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.
Furthermore, the critic network optimizes the mean square error between the rounded matrix produced by the Hungarian algorithm @cite_9 and the output of the Sinkhorn layer. The Sinkhorn layer generates a doubly stochastic permutation matrix from the output of the correlation mappings. Similar to the DDPG @cite_17 algorithm, the critic network takes action and a representation of the learned state generated by actor net. TauRieL does not enforce any double stochasticity and can generate permutations from the transition matrix either stochastically or deterministically.
@cite_9 @cite_17
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Many problems at the intersection of combinatorics and computer science require solving for a permutation that optimally matches, ranks, or sorts some data. These problems usually have a task-specific, often non-differentiable objective function that data-driven algorithms can use as a learning signal. In this paper, we propose the Sinkhorn Policy Gradient (SPG) algorithm for learning policies on permutation matrices. The actor-critic neural network architecture we introduce for SPG uniquely decouples representation learning of the state space from the highly-structured action space of permutations with a temperature-controlled Sinkhorn layer. The Sinkhorn layer produces continuous relaxations of permutation matrices so that the actor-critic architecture can be trained end-to-end. Our empirical results show that agents trained with SPG can perform competitively on sorting, the Euclidean TSP, and matching tasks. We also observe that SPG is significantly more data efficient at the matching task than the baseline methods, which indicates that SPG is conducive to learning representations that are useful for reasoning about permutations. The Sudoku puzzle is a discrete constraint satisfaction problem, as is the error correction decoding problem. We propose here an algorithm for solution to the Sinkhorn puzzle based on Sinkhorn balancing. Sinkhorn balancing is an algorithm for projecting a matrix onto the space of doubly stochastic matrices. The Sinkhorn balancing solver is capable of solving all but the most difficult puzzles. A proof of convergence is presented, with some information theoretic connections. A random generalization of the Sudoku puzzle is presented, for which the Sinkhorn-based solver is also very effective. We give an algorithm for the on-line learning of permutations. The algorithm maintains its uncertainty about the target permutation as a doubly stochastic weight matrix, and makes predictions using an efficient method for decomposing the weight matrix into a convex combination of permutations. The weight matrix is updated by multiplying the current matrix entries by exponential factors, and an iterative procedure is needed to restore double stochasticity. Even though the result of this procedure does not have a closed form, a new analysis approach allows us to prove an optimal (up to small constant factors) bound on the regret of our algorithm. This regret bound is significantly better than that of either Kalai and Vempala's more efficient Follow the Perturbed Leader algorithm or the computationally expensive method of explicitly representing each permutation as an expert. We transform the outputs of each hidden neuron in a multi-layer perceptron network to have zero output and zero slope on average, and use separate shortcut connections to model the linear dependencies instead. This transformation aims at separating the problems of learning the linear and nonlinear parts of the whole input-output mapping, which has many benefits. We study the theoretical properties of the transformation by noting that they make the Fisher information matrix closer to a diagonal matrix, and thus standard gradient closer to the natural gradient. We experimentally confirm the usefulness of the transformations by noting that they make basic stochastic gradient learning competitive with state-of-the-art learning algorithms in speed, and that they seem also to help find solutions that generalize better. The experiments include both classification of small images and learning a lowdimensional representation for images by using a deep unsupervised auto-encoder network. The transformations were beneficial in all cases, with and without regularization and with networks from two to five hidden layers.