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The framework of decision-making, modeled as a Markov Decision Process (MDP), typically assumes a single objective. However, most practical scenarios involve considering tradeoffs between multiple objectives. With that as the motivation, we consider the task of finding the Pareto front of achievable tradeoffs in the context of Linear Quadratic Regulator (LQR), a canonical example of a continuous, infinite horizon MDP. As our first contribution, we establish that the Pareto front for LQR is characterized by linear scalarization, wherein a linear combination of the objectives creates a single objective, and by varying the weight of the linear combination one achieves different possible tradeoffs. That is, each tradeoff point on the Pareto front of multi-objective LQR turns out to be a single objective LQR where the objective is a convex combination of the multiple objectives. Intellectually, our work provides an important example of linear scalarization being sufficient for a non-convex multi-objective problem. As our second contribution, we establish the smoothness of the Pareto front, showing that the optimal control to an $\epsilon$-perturbation to a scalarization parameter yields an $O(\epsilon)$-approximation to its objective performance. Together these results highlight a simple algorithm to approximate the continuous Pareto front by optimizing over a grid of scalarization parameters. Unlike other scalarization methods, each individual optimization problem retains the structure of a single objective LQR problem, making them computationally feasible. Lastly, we extend the results to consider certainty equivalence, where the unknown dynamics are replaced with estimates., Comment: 34 pages, 2 figures

The purpose of this article is to infer patient level outcomes from population level randomized control trials (RCTs). In this pursuit, we utilize the recently proposed synthetic nearest neighbors (SNN) estimator. At its core, SNN leverages information across patients to impute missing data associated with each patient of interest. We focus on two types of missing data: (i) unrecorded outcomes from discontinuing the assigned treatments and (ii) unobserved outcomes associated with unassigned treatments. Data imputation in the former powers and de-biases RCTs, while data imputation in the latter simulates "synthetic RCTs" to predict the outcomes for each patient under every treatment. The SNN estimator is interpretable, transparent, and causally justified under a broad class of missing data scenarios. Relative to several standard methods, we empirically find that SNN performs well for the above two applications using Phase 3 clinical trial data on patients with Alzheimer's Disease. Our findings directly suggest that SNN can tackle a current pain point within the clinical trial workflow on patient dropouts and serve as a new tool towards the development of precision medicine. Building on our insights, we discuss how SNN can further generalize to real-world applications.

We are interested in developing a data-driven method to evaluate race-induced biases in law enforcement systems. While the recent works have addressed this question in the context of police-civilian interactions using police stop data, they have two key limitations. First, bias can only be properly quantified if true criminality is accounted for in addition to race, but it is absent in prior works. Second, law enforcement systems are multi-stage and hence it is important to isolate the true source of bias within the "causal chain of interactions" rather than simply focusing on the end outcome; this can help guide reforms. In this work, we address these challenges by presenting a multi-stage causal framework incorporating criminality. We provide a theoretical characterization and an associated data-driven method to evaluate (a) the presence of any form of racial bias, and (b) if so, the primary source of such a bias in terms of race and criminality. Our framework identifies three canonical scenarios with distinct characteristics: in settings like (1) airport security, the primary source of observed bias against a race is likely to be bias in law enforcement against innocents of that race; (2) AI-empowered policing, the primary source of observed bias against a race is likely to be bias in law enforcement against criminals of that race; and (3) police-civilian interaction, the primary source of observed bias against a race could be bias in law enforcement against that race or bias from the general public in reporting against the other race. Through an extensive empirical study using police-civilian interaction data and 911 call data, we find an instance of such a counter-intuitive phenomenon: in New Orleans, the observed bias is against the majority race and the likely reason for it is the over-reporting (via 911 calls) of incidents involving the minority race by the general public.

We introduce a novel framework for incorporating human expertise into algorithmic predictions. Our approach focuses on the use of human judgment to distinguish inputs which `look the same' to any feasible predictive algorithm. We argue that this framing clarifies the problem of human/AI collaboration in prediction tasks, as experts often have access to information -- particularly subjective information -- which is not encoded in the algorithm's training data. We use this insight to develop a set of principled algorithms for selectively incorporating human feedback only when it improves the performance of any feasible predictor. We find empirically that although algorithms often outperform their human counterparts on average, human judgment can significantly improve algorithmic predictions on specific instances (which can be identified ex-ante). In an X-ray classification task, we find that this subset constitutes nearly 30% of the patient population. Our approach provides a natural way of uncovering this heterogeneity and thus enabling effective human-AI collaboration., Comment: 35 pages, 13 figures

The study of ground reaction forces (GRF) is used to characterize the mechanical loading experienced by individuals in movements such as running, which is clinically applicable to identify athletes at risk for stress-related injuries. Our aim in this paper is to determine if data collected with inertial measurement units (IMUs), that can be worn by athletes during outdoor runs, can be used to predict GRF with sufficient accuracy to allow the analysis of its derived biomechanical variables (e.g., contact time and loading rate). In this paper, we consider lightweight approaches in contrast to state-of-the-art prediction using LSTM neural networks. Specifically, we compare use of LSTMs to k-Nearest Neighbors (KNN) regression as well as propose a novel solution, SVD Embedding Regression (SER), using linear regression between singular value decomposition embeddings of IMUs data (input) and GRF data (output). We evaluate the accuracy of these techniques when using training data collected from different athletes, from the same athlete, or both, and we explore the use of acceleration and angular velocity data from sensors at different locations (sacrum and shanks). Our results illustrate that simple machine learning methods such as SER and KNN can be similarly accurate or more accurate than LSTM neural networks, with much faster training times and hyperparameter optimization; in particular, SER and KNN are more accurate when personal training data are available, and KNN comes with benefit of providing provenance of prediction. Notably, the use of personal data reduces prediction errors of all methods for most biomechanical variables.

