We argue that regularizing terms in standard regression methods not only help against overfitting finite data, but sometimes also help in getting better causal models. We first consider a multi-dimensional variable linearly influencing a target variable with some multi-dimensional unobserved common cause, where the confounding effect can be decreased by keeping the penalizing term in Ridge and Lasso regression even in the population limit. The reason is a close analogy between overfitting and confounding observed for our toy model. In the case of overfitting, we can choose regularization constants via cross validation, but here we choose the regularization constant by first estimating the strength of confounding, which yielded reasonable results for simulated and real data. Further, we show a'causal generalization bound' which states (subject to our particular model of confounding) that the error made by interpreting any non-linear regression as causal model can be bounded from above whenever functions are taken from a not too rich class.

We consider a stochastic linear bandit model in which the available actions correspond to arbitrary context vectors whose associated rewards follow a non-stationary linear regression model. In this setting, the unknown regression parameter is allowed to vary in time. To address this problem, we propose D-LinUCB, a novel optimistic algorithm based on discounted linear regression, where exponential weights are used to smoothly forget the past. This involves studying the deviations of the sequential weighted least-squares estimator under generic assumptions. As a by-product, we obtain novel deviation results that can be used beyond non-stationary environments.

Logistic regression is commonly used for modeling dichotomous outcomes. In the classical setting, where the number of observations is much larger than the number of parameters, properties of the maximum likelihood estimator in logistic regression are well understood. Recently, Sur and Candes \cite{sur2018modern} have studied logistic regression in the high-dimensional regime, where the number of observations and parameters are comparable, and show, among other things, that the maximum likelihood estimator is biased. In the high-dimensional regime the underlying parameter vector is often structured (sparse, block-sparse, finite-alphabet, etc.) and so in this paper we study regularized logistic regression (RLR), where a convex regularizer that encourages the desired structure is added to the negative of the log-likelihood function. An advantage of RLR is that it allows parameter recovery even for instances where the (unconstrained) maximum likelihood estimate does not exist.

We consider the variable selection problem of generalized linear models (GLMs). Stability selection (SS) is a promising method proposed for solving this problem. Although SS provides practical variable selection criteria, it is computationally demanding because it needs to fit GLMs to many re-sampled datasets. We propose a novel approximate inference algorithm that can conduct SS without the repeated fitting. The algorithm is based on the replica method of statistical mechanics and vector approximate message passing of information theory. For datasets characterized by rotation-invariant matrix ensembles, we derive state evolution equations that macroscopically describe the dynamics of the proposed algorithm. We also show that their fixed points are consistent with the replica symmetric solution obtained by the replica method. Numerical experiments indicate that the algorithm exhibits fast convergence and high approximation accuracy for both synthetic and real-world data.

We characterize the effectiveness of a classical algorithm for recovering the Markov graph of a general discrete pairwise graphical model from i.i.d. The algorithm is (appropriately regularized) maximum conditional log-likelihood, which involves solving a convex program for each node; for Ising models this is $\ell_1$-constrained logistic regression, while for more general alphabets an $\ell_{2,1}$ group-norm constraint needs to be used. We show that this algorithm can recover any arbitrary discrete pairwise graphical model, and also characterize its sample complexity as a function of model width, alphabet size, edge parameter accuracy, and the number of variables. We show that along every one of these axes, it matches or improves on all existing results and algorithms for this problem. Our analysis applies a sharp generalization error bound for logistic regression when the weight vector has an $\ell_1$ (or $\ell_{2,1}$) constraint and the sample vector has an $\ell_{\infty}$ (or $\ell_{2, \infty}$) constraint.

We give the first polynomial-time algorithm for robust regression in the list-decodable setting where an adversary can corrupt a greater than 1/2 fraction of examples. It outputs a list L of size O(1/\alpha) - a fixed constant - that contains an \ell that is close to \ell *. Our algorithm succeeds whenever the inliers are chosen from a certifiably anti-concentrated distribution D. In particular, this gives a (d/\alpha) {O(1/\alpha 8)} time algorithm to find a O(1/\alpha) size list when the inlier distribution is a standard Gaussian. For discrete product distributions that are anti-concentrated only in regular directions, we give an algorithm that achieves similar guarantee under the promise that \ell * has all coordinates of the same magnitude. To complement our result, we prove that the anti-concentration assumption on the inliers is information-theoretically necessary.

Given a linear regression setting, Iterative Least Trimmed Squares (ILTS) involves alternating between (a) selecting the subset of samples with lowest current loss, and (b) re-fitting the linear model only on that subset. Both steps are very fast and simple. In this paper, we analyze ILTS in the setting of mixed linear regression with corruptions (MLR-C). We first establish deterministic conditions (on the features etc.) under which the ILTS iterate converges linearly to the closest mixture component. We also provide a global algorithm that uses ILTS as a subroutine, to fully solve mixed linear regressions with corruptions.

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. Specifically, we propose a novel linearized proximal ADMM to solve the DRLR problem, whose objective is convex but consists of a smooth term plus two non-separable non-smooth terms.

We consider first order expansions of convex penalized estimators in high-dimensional regression problems with random designs. Our setting includes linear regression and logistic regression as special cases. For a given penalty function $h$ and the corresponding penalized estimator $\hbeta$, we construct a quantity $\eta$, the first order expansion of $\hbeta$, such that the distance between $\hbeta$ and $\eta$ is an order of magnitude smaller than the estimation error $\ \hat{\beta} - \beta *\ $. In this sense, the first order expansion $\eta$ can be thought of as a generalization of influence functions from the mathematical statistics literature to regularized estimators in high-dimensions. Such first order expansion implies that the risk of $\hat{\beta}$ is asymptotically the same as the risk of $\eta$ which leads to a precise characterization of the MSE of $\hbeta$; this characterization takes a particularly simple form for isotropic design.

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.