arXiv subjects

Devavrat Shah

Publications and source records attributed to Devavrat Shah.

At least 73 records · Page 4Linked to original sources

arXiv2020

Two Burning Questions on COVID-19: Did shutting down the economy help? Can we (partially) reopen the economy without risking the second wave?

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.

econ.EM↗

arXiv2020

Deconvolution with Unknown Error Distribution Interpreted as Blind Isotonic Regression

Deconvolution is a statistical inverse problem to estimate the distribution of a random variable based on its noisy observations. Despite the extensive studies on the topic, deconvolution with unknown noise distribution remains as a notoriously hard problem. We propose a matrix-based viewpoint for collective deconvolution that subsumes the setup with repeated measurements as a special case. As the main result, we describe a simple algorithm that partially utilizes matrix structure to solve deconvolution problem and provide non-asymptotic error analysis for the algorithm. We show that the proposed algorithm achieves the minimax optimal rate for deconvolution in a restricted sense. We also remark the connection between the collective deconvolution and the so-called statistical seriation as a byproduct or our matrix viewpoint. We conjecture that the link suggests that collective deconvolution, as well as deconvolution with repeated measurements, is intrinsically much easier than usual deconvolution of a single distribution.

math.ST↗

arXiv2020

Learning RUMs: Reducing Mixture to Single Component via PCA

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.

stat.ML↗

arXiv2020

On Reinforcement Learning for Turn-based Zero-sum Markov Games

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 $\widetildeΩ(\varepsilon^{-(d+2)})$ for any policy.

cs.LG↗

arXiv2020

Non-Asymptotic Analysis of Monte Carlo Tree Search

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 $\widetildeΩ\big(\varepsilon^{-(d+2)}\big)$.

stat.ML↗

arXiv2019

Short and Wide Network Paths

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.

cs.NE↗

arXiv2019

Nearest Neighbors for Matrix Estimation Interpreted as Blind Regression for Latent Variable Model

We consider the setup of nonparametric {\em blind regression} for estimating the entries of a large $m \times n$ matrix, when provided with a small, random fraction of noisy measurements. We assume that all rows $u \in [m]$ and columns $i \in [n]$ of the matrix are associated to latent features $x_{\text{row}}(u)$ and $x_{\text{col}}(i)$ respectively, and the $(u,i)$-th entry of the matrix, $A(u, i)$ is equal to $f(x_{\text{row}}(u), x_{\text{col}}(i))$ for a latent function $f$. Given noisy observations of a small, random subset of the matrix entries, our goal is to estimate the unobserved entries of the matrix as well as to "de-noise" the observed entries. As the main result of this work, we introduce a nearest-neighbor-based estimation algorithm, and establish its consistency when the underlying latent function $f$ is Lipschitz, the underlying latent space is a bounded diameter Polish space, and the random fraction of observed entries in the matrix is at least $\max \left( m^{-1 + δ}, n^{-1/2 + δ} \right)$, for any $δ> 0$. As an important byproduct, our analysis sheds light into the performance of the classical collaborative filtering algorithm for matrix completion, which has been widely utilized in practice. Experiments with the MovieLens and Netflix datasets suggest that our algorithm provides a principled improvement over basic collaborative filtering and is competitive with matrix factorization methods. Our algorithm has a natural extension to the setting of tensor completion via flattening the tensor to matrix. When applied to the setting of image in-painting, which is a $3$-order tensor, we find that our approach is competitive with respect to state-of-art tensor completion algorithms across benchmark images.

math.ST↗

arXiv2019

mRSC: Multi-dimensional Robust Synthetic Control

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).

stat.ME↗

arXiv2019

Leaders, Followers, and Community Detection

Communities in social networks or graphs are sets of well-connected, overlapping vertices. The effectiveness of a community detection algorithm is determined by accuracy in finding the ground-truth communities and ability to scale with the size of the data. In this work, we provide three contributions. First, we show that a popular measure of accuracy known as the F1 score, which is between 0 and 1, with 1 being perfect detection, has an information lower bound is 0.5. We provide a trivial algorithm that produces communities with an F1 score of 0.5 for any graph! Somewhat surprisingly, we find that popular algorithms such as modularity optimization, BigClam and CESNA have F1 scores less than 0.5 for the popular IMDB graph. To rectify this, as the second contribution we propose a generative model for community formation, the sequential community graph, which is motivated by the formation of social networks. Third, motivated by our generative model, we propose the leader-follower algorithm (LFA). We prove that it recovers all communities for sequential community graphs by establishing a structural result that sequential community graphs are chordal. For a large number of popular social networks, it recovers communities with a much higher F1 score than other popular algorithms. For the IMDB graph, it obtains an F1 score of 0.81. We also propose a modification to the LFA called the fast leader-follower algorithm (FLFA) which in addition to being highly accurate, is also fast, with a scaling that is almost linear in the network size.

