When we are interested in high-dimensional system and focus on classification performance, the $\ell_{1}$-penalized logistic regression is becoming important and popular. However, the Lasso estimates could be problematic when penalties of different coefficients are all the same and not related to the data. We proposed two types of weighted Lasso estimates depending on covariates by the McDiarmid inequality. Given sample size $n$ and dimension of covariates $p$, the finite sample behavior of our proposed methods with a diverging number of predictors is illustrated by non-asymptotic oracle inequalities such as $\ell_{1}$-estimation error and squared prediction error of the unknown parameters. We compare the performance of our methods with former weighted estimates on simulated data, then apply these methods to do real data analysis.

We consider the question of learning $Q$-function in a sample efficient manner for reinforcement learning with continuous state and action spaces under a generative model. If $Q$-function is Lipschitz continuous, then the minimal sample complexity for estimating $\epsilon$-optimal $Q$-function is known to scale as ${\Omega}(\frac{1}{\epsilon^{d_1+d_2 +2}})$ per classical non-parametric learning theory, where $d_1$ and $d_2$ denote the dimensions of the state and action spaces respectively. The $Q$-function, when viewed as a kernel, induces a Hilbert-Schmidt operator and hence possesses square-summable spectrum. This motivates us to consider a parametric class of $Q$-functions parameterized by its "rank" $r$, which contains all Lipschitz $Q$-functions as $r \to \infty$. As our key contribution, we develop a simple, iterative learning algorithm that finds $\epsilon$-optimal $Q$-function with sample complexity of $\widetilde{O}(\frac{1}{\epsilon^{\max(d_1, d_2)+2}})$ when the optimal $Q$-function has low rank $r$ and the discounting factor $\gamma$ is below a certain threshold. Thus, this provides an exponential improvement in sample complexity. To enable our result, we develop a novel Matrix Estimation algorithm that faithfully estimates an unknown low-rank matrix in the $\ell_\infty$ sense even in the presence of arbitrary bounded noise, which might be of interest in its own right. Empirical results on several stochastic control tasks confirm the efficacy of our "low-rank" algorithms.

A crucial ability of human intelligence is to build up models of individual 3D objects from partial scene observations. Recent works achieve object-centric generation but without the ability to infer the representation, or achieve 3D scene representation learning but without object-centric compositionality. Therefore, learning to represent and render 3D scenes with object-centric compositionality remains elusive. In this paper, we propose a probabilistic generative model for learning to build modular and compositional 3D object models from partial observations of a multi-object scene. The proposed model can (i) infer the 3D object representations by learning to search and group object areas and also (ii) render from an arbitrary viewpoint not only individual objects but also the full scene by compositing the objects. The entire learning process is unsupervised and end-to-end. In experiments, in addition to generation quality, we also demonstrate that the learned representation permits object-wise manipulation and novel scene generation, and generalizes to various settings. Results can be found on our project website: https://sites.google.com/view/roots3d

Stochastic Gradient Algorithms (SGAs) are ubiquitous in computational statistics, machine learning and optimisation. Recent years have brought an influx of interest in SGAs and the non-asymptotic analysis of their bias is by now well-developed. However, in order to fully understand the efficiency of Monte Carlo algorithms utilizing stochastic gradients, one also needs to carry out the analysis of their variance, which turns out to be problem-specific. For this reason, there is no systematic theory that would specify the optimal choice of the random approximation of the gradient in SGAs for a given data regime. Furthermore, while there have been numerous attempts to reduce the variance of SGAs, these typically exploit a particular structure of the sampled distribution. In this paper we use the Multi-index Monte Carlo apparatus combined with the antithetic approach to construct the Multi-index Antithetic Stochastic Gradient Algorithm (MASGA), which can be used to sample from any probability distribution. This, to our knowledge, is the first SGA that, for all data regimes and without relying on any specific structure of the target measure, achieves performance on par with Monte Carlo estimators that have access to unbiased samples from the distribution of interest. In other words, MASGA is an optimal estimator from the error-computational cost perspective within the class of Monte Carlo estimators.

We analyze in a closed form the learning dynamics of stochastic gradient descent (SGD) for a single layer neural network classifying a high-dimensional Gaussian mixture where each cluster is assigned one of two labels. This problem provides a prototype of a non-convex loss landscape with interpolating regimes and a large generalization gap. We define a particular stochastic process for which SGD can be extended to a continuous-time limit that we call stochastic gradient flow. In the full-batch limit we recover the standard gradient flow. We apply dynamical mean-field theory from statistical physics to track the dynamics of the algorithm in the high-dimensional limit via a self-consistent stochastic process. We explore the performance of the algorithm as a function of control parameters shedding light on how it navigates the loss landscape.

