The key assumption underlying linear Markov Decision Processes (MDPs) is that the learner has access to a known feature map $\phi(x, a)$ that maps state-action pairs to $d$-dimensional vectors, and that the rewards and transitions are linear functions in this representation. But where do these features come from? In the absence of expert domain knowledge, a tempting strategy is to use the ``kitchen sink" approach and hope that the true features are included in a much larger set of potential features. In this paper we revisit linear MDPs from the perspective of feature selection. In a $k$-sparse linear MDP, there is an unknown subset $S \subset [d]$ of size $k$ containing all the relevant features, and the goal is to learn a near-optimal policy in only poly$(k,\log d)$ interactions with the environment. Our main result is the first polynomial-time algorithm for this problem. In contrast, earlier works either made prohibitively strong assumptions that obviated the need for exploration, or required solving computationally intractable optimization problems. Along the way we introduce the notion of an emulator: a succinct approximate representation of the transitions that suffices for computing certain Bellman backups. Since linear MDPs are a non-parametric model, it is not even obvious whether polynomial-sized emulators exist. We show that they do exist and can be computed efficiently via convex programming. As a corollary of our main result, we give an algorithm for learning a near-optimal policy in block MDPs whose decoding function is a low-depth decision tree; the algorithm runs in quasi-polynomial time and takes a polynomial number of samples. This can be seen as a reinforcement learning analogue of classic results in computational learning theory. Furthermore, it gives a natural model where improving the sample complexity via representation learning is computationally feasible.

Recently Chen and Poor initiated the study of learning mixtures of linear dynamical systems. While linear dynamical systems already have wide-ranging applications in modeling time-series data, using mixture models can lead to a better fit or even a richer understanding of underlying subpopulations represented in the data. In this work we give a new approach to learning mixtures of linear dynamical systems that is based on tensor decompositions. As a result, our algorithm succeeds without strong separation conditions on the components, and can be used to compete with the Bayes optimal clustering of the trajectories. Moreover our algorithm works in the challenging partially-observed setting. Our starting point is the simple but powerful observation that the classic Ho-Kalman algorithm is a close relative of modern tensor decomposition methods for learning latent variable models. This gives us a playbook for how to extend it to work with more complicated generative models.

Score matching is an alternative to maximum likelihood (ML) for estimating a probability distribution parametrized up to a constant of proportionality. By fitting the ''score'' of the distribution, it sidesteps the need to compute this constant of proportionality (which is often intractable). While score matching and variants thereof are popular in practice, precise theoretical understanding of the benefits and tradeoffs with maximum likelihood -- both computational and statistical -- are not well understood. In this work, we give the first example of a natural exponential family of distributions such that the score matching loss is computationally efficient to optimize, and has a comparable statistical efficiency to ML, while the ML loss is intractable to optimize using a gradient-based method. The family consists of exponentials of polynomials of fixed degree, and our result can be viewed as a continuous analogue of recent developments in the discrete setting. Precisely, we show: (1) Designing a zeroth-order or first-order oracle for optimizing the maximum likelihood loss is NP-hard. (2) Maximum likelihood has a statistical efficiency polynomial in the ambient dimension and the radius of the parameters of the family. (3) Minimizing the score matching loss is both computationally and statistically efficient, with complexity polynomial in the ambient dimension.

Linear dynamical systems are the foundational statistical model upon which control theory is built. Both the celebrated Kalman filter and the linear quadratic regulator require knowledge of the system dynamics to provide analytic guarantees. Naturally, learning the dynamics of a linear dynamical system from linear measurements has been intensively studied since Rudolph Kalman's pioneering work in the 1960's. Towards these ends, we provide the first polynomial time algorithm for learning a linear dynamical system from a polynomial length trajectory up to polynomial error in the system parameters under essentially minimal assumptions: observability, controllability, and marginal stability. Our algorithm is built on a method of moments estimator to directly estimate Markov parameters from which the dynamics can be extracted. Furthermore, we provide statistical lower bounds when our observability and controllability assumptions are violated.

The composition of the training dataset has key implications for machine learning models' behavior [Fel19; CLK+19; KL17; GZ19; IPE+22], especially as the training environments often deviate from deployment conditions [RGL19; KSM+20; HBM+20]. For example, a model might struggle on specific subpopulations in the data if that subpopulation was mislabeled [NAM21; SC18; BHK+20; VCG+22], underrepresented [SKH+20; STM21], or corrupted [HD19; HBM+20]. More broadly, the training dataset might contain spurious correlations, encouraging the model to depend on prediction rules that do not generalize to deployment [XEI+20; GJM+20; DJL21]. Moreover, identifying meaningful subpopulations within data allows for dataset refinement (such as filtering or relabeling) [YQF+19; SC18], and training more fair [KGZ19; DYZ+21] or accurate [JFK+20; SHL20] models. However, dominant approaches to such identification of biases and difficult subpopulations within datasets often require human intervention, which is typically labor intensive and thus not conducive to routine usage.

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.

Tensor completion is a natural higher-order generalization of matrix completion where the goal is to recover a low-rank tensor from sparse observations of its entries. Existing algorithms are either heuristic without provable guarantees, based on solving large semidefinite programs which are impractical to run, or make strong assumptions such as requiring the factors to be nearly orthogonal. In this paper we introduce a new variant of alternating minimization, which in turn is inspired by understanding how the progress measures that guide convergence of alternating minimization in the matrix setting need to be adapted to the tensor setting. We show strong provable guarantees, including showing that our algorithm converges linearly to the true tensors even when the factors are highly correlated and can be implemented in nearly linear time. Moreover our algorithm is also highly practical and we show that we can complete third order tensors with a thousand dimensions from observing a tiny fraction of its entries. In contrast, and somewhat surprisingly, we show that the standard version of alternating minimization, without our new twist, can converge at a drastically slower rate in practice.

In this work we study the problem of robustly learning a Mallows model. We give an algorithm that can accurately estimate the central ranking even when a constant fraction of its samples are arbitrarily corrupted. Moreover our robustness guarantees are dimension-independent in the sense that our overall accuracy does not depend on the number of alternatives being ranked. Our work can be thought of as a natural infusion of perspectives from algorithmic robust statistics into one of the central inference problems in voting and information-aggregation. Specifically, our voting rule is efficiently computable and its outcome cannot be changed by much by a large group of colluding voters.