Stochastic gradient-based Monte Carlo methods such as stochastic gradient Langevin dynamics are useful tools for posterior inference on large scale datasets in many machine learning applications. These methods scale to large datasets by using noisy gradients calculated using a mini-batch or subset of the dataset. However, the high variance inherent in these noisy gradients degrades performance and leads to slower mixing. In this paper, we present techniques for reducing variance in stochastic gradient Langevin dynamics, yielding novel stochastic Monte Carlo methods that improve performance by reducing the variance in the stochastic gradient. We show that our proposed method has better theoretical guarantees on convergence rate than stochastic Langevin dynamics. This is complemented by impressive empirical results obtained on a variety of real world datasets, and on four different machine learning tasks (regression, classification, independent component analysis and mixture modeling). These theoretical and empirical contributions combine to make a compelling case for using variance reduction in stochastic Monte Carlo methods.

We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gradient method with constant minibatch converges to a stationary point. To tackle this issue, we develop fast stochastic algorithms that provably converge to a stationary point for constant minibatches. Furthermore, using a variant of these algorithms, we obtain provably faster convergence than batch proximal gradient descent. Our results are based on the recent variance reduction techniques for convex optimization but with a novel analysis for handling nonconvex and nonsmooth functions. We also prove global linear convergence rate for an interesting subclass of nonsmooth nonconvex functions, which subsumes several recent works.

We study the problem of designing models for machine learning tasks defined on sets. In contrast to traditional approach of operating on fixed dimensional vectors, we consider objective functions defined on sets that are invariant to permutations. Such problems are widespread, ranging from estimation of population statistics [1], to anomaly detection in piezometer data of embankment dams [2], to cosmology [3, 4]. Our main theorem characterizes the permutation invariant functions and provides a family of functions to which any permutation invariant objective function must belong. This family of functions has a special structure which enables us to design a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks. We also derive the necessary and sufficient conditions for permutation equivariance in deep models. We demonstrate the applicability of our method on population statistic estimation, point cloud classification, set expansion, and outlier detection.

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