We consider the classical problem of learning, with arbitrary accuracy, the natural parameters of a $k$-parameter truncated \textit{minimal} exponential family from i.i.d. samples in a computationally and statistically efficient manner. We focus on the setting where the support as well as the natural parameters are appropriately bounded. While the traditional maximum likelihood estimator for this class of exponential family is consistent, asymptotically normal, and asymptotically efficient, evaluating it is computationally hard. In this work, we propose a novel loss function and a computationally efficient estimator that is consistent as well as asymptotically normal under mild conditions. We show that, at the population level, our method can be viewed as the maximum likelihood estimation of a re-parameterized distribution belonging to the same class of exponential family. Further, we show that our estimator can be interpreted as a solution to minimizing a particular Bregman score as well as an instance of minimizing the \textit{surrogate} likelihood. We also provide finite sample guarantees to achieve an error (in $\ell_2$-norm) of $\alpha$ in the parameter estimation with sample complexity $O({\sf poly}(k)/\alpha^2)$. Our method achives the order-optimal sample complexity of $O({\sf log}(k)/\alpha^2)$ when tailored for node-wise-sparse Markov random fields. Finally, we demonstrate the performance of our estimator via numerical experiments., Comment: An earlier version of this work arXiv:2110.15397 was presented at the Neural Information Processing Systems Conference in December 2021 titled "A Computationally Efficient Method for Learning Exponential Family Distributions"

We consider a variant of matrix completion where entries are revealed in a biased manner, adopting a model akin to that introduced by Ma and Chen. Instead of treating this observation bias as a disadvantage, as is typically the case, the goal is to exploit the shared information between the bias and the outcome of interest to improve predictions. Towards this, we consider a natural model where the observation pattern and outcome of interest are driven by the same set of underlying latent or unobserved factors. This leads to a two stage matrix completion algorithm: first, recover (distances between) the latent factors by utilizing matrix completion for the fully observed noisy binary matrix corresponding to the observation pattern; second, utilize the recovered latent factors as features and sparsely observed noisy outcomes as labels to perform non-parametric supervised learning. The finite-sample error rates analysis suggests that, ignoring logarithmic factors, this approach is competitive with the corresponding supervised learning parametric rates. This implies the two-stage method has performance that is comparable to having access to the unobserved latent factors through exploiting the shared information between the bias and outcomes. Through empirical evaluation using a real-world dataset, we find that with this two-stage algorithm, the estimates have 30x smaller mean squared error compared to traditional matrix completion methods, suggesting the utility of the model and the method proposed in this work.

High-stakes prediction tasks (e.g., patient diagnosis) are often handled by trained human experts. A common source of concern about automation in these settings is that experts may exercise intuition that is difficult to model and/or have access to information (e.g., conversations with a patient) that is simply unavailable to a would-be algorithm. This raises a natural question whether human experts add value which could not be captured by an algorithmic predictor. We develop a statistical framework under which we can pose this question as a natural hypothesis test. Indeed, as our framework highlights, detecting human expertise is more subtle than simply comparing the accuracy of expert predictions to those made by a particular learning algorithm. Instead, we propose a simple procedure which tests whether expert predictions are statistically independent from the outcomes of interest after conditioning on the available inputs (`features'). A rejection of our test thus suggests that human experts may add value to any algorithm trained on the available data, and has direct implications for whether human-AI `complementarity' is achievable in a given prediction task. We highlight the utility of our procedure using admissions data collected from the emergency department of a large academic hospital system, where we show that physicians' admit/discharge decisions for patients with acute gastrointestinal bleeding (AGIB) appear to be incorporating information that is not available to a standard algorithmic screening tool. This is despite the fact that the screening tool is arguably more accurate than physicians' discretionary decisions, highlighting that -- even absent normative concerns about accountability or interpretability -- accuracy is insufficient to justify algorithmic automation., Comment: 30 pages, 10 figures. To appear in the proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS 2023)

The well-established practice of time series analysis involves estimating deterministic, non-stationary trend and seasonality components followed by learning the residual stochastic, stationary components. Recently, it has been shown that one can learn the deterministic non-stationary components accurately using multivariate Singular Spectrum Analysis (mSSA) in the absence of a correlated stationary component; meanwhile, in the absence of deterministic non-stationary components, the Autoregressive (AR) stationary component can also be learnt readily, e.g. via Ordinary Least Squares (OLS). However, a theoretical underpinning of multi-stage learning algorithms involving both deterministic and stationary components has been absent in the literature despite its pervasiveness. We resolve this open question by establishing desirable theoretical guarantees for a natural two-stage algorithm, where mSSA is first applied to estimate the non-stationary components despite the presence of a correlated stationary AR component, which is subsequently learned from the residual time series. We provide a finite-sample forecasting consistency bound for the proposed algorithm, SAMoSSA, which is data-driven and thus requires minimal parameter tuning. To establish theoretical guarantees, we overcome three hurdles: (i) we characterize the spectra of Page matrices of stable AR processes, thus extending the analysis of mSSA; (ii) we extend the analysis of AR process identification in the presence of arbitrary bounded perturbations; (iii) we characterize the out-of-sample or forecasting error, as opposed to solely considering model identification. Through representative empirical studies, we validate the superior performance of SAMoSSA compared to existing baselines. Notably, SAMoSSA's ability to account for AR noise structure yields improvements ranging from 5% to 37% across various benchmark datasets.

People form judgments and make decisions based on the information that they observe. A growing portion of that information is not only provided, but carefully curated by social media platforms. Although lawmakers largely agree that platforms should not operate without any oversight, there is little consensus on how to regulate social media. There is consensus, however, that creating a strict, global standard of "acceptable" content is untenable (e.g., in the US, it is incompatible with Section 230 of the Communications Decency Act and the First Amendment). In this work, we propose that algorithmic filtering should be regulated with respect to a flexible, user-driven baseline. We provide a concrete framework for regulating and auditing a social media platform according to such a baseline. In particular, we introduce the notion of a baseline feed: the content that a user would see without filtering (e.g., on Twitter, this could be the chronological timeline). We require that the feeds a platform filters contain "similar" informational content as their respective baseline feeds, and we design a principled way to measure similarity. This approach is motivated by related suggestions that regulations should increase user agency. We present an auditing procedure that checks whether a platform honors this requirement. Notably, the audit needs only black-box access to a platform's filtering algorithm, and it does not access or infer private user information. We provide theoretical guarantees on the strength of the audit. We further show that requiring closeness between filtered and baseline feeds does not impose a large performance cost, nor does it create echo chambers., Comment: 21 pages, 4 figures

We study counterfactual identifiability in causal models with bijective generation mechanisms (BGM), a class that generalizes several widely-used causal models in the literature. We establish their counterfactual identifiability for three common causal structures with unobserved confounding, and propose a practical learning method that casts learning a BGM as structured generative modeling. Learned BGMs enable efficient counterfactual estimation and can be obtained using a variety of deep conditional generative models. We evaluate our techniques in a visual task and demonstrate its application in a real-world video streaming simulation task.

