In this paper, we propose TauRieL and target Traveling Salesman Problem (TSP) since it has broad applicability in theoretical and applied sciences. TauRieL utilizes an actor-critic inspired architecture that adopts ordinary feedforward nets to obtain a policy update vector @math . Then, we use @math to improve the state transition matrix from which we generate the policy. Also, the state transition matrix allows the solver to initialize from precomputed solutions such as nearest neighbors. In an online learning setting, TauRieL unifies the training and the search where it can generate near-optimal results in seconds. The input to the neural nets in the actor-critic architecture are raw 2-D inputs, and the design idea behind this decision is to keep neural nets relatively smaller than the architectures with wide embeddings with the tradeoff of omitting any distributed representations of the embeddings. Consequently, TauRieL generates TSP solutions two orders of magnitude faster per TSP instance as compared to state-of-the-art offline techniques with a performance impact of 6.1 in the worst case.
@cite_19 proposed a TSP solver RL framework based on neural nets. Pointer-nets have been employed for policy gradient and expected tour length prediction. The sequential nature of the framework resembles tour construction algorithms. Stochastic sampling and the actor-critic architecture updates the expected tour length with the current policy (on-policy) and the gradient updates are worked out using reinforce algorithm @cite_38 . In addition, a separate neural net for tour exploration incorporate an expected reward based value iteration approximation. All neural nets are constructed using pointer nets @cite_26 . Both pre-training and random initialization of the weights are realized as the initial starting state. The framework presents improved tour lengths and execution times compared to Christofides and supervised learning methods @cite_29 @cite_26 .
@cite_19 @cite_29 @cite_26 @cite_38
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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 much engineering and heuristic designing, Neural Combinatorial Optimization achieves close to optimal results on 2D Euclidean graphs with up to 100 nodes. Applied to the KnapSack, another NP-hard problem, the same method obtains optimal solutions for instances with up to 200 items. Policy gradient reinforcement learning has been applied to two-player alternate-turn zero-sum games, e.g., in AlphaGo, self-play REINFORCE was used to improve the neural net model after supervised learning. In this paper, we emphasize that two-player zero-sum games with alternating turns, which have been previously formulated as Alternating Markov Games (AMGs), are different from standard MDP because of their two-agent nature. We exploit the difference in associated Bellman equations, which leads to different policy iteration algorithms. As policy gradient method is a kind of generalized policy iteration, we show how these differences in policy iteration are reflected in policy gradient for AMGs. We formulate an adversarial policy gradient and discuss potential possibilities for developing better policy gradient methods other than self-play REINFORCE. The core idea is to estimate the minimum rather than the mean for the “critic”. Experimental results on the game of Hex show the modified Monte Carlo policy gradient methods are able to learn better pure neural net policies than the REINFORCE variants. To apply learned neural weights to multiple board sizes Hex, we describe a board-size independent neural net architecture. We show that when combined with search, using a single neural net model, the resulting program consistently beats MoHex 2.0, the state-of-the-art computer Hex player, on board sizes from 9×9 to 13×13. 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 neural 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 neural 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. Without much engineering and heuristic designing, Neural Combinatorial Optimization achieves close to optimal results on 2D Euclidean graphs with up to 100 nodes. Applied to the KnapSack, another NP-hard problem, the same method obtains optimal solutions for instances with up to 200 items. These results, albeit still far from state-of-the-art, give insights into how neural networks can be used as a general tool for tackling combinatorial optimization problems. We propose a framework for solving combinatorial optimization problems of which the output can be represented as a sequence of input elements. As an alternative to the Pointer Network, we parameterize a policy by a model based entirely on (graph) attention layers, and train it efficiently using REINFORCE with a simple and robust baseline based on a deterministic (greedy) rollout of the best policy found during training. We significantly improve over state-of-the-art results for learning algorithms for the 2D Euclidean TSP, reducing the optimality gap for a single tour construction by more than 75 (to 0.33 ) and 50 (to 2.28 ) for instances with 20 and 50 nodes respectively. Model-free deep reinforcement learning (RL) algorithms have been demonstrated on a range of challenging decision making and control tasks. However, these methods typically suffer from two major challenges: very high sample complexity and brittle convergence properties, which necessitate meticulous hyperparameter tuning. Both of these challenges severely limit the applicability of such methods to complex, real-world domains. In this paper, we propose soft actor-critic, an off-policy actor-critic deep RL algorithm based on the maximum entropy reinforcement learning framework. In this framework, the actor aims to