The natural gradient method is widely used in statistical optimization, but its standard formulation assumes a Euclidean parameter space. This paper proposes an inversion-free stochastic natural gradient method for probability distributions whose parameters lie on a Riemannian manifold. The manifold setting offers several advantages: one can implicitly enforce parameter constraints such as positive definiteness and orthogonality, ensure parameters are identifiable, or guarantee regularity properties of the objective like geodesic convexity. Building on an intrinsic formulation of the Fisher information matrix (FIM) on a manifold, our method maintains an online approximation of the inverse FIM, which is efficiently updated at quadratic cost using score vectors sampled at successive iterates. In the Riemannian setting, these score vectors belong to different tangent spaces and must be combined using transport operations. We prove almost-sure convergence rates of $O(\log{s}/s^α)$ for the squared distance to the minimizer when the step size exponent $α>2/3$. We also establish almost-sure rates for the approximate FIM, which now accumulates transport-based errors. A limited-memory variant of the algorithm with sub-quadratic storage complexity is proposed. Finally, we demonstrate the effectiveness of our method relative to its Euclidean counterparts on variational Bayes with Gaussian approximations and normalizing flows.

In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. More specifically, we prove that for monotone continuous DR-submodular functions, all fixed points of projected gradient ascent provide a factor $1/2$ approximation to the global maxima. We also study stochastic gradient methods and show that after $\mathcal{O}(1/\epsilon^2)$ iterations these methods reach solutions which achieve in expectation objective values exceeding $(\frac{\text{OPT}}{2}-\epsilon)$. An immediate application of our results is to maximize submodular functions that are defined stochastically, i.e. the submodular function is defined as an expectation over a family of submodular functions with an unknown distribution. We will show how stochastic gradient methods are naturally well-suited for this setting, leading to a factor $1/2$ approximation when the function is monotone. In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient ascent on a continuous relaxation, directly connecting the discrete and continuous domains. Finally, experiments on real data demonstrate that our projected gradient methods consistently achieve the best utility compared to other continuous baselines while remaining competitive in terms of computational effort.

Policy gradient methods are widely used for control in reinforcement learning, particularly for the continuous action setting. There have been a host of theoretically sound algorithms proposed for the on-policy setting, due to the existence of the policy gradient theorem which provides a simplified form for the gradient. In off-policy learning, however, where the behaviour policy is not necessarily attempting to learn and follow the optimal policy for the given task, the existence of such a theorem has been elusive. In this work, we solve this open problem by providing the first off-policy policy gradient theorem. The key to the derivation is the use of emphatic weightings. We develop a new actor-critic algorithm--called Actor Critic with Emphatic weightings (ACE)--that approximates the simplified gradients provided by the theorem. We demonstrate in a simple counterexample that previous off-policy policy gradient methods--particularly OffPAC and DPG--converge to the wrong solution whereas ACE finds the optimal solution.

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

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

The technique of regularization is ubiquitous in machine learning as it can effectively prevent overfitting and yield better generalization.

Recently, there has been a marked increase in interest regarding the generalization capabilities of unregularized gradient-based learning methods.

Recently, there has been a marked increase in interest regarding the generalization capabilities of unregularized gradient-based learning methods.

Consequently, our framework transforms the task of proving convergence rate into verifying the positive semidefiniteness of a specific integral kernel.