In recent years, multiple notions of algorithmic fairness have arisen. One such notion is individual fairness (IF), which requires that individuals who are similar receive similar treatment. In parallel, matrix estimation (ME) has emerged as a natural paradigm for handling noisy data with missing values. In this work, we connect the two concepts. We show that pre-processing data using ME can improve an algorithm's IF without sacrificing performance. Specifically, we show that using a popular ME method known as singular value thresholding (SVT) to pre-process the data provides a strong IF guarantee under appropriate conditions. We then show that, under analogous conditions, SVT pre-processing also yields estimates that are consistent and approximately minimax optimal. As such, the ME pre-processing step does not, under the stated conditions, increase the prediction error of the base algorithm, i.e., does not impose a fairness-performance trade-off. We verify these results on synthetic and real data., Comment: 23 pages, 3 figures, ICML 2023

We introduce and analyze an improved variant of nearest neighbors (NN) for estimation with missing data in latent factor models. We consider a matrix completion problem with missing data, where the $(i, t)$-th entry, when observed, is given by its mean $f(u_i, v_t)$ plus mean-zero noise for an unknown function $f$ and latent factors $u_i$ and $v_t$. Prior NN strategies, like unit-unit NN, for estimating the mean $f(u_i, v_t)$ relies on existence of other rows $j$ with $u_j \approx u_i$. Similarly, time-time NN strategy relies on existence of columns $t'$ with $v_{t'} \approx v_t$. These strategies provide poor performance respectively when similar rows or similar columns are not available. Our estimate is doubly robust to this deficit in two ways: (1) As long as there exist either good row or good column neighbors, our estimate provides a consistent estimate. (2) Furthermore, if both good row and good column neighbors exist, it provides a (near-)quadratic improvement in the non-asymptotic error and admits a significantly narrower asymptotic confidence interval when compared to both unit-unit or time-time NN.

Given an observational study with $n$ independent but heterogeneous units, our goal is to learn the counterfactual distribution for each unit using only one $p$-dimensional sample per unit containing covariates, interventions, and outcomes. Specifically, we allow for unobserved confounding that introduces statistical biases between interventions and outcomes as well as exacerbates the heterogeneity across units. Modeling the conditional distribution of the outcomes as an exponential family, we reduce learning the unit-level counterfactual distributions to learning $n$ exponential family distributions with heterogeneous parameters and only one sample per distribution. We introduce a convex objective that pools all $n$ samples to jointly learn all $n$ parameter vectors, and provide a unit-wise mean squared error bound that scales linearly with the metric entropy of the parameter space. For example, when the parameters are $s$-sparse linear combination of $k$ known vectors, the error is $O(s\log k/p)$. En route, we derive sufficient conditions for compactly supported distributions to satisfy the logarithmic Sobolev inequality. As an application of the framework, our results enable consistent imputation of sparsely missing covariates.

We propose a generalization of the synthetic controls and synthetic interventions methodology to incorporate network interference. We consider the estimation of unit-specific potential outcomes from panel data in the presence of spillover across units and unobserved confounding. Key to our approach is a novel latent factor model that takes into account network interference and generalizes the factor models typically used in panel data settings. We propose an estimator, Network Synthetic Interventions (NSI), and show that it consistently estimates the mean outcomes for a unit under an arbitrary set of counterfactual treatments for the network. We further establish that the estimator is asymptotically normal. We furnish two validity tests for whether the NSI estimator reliably generalizes to produce accurate counterfactual estimates. We provide a novel graph-based experiment design that guarantees the NSI estimator produces accurate counterfactual estimates, and also analyze the sample complexity of the proposed design. We conclude with simulations that corroborate our theoretical findings., Comment: 49 pages, 6 figures

Several recent empirical studies demonstrate that important machine learning tasks, e.g., training deep neural networks, exhibit low-rank structure, where the loss function varies significantly in only a few directions of the input space. In this paper, we leverage such low-rank structure to reduce the high computational cost of canonical gradient-based methods such as gradient descent (GD). Our proposed \emph{Low-Rank Gradient Descent} (LRGD) algorithm finds an $\epsilon$-approximate stationary point of a $p$-dimensional function by first identifying $r \leq p$ significant directions, and then estimating the true $p$-dimensional gradient at every iteration by computing directional derivatives only along those $r$ directions. We establish that the "directional oracle complexities" of LRGD for strongly convex and non-convex objective functions are $\mathcal{O}(r \log(1/\epsilon) + rp)$ and $\mathcal{O}(r/\epsilon^2 + rp)$, respectively. When $r \ll p$, these complexities are smaller than the known complexities of $\mathcal{O}(p \log(1/\epsilon))$ and $\mathcal{O}(p/\epsilon^2)$ of {\gd} in the strongly convex and non-convex settings, respectively. Thus, LRGD significantly reduces the computational cost of gradient-based methods for sufficiently low-rank functions. In the course of our analysis, we also formally define and characterize the classes of exact and approximately low-rank functions., Comment: 26 pages, 2 figures

Successful predictive modeling of epidemics requires an understanding of the implicit feedback control strategies which are implemented by populations to modulate the spread of contagion. While this task of capturing endogenous behavior can be achieved through intricate modeling assumptions, we find that a population's reaction to case counts can be described through a second order affine dynamical system with linear control which fits well to the data across different regions and times throughout the COVID-19 pandemic. The model fits the data well both in and out of sample across the 50 states of the United States, with comparable $R^2$ scores to state of the art ensemble predictions. In contrast to recent models of epidemics, rather than assuming that individuals directly control the contact rate which governs the spread of disease, we assume that individuals control the rate at which they vary their number of interactions, i.e. they control the derivative of the contact rate. We propose an implicit feedback law for this control input and verify that it correlates with policies taken throughout the pandemic. A key takeaway of the dynamical model is that the "stable" point of case counts is non-zero, i.e. COVID-19 will not be eradicated under the current collection of policies and strategies, and additional policies are needed to fully eradicate it quickly. Hence, we suggest alternative implicit policies which focus on making interventions (such as vaccinations and mobility restrictions) a function of cumulative case counts, for which our results suggest a better possibility of eradicating COVID-19.