maximize expected reward while also maximizing entropy - that is, succeed at the task while acting as randomly as possible. Prior deep RL methods based on this framework have been formulated as either off-policy Q-learning, or on-policy policy gradient methods. By combining off-policy updates with a stable stochastic actor-critic formulation, our method achieves state-of-the-art performance on a range of continuous control benchmark tasks, outperforming prior on-policy and off-policy methods. Furthermore, we demonstrate that, in contrast to other off-policy algorithms, our approach is very stable, achieving very similar performance across different random seeds. We present a new optical method for solving bounded (input-length-restricted) NP-complete combinatorial problems. We have chosen to demonstrate the method with an NP-complete problem called the traveling salesman problem (TSP). The power of optics in this method is realized by using a fast matrix-vector multiplication between a binary matrix, representing all feasible TSP tours, and a gray-scale vector, representing the weights among the TSP cities. The multiplication is performed optically by using an optical correlator. To synthesize the initial binary matrix representing all feasible tours, an efficient algorithm is provided. Simulations and experimental results prove the validity of the new method. Model-free deep reinforcement learning (RL) algorithms have been demonstrated on a range of challenging decision making and control tasks. However, these methods typically suffer from two major challenges: very high sample complexity and brittle convergence properties, which necessitate meticulous hyperparameter tuning. Both of these challenges severely limit the applicability of such methods to complex, real-world domains. In this paper, we propose soft actor-critic, an off-policy actor-critic deep RL algorithm based on the maximum entropy reinforcement learning framework. In this framework, the actor aims to maximize expected reward while also maximizing entropy - that is, succeed at the task while acting as randomly as possible. Prior deep RL methods based on this framework have been formulated as Q-learning methods. By combining off-policy updates with a stable stochastic actor-critic formulation, our method achieves state-of-the-art performance on a range of continuous control benchmark tasks, outperforming prior on-policy and off-policy methods. Furthermore, we demonstrate that, in contrast to other off-policy algorithms, our approach is very stable, achieving very similar performance across different random seeds. This paper presents MAPO: a novel policy optimization formulation that incorporates a memory buffer of promising trajectories to reduce the variance of policy gradient estimates for deterministic environments with discrete actions. The formulation expresses the expected return objective as a weighted sum of two terms: an expectation over a memory of trajectories with high rewards, and a separate expectation over the trajectories outside the memory. We propose 3 techniques to make an efficient training algorithm for MAPO: (1) distributed sampling from inside and outside memory with an actor-learner architecture; (2) a marginal likelihood constraint over the memory to initiate training; (3) systematic exploration to discover new high reward trajectories. MAPO improves the sample efficiency and robustness of policy gradient, especially on tasks with a sparse reward. We evaluate MAPO on weakly supervised program synthesis from natural language semantic parsing. On the WikiTableQuestions benchmark we improve the state-of-the-art by 2.5 , achieving an accuracy of 46.2 , and on the WikiSQL benchmark, MAPO achieves an accuracy of 74.9 with only weak supervision, outperforming the state-of-the-art with full supervision. Function approximation is essential to reinforcement learning, but the standard approach of approximating a value function and determining a policy from it has so far proven theoretically intractable. In this paper we explore an alternative approach in which the policy is explicitly represented by its own function approximator, independent of the value function, and is updated according to the gradient of expected reward with respect to the policy parameters. Williams's REINFORCE method and actor-critic methods are examples of this approach. Our main new result is to show that the gradient can be written in a form suitable for estimation from experience aided by an approximate action-value or advantage function. Using this result, we prove for the first time that a version of policy iteration with arbitrary differentiable function approximation is convergent to a locally optimal policy. As a new way of training generative models, Generative Adversarial Net (GAN) that uses a discriminative model to guide the training of the generative model has enjoyed considerable success in generating real-valued data. However, it has limitations when the goal is for generating sequences of discrete tokens. A major reason lies in that the discrete outputs from the generative model make it difficult to pass the gradient update from the discriminative model to the generative model. Also, the discriminative model can only assess a complete sequence, while for a partially generated sequence, it is nontrivial to balance its current score and the future one once the entire sequence has been generated. In this paper, we propose a sequence generation framework, called SeqGAN, to solve the problems. Modeling the data generator as a stochastic policy in reinforcement learning (RL), SeqGAN bypasses the generator differentiation problem by directly performing gradient policy update. The RL reward signal comes from the GAN