We propose a Distributionally Robust Optimization (DRO) formulation with a Wasserstein-based uncertainty set for selecting grouped variables under perturbations on the data for both linear regression and classification problems. The resulting model offers robustness explanations for Grouped Least Absolute Shrinkage and Selection Operator (GLASSO) algorithms and highlights the connection between robustness and regularization. We prove probabilistic bounds on the out-of-sample loss and the estimation bias, and establish the grouping effect of our estimator, showing that coefficients in the same group converge to the same value as the sample correlation between covariates approaches 1. Based on this result, we propose to use the spectral clustering algorithm with the Gaussian similarity function to perform grouping on the predictors, which makes our approach applicable without knowing the grouping structure a priori. We compare our approach to an array of alternatives and provide extensive numerical results on both synthetic data and a real large dataset of surgery-related medical records, showing that our formulation produces an interpretable and parsimonious model that encourages sparsity at a group level and is able to achieve better prediction and estimation performance in the presence of outliers.

We develop Distributionally Robust Optimization (DRO) formulations for Multivariate Linear Regression (MLR) and Multiclass Logistic Regression (MLG) when both the covariates and responses/labels may be contaminated by outliers. The DRO framework uses a probabilistic ambiguity set defined as a ball of distributions that are close to the empirical distribution of the training set in the sense of the Wasserstein metric. We relax the DRO formulation into a regularized learning problem whose regularizer is a norm of the coefficient matrix. We establish out-of-sample performance guarantees for the solutions to our model, offering insights on the role of the regularizer in controlling the prediction error. Experimental results show that our approach improves the predictive error by 7% -- 37% for MLR, and a metric of robustness by 100% for MLG.

Mixup is a data augmentation technique that creates new examples as convex combinations of training points and labels. This simple technique has empirically shown to improve the accuracy of many state-of-the-art models in different settings and applications, but the reasons behind this empirical success remain poorly understood. In this paper we take a substantial step in explaining the theoretical foundations of Mixup, by clarifying its regularization effects. We show that Mixup can be interpreted as standard empirical risk minimization estimator subject to a combination of data transformation and random perturbation of the transformed data. We further show that these transformations and perturbations induce multiple known regularization schemes, including label smoothing and reduction of the Lipschitz constant of the estimator, and that these schemes interact synergistically with each other, resulting in a self calibrated and effective regularization effect that prevents overfitting and overconfident predictions. We illustrate our theoretical analysis by experiments that empirically support our conclusions.

We propose a model-independent definition of $n$-point interaction within a system of binary and categorical random variables from first principles, via the non-parametric framework of Targeted Learning, a subfield of mathematical statistics. This definition provides an interpretation for both magnitude and sign of $2$-point, $3$-point, and general $n$-point interactions. We show that the sign of an $n$-point interaction is interpretable relative to an $(n-1)$-point interaction obtained by fixing any one of the $n$ variables. The non-parametric definition of interaction is fundamentally unbiased and reduces to familiar notions of interaction in parametric statistical physics models. Moreover, by taking into account information on conditional independence and without any further assumptions, the accuracy of interactions estimated directly from data is substantially increased whilst the number of samples required and the computational run time are both reduced. We illustrate these concepts both analytically and numerically on (i) the $2$-dimensional Ising model, (ii) an Ising-like model with non-zero $2$-point, $3$-point, and $4$-point interactions, (iii) the Restricted Boltzmann Machine (RBM), and argue that the formulation applies to energy-based models more generally. The non-parametric formulation allows for the direct reconstruction of the Hamiltonian from the data it generated. Finally, we discuss novel applications of this work, namely estimating causal molecular interactions leading to physiological outcomes, in population biomedicine.

We consider sampling from composite densities on $\mathbb{R}^d$ of the form $d\pi(x) \propto \exp(-f(x) - g(x))dx$ for well-conditioned $f$ and convex (but possibly non-smooth) $g$, a family generalizing restrictions to a convex set, through the abstraction of a restricted Gaussian oracle. For $f$ with condition number $\kappa$, our algorithm runs in $O \left(\kappa^2 d \log^2\tfrac{\kappa d}{\epsilon}\right)$ iterations, each querying a gradient of $f$ and a restricted Gaussian oracle, to achieve total variation distance $\epsilon$. The restricted Gaussian oracle, which draws samples from a distribution whose negative log-likelihood sums a quadratic and $g$, has been previously studied and is a natural extension of the proximal oracle used in composite optimization. Our algorithm is conceptually simple and obtains stronger provable guarantees and greater generality than existing methods for composite sampling. We conduct experiments showing our algorithm vastly improves upon the hit-and-run algorithm for sampling the restriction of a (non-diagonal) Gaussian to the positive orthant.