stat.ML↗

arXiv2019

Reducing Crowdsourcing to Graphon Estimation, Statistically

Inferring the correct answers to binary tasks based on multiple noisy answers in an unsupervised manner has emerged as the canonical question for micro-task crowdsourcing or more generally aggregating opinions. In graphon estimation, one is interested in estimating edge intensities or probabilities between nodes using a single snapshot of a graph realization. In the recent literature, there has been exciting development within both of these topics. In the context of crowdsourcing, the key intellectual challenge is to understand whether a given task can be more accurately denoised by aggregating answers collected from other different tasks. In the context of graphon estimation, precise information limits and estimation algorithms remain of interest. In this paper, we utilize a statistical reduction from crowdsourcing to graphon estimation to advance the state-of-art for both of these challenges. We use concepts from graphon estimation to design an algorithm that achieves better performance than the {\em majority voting} scheme for a setup that goes beyond the {\em rank one} models considered in the literature. We use known explicit lower bounds for crowdsourcing to provide refined lower bounds for graphon estimation.

stat.ML↗

arXiv2019

Zorro: A Model Agnostic System to Price Consumer Data

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.

cs.SI↗

arXiv2019

Model Agnostic Time Series Analysis via Matrix Estimation

We propose an algorithm to impute and forecast a time series by transforming the observed time series into a matrix, utilizing matrix estimation to recover missing values and de-noise observed entries, and performing linear regression to make predictions. At the core of our analysis is a representation result, which states that for a large model class, the transformed time series matrix is (approximately) low-rank. In effect, this generalizes the widely used Singular Spectrum Analysis (SSA) in time series literature, and allows us to establish a rigorous link between time series analysis and matrix estimation. The key to establishing this link is constructing a Page matrix with non-overlapping entries rather than a Hankel matrix as is commonly done in the literature (e.g., SSA). This particular matrix structure allows us to provide finite sample analysis for imputation and prediction, and prove the asymptotic consistency of our method. Another salient feature of our algorithm is that it is model agnostic with respect to both the underlying time dynamics and the noise distribution in the observations. The noise agnostic property of our approach allows us to recover the latent states when only given access to noisy and partial observations a la a Hidden Markov Model; e.g., recovering the time-varying parameter of a Poisson process without knowing that the underlying process is Poisson. Furthermore, since our forecasting algorithm requires regression with noisy features, our approach suggests a matrix estimation based method - coupled with a novel, non-standard matrix estimation error metric - to solve the error-in-variable regression problem, which could be of interest in its own right. Through synthetic and real-world datasets, we demonstrate that our algorithm outperforms standard software packages (including R libraries) in the presence of missing data as well as high levels of noise.

cs.LG↗

arXiv2019

Asynchronous Approximation of a Single Component of the Solution to a Linear System

We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations $Ax = b$, where $A$ is a positive definite real matrix, and $b \in \mathbb{R}^n$. This is equivalent to solving for $x_i$ in $x = Gx + z$ for some $G$ and $z$ such that the spectral radius of $G$ is less than 1. Our algorithm relies on the Neumann series characterization of the component $x_i$, and is based on residual updates. We analyze our algorithm within the context of a cloud computation model, in which the computation is split into small update tasks performed by small processors with shared access to a distributed file system. We prove a robust asymptotic convergence result when the spectral radius $ρ(|G|) < 1$, regardless of the precise order and frequency in which the update tasks are performed. We provide convergence rate bounds which depend on the order of update tasks performed, analyzing both deterministic update rules via counting weighted random walks, as well as probabilistic update rules via concentration bounds. The probabilistic analysis requires analyzing the product of random matrices which are drawn from distributions that are time and path dependent. We specifically consider the setting where $n$ is large, yet $G$ is sparse, e.g., each row has at most $d$ nonzero entries. This is motivated by applications in which $G$ is derived from the edge structure of an underlying graph. Our results prove that if the local neighborhood of the graph does not grow too quickly as a function of $n$, our algorithm can provide significant reduction in computation cost as opposed to any algorithm which computes the global solution vector $x$. Our algorithm obtains an $ε\|x\|_2$ additive approximation for $x_i$ in constant time with respect to the size of the matrix when the maximum row sparsity $d = O(1)$ and $1/(1-\|G\|_2) = O(1)$.

