Sparse linear regression is a central problem in high-dimensional statistics. We study the correlated random design setting, where the covariates are drawn from a multivariate Gaussian $N(0,\Sigma)$, and we seek an estimator with small excess risk. If the true signal is $t$-sparse, information-theoretically, it is possible to achieve strong recovery guarantees with only $O(t\log n)$ samples. However, computationally efficient algorithms have sample complexity linear in (some variant of) the condition number of $\Sigma$. Classical algorithms such as the Lasso can require significantly more samples than necessary even if there is only a single sparse approximate dependency among the covariates. We provide a polynomial-time algorithm that, given $\Sigma$, automatically adapts the Lasso to tolerate a small number of approximate dependencies. In particular, we achieve near-optimal sample complexity for constant sparsity and if $\Sigma$ has few ``outlier'' eigenvalues. Our algorithm fits into a broader framework of feature adaptation for sparse linear regression with ill-conditioned covariates. With this framework, we additionally provide the first polynomial-factor improvement over brute-force search for constant sparsity $t$ and arbitrary covariance $\Sigma$.

Much of reinforcement learning theory is built on top of oracles that are computationally hard to implement. Specifically for learning near-optimal policies in Partially Observable Markov Decision Processes (POMDPs), existing algorithms either need to make strong assumptions about the model dynamics (e.g. deterministic transitions) or assume access to an oracle for solving a hard optimistic planning or estimation problem as a subroutine. In this work we develop the first oracle-free learning algorithm for POMDPs under reasonable assumptions. Specifically, we give a quasipolynomial-time end-to-end algorithm for learning in "observable" POMDPs, where observability is the assumption that well-separated distributions over states induce well-separated distributions over observations. Our techniques circumvent the more traditional approach of using the principle of optimism under uncertainty to promote exploration, and instead give a novel application of barycentric spanners to constructing policy covers.

Measuring the stability of conclusions derived from Ordinary Least Squares linear regression is critically important, but most metrics either only measure local stability (i.e. against infinitesimal changes in the data), or are only interpretable under statistical assumptions. Recent work proposes a simple, global, finite-sample stability metric: the minimum number of samples that need to be removed so that rerunning the analysis overturns the conclusion, specifically meaning that the sign of a particular coefficient of the estimated regressor changes. However, besides the trivial exponential-time algorithm, the only approach for computing this metric is a greedy heuristic that lacks provable guarantees under reasonable, verifiable assumptions; the heuristic provides a loose upper bound on the stability and also cannot certify lower bounds on it. We show that in the low-dimensional regime where the number of covariates is a constant but the number of samples is large, there are efficient algorithms for provably estimating (a fractional version of) this metric. Applying our algorithms to the Boston Housing dataset, we exhibit regression analyses where we can estimate the stability up to a factor of $3$ better than the greedy heuristic, and analyses where we can certify stability to dropping even a majority of the samples.

Partially Observable Markov Decision Processes (POMDPs) are a natural and general model in reinforcement learning that take into account the agent's uncertainty about its current state. In the literature on POMDPs, it is customary to assume access to a planning oracle that computes an optimal policy when the parameters are known, even though the problem is known to be computationally hard. Almost all existing planning algorithms either run in exponential time, lack provable performance guarantees, or require placing strong assumptions on the transition dynamics under every possible policy. In this work, we revisit the planning problem and ask: are there natural and well-motivated assumptions that make planning easy? Our main result is a quasipolynomial-time algorithm for planning in (one-step) observable POMDPs. Specifically, we assume that well-separated distributions on states lead to well-separated distributions on observations, and thus the observations are at least somewhat informative in each step. Crucially, this assumption places no restrictions on the transition dynamics of the POMDP; nevertheless, it implies that near-optimal policies admit quasi-succinct descriptions, which is not true in general (under standard hardness assumptions). Our analysis is based on new quantitative bounds for filter stability -- i.e. the rate at which an optimal filter for the latent state forgets its initialization. Furthermore, we prove matching hardness for planning in observable POMDPs under the Exponential Time Hypothesis.

