We analyze principal component regression (PCR) in a high-dimensional error-in-variables setting with fixed design. Under suitable conditions, we show that PCR consistently identifies the unique model with minimum $\ell_2$-norm. These results enable us to establish non-asymptotic out-of-sample prediction guarantees that improve upon the best known rates. In the course of our analysis, we introduce a natural linear algebraic condition between the in- and out-of-sample covariates, which allows us to avoid distributional assumptions for out-of-sample predictions. Our simulations illustrate the importance of this condition for generalization, even under covariate shifts. Accordingly, we construct a hypothesis test to check when this condition holds in practice. As a byproduct, our results also lead to novel results for the synthetic controls literature, a leading approach for policy evaluation. To the best of our knowledge, our prediction guarantees for the fixed design setting have been elusive in both the high-dimensional error-in-variables and synthetic controls literatures.

Online experimentation with interference is a common challenge in modern applications such as e-commerce and adaptive clinical trials in medicine. For example, in online marketplaces, the revenue of a good depends on discounts applied to competing goods. Statistical inference with interference is widely studied in the offline setting, but far less is known about how to adaptively assign treatments to minimize regret. We address this gap by studying a multi-armed bandit (MAB) problem where a learner (e-commerce platform) sequentially assigns one of possible $\mathcal{A}$ actions (discounts) to $N$ units (goods) over $T$ rounds to minimize regret (maximize revenue). Unlike traditional MAB problems, the reward of each unit depends on the treatments assigned to other units, i.e., there is *interference* across the underlying network of units. With $\mathcal{A}$ actions and $N$ units, minimizing regret is combinatorially difficult since the action space grows as $\mathcal{A}^N$. To overcome this issue, we study a *sparse network interference* model, where the reward of a unit is only affected by the treatments assigned to $s$ neighboring units. We use tools from discrete Fourier analysis to develop a sparse linear representation of the unit-specific reward $r_n: [\mathcal{A}]^N \rightarrow \mathbb{R} $, and propose simple, linear regression-based algorithms to minimize regret. Importantly, our algorithms achieve provably low regret both when the learner observes the interference neighborhood for all units and when it is unknown. This significantly generalizes other works on this topic which impose strict conditions on the strength of interference on a *known* network, and also compare regret to a markedly weaker optimal action. Empirically, we corroborate our theoretical findings via numerical simulations.

Matrix completion is the study of recovering an underlying matrix from a sparse subset of noisy observations. Traditionally, it is assumed that the entries of the matrix are “missing completely at random” (MCAR), i.e., each entry is revealed at random, independent of everything else, with uniform probability. This is likely unrealistic due to the presence of “latent confounders”, i.e., unobserved factors that determine both the entries of the underlying matrix and the missingness pattern in the observed matrix. For example, in the context of movie recommender systems—a canonical application for matrix completion—a user who vehemently dislikes horror films is unlikely to ever watch horror films. In general, these confounders yield “missing not at random” (MNAR) data, which can severely impact any inference procedure that does not correct for this bias. We develop a formal causal model for matrix completion through the language of potential outcomes, and provide novel identification arguments for a variety of causal estimands of interest. We design a procedure, which we call “synthetic nearest neighbors” (SNN), to estimate these causal estimands. We prove finite-sample consistency and asymptotic normality of our estimator. Our analysis also leads to new theoretical results for the matrix completion literature. In particular, we establish entry-wise, i.e., max-norm, finite-sample consistency and asymptotic normality results for matrix completion with MNAR data. As a special case, this also provides entry-wise bounds for matrix completion with MCAR data. Across simulated and real data, we demonstrate the efficacy of our proposed estimator.

We consider a setting where there are $N$ heterogeneous units and $p$ interventions. Our goal is to learn unit-specific potential outcomes for any combination of these $p$ interventions, i.e., $N \times 2^p$ causal parameters. Choosing a combination of interventions is a problem that naturally arises in a variety of applications such as factorial design experiments and recommendation engines (e.g., showing a set of movies that maximizes engagement for a given user). Running $N \times 2^p$ experiments to estimate the various parameters is likely expensive and/or infeasible as $N$ and $p$ grow. Further, with observational data there is likely confounding, i.e., whether or not a unit is seen under a combination is correlated with its potential outcome under that combination. We study this problem under a novel model that imposes latent structure across both units and combinations of interventions. Specifically, we assume latent similarity in potential outcomes across units (i.e., the matrix of potential outcomes is approximately rank $r$) and regularity in how combinations of interventions interact (i.e., the coefficients in the Fourier expansion of the potential outcomes is approximately $s$ sparse). We establish identification for all $N \times 2^p$ parameters despite unobserved confounding. We propose an estimation procedure, Synthetic Combinations, and establish finite-sample consistency under precise conditions on the observation pattern. We show that Synthetic Combinations is able to consistently estimate unit-specific potential outcomes given a total of $\text{poly}(r) \times \left( N + s^2p\right)$ observations. In comparison, previous methods that do not exploit structure across both units and combinations have poorer sample complexity scaling as $\min(N \times s^2p, \ \ r \times (N + 2^p))$.