We consider after-study statistical inference for sequentially designed experiments wherein multiple units are assigned treatments for multiple time points using treatment policies that adapt over time. Our goal is to provide inference guarantees for the counterfactual mean at the smallest possible scale -- mean outcome under different treatments for each unit and each time -- with minimal assumptions on the adaptive treatment policy. Without any structural assumptions on the counterfactual means, this challenging task is infeasible due to more unknowns than observed data points. To make progress, we introduce a latent factor model over the counterfactual means that serves as a non-parametric generalization of the non-linear mixed effects model and the bilinear latent factor model considered in prior works. For estimation, we use a non-parametric method, namely a variant of nearest neighbors, and establish a non-asymptotic high probability error bound for the counterfactual mean for each unit and each time. Under regularity conditions, this bound leads to asymptotically valid confidence intervals for the counterfactual mean as the number of units and time points grows to $\infty$ together at suitable rates. We illustrate our theory via several simulations and a case study involving data from a mobile health clinical trial HeartSteps.

The COVID-19 pandemic has emphasized the need for a robust understanding of epidemic models. Current models of epidemics are classified as either mechanistic or non-mechanistic: mechanistic models make explicit assumptions on the dynamics of disease, whereas non-mechanistic models make assumptions on the form of observed time series. Here, we introduce a simple mixture-based model which bridges the two approaches while retaining benefits of both. The model represents time series of cases and fatalities as a mixture of Gaussian curves, providing a flexible function class to learn from data compared to traditional mechanistic models. Although the model is non-mechanistic, we show that it arises as the natural outcome of a stochastic process based on a networked SIR framework. This allows learned parameters to take on a more meaningful interpretation compared to similar non-mechanistic models, and we validate the interpretations using auxiliary mobility data collected during the COVID-19 pandemic. We provide a simple learning algorithm to identify model parameters and establish theoretical results which show the model can be efficiently learned from data. Empirically, we find the model to have low prediction error. The model is available live at covidpredictions.mit.edu. Ultimately, this allows us to systematically understand the impacts of interventions on COVID-19, which is critical in developing data-driven solutions to controlling epidemics.

In this work, we study empirical risk minimization (ERM) within a federated learning framework, where a central server minimizes an ERM objective function using training data that is stored across $m$ clients. In this setting, the Federated Averaging (FedAve) algorithm is the staple for determining $\epsilon$-approximate solutions to the ERM problem. Similar to standard optimization algorithms, the convergence analysis of FedAve only relies on smoothness of the loss function in the optimization parameter. However, loss functions are often very smooth in the training data too. To exploit this additional smoothness, we propose the Federated Low Rank Gradient Descent (FedLRGD) algorithm. Since smoothness in data induces an approximate low rank structure on the loss function, our method first performs a few rounds of communication between the server and clients to learn weights that the server can use to approximate clients' gradients. Then, our method solves the ERM problem at the server using inexact gradient descent. To show that FedLRGD can have superior performance to FedAve, we present a notion of federated oracle complexity as a counterpart to canonical oracle complexity. Under some assumptions on the loss function, e.g., strong convexity in parameter, $\eta$-H\"older smoothness in data, etc., we prove that the federated oracle complexity of FedLRGD scales like $\phi m(p/\epsilon)^{\Theta(d/\eta)}$ and that of FedAve scales like $\phi m(p/\epsilon)^{3/4}$ (neglecting sub-dominant factors), where $\phi\gg 1$ is a "communication-to-computation ratio," $p$ is the parameter dimension, and $d$ is the data dimension. Then, we show that when $d$ is small and the loss function is sufficiently smooth in the data, FedLRGD beats FedAve in federated oracle complexity. Finally, in the course of analyzing FedLRGD, we also establish a result on low rank approximation of latent variable models., Comment: 31 pages, double column format, 2 figures

We present CausalSim, a causal framework for unbiased trace-driven simulation. Current trace-driven simulators assume that the interventions being simulated (e.g., a new algorithm) would not affect the validity of the traces. However, real-world traces are often biased by the choices algorithms make during trace collection, and hence replaying traces under an intervention may lead to incorrect results. CausalSim addresses this challenge by learning a causal model of the system dynamics and latent factors capturing the underlying system conditions during trace collection. It learns these models using an initial randomized control trial (RCT) under a fixed set of algorithms, and then applies them to remove biases from trace data when simulating new algorithms. Key to CausalSim is mapping unbiased trace-driven simulation to a tensor completion problem with extremely sparse observations. By exploiting a basic distributional invariance property present in RCT data, CausalSim enables a novel tensor completion method despite the sparsity of observations. Our extensive evaluation of CausalSim on both real and synthetic datasets, including more than ten months of real data from the Puffer video streaming system shows it improves simulation accuracy, reducing errors by 53% and 61% on average compared to expert-designed and supervised learning baselines. Moreover, CausalSim provides markedly different insights about ABR algorithms compared to the biased baseline simulator, which we validate with a real deployment., Comment: NSDI'23 Best Paper Award

We revisit a model for time-varying linear regression that assumes the unknown parameters evolve according to a linear dynamical system. Counterintuitively, we show that when the underlying dynamics are stable the parameters of this model can be estimated from data by combining just two ordinary least squares estimates. We offer a finite sample guarantee on the estimation error of our method and discuss certain advantages it has over Expectation-Maximization (EM), which is the main approach proposed by prior work., Comment: 22 pages

We consider the question of learning the natural parameters of a $k$ parameter minimal exponential family from i.i.d. samples in a computationally and statistically efficient manner. We focus on the setting where the support as well as the natural parameters are appropriately bounded. While the traditional maximum likelihood estimator for this class of exponential family is consistent, asymptotically normal, and asymptotically efficient, evaluating it is computationally hard. In this work, we propose a computationally efficient estimator that is consistent as well as asymptotically normal under mild conditions. We provide finite sample guarantees to achieve an ($\ell_2$) error of $\alpha$ in the parameter estimation with sample complexity $O(\mathrm{poly}(k/\alpha))$ and computational complexity ${O}(\mathrm{poly}(k/\alpha))$. To establish these results, we show that, at the population level, our method can be viewed as the maximum likelihood estimation of a re-parameterized distribution belonging to the same class of exponential family.