For many inference problems in statistics and econometrics, the unknown parameter is identified by a set of moment conditions. A generic method of solving moment conditions is the Generalized Method of Moments (GMM). However, classical GMM estimation is potentially very sensitive to outliers. Robustified GMM estimators have been developed in the past, but suffer from several drawbacks: computational intractability, poor dimension-dependence, and no quantitative recovery guarantees in the presence of a constant fraction of outliers. In this work, we develop the first computationally efficient GMM estimator (under intuitive assumptions) that can tolerate a constant $\epsilon$ fraction of adversarially corrupted samples, and that has an $\ell_2$ recovery guarantee of $O(\sqrt{\epsilon})$. To achieve this, we draw upon and extend a recent line of work on algorithmic robust statistics for related but simpler problems such as mean estimation, linear regression and stochastic optimization. As two examples of the generality of our algorithm, we show how our estimation algorithm and assumptions apply to instrumental variables linear and logistic regression. Moreover, we experimentally validate that our estimator outperforms classical IV regression and two-stage Huber regression on synthetic and semi-synthetic datasets with corruption.

Sparse linear regression is a fundamental problem in high-dimensional statistics, but strikingly little is known about how to efficiently solve it without restrictive conditions on the design matrix. We consider the (correlated) random design setting, where the covariates are independently drawn from a multivariate Gaussian $N(0,\Sigma)$ with $\Sigma : n \times n$, and seek estimators $\hat{w}$ minimizing $(\hat{w}-w^*)^T\Sigma(\hat{w}-w^*)$, where $w^*$ is the $k$-sparse ground truth. Information theoretically, one can achieve strong error bounds with $O(k \log n)$ samples for arbitrary $\Sigma$ and $w^*$; however, no efficient algorithms are known to match these guarantees even with $o(n)$ samples, without further assumptions on $\Sigma$ or $w^*$. As far as hardness, computational lower bounds are only known with worst-case design matrices. Random-design instances are known which are hard for the Lasso, but these instances can generally be solved by Lasso after a simple change-of-basis (i.e. preconditioning). In this work, we give upper and lower bounds clarifying the power of preconditioning in sparse linear regression. First, we show that the preconditioned Lasso can solve a large class of sparse linear regression problems nearly optimally: it succeeds whenever the dependency structure of the covariates, in the sense of the Markov property, has low treewidth -- even if $\Sigma$ is highly ill-conditioned. Second, we construct (for the first time) random-design instances which are provably hard for an optimally preconditioned Lasso. In fact, we complete our treewidth classification by proving that for any treewidth-$t$ graph, there exists a Gaussian Markov Random Field on this graph such that the preconditioned Lasso, with any choice of preconditioner, requires $\Omega(t^{1/20})$ samples to recover $O(\log n)$-sparse signals when covariates are drawn from this model.

As in standard linear regression, in truncated linear regression, we are given access to observations $(A_i, y_i)_i$ whose dependent variable equals $y_i= A_i^{\rm T} \cdot x^* + \eta_i$, where $x^*$ is some fixed unknown vector of interest and $\eta_i$ is independent noise; except we are only given an observation if its dependent variable $y_i$ lies in some "truncation set" $S \subset \mathbb{R}$. The goal is to recover $x^*$ under some favorable conditions on the $A_i$'s and the noise distribution. We prove that there exists a computationally and statistically efficient method for recovering $k$-sparse $n$-dimensional vectors $x^*$ from $m$ truncated samples, which attains an optimal $\ell_2$ reconstruction error of $O(\sqrt{(k \log n)/m})$. As a corollary, our guarantees imply a computationally efficient and information-theoretically optimal algorithm for compressed sensing with truncation, which may arise from measurement saturation effects. Our result follows from a statistical and computational analysis of the Stochastic Gradient Descent (SGD) algorithm for solving a natural adaptation of the LASSO optimization problem that accommodates truncation. This generalizes the works of both: (1) [Daskalakis et al. 2018], where no regularization is needed due to the low-dimensionality of the data, and (2) [Wainright 2009], where the objective function is simple due to the absence of truncation. In order to deal with both truncation and high-dimensionality at the same time, we develop new techniques that not only generalize the existing ones but we believe are of independent interest.