However, it has been recently demonstrated that such methods can fail to converge even in simple convex optimization settings. In this work, we provide a new analysis of such methods applied to nonconvex stochastic optimization problems, characterizing the effect of increasing minibatch size. Our analysis shows that under this scenario such methods do converge to stationarity up to the statistical limit of variance in the stochastic gradients (scaled by a constant factor). In particular, our result implies that increasing minibatch sizes enables convergence, thus providing a way to circumvent the nonconvergence issues.

We study minimax convergence rates of nonparametric density estimation under a large class of loss functions called "adversarial losses", which, besides classical L

However, it has been recently demonstrated that such methods can fail to converge even in simple convex optimization settings. In this work, we provide a new analysis of such methods applied to nonconvex stochastic optimization problems, characterizing the effect of increasing minibatch size. Our analysis shows that under this scenario such methods do converge to stationarity up to the statistical limit of variance in the stochastic gradients (scaled by a constant factor). In particular, our result implies that increasing minibatch sizes enables convergence, thus providing a way to circumvent the nonconvergence issues.

Neural Information Processing Systems http://nips.cc/

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.

Neural Information Processing Systems http://nips.cc/