We consider training over-parameterized two-layer neural networks with Rectified Linear Unit (ReLU) using gradient descent (GD) method. Inspired by a recent line of work, we study the evolutions of network prediction errors across GD iterations, which can be neatly described in a matrix form. When the network is sufficiently over-parameterized, these matrices individually approximate {\em an} integral operator which is determined by the feature vector distribution $\rho$ only. Consequently, GD method can be viewed as {\em approximately} applying the powers of this integral operator on the underlying/target function $f *$ that generates the responses/labels. We show that if $f *$ admits a low-rank approximation with respect to the eigenspaces of this integral operator, then the empirical risk decreases to this low rank approximation error at a linear rate which is determined by $f *$ and $\rho$ only, i.e., the rate is independent of the sample size $n$.

Despite the non-convex nature of their loss functions, deep neural networks are known to generalize well when optimized with stochastic gradient descent (SGD). Recent work conjectures that SGD with proper configuration is able to find wide and flat local minima, which are correlated with good generalization performance. In this paper, we observe that local minima of modern deep networks are more than being flat or sharp. Instead, at a local minimum there exist many asymmetric directions such that the loss increases abruptly along one side, and slowly along the opposite side – we formally define such minima as asymmetric valleys. Under mild assumptions, we first prove that for asymmetric valleys, a solution biased towards the flat side generalizes better than the exact empirical minimizer.

Motivated by the application of real-time pricing in e-commerce platforms, we consider the problem of revenue-maximization in a setting where the seller can leverage contextual information describing the customer's history and the product's type to predict her valuation of the product. However, her true valuation is unobservable to the seller, only binary outcome in the form of success-failure of a transaction is observed. Unlike in usual contextual bandit settings, the optimal price/arm given a covariate in our setting is sensitive to the detailed characteristics of the residual uncertainty distribution. We develop a semi-parametric model in which the residual distribution is non-parametric and provide the first algorithm which learns both regression parameters and residual distribution with $\tilde O(\sqrt{n})$ regret. We empirically test a scalable implementation of our algorithm and observe good performance. Papers published at the Neural Information Processing Systems Conference.

We propose a novel active learning (AL) model that integrates Bayesian and discriminative kernel machines for fast and accurate multi-class data sampling. By joining a sparse Bayesian model and a maximum margin machine under a unified kernel machine committee (KMC), the proposed model is able to identify a small number of data samples that best represent the overall data space while accurately capturing the decision boundaries. The integration is conducted using the maximum entropy discrimination framework, resulting in a joint objective function that contains generalized entropy as a regularizer. Such a property allows the proposed AL model to choose data samples that more effectively handle non-separable classification problems. Parameter learning is achieved through a principled optimization framework that leverages convex duality and sparse structure of KMC to efficiently optimize the joint objective function.

This paper proposes a partitioning structure learning method for segmented linear regression trees (SLRT), which assigns linear predictors over the terminal nodes. The recursive partitioning process is driven by an adaptive split selection algorithm that maximizes, at each node, a criterion function based on a conditional Kendall's τ statistic that measures the rank dependence between the regressors and the fit- ted linear residuals. Theoretical analysis shows that the split selection algorithm permits consistent identification and estimation of the unknown segments. A suffi- ciently large tree is induced by applying the split selection algorithm recursively. Then the minimal cost-complexity tree pruning procedure is applied to attain the right-sized tree, that ensures (i) the nested structure of pruned subtrees and (ii) consistent estimation to the number of segments.

Sampling from log-concave distributions is a well researched problem that has many applications in statistics and machine learning. In our paper, we propose a Markov chain Monte Carlo (MCMC) algorithm based on the underdamped Langevin diffusion (ULD). Our algorithm performs significantly faster than the previously best known algorithm for solving this problem, which requires $\tilde{O}\left(\kappa {1.5}/\epsilon\right)$ steps \cite{chen2019optimal,dalalyan2018sampling}. Moreover, our algorithm can be easily parallelized to require only $O(\kappa\log\frac{1}{\epsilon})$ parallel steps. To solve the sampling problem, we propose a new framework to discretize stochastic differential equations. We apply this framework to discretize and simulate ULD, which converges to the target distribution $p {*}$.

A recent line of research has shown that gradient-based algorithms with random initialization can converge to the global minima of the training loss for over-parameterized (i.e., sufficiently wide) deep neural networks. However, the condition on the width of the neural network to ensure the global convergence is very stringent, which is often a high-degree polynomial in the training sample size $n$ (e.g., $O(n {24})$). In this paper, we provide an improved analysis of the global convergence of (stochastic) gradient descent for training deep neural networks, which only requires a milder over-parameterization condition than previous work in terms of the training sample size and other problem-dependent parameters. The main technical contributions of our analysis include (a) a tighter gradient lower bound that leads to a faster convergence of the algorithm, and (b) a sharper characterization of the trajectory length of the algorithm. By specializing our result to two-layer (i.e., one-hidden-layer) neural networks, it also provides a milder over-parameterization condition than the best-known result in prior work. Papers published at the Neural Information Processing Systems Conference.

Sparse Group Lasso is a method of linear regression analysis that finds sparse parameters in terms of both feature groups and individual features. Block Coordinate Descent is a standard approach to obtain the parameters of Sparse Group Lasso, and iteratively updates the parameters for each parameter group. However, as an update of only one parameter group depends on all the parameter groups or data points, the computation cost is high when the number of the parameters or data points is large. This paper proposes a fast Block Coordinate Descent for Sparse Group Lasso. It efficiently skips the updates of the groups whose parameters must be zeros by using the parameters in one group. In addition, it preferentially updates parameters in a candidate group set, which contains groups whose parameters must not be zeros.

We analyze stochastic gradient algorithms for optimizing nonconvex problems. In particular, our goal is to find local minima (second-order stationary points) instead of just finding first-order stationary points which may be some bad unstable saddle points. We show that a simple perturbed version of stochastic recursive gradient descent algorithm (called SSRGD) can find an $(\epsilon,\delta)$-second-order stationary point with $\widetilde{O}(\sqrt{n}/\epsilon 2 \sqrt{n}/\delta 4 n/\delta 3)$ stochastic gradient complexity for nonconvex finite-sum problems. As a by-product, SSRGD finds an $\epsilon$-first-order stationary point with $O(n \sqrt{n}/\epsilon 2)$ stochastic gradients. These results are almost optimal since Fang et al. [2018] provided a lower bound $\Omega(\sqrt{n}/\epsilon 2)$ for finding even just an $\epsilon$-first-order stationary point.

Given a sequence of convex functions $f_0, f_1, \ldots, f_T$, we study the problem of sampling from the Gibbs distribution $\pi_t \propto e {-\sum_{k 0} t f_k}$ for each epoch $t$ in an {\em online} manner. Interest in this problem derives from applications in machine learning, Bayesian statistics, and optimization where, rather than obtaining all the observations at once, one constantly acquires new data, and must continuously update the distribution. Our main result is an algorithm that generates roughly independent samples from $\pi_t$ for every epoch $t$ and, under mild assumptions, makes $\mathrm{polylog}(T)$ gradient evaluations per epoch. All previous results imply a bound on the number of gradient or function evaluations which is at least linear in $T$. Motivated by real-world applications, we assume that functions are smooth, their associated distributions have a bounded second moment, and their minimizer drifts in a bounded manner, but do not assume they are strongly convex.