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.

Variational approximation, such as mean-field (MF) and tree-reweighted (TRW), provide a computationally efficient approximation of the log-partition function for a generic graphical model. TRW provably provides an upper bound, but the approximation ratio is generally not quantified. As the primary contribution of this work, we provide an approach to quantify the approximation ratio through the property of the underlying graph structure. Specifically, we argue that (a variant of) TRW produces an estimate that is within factor $\frac{1}{\sqrt{\kappa(G)}}$ of the true log-partition function for any discrete pairwise graphical model over graph $G$, where $\kappa(G) \in (0,1]$ captures how far $G$ is from tree structure with $\kappa(G) = 1$ for trees and $2/N$ for the complete graph over $N$ vertices. As a consequence, the approximation ratio is $1$ for trees, $\sqrt{(d+1)/2}$ for any graph with maximum average degree $d$, and $\stackrel{\beta\to\infty}{\approx} 1+1/(2\beta)$ for graphs with girth (shortest cycle) at least $\beta \log N$. In general, $\kappa(G)$ is the solution of a max-min problem associated with $G$ that can be evaluated in polynomial time for any graph. Using samples from the uniform distribution over the spanning trees of G, we provide a near linear-time variant that achieves an approximation ratio equal to the inverse of square-root of minimal (across edges) effective resistance of the graph. We connect our results to the graph partition-based approximation method and thus provide a unified perspective. Keywords: variational inference, log-partition function, spanning tree polytope, minimum effective resistance, min-max spanning tree, local inference

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.

Making an informed decision -- for example, when choosing a career or housing -- requires knowledge about the available options. Such knowledge is generally acquired through costly trial and error, but this learning process can be disrupted by competition. In this work, we study how competition affects the long-term outcomes of individuals as they learn. We build on a line of work that models this setting as a two-sided matching market with bandit learners. A recent result in this area states that it is impossible to simultaneously guarantee two natural desiderata: stability and low optimal regret for all agents. Resource-allocating platforms can point to this result as a justification for assigning good long-term outcomes to some agents and poor ones to others. We show that this impossibility need not hold true. In particular, by modeling two additional components of competition -- namely, costs and transfers -- we prove that it is possible to simultaneously guarantee four desiderata: stability, low optimal regret, fairness in the distribution of regret, and high social welfare., Comment: 31 pages, 1 figure, 25th International Conference on Artificial Intelligence and Statistics (AISTATS) 2022

Consider the problem of determining the effect of a compound on a specific cell type. To answer this question, researchers traditionally need to run an experiment applying the drug of interest to that cell type. This approach is not scalable: given a large number of different actions (compounds) and a large number of different contexts (cell types), it is infeasible to run an experiment for every action-context pair. In such cases, one would ideally like to predict the outcome for every pair while only having to perform experiments on a small subset of pairs. This task, which we label "causal imputation", is a generalization of the causal transportability problem. To address this challenge, we extend the recently introduced synthetic interventions (SI) estimator to handle more general data sparsity patterns. We prove that, under a latent factor model, our estimator provides valid estimates for the causal imputation task. We motivate this model by establishing a connection to the linear structural causal model literature. Finally, we consider the prominent CMAP dataset in predicting the effects of compounds on gene expression across cell types. We find that our estimator outperforms standard baselines, thus confirming its utility in biological applications.

In this paper, we consider the problem of empirical risk minimization (ERM) of smooth, strongly convex loss functions using iterative gradient-based methods. A major goal of this literature has been to compare different algorithms, such as gradient descent (GD) or stochastic gradient descent (SGD), by analyzing their rates of convergence to $\epsilon$-approximate solutions. For example, the oracle complexity of GD is $O(n\log(\epsilon^{-1}))$, where $n$ is the number of training samples. When $n$ is large, this can be expensive in practice, and SGD is preferred due to its oracle complexity of $O(\epsilon^{-1})$. Such standard analyses only utilize the smoothness of the loss function in the parameter being optimized. In contrast, we demonstrate that when the loss function is smooth in the data, we can learn the oracle at every iteration and beat the oracle complexities of both GD and SGD in important regimes. Specifically, at every iteration, our proposed algorithm performs local polynomial regression to learn the gradient of the loss function, and then estimates the true gradient of the ERM objective function. We establish that the oracle complexity of our algorithm scales like $\tilde{O}((p \epsilon^{-1})^{d/(2\eta)})$ (neglecting sub-dominant factors), where $d$ and $p$ are the data and parameter space dimensions, respectively, and the gradient of the loss function belongs to a $\eta$-H\"{o}lder class with respect to the data. Our proof extends the analysis of local polynomial regression in non-parametric statistics to provide interpolation guarantees in multivariate settings, and also exploits tools from the inexact GD literature. Unlike GD and SGD, the complexity of our method depends on $d$ and $p$. However, when $d$ is small and the loss function exhibits modest smoothness in the data, our algorithm beats GD and SGD in oracle complexity for a very broad range of $p$ and $\epsilon$., Comment: 34 pages

We consider learning a sparse pairwise Markov Random Field (MRF) with continuous-valued variables from i.i.d samples. We adapt the algorithm of Vuffray et al. (2019) to this setting and provide finite-sample analysis revealing sample complexity scaling logarithmically with the number of variables, as in the discrete and Gaussian settings. Our approach is applicable to a large class of pairwise MRFs with continuous variables and also has desirable asymptotic properties, including consistency and normality under mild conditions. Further, we establish that the population version of the optimization criterion employed in Vuffray et al. (2019) can be interpreted as local maximum likelihood estimation (MLE). As part of our analysis, we introduce a robust variation of sparse linear regression a` la Lasso, which may be of interest in its own right.

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 conditions 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.