Principal component regression (PCR) is a popular technique for fixed-design error-in-variables regression, a generalization of the linear regression setting in which the observed covariates are corrupted with random noise. We provide the first time-uniform finite sample guarantees for online (regularized) PCR whenever data is collected adaptively. Since the proof techniques for PCR in the fixed design setting do not readily extend to the online setting, our results rely on adapting tools from modern martingale concentration to the error-in-variables setting. As an application of our bounds, we provide a framework for counterfactual estimation of unit-specific treatment effects in panel data settings when interventions are assigned adaptively. Our framework may be thought of as a generalization of the synthetic interventions framework where data is collected via an adaptive intervention assignment policy.

We consider offline reinforcement learning (RL) with heterogeneous agents under severe data scarcity, i.e., we only observe a single historical trajectory for every agent under an unknown, potentially sub-optimal policy. We find that the performance of state-of-the-art offline and model-based RL methods degrade significantly given such limited data availability, even for commonly perceived "solved" benchmark settings such as "MountainCar" and "CartPole". To address this challenge, we propose PerSim, a model-based offline RL approach which first learns a personalized simulator for each agent by collectively using the historical trajectories across all agents, prior to learning a policy. We do so by positing that the transition dynamics across agents can be represented as a latent function of latent factors associated with agents, states, and actions; subsequently, we theoretically establish that this function is well-approximated by a "low-rank" decomposition of separable agent, state, and action latent functions. This representation suggests a simple, regularized neural network architecture to effectively learn the transition dynamics per agent, even with scarce, offline data. We perform extensive experiments across several benchmark environments and RL methods. The consistent improvement of our approach, measured in terms of both state dynamics prediction and eventual reward, confirms the efficacy of our framework in leveraging limited historical data to simultaneously learn personalized policies across agents.

Consider the setting of Linear Regression where the observed response variables, in expectation, are linear functions of the p-dimensional covariates. Then to achieve vanishing prediction error, the number of required samples scales faster than pσ2, where σ2 is a bound on the noise variance. In a high-dimensional setting where p is large but the covariates admit a low-dimensional representation (say r ≪ p), then Principal Component Regression (PCR), cf. [36], is an effective approach; here, the response variables are regressed with respect to the principal components of the covariates. The resulting number of required samples to achieve vanishing prediction error now scales faster than rσ2(≪ pσ2). Despite the tremendous utility of PCR, its ability to handle settings with noisy, missing, and mixed (discrete and continuous) valued covariates is not understood and remains an important open challenge, cf. [24]. As the main contribution of this work, we address this challenge by rigorously establishing that PCR is robust to noisy, sparse, and possibly mixed valued covariates. Specifically, under PCR, vanishing prediction error is achieved with the number of samples scaling as r max(σ2, ρ−4 log5(p)), where ρ denotes the fraction of observed (noisy) covariates. We establish generalization error bounds on the performance of PCR, which provides a systematic approach in selecting the correct number of components r in a data-driven manner. The key to our result is a simple, but powerful equivalence between (i) PCR and (ii) Linear Regression with covariate pre-processing via Hard Singular Value Thresholding (HSVT). From a technical standpoint, this work advances the state-of-the-art analysis for HSVT by establishing stronger guarantees with respect to the ∥·∥2,∞-error for the estimated matrix rather than the Frobenius norm/mean-squared error (MSE) as is commonly done in the matrix estimation / completion literature.

Making use of predictions is a crucial, but under-explored, area of online algorithms. This paper studies a class of online optimization problems where we have external noisy predictions available. We propose a stochastic prediction error model that generalizes prior models in the learning and stochastic control communities, incorporates correlation among prediction errors, and captures the fact that predictions improve as time passes. We prove that achieving sublinear regret and constant competitive ratio for online algorithms requires the use of an unbounded prediction window in adversarial settings, but that under more realistic stochastic prediction error models it is possible to use Averaging Fixed Horizon Control (AFHC) to simultaneously achieve sublinear regret and constant competitive ratio in expectation using only a constant-sized prediction window. Furthermore, we show that the performance of AFHC is tightly concentrated around its mean.