discriminator judged on a complete sequence, and is passed back to the intermediate state-action steps using Monte Carlo search. Extensive experiments on synthetic data and real-world tasks demonstrate significant improvements over strong baselines. We present the first class of policy-gradient algorithms that work with both state-value and policy function-approximation, and are guaranteed to converge under off-policy training. Our solution targets problems in reinforcement learning where the action representation adds to the-curse-of-dimensionality; that is, with continuous or large action sets, thus making it infeasible to estimate state-action value functions (Q functions). Using state-value functions helps to lift the curse and as a result naturally turn our policy-gradient solution into classical Actor-Critic architecture whose Actor uses state-value function for the update. Our algorithms, Gradient Actor-Critic and Emphatic Actor-Critic, are derived based on the exact gradient of averaged state-value function objective and thus are guaranteed to converge to its optimal solution, while maintaining all the desirable properties of classical Actor-Critic methods with no additional hyper-parameters. To our knowledge, this is the first time that convergent off-policy learning methods have been extended to classical Actor-Critic methods with function approximation. As a new way of training generative models, Generative Adversarial Nets (GAN) that uses a discriminative model to guide the training of the generative model has enjoyed considerable success in generating real-valued data. However, it has limitations when the goal is for generating sequences of discrete tokens. A major reason lies in that the discrete outputs from the generative model make it difficult to pass the gradient update from the discriminative model to the generative model. Also, the discriminative model can only assess a complete sequence, while for a partially generated sequence, it is non-trivial to balance its current score and the future one once the entire sequence has been generated. In this paper, we propose a sequence generation framework, called SeqGAN, to solve the problems. Modeling the data generator as a stochastic policy in reinforcement learning (RL), SeqGAN bypasses the generator differentiation problem by directly performing gradient policy update. The RL reward signal comes from the GAN discriminator judged on a complete sequence, and is passed back to the intermediate state-action steps using Monte Carlo search. Extensive experiments on synthetic data and real-world tasks demonstrate significant improvements over strong baselines. Deep reinforcement learning dialogue systems are attractive because they can jointly learn their feature representations and policies without manual feature engineering. But its application is challenging due to slow learning. We propose a two-stage method for accelerating the induction of single or multi-domain dialogue policies. While the first stage reduces the amount of weight updates over time, the second stage uses very limited minibatches (of as much as two learning experiences) sampled from experience replay memories. The former frequently updates the weights of the neural nets at early stages of training, and decreases the amount of updates as training progresses by performing updates during exploration and by skipping updates during exploitation. The learning process is thus accelerated through less weight updates in both stages. An empirical evaluation in three domains (restaurants, hotels and tv guide) confirms that the proposed method trains policies 5 times faster than a baseline without the proposed method. Our findings are useful for training larger-scale neural-based spoken dialogue systems. In this work, we study the 1-bit convolutional neural networks (CNNs), of which both the weights and activations are binary. While being efficient, the classification accuracy of the current 1-bit CNNs is much worse compared to their counterpart real-valued CNN models on the large-scale dataset, like ImageNet. To minimize the performance gap between the 1-bit and real-valued CNN models, we propose a novel model, dubbed Bi-Real net, which connects the real activations (after the 1-bit convolution and or BatchNorm layer, before the sign function) to activations of the consecutive block, through an identity shortcut. Consequently, compared to the standard 1-bit CNN, the representational capability of the Bi-Real net is significantly enhanced and the additional cost on computation is negligible. Moreover, we develop a specific training algorithm including three technical novelties for 1- bit CNNs. Firstly, we derive a tight approximation to the derivative of the non-differentiable sign function with respect to activation. Secondly, we propose a magnitude-aware gradient with respect to the weight for updating the weight parameters. Thirdly, we pre-train the real-valued CNN model with a clip function, rather than the ReLU function, to better initialize the Bi-Real net. Experiments on ImageNet show that the Bi-Real net with the proposed training algorithm achieves 56.4 and 62.2 top-1 accuracy with 18 layers and 34 layers, respectively. Compared to the state-of-the-arts (e.g., XNOR Net), Bi-Real net achieves up to 10 higher top-1 accuracy with more memory saving and lower computational cost. Keywords: binary neural network, 1-bit CNNs, 1-layer-per-block We consider the metric Traveling Salesman Problem (Δ-TSP for short) and study how stability (as defined by Bilu and Linial [3]) influences the complexity of the problem. On an intuitive level, an instance of Δ-TSP is γ-stable (γ > 1), if there is a unique optimum Hamiltonian tour and any perturbation of arbitrary edge