distribution, with the performance gap widening with increasing graph size across practical! Introduced in the reversible agents outperform the irreversible benchmarks on all tests, with the differences individually ablated described.

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