We introduce and analyze a variant of multivariate singular spectrum analysis (mSSA), a popular time series method to impute and forecast a multivariate time series. Under a spatio-temporal factor model we introduce, given $N$ time series and $T$ observations per time series, we establish prediction mean-squared-error for both imputation and out-of-sample forecasting effectively scale as $1 / \sqrt{\min(N, T )T}$. This is an improvement over: (i) $1 /\sqrt{T}$ error scaling of SSA, the restriction of mSSA to a univariate time series; (ii) $1/\min(N, T)$ error scaling for matrix estimation methods which do not exploit temporal structure in the data. The spatio-temporal model we introduce includes any finite sum and products of: harmonics, polynomials, differentiable periodic functions, and Holder continuous functions. Our out-of-sample forecasting result could be of independent interest for online learning under a spatio-temporal factor model. Empirically, on benchmark datasets, our variant of mSSA performs competitively with state-of-the-art neural-network time series methods (e.g. DeepAR, LSTM) and significantly outperforms classical methods such as vector autoregression (VAR). Finally, we propose extensions of mSSA: (i) a variant to estimate time-varying variance of a time series; (ii) a tensor variant which has better sample complexity for certain regimes of $N$ and $T$.

By filtering the content that users see, social media platforms have the ability to influence users' perceptions and decisions, from their dining choices to their voting preferences. This influence has drawn scrutiny, with many calling for regulations on filtering algorithms, but designing and enforcing regulations remains challenging. In this work, we examine three questions. First, given a regulation, how would one design an audit to enforce it? Second, does the audit impose a performance cost on the platform? Third, how does the audit affect the content that the platform is incentivized to filter? In response, we propose a method such that, given a regulation, an auditor can test whether that regulation is met with only black-box access to the filtering algorithm. We then turn to the platform's perspective. The platform's goal is to maximize an objective function while meeting regulation. We find that there are conditions under which the regulation does not place a high performance cost on the platform and, notably, that content diversity can play a key role in aligning the interests of the platform and regulators., Comment: 23 pages, 3 figures

In this paper, we study the problem of learning the skill distribution of a population of agents from observations of pairwise games in a tournament. These games are played among randomly drawn agents from the population. The agents in our model can be individuals, sports teams, or Wall Street fund managers. Formally, we postulate that the likelihoods of game outcomes are governed by the Bradley-Terry-Luce (or multinomial logit) model, where the probability of an agent beating another is the ratio between its skill level and the pairwise sum of skill levels, and the skill parameters are drawn from an unknown skill density of interest. The problem is, in essence, to learn a distribution from noisy, quantized observations. We propose a simple and tractable algorithm that learns the skill density with near-optimal minimax mean squared error scaling as $n^{-1+\varepsilon}$, for any $\varepsilon>0$, when the density is smooth. Our approach brings together prior work on learning skill parameters from pairwise comparisons with kernel density estimation from non-parametric statistics. Furthermore, we prove minimax lower bounds which establish minimax optimality of the skill parameter estimation technique used in our algorithm. These bounds utilize a continuum version of Fano's method along with a covering argument. We apply our algorithm to various soccer leagues and world cups, cricket world cups, and mutual funds. We find that the entropy of a learnt distribution provides a quantitative measure of skill, which provides rigorous explanations for popular beliefs about perceived qualities of sporting events, e.g., soccer league rankings. Finally, we apply our method to assess the skill distributions of mutual funds. Our results shed light on the abundance of low quality funds prior to the Great Recession of 2008, and the domination of the industry by more skilled funds after the financial crisis., Comment: 37 pages, 1 figure

Consider a setting with $N$ heterogeneous units (e.g., individuals, sub-populations) and $D$ interventions (e.g., socio-economic policies). Our goal is to learn the expected potential outcome associated with every intervention on every unit, totaling $N \times D$ causal parameters. Towards this, we present a causal framework, synthetic interventions (SI), to infer these $N \times D$ causal parameters while only observing each of the $N$ units under at most two interventions, independent of $D$. This can be significant as the number of interventions, i.e., level of personalization, grows. Under a novel tensor factor model across units, outcomes, and interventions, we prove an identification result for each of these $N \times D$ causal parameters, establish finite-sample consistency of our estimator along with asymptotic normality under additional conditions. Importantly, our estimator also allows for latent confounders that determine how interventions are assigned. The estimator is further furnished with data-driven tests to examine its suitability. Empirically, we validate our framework through a large-scale A/B test performed on an e-commerce platform. We believe our results could have implications for the design of data-efficient randomized experiments (e.g., randomized control trials) with heterogeneous units and multiple interventions.

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.

We consider the problem of reinforcement learning (RL) with unbounded state space motivated by the classical problem of scheduling in a queueing network. Traditional policies as well as error metric that are designed for finite, bounded or compact state space, require infinite samples for providing any meaningful performance guarantee (e.g. $\ell_\infty$ error) for unbounded state space. That is, we need a new notion of performance metric. As the main contribution of this work, inspired by the literature in queuing systems and control theory, we propose stability as the notion of "goodness": the state dynamics under the policy should remain in a bounded region with high probability. As a proof of concept, we propose an RL policy using Sparse-Sampling-based Monte Carlo Oracle and argue that it satisfies the stability property as long as the system dynamics under the optimal policy respects a Lyapunov function. The assumption of existence of a Lyapunov function is not restrictive as it is equivalent to the positive recurrence or stability property of any Markov chain, i.e., if there is any policy that can stabilize the system then it must possess a Lyapunov function. And, our policy does not utilize the knowledge of the specific Lyapunov function. To make our method sample efficient, we provide an improved, sample efficient Sparse-Sampling-based Monte Carlo Oracle with Lipschitz value function that may be of interest in its own right. Furthermore, we design an adaptive version of the algorithm, based on carefully constructed statistical tests, which finds the correct tuning parameter automatically.

As we reach the apex of the COVID-19 pandemic, the most pressing question facing us is: can we even partially reopen the economy without risking a second wave? We first need to understand if shutting down the economy helped. And if it did, is it possible to achieve similar gains in the war against the pandemic while partially opening up the economy? To do so, it is critical to understand the effects of the various interventions that can be put into place and their corresponding health and economic implications. Since many interventions exist, the key challenge facing policy makers is understanding the potential trade-offs between them, and choosing the particular set of interventions that works best for their circumstance. In this memo, we provide an overview of Synthetic Interventions (a natural generalization of Synthetic Control), a data-driven and statistically principled method to perform what-if scenario planning, i.e., for policy makers to understand the trade-offs between different interventions before having to actually enact them. In essence, the method leverages information from different interventions that have already been enacted across the world and fits it to a policy maker's setting of interest, e.g., to estimate the effect of mobility-restricting interventions on the U.S., we use daily death data from countries that enforced severe mobility restrictions to create a "synthetic low mobility U.S." and predict the counterfactual trajectory of the U.S. if it had indeed applied a similar intervention. Using Synthetic Interventions, we find that lifting severe mobility restrictions and only retaining moderate mobility restrictions (at retail and transit locations), seems to effectively flatten the curve. We hope this provides guidance on weighing the trade-offs between the safety of the population, strain on the healthcare system, and impact on the economy.