cs.DS↗

arXiv2018

Q-learning with Nearest Neighbors

We consider model-free reinforcement learning for infinite-horizon discounted Markov Decision Processes (MDPs) with a continuous state space and unknown transition kernel, when only a single sample path under an arbitrary policy of the system is available. We consider the Nearest Neighbor Q-Learning (NNQL) algorithm to learn the optimal Q function using nearest neighbor regression method. As the main contribution, we provide tight finite sample analysis of the convergence rate. In particular, for MDPs with a $d$-dimensional state space and the discounted factor $γ\in (0,1)$, given an arbitrary sample path with "covering time" $ L $, we establish that the algorithm is guaranteed to output an $\varepsilon$-accurate estimate of the optimal Q-function using $\tilde{O}\big(L/(\varepsilon^3(1-γ)^7)\big)$ samples. For instance, for a well-behaved MDP, the covering time of the sample path under the purely random policy scales as $ \tilde{O}\big(1/\varepsilon^d\big),$ so the sample complexity scales as $\tilde{O}\big(1/\varepsilon^{d+3}\big).$ Indeed, we establish a lower bound that argues that the dependence of $ \tildeΩ\big(1/\varepsilon^{d+2}\big)$ is necessary.

cs.LG↗

arXiv2018

Regret vs. Bandwidth Trade-off for Recommendation Systems

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.

cs.IR↗

arXiv2017

Robust Synthetic Control

We present a robust generalization of the synthetic control method for comparative case studies. Like the classical method, we present an algorithm to estimate the unobservable counterfactual of a treatment unit. A distinguishing feature of our algorithm is that of de-noising the data matrix via singular value thresholding, which renders our approach robust in multiple facets: it automatically identifies a good subset of donors, overcomes the challenges of missing data, and continues to work well in settings where covariate information may not be provided. To begin, we establish the condition under which the fundamental assumption in synthetic control-like approaches holds, i.e. when the linear relationship between the treatment unit and the donor pool prevails in both the pre- and post-intervention periods. We provide the first finite sample analysis for a broader class of models, the Latent Variable Model, in contrast to Factor Models previously considered in the literature. Further, we show that our de-noising procedure accurately imputes missing entries, producing a consistent estimator of the underlying signal matrix provided $p = Ω( T^{-1 + ζ})$ for some $ζ> 0$; here, $p$ is the fraction of observed data and $T$ is the time interval of interest. Under the same setting, we prove that the mean-squared-error (MSE) in our prediction estimation scales as $O(σ^2/p + 1/\sqrt{T})$, where $σ^2$ is the noise variance. Using a data aggregation method, we show that the MSE can be made as small as $O(T^{-1/2+γ})$ for any $γ\in (0, 1/2)$, leading to a consistent estimator. We also introduce a Bayesian framework to quantify the model uncertainty through posterior probabilities. Our experiments, using both real-world and synthetic datasets, demonstrate that our robust generalization yields an improvement over the classical synthetic control method.

econ.EM↗

arXiv2017

Centralized Congestion Control and Scheduling in a Datacenter

We consider the problem of designing a packet-level congestion control and scheduling policy for datacenter networks. Current datacenter networks primarily inherit the principles that went into the design of Internet, where congestion control and scheduling are distributed. While distributed architecture provides robustness, it suffers in terms of performance. Unlike Internet, data center is fundamentally a "controlled" environment. This raises the possibility of designing a centralized architecture to achieve better performance. Recent solutions such as Fastpass and Flowtune have provided the proof of this concept. This raises the question: what is theoretically optimal performance achievable in a data center? We propose a centralized policy that guarantees a per-flow end-to-end flow delay bound of $O$(#hops $\times$ flow-size $/$ gap-to-capacity). Effectively such an end-to-end delay will be experienced by flows even if we removed congestion control and scheduling constraints as the resulting queueing networks can be viewed as the classical reversible multi-class queuing network, which has a product-form stationary distribution. In the language of Harrison et al., we establish that baseline performance for this model class is achievable. Indeed, as the key contribution of this work, we propose a method to emulate such a reversible queuing network while satisfying congestion control and scheduling constraints. Precisely, our policy is an emulation of Store-and-Forward (SFA) congestion control in conjunction with Last-Come-First-Serve Preemptive-Resume (LCFS-PR) scheduling policy.

cs.NI↗

arXiv2016

Regret Guarantees for Item-Item Collaborative Filtering

There is much empirical evidence that item-item collaborative filtering works well in practice. Motivated to understand this, we provide a framework to design and analyze various recommendation algorithms. The setup amounts to online binary matrix completion, where at each time a random user requests a recommendation and the algorithm chooses an entry to reveal in the user's row. The goal is to minimize regret, or equivalently to maximize the number of +1 entries revealed at any time. We analyze an item-item collaborative filtering algorithm that can achieve fundamentally better performance compared to user-user collaborative filtering. The algorithm achieves good "cold-start" performance (appropriately defined) by quickly making good recommendations to new users about whom there is little information.

cs.LG↗