We study the problem of model selection in bandit scenarios in the presence of nested policy classes, with the goal of obtaining simultaneous adversarial and stochastic ( best of both worlds") high-probability regret guarantees. Our approach requires that each base learner comes with a candidate regret bound that may or may not hold, while our meta algorithm plays each base learner according to a schedule that keeps the base learner's candidate regret bounds balanced until they are detected to violate their guarantees. We develop careful mis-specification tests specifically designed to blend the above model selection criterion with the ability to leverage the (potentially benign) nature of the environment. We recover the model selection guarantees of the CORRAL algorithm for adversarial environments, but with the additional benefit of achieving high probability regret bounds. More importantly, our model selection results also hold simultaneously in stochastic environments under gap assumptions.

Natural gradient descent has proven very effective at mitigating the catastrophic effects of pathological curvature in the objective function, but little is known theoretically about its convergence properties, especially for \emph{non-linear} networks. In this work, we analyze for the first time the speed of convergence to global optimum for natural gradient descent on non-linear neural networks with the squared error loss. We identify two conditions which guarantee the global convergence: (1) the Jacobian matrix (of network's output for all training cases w.r.t the parameters) is full row rank and (2) the Jacobian matrix is stable for small perturbations around the initialization. For two-layer ReLU neural networks (i.e. with one hidden layer), we prove that these two conditions do hold throughout the training under the assumptions that the inputs do not degenerate and the network is over-parameterized. We further extend our analysis to more general loss function with similar convergence property.

Out of the rich family of generalized linear bandits, perhaps the most well studied ones are logistic bandits that are used in problems with binary rewards: for instance, when the learner aims to maximize the profit over a user that can select one of two possible outcomes (e.g., click' vs no-click'). Despite remarkable recent progress and improved algorithms for logistic bandits, existing works do not address practical situations where the number of outcomes that can be selected by the user is larger than two (e.g., click', show me later', never show again', no click'). In this paper, we study such an extension. We use multinomial logit (MNL) to model the probability of each one of K 1\geq 2 possible outcomes ( 1 stands for the not click' outcome): we assume that for a learner's action \mathbf{x}_t, the user selects one of K 1\geq 2 outcomes, say outcome i, with a MNL probabilistic model with corresponding unknown parameter \bar{\boldsymbol{\theta}}_{\ast i} . Each outcome i is also associated with a revenue parameter \rho_i and the goal is to maximize the expected revenue.

We study efficient distributed algorithms for the fundamental problem of principal component analysis and leading eigenvector computation on the sphere, when the data are randomly distributed among a set of computational nodes. We propose a new quantized variant of Riemannian gradient descent to solve this problem, and prove that the algorithm converges with high probability under a set of necessary spherical-convexity properties. We give bounds on the number of bits transmitted by the algorithm under common initialization schemes, and investigate the dependency on the problem dimension in each case.

Stein Variational Gradient Descent (SVGD), a popular sampling algorithm, is often described as the kernelized gradient flow for the Kullback-Leibler divergence in the geometry of optimal transport. We introduce a new perspective on SVGD that instead views SVGD as the kernelized gradient flow of the chi-squared divergence. Motivated by this perspective, we provide a convergence analysis of the chi-squared gradient flow. We also show that our new perspective provides better guidelines for choosing effective kernels for SVGD.

The lottery ticket hypothesis (LTH) states that learning on a properly pruned network (the winning ticket) has improved test accuracy over the original unpruned network. Although LTH has been justified empirically in a broad range of deep neural network (DNN) involved applications like computer vision and natural language processing, the theoretical validation of the improved generalization of a winning ticket remains elusive. To the best of our knowledge, our work, for the first time, characterizes the performance of training a pruned neural network by analyzing the geometric structure of the objective function and the sample complexity to achieve zero generalization error. We show that the convex region near a desirable model with guaranteed generalization enlarges as the neural network model is pruned, indicating the structural importance of a winning ticket. Moreover, as the algorithm for training a pruned neural network is specified as an (accelerated) stochastic gradient descent algorithm, we theoretically show that the number of samples required for achieving zero generalization error is proportional to the number of the non-pruned weights in the hidden layer.

Current theoretical results on optimization trajectories of neural networks trained by gradient descent typically have the form of rigorous but potentially loose bounds on the loss values. In the present work we take a different approach and show that the learning trajectory of a wide network in a lazy training regime can be characterized by an explicit asymptotic at large training times. Specifically, the leading term in the asymptotic expansion of the loss behaves as a power law L(t) \sim C t {-\xi} with exponent \xi expressed only through the data dimension, the smoothness of the activation function, and the class of function being approximated. Our results are based on spectral analysis of the integral operator representing the linearized evolution of a large network trained on the expected loss. Importantly, the techniques we employ do not require a specific form of the data distribution, for example Gaussian, thus making our findings sufficiently universal.

Wasserstein distance-based distributionally robust optimization (DRO) has received much attention lately due to its ability to provide a robustness interpretation of various learning models. Moreover, many of the DRO problems that arise in the learning context admits exact convex reformulations and hence can be tackled by off-the-shelf solvers. Nevertheless, the use of such solvers severely limits the applicability of DRO in large-scale learning problems, as they often rely on general purpose interior-point algorithms. On the other hand, there are very few works that attempt to develop fast iterative methods to solve these DRO problems, which typically possess complicated structures. In this paper, we take a first step towards resolving the above difficulty by developing a first-order algorithmic framework for tackling a class of Wasserstein distance-based distributionally robust logistic regression (DRLR) problem.

The curse of dimensionality is a longstanding challenge in Bayesian inference in high dimensions. In this work, we propose a {projected Stein variational gradient descent} (pSVGD) method to overcome this challenge by exploiting the fundamental property of intrinsic low dimensionality of the data informed subspace stemming from ill-posedness of such problems. We adaptively construct the subspace using a gradient information matrix of the log-likelihood, and apply pSVGD to the much lower-dimensional coefficients of the parameter projection. The method is demonstrated to be more accurate and efficient than SVGD. It is also shown to be more scalable with respect to the number of parameters, samples, data points, and processor cores via experiments with parameters dimensions ranging from the hundreds to the tens of thousands.

Quantum information is a promising new paradigm for fast computations that can provide substantial speedups for many algorithms we use today. Among them, quantum machine learning is one of the most exciting applications of quantum computers. In this paper, we introduce q-means, a new quantum algorithm for clustering. It is a quantum version of a robust k-means algorithm, with similar convergence and precision guarantees. We also design a method to pick the initial centroids equivalent to the classical k-means method.