We consider the problem of finding Nash equilibrium for two-player turn-based zero-sum games. Inspired by the AlphaGo Zero (AGZ) algorithm, we develop a Reinforcement Learning based approach. Specifically, we propose Explore-Improve-Supervise (EIS) method that combines "exploration", "policy improvement"' and "supervised learning" to find the value function and policy associated with Nash equilibrium. We identify sufficient conditions for convergence and correctness for such an approach. For a concrete instance of EIS where random policy is used for "exploration", Monte-Carlo Tree Search is used for "policy improvement" and Nearest Neighbors is used for "supervised learning", we establish that this method finds an $\varepsilon$-approximate value function of Nash equilibrium in $\widetilde{O}(\varepsilon^{-(d+4)})$ steps when the underlying state-space of the game is continuous and $d$-dimensional. This is nearly optimal as we establish a lower bound of $\widetilde{\Omega}(\varepsilon^{-(d+2)})$ for any policy.

Network flow is a powerful mathematical framework to systematically explore the relationship between structure and function in biological, social, and technological networks. We introduce a new pipelining model of flow through networks where commodities must be transported over single paths rather than split over several paths and recombined. We show this notion of pipelined network flow is optimized using network paths that are both short and wide, and develop efficient algorithms to compute such paths for given pairs of nodes and for all-pairs. Short and wide paths are characterized for many real-world networks. To further demonstrate the utility of this network characterization, we develop novel information-theoretic lower bounds on computation speed in nervous systems due to limitations from anatomical connectivity and physical noise. For the nematode Caenorhabditis elegans, we find these bounds are predictive of biological timescales of behavior. Further, we find the particular C. elegans connectome is globally less efficient for information flow than random networks, but the hub-and-spoke architecture of functional subcircuits is optimal under constraint on number of synapses. This suggests functional subcircuits are a primary organizational principle of this small invertebrate nervous system.

Consider the task of estimating a 3-order $n \times n \times n$ tensor from noisy observations of randomly chosen entries in the sparse regime. We introduce a similarity based collaborative filtering algorithm for estimating a tensor from sparse observations and argue that it achieves sample complexity that nearly matches the conjectured computationally efficient lower bound on the sample complexity for the setting of low-rank tensors. Our algorithm uses the matrix obtained from the flattened tensor to compute similarity, and estimates the tensor entries using a nearest neighbor estimator. We prove that the algorithm recovers a finite rank tensor with maximum entry-wise error (MEE) and mean-squared-error (MSE) decaying to $0$ as long as each entry is observed independently with probability $p = \Omega(n^{-3/2 + \kappa})$ for any arbitrarily small $\kappa > 0$. More generally, we establish robustness of the estimator, showing that when arbitrary noise bounded by $\varepsilon \geq 0$ is added to each observation, the estimation error with respect to MEE and MSE degrades by $\text{poly}(\varepsilon)$. Consequently, even if the tensor may not have finite rank but can be approximated within $\varepsilon \geq 0$ by a finite rank tensor, then the estimation error converges to $\text{poly}(\varepsilon)$. Our analysis sheds insight into the conjectured sample complexity lower bound, showing that it matches the connectivity threshold of the graph used by our algorithm for estimating similarity between coordinates.

Personal data is essential in showing users targeted ads - the economic backbone of the web. Still, there are major inefficiencies in how data is transacted online: (1) users don't decide what information is released nor get paid for this privacy loss; (2) algorithmic advertisers are stuck in inefficient long-term contracts where they purchase user data without knowing the value it provides. This paper proposes a system, Zorro, which aims to rectify aforementioned two problems. As the main contribution, we provide a natural, 'absolute' definition of 'Value of Data' (VoD) - for any quantity of interest, it is the delta between an individual's value and population mean. The challenge remains how to operationalize this definition, independently of a buyer's model for VoD. We propose a model-agnostic solution, relying on matrix estimation, and use it to estimate click-through-rate (CTR), as an example. Regarding (2), Zorro empowers advertisers to measure value of user data on a query-by-query basis and based only on the increase in accuracy it provides in estimating CTR. In contrast advertisers currently engage in inefficient long-term data contracts with third party data sellers. We highlight two results on a large ad-click dataset: (i) our system has R^2=0.58, in line with best-in-class results for related problems (e.g. content recommendation). Crucially, our system is model-agnostic - we estimate CTR without accessing an advertiser's proprietary models, a required property of any such pricing system;(ii) our experiments show selling user data has incremental value ranging from 30%-69% depending on ad category. Roughly, this translates to at least USD 16 Billion loss in value for advertisers if user data is not provided. Regarding (1), in addition to allowing users to get paid for data sharing, we extend our mathematical framework to when users provide explicit intent.

When evaluating the impact of a policy on a metric of interest, it may not be possible to conduct a randomized control trial. In settings where only observational data is available, Synthetic Control (SC) methods provide a popular data-driven approach to estimate a "synthetic" control by combining measurements of "similar" units (donors). Recently, Robust SC (RSC) was proposed as a generalization of SC to overcome the challenges of missing data high levels of noise, while removing the reliance on domain knowledge for selecting donors. However, SC, RSC, and their variants, suffer from poor estimation when the pre-intervention period is too short. As the main contribution, we propose a generalization of unidimensional RSC to multi-dimensional RSC, mRSC. Our proposed mechanism incorporates multiple metrics to estimate a synthetic control, thus overcoming the challenge of poor inference from limited pre-intervention data. We show that the mRSC algorithm with $K$ metrics leads to a consistent estimator of the synthetic control for the target unit under any metric. Our finite-sample analysis suggests that the prediction error decays to zero at a rate faster than the RSC algorithm by a factor of $K$ and $\sqrt{K}$ for the training and testing periods (pre- and post-intervention), respectively. Additionally, we provide a diagnostic test that evaluates the utility of including additional metrics. Moreover, we introduce a mechanism to validate the performance of mRSC: time series prediction. That is, we propose a method to predict the future evolution of a time series based on limited data when the notion of time is relative and not absolute, i.e., we have access to a donor pool that has undergone the desired future evolution. Finally, we conduct experimentation to establish the efficacy of mRSC on synthetic data and two real-world case studies (retail and Cricket).