weights by at most γ does not change the edge set of the optimal solution (i.e., there is a significant gap between the optimum tour and all other tours). We show that for γ ≥ 1.8 a simple greedy algorithm (resembling Prim's algorithm for constructing a minimum spanning tree) computes the optimum Hamiltonian tour for every γ-stable instance of the Δ-TSP, whereas a simple local search algorithm can fail to find the optimum even if γ is arbitrary. We further show that there are γ-stable instances of Δ-TSP for every 1 < γ < 2. These results provide a different view on the hardness of the Δ-TSP and give rise to a new class of problem instances which are substantially easier to solve than instances of the general Δ-TSP. Automatically generating coherent and semantically meaningful text has many applications in machine translation, dialogue systems, image captioning, etc. Recently, by combining with policy gradient, Generative Adversarial Nets (GAN) that use a discriminative model to guide the training of the generative model as a reinforcement learning policy has shown promising results in text generation. However, the scalar guiding signal is only available after the entire text has been generated and lacks intermediate information about text structure during the generative process. As such, it limits its success when the length of the generated text samples is long (more than 20 words). In this paper, we propose a new framework, called LeakGAN, to address the problem for long text generation. We allow the discriminative net to leak its own high-level extracted features to the generative net to further help the guidance. The generator incorporates such informative signals into all generation steps through an additional Manager module, which takes the extracted features of current generated words and outputs a latent vector to guide the Worker module for next-word generation. Our extensive experiments on synthetic data and various real-world tasks with Turing test demonstrate that LeakGAN is highly effective in long text generation and also improves the performance in short text generation scenarios. More importantly, without any supervision, LeakGAN would be able to implicitly learn sentence structures only through the interaction between Manager and Worker. The aim of the study is to provide interesting insights on how efficient machine learning algorithms could be adapted to solve combinatorial optimization problems in conjunction with existing heuristic procedures. More specifically, we extend the neural combinatorial optimization framework to solve the traveling salesman problem (TSP). In this framework, the city coordinates are used as inputs and the neural network is trained using reinforcement learning to predict a distribution over city permutations. Our proposed framework differs from the one in [1] since we do not make use of the Long Short-Term Memory (LSTM) architecture and we opted to design our own critic to compute a baseline for the tour length which results in more efficient learning. More importantly, we further enhance the solution approach with the well-known 2-opt heuristic. The results show that the performance of the proposed framework alone is generally as good as high performance heuristics (OR-Tools). When the framework is equipped with a simple 2-opt procedure, it could outperform such heuristics and achieve close to optimal results on 2D Euclidean graphs. This demonstrates that our approach based on machine learning techniques could learn good heuristics which, once being enhanced with a simple local search, yield promising results. We consider the asymmetric traveling salesman problem for which the triangular inequality is satisfied. For various heuristics we construct examples to show that the worst-case ratio of length of tour found to minimum length tour is (n) for n city problems. We also provide a new O([log2n]) heuristic. We present an on-line strategy that enables a mobile robot with vision to explore an unknown simple polygon. We prove that the resulting tour is less than 26.5 times as long as the shortest watchman tour that could be computed off-line. Our analysis is doubly founded on a novel geometric structure called angle hull. Let D be a connected region inside a simple polygon, P. We define the angle hull of D, @math , to be the set of all points in P that can see two points of D at a right angle. We show that the perimeter of @math cannot exceed in length the perimeter of D by more than a factor of 2. This upper bound is tight. Abstract In 1985, Jaillet introduced the probabilistic traveling salesman problem (PTSP), a variant of the classical TSP in which only a subset of the nodes may be present in any given instance of the problem. The goal is to find an a priori tour of minimal expected length, with the strategy of visiting the present nodes in a particular instance in the same order as they appear in the a priori tour. In this paper we reexamine the PTSP using a variety of theoretical and computational approaches. We sharpen the best known bounds for the PTSP, derive several asymptotic relations, and compare from various veiwpoints the PTSP with the re-optimization strategy, i.e., finding an optimal tour in every problem instance. When a Euclidean metric is used and the nodes are uniformly distributed in the unit square, a heuristic for the PTSP is shown to be very close to the re-optimization strategy. We examine some PTSP heuristics with provable worst-case performance, and address the question of finding constant-guarantee heuristics. Implementations of various heuristics, some based on sorting and some on local optimality, permit us to discuss the qualitative and quantitative properties of computational problems with up to 5000 nodes.