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Register a new userStochastic Optimization of Sorting Networks via Continuous Relaxations
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Sorting input objects is an important step in many machine learning pipelines. However, the sorting operator is non-differentiable with respect to its inputs, which prohibits end-to-end gradient-based optimization. In this work, we propose NeuralSort, a general-purpose continuous relaxation of the output of the sorting operator from permutation matrices to the set of unimodal row-stochastic matrices, where every row sums to one and has a distinct arg max. This relaxation permits straight-through optimization of any computational graph involve a sorting operation. Further, we use this relaxation to enable gradient-based stochastic optimization over the combinatorially large space of permutations by deriving a reparameterized gradient estimator for the Plackett-Luce family of distributions over permutations. We demonstrate the usefulness of our framework on three tasks that require learning semantic orderings of high-dimensional objects, including a fully differentiable, parameterized extension of the k-nearest neighbors algorithm.
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This paper proposes a new family of algorithms for training neural networks (NNs). These are based on recent developments in the field of non-convex optimization, going under the general name of successive convex approximation (SCA) techniques. The basic idea is to iteratively replace the original (non-convex, highly dimensional) learning problem with a sequence of (strongly convex) approximations, which are both accurate and simple to optimize. Differently from similar ideas (e.g., quasi-Newton algorithms), the approximations can be constructed using only first-order information of the neural network function, in a stochastic fashion, while exploiting the overall structure of the learning problem for a faster convergence. We discuss several use cases, based on different choices for the loss function (e.g., squared loss and cross-entropy loss), and for the regularization of the NNs weights. We experiment on several medium-sized benchmark problems, and on a large-scale dataset involving simulated physical data. The results show how the algorithm outperforms state-of-the-art techniques, providing faster convergence to a better minimum. Additionally, we show how the algorithm can be easily parallelized over multiple computational units without hindering its performance. In particular, each computational unit can optimize a tailored surrogate function defined on a randomly assigned subset of the input variables, whose dimension can be selected depending entirely on the available computational power.
This paper proposes SplitSGD, a new dynamic learning rate schedule for stochastic optimization. This method decreases the learning rate for better adaptation to the local geometry of the objective function whenever a stationary phase is detected, that is, the iterates are likely to bounce at around a vicinity of a local minimum. The detection is performed by splitting the single thread into two and using the inner product of the gradients from the two threads as a measure of stationarity. Owing to this simple yet provably valid stationarity detection, SplitSGD is easy-to-implement and essentially does not incur additional computational cost than standard SGD. Through a series of extensive experiments, we show that this method is appropriate for both convex problems and training (non-convex) neural networks, with performance compared favorably to other stochastic optimization methods. Importantly, this method is observed to be very robust with a set of default parameters for a wide range of problems and, moreover, yields better generalization performance than other adaptive gradient methods such as Adam.
Control variates are a well-established tool to reduce the variance of Monte Carlo estimators. However, for large-scale problems including high-dimensional and large-sample settings, their advantages can be outweighed by a substantial computational cost. This paper considers control variates based on Stein operators, presenting a framework that encompasses and generalizes existing approaches that use polynomials, kernels and neural networks. A learning strategy based on minimising a variational objective through stochastic optimization is proposed, leading to scalable and effective control variates. Novel theoretical results are presented to provide insight into the variance reduction that can be achieved, and an empirical assessment, including applications to Bayesian inference, is provided in support.
Variational Optimization forms a differentiable upper bound on an objective. We show that approaches such as Natural Evolution Strategies and Gaussian Perturbation, are special cases of Variational Optimization in which the expectations are approximated by Gaussian sampling. These approaches are of particular interest because they are parallelizable. We calculate the approximate bias and variance of the corresponding gradient estimators and demonstrate that using antithetic sampling or a baseline is crucial to mitigate their problems. We contrast these methods with an alternative parallelizable method, namely Directional Derivatives. We conclude that, for differentiable objectives, using Directional Derivatives is preferable to using Variational Optimization to perform parallel Stochastic Gradient Descent.
Many machine learning algorithms minimize a regularized risk, and stochastic optimization is widely used for this task. When working with massive data, it is desirable to perform stochastic optimization in parallel. Unfortunately, many existing stochastic optimization algorithms cannot be parallelized efficiently. In this paper we show that one can rewrite the regularized risk minimization problem as an equivalent saddle-point problem, and propose an efficient distributed stochastic optimization (DSO) algorithm. We prove the algorithms rate of convergence; remarkably, our analysis shows that the algorithm scales almost linearly with the number of processors. We also verify with empirical evaluations that the proposed algorithm is competitive with other parallel, general purpose stochastic and batch optimization algorithms for regularized risk minimization.
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