A major bottleneck of the current Machine Learning (ML) workflow is the time consuming, error prone engineering required to get data from a datastore or a database (DB) to the point an ML algorithm can be applied to it. Hence, we explore the feasibility of directly integrating prediction functionality on top of a data store or DB. Such a system ideally: (i) provides an intuitive prediction query interface which alleviates the unwieldy data engineering; (ii) provides state-of-the-art statistical accuracy while ensuring incremental model update, low model training time and low latency for making predictions. As the main contribution we explicitly instantiate a proof-of-concept, tspDB, which directly integrates with PostgreSQL. We rigorously test tspDB's statistical and computational performance against the state-of-the-art time series algorithms, including a Long-Short-Term-Memory (LSTM) neural network and DeepAR (industry standard deep learning library by Amazon). Statistically, on standard time series benchmarks, tspDB outperforms LSTM and DeepAR with 1.1-1.3x higher relative accuracy. Computationally, tspDB is 59-62x and 94-95x faster compared to LSTM and DeepAR in terms of median ML model training time and prediction query latency, respectively. Further, compared to PostgreSQL's bulk insert time and its SELECT query latency, tspDB is slower only by 1.3x and 2.6x respectively. That is, tspDB is a real-time prediction system in that its model training / prediction query time is similar to just inserting / reading data from a DB. As an algorithmic contribution, we introduce an incremental multivariate matrix factorization based time series method, which tspDB is built off. We show this method also allows one to produce reliable prediction intervals by accurately estimating the time-varying variance of a time series, thereby addressing an important problem in time series analysis.

Principal component regression (PCR) is a simple, but powerful and ubiquitously utilized method. Its effectiveness is well established when the covariates exhibit low-rank structure. However, its ability to handle settings with noisy, missing, and mixed-valued, i.e., discrete and continuous, covariates is not understood and remains an important open challenge. As the main contribution of this work we establish the robustness of PCR, without any change, in this respect and provide meaningful finite-sample analysis. To do so, we establish that PCR is equivalent to performing linear regression after pre-processing the covariate matrix via hard singular value thresholding (HSVT). As a result, in the context of counterfactual analysis using observational data, we show PCR is equivalent to the recently proposed robust variant of the synthetic control method, known as robust synthetic control (RSC). As an immediate consequence, we obtain finite-sample analysis of the RSC estimator that was previously absent. As an important contribution to the synthetic controls literature, we establish that an (approximate) linear synthetic control exists in the setting of a generalized factor model or latent variable model; traditionally in the literature, the existence of a synthetic control needs to be assumed to exist as an axiom. We further discuss a surprising implication of the robustness property of PCR with respect to noise, i.e., PCR can learn a good predictive model even if the covariates are tactfully transformed to preserve differential privacy. Finally, this work advances the state-of-the-art analysis for HSVT by establishing stronger guarantees with respect to the $\ell_{2, \infty}$-norm rather than the Frobenius norm as is commonly done in the matrix estimation literature, which may be of interest in its own right.

In this work, we consider the popular tree-based search strategy within the framework of reinforcement learning, the Monte Carlo Tree Search (MCTS), in the context of infinite-horizon discounted cost Markov Decision Process (MDP). While MCTS is believed to provide an approximate value function for a given state with enough simulations, the claimed proof in the seminal works is incomplete. This is due to the fact that the variant, the Upper Confidence Bound for Trees (UCT), analyzed in prior works utilizes "logarithmic" bonus term for balancing exploration and exploitation within the tree-based search, following the insights from stochastic multi-arm bandit (MAB) literature. In effect, such an approach assumes that the regret of the underlying recursively dependent non-stationary MABs concentrates around their mean exponentially in the number of steps, which is unlikely to hold as pointed out in literature, even for stationary MABs. As the key contribution of this work, we establish polynomial concentration property of regret for a class of non-stationary MAB. This in turn establishes that the MCTS with appropriate polynomial rather than logarithmic bonus term in UCB has the claimed property. Using this as a building block, we argue that MCTS, combined with nearest neighbor supervised learning, acts as a "policy improvement" operator: it iteratively improves value function approximation for all states, due to combining with supervised learning, despite evaluating at only finitely many states. In effect, we establish that to learn an $\varepsilon$ approximation of the value function with respect to $\ell_\infty$ norm, MCTS combined with nearest neighbor requires a sample size scaling as $\widetilde{O}\big(\varepsilon^{-(d+4)}\big)$, where $d$ is the dimension of the state space. This is nearly optimal due to a minimax lower bound of $\widetilde{\Omega}\big(\varepsilon^{-(d+2)}\big)$.

We consider the problem of learning a mixture of Random Utility Models (RUMs). Despite the success of RUMs in various domains and the versatility of mixture RUMs to capture the heterogeneity in preferences, there has been only limited progress in learning a mixture of RUMs from partial data such as pairwise comparisons. In contrast, there have been significant advances in terms of learning a single component RUM using pairwise comparisons. In this paper, we aim to bridge this gap between mixture learning and single component learning of RUM by developing a `reduction' procedure. We propose to utilize PCA-based spectral clustering that simultaneously `de-noises' pairwise comparison data. We prove that our algorithm manages to cluster the partial data correctly (i.e., comparisons from the same RUM component are grouped in the same cluster) with high probability even when data is generated from a possibly {\em heterogeneous} mixture of well-separated {\em generic} RUMs. Both the time and the sample complexities scale polynomially in model parameters including the number of items. Two key features in the analysis are in establishing (1) a meaningful upper bound on the sub-Gaussian norm for RUM components embedded into the vector space of pairwise marginals and (2) the robustness of PCA with missing values in the $L_{2, \infty}$ sense, which might be of interest in their own right., Comment: 28 pages, 4 figures

We consider recommendation systems that need to operate under wireless bandwidth constraints, measured as number of broadcast transmissions, and demonstrate a (tight for some instances) tradeoff between regret and bandwidth for two scenarios: the case of multi-armed bandit with context, and the case where there is a latent structure in the message space that we can exploit to reduce the learning phase.