We investigate the problem of learning a Lipschitz function from binary feedback. In this problem, a learner is trying to learn a Lipschitz function $f:[0,1]^d \rightarrow [0,1]$ over the course of $T$ rounds. On round $t$, an adversary provides the learner with an input $x_t$, the learner submits a guess $y_t$ for $f(x_t)$, and learns whether $y_t > f(x_t)$ or $y_t \leq f(x_t)$. The learner's goal is to minimize their total loss $\sum_t\ell(f(x_t), y_t)$ (for some loss function $\ell$). The problem is motivated by \textit{contextual dynamic pricing}, where a firm must sell a stream of differentiated products to a collection of buyers with non-linear valuations for the items and observes only whether the item was sold or not at the posted price.

An Euler discretization of the Langevin diffusion is known to converge to the global minimizers of certain convex and non-convex optimization problems. We show that this property holds for any suitably smooth diffusion and that different diffusions are suitable for optimizing different classes of convex and non-convex functions. This allows us to design diffusions suitable for globally optimizing convex and non-convex functions not covered by the existing Langevin theory. Our non-asymptotic analysis delivers computable optimization and integration error bounds based on easily accessed properties of the objective and chosen diffusion. Central to our approach are new explicit Stein factor bounds on the solutions of Poisson equations. We complement these results with improved optimization guarantees for targets other than the standard Gibbs measure.

Consider the situation where you are given an existing $k$-way clustering $\pi$. A challenge for explainable AI is to find a compact and distinct explanations of each cluster which in this paper is using instance-level descriptors/tags from a common dictionary. Since the descriptors/tags were not given to the clustering method, this is not a semi-supervised learning situation. We show that the \emph{feasibility} problem of just testing whether any distinct description (not the most compact) exists is generally intractable for just two clusters. This means that unless \textbf{P} = \cnp, there cannot exist an efficient algorithm for the cluster description problem. Hence, we explore ILP formulations for smaller problems and a relaxed but restricted setting that leads to a polynomial time algorithm for larger problems. We explore several extension to the basic setting such as the ability to ignore some instances and composition constraints on the descriptions of the clusters. We show our formulation's usefulness on Twitter data where the communities were found using social connectivity (i.e.

Applications of optimal transport have recently gained remarkable attention as a result of the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation to the Wasserstein distance is replaced by a regularized version that is less accurate but easy to differentiate. In this work we characterize the differential properties of the original Sinkhorn approximation, proving that it enjoys the same smoothness as its regularized version and we explicitly provide an efficient algorithm to compute its gradient. We show that this result benefits both theory and applications: on one hand, high order smoothness confers statistical guarantees to learning with Wasserstein approximations. On the other hand, the gradient formula allows to efficiently solve learning and optimization problems in practice. Promising preliminary experiments complement our analysis.

Spiking neural networks (SNNs) are positioned to enable spatio-temporal information processing and ultra-low power event-driven neuromorphic hardware. However, SNNs are yet to reach the same performances of conventional deep artificial neural networks (ANNs), a long-standing challenge due to complex dynamics and non-differentiable spike events encountered in training. The existing SNN error backpropagation (BP) methods are limited in terms of scalability, lack of proper handling of spiking discontinuities, and/or mismatch between the rate-coded loss function and computed gradient. We present a hybrid macro/micro level backpropagation (HM2-BP) algorithm for training multi-layer SNNs. The temporal effects are precisely captured by the proposed spike-train level post-synaptic potential (S-PSP) at the microscopic level.

Recent advances in recording technologies have allowed neuroscientists to record simultaneous spiking activity from hundreds to thousands of neurons in multiple brain regions. Such large-scale recordings pose a major challenge to existing statistical methods for neural data analysis. Here we develop highly scalable approximate inference methods for Poisson generalized linear models (GLMs) that require only a single pass over the data. Our approach relies on a recently proposed method for obtaining approximate sufficient statistics for GLMs using polynomial approximations [Huggins et al., 2017], which we adapt to the Poisson GLM setting. We focus on inference using quadratic approximations to nonlinear terms in the Poisson GLM log-likelihood with Gaussian priors, for which we derive closed-form solutions to the approximate maximum likelihood and MAP estimates, posterior distribution, and marginal likelihood. We introduce an adaptive procedure to select the polynomial approximation interval and show that the resulting method allows for efficient and accurate inference and regularization of high-dimensional parameters. We use the quadratic estimator to fit a fully-coupled Poisson GLM to spike train data recorded from 831 neurons across five regions of the mouse brain for a duration of 41 minutes, binned at 1 ms resolution. Across all neurons, this model is fit to over 2 billion spike count bins and identifies fine-timescale statistical dependencies between neurons within and across cortical and subcortical areas.

Multiple-step lookahead policies have demonstrated high empirical competence in Reinforcement Learning, via the use of Monte Carlo Tree Search or Model Predictive Control. In a recent work (Efroni et al., 2018), multiple-step greedy policies and their use in vanilla Policy Iteration algorithms were proposed and analyzed. In this work, we study multiple-step greedy algorithms in more practical setups. We begin by highlighting a counter-intuitive difficulty, arising with soft-policy updates: even in the absence of approximations, and contrary to the 1-step-greedy case, monotonic policy improvement is not guaranteed unless the update stepsize is sufficiently large. Taking particular care about this difficulty, we formulate and analyze online and approximate algorithms that use such a multi-step greedy operator.

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.

We suggest a general oracle-based framework that captures parallel stochastic optimization in different parallelization settings described by a dependency graph, and derive generic lower bounds in terms of this graph. We then use the framework and derive lower bounds to study several specific parallel optimization settings, including delayed updates and parallel processing with intermittent communication. We highlight gaps between lower and upper bounds on the oracle complexity, and cases where the ``natural'' algorithms are not known to be optimal.

Estimating how uncertain an AI system is in its predictions is important to improve the safety of such systems. Uncertainty in predictive can result from uncertainty in model parameters, irreducible \emph{data uncertainty} and uncertainty due to distributional mismatch between the test and training data distributions. Different actions might be taken depending on the source of the uncertainty so it is important to be able to distinguish between them. Recently, baseline tasks and metrics have been defined and several practical methods to estimate uncertainty developed. These methods, however, attempt to model uncertainty due to distributional mismatch either implicitly through \emph{model uncertainty} or as \emph{data uncertainty}. This work proposes a new framework for modeling predictive uncertainty called Prior Networks (PNs) which explicitly models \emph{distributional uncertainty}. PNs do this by parameterizing a prior distribution over predictive distributions. This work focuses on uncertainty for classification and evaluates PNs on the tasks of identifying out-of-distribution (OOD) samples and detecting misclassification on the MNIST and CIFAR-10 datasets, where they are found to outperform previous methods. Experiments on synthetic and MNIST and CIFAR-10 data show that unlike previous non-Bayesian methods PNs are able to distinguish between data and distributional uncertainty.