arXiv subjects

Arnab Bhattacharyya

Publications and source records attributed to Arnab Bhattacharyya.

At least 37 records · Page 2Linked to original sources

arXiv2023

On Approximating Total Variation Distance

Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the problem of computing the TV distance of two product distributions over the domain $\{0,1\}^n$. In particular, we establish the following results. 1. The problem of exactly computing the TV distance of two product distributions is $\#\mathsf{P}$-complete. This is in stark contrast with other distance measures such as KL, Chi-square, and Hellinger which tensorize over the marginals leading to efficient algorithms. 2. There is a fully polynomial-time deterministic approximation scheme (FPTAS) for computing the TV distance of two product distributions $P$ and $Q$ where $Q$ is the uniform distribution. This result is extended to the case where $Q$ has a constant number of distinct marginals. In contrast, we show that when $P$ and $Q$ are Bayes net distributions, the relative approximation of their TV distance is $\mathsf{NP}$-hard.

cs.DS↗

arXiv2023

Active causal structure learning with advice

We introduce the problem of active causal structure learning with advice. In the typical well-studied setting, the learning algorithm is given the essential graph for the observational distribution and is asked to recover the underlying causal directed acyclic graph (DAG) $G^*$ while minimizing the number of interventions made. In our setting, we are additionally given side information about $G^*$ as advice, e.g. a DAG $G$ purported to be $G^*$. We ask whether the learning algorithm can benefit from the advice when it is close to being correct, while still having worst-case guarantees even when the advice is arbitrarily bad. Our work is in the same space as the growing body of research on algorithms with predictions. When the advice is a DAG $G$, we design an adaptive search algorithm to recover $G^*$ whose intervention cost is at most $O(\max\{1, \log ψ\})$ times the cost for verifying $G^*$; here, $ψ$ is a distance measure between $G$ and $G^*$ that is upper bounded by the number of variables $n$, and is exactly 0 when $G=G^*$. Our approximation factor matches the state-of-the-art for the advice-less setting.

cs.LG↗

arXiv2023

Near-Optimal Degree Testing for Bayes Nets

This paper considers the problem of testing the maximum in-degree of the Bayes net underlying an unknown probability distribution $P$ over $\{0,1\}^n$, given sample access to $P$. We show that the sample complexity of the problem is $\tildeΘ(2^{n/2}/\varepsilon^2)$. Our algorithm relies on a testing-by-learning framework, previously used to obtain sample-optimal testers; in order to apply this framework, we develop new algorithms for ``near-proper'' learning of Bayes nets, and high-probability learning under $χ^2$ divergence, which are of independent interest.

cs.LG↗

arXiv2023

Constraint Optimization over Semirings

Interpretations of logical formulas over semirings have applications in various areas of computer science including logic, AI, databases, and security. Such interpretations provide richer information beyond the truth or falsity of a statement. Examples of such semirings include Viterbi semiring, min-max or access control semiring, tropical semiring, and fuzzy semiring. The present work investigates the complexity of constraint optimization problems over semirings. The generic optimization problem we study is the following: Given a propositional formula $φ$ over $n$ variable and a semiring $(K,+,\cdot,0,1)$, find the maximum value over all possible interpretations of $φ$ over $K$. This can be seen as a generalization of the well-known satisfiability problem. A related problem is to find an interpretation that achieves the maximum value. In this work, we first focus on these optimization problems over the Viterbi semiring, which we call optConfVal and optConf. We show that for general propositional formulas in negation normal form, optConfVal and optConf are in ${\mathrm{FP}}^{\mathrm{NP}}$. We investigate optConf when the input formula $φ$ is represented as a CNF. For CNF formulae, we first derive an upper bound on optConfVal as a function of the number of maximum satisfiable clauses. In particular, we show that if $r$ is the maximum number of satisfiable clauses in a CNF formula with $m$ clauses, then its optConfVal is at most $1/4^{m-r}$. Building on this we establish that optConfVal for CNF formulae is hard for the complexity class ${\mathrm{FP}}^{\mathrm{NP}[\log]}$. We also design polynomial-time approximation algorithms and establish an inapproximability for optConfVal. We establish similar complexity results for these optimization problems over other semirings including tropical, fuzzy, and access control semirings.

cs.LO↗

arXiv2023

On the Interventional Kullback-Leibler Divergence

Modern machine learning approaches excel in static settings where a large amount of i.i.d. training data are available for a given task. In a dynamic environment, though, an intelligent agent needs to be able to transfer knowledge and re-use learned components across domains. It has been argued that this may be possible through causal models, aiming to mirror the modularity of the real world in terms of independent causal mechanisms. However, the true causal structure underlying a given set of data is generally not identifiable, so it is desirable to have means to quantify differences between models (e.g., between the ground truth and an estimate), on both the observational and interventional level. In the present work, we introduce the Interventional Kullback-Leibler (IKL) divergence to quantify both structural and distributional differences between models based on a finite set of multi-environment distributions generated by interventions from the ground truth. Since we generally cannot quantify all differences between causal models for every finite set of interventional distributions, we propose a sufficient condition on the intervention targets to identify subsets of observed variables on which the models provably agree or disagree.

cs.LG↗

arXiv2023

Independence Testing for Bounded Degree Bayesian Network

We study the following independence testing problem: given access to samples from a distribution $P$ over $\{0,1\}^n$, decide whether $P$ is a product distribution or whether it is $\varepsilon$-far in total variation distance from any product distribution. For arbitrary distributions, this problem requires $\exp(n)$ samples. We show in this work that if $P$ has a sparse structure, then in fact only linearly many samples are required. Specifically, if $P$ is Markov with respect to a Bayesian network whose underlying DAG has in-degree bounded by $d$, then $\tildeΘ(2^{d/2}\cdot n/\varepsilon^2)$ samples are necessary and sufficient for independence testing.

cs.DS↗

arXiv2023

An Adaptive Kernel Approach to Federated Learning of Heterogeneous Causal Effects

We propose a new causal inference framework to learn causal effects from multiple, decentralized data sources in a federated setting. We introduce an adaptive transfer algorithm that learns the similarities among the data sources by utilizing Random Fourier Features to disentangle the loss function into multiple components, each of which is associated with a data source. The data sources may have different distributions; the causal effects are independently and systematically incorporated. The proposed method estimates the similarities among the sources through transfer coefficients, and hence requiring no prior information about the similarity measures. The heterogeneous causal effects can be estimated with no sharing of the raw training data among the sources, thus minimizing the risk of privacy leak. We also provide minimax lower bounds to assess the quality of the parameters learned from the disparate sources. The proposed method is empirically shown to outperform the baselines on decentralized data sources with dissimilar distributions.

cs.LG↗

arXiv2022

Learning Sparse Fixed-Structure Gaussian Bayesian Networks

Gaussian Bayesian networks (a.k.a. linear Gaussian structural equation models) are widely used to model causal interactions among continuous variables. In this work, we study the problem of learning a fixed-structure Gaussian Bayesian network up to a bounded error in total variation distance. We analyze the commonly used node-wise least squares regression (LeastSquares) and prove that it has a near-optimal sample complexity. We also study a couple of new algorithms for the problem: - BatchAvgLeastSquares takes the average of several batches of least squares solutions at each node, so that one can interpolate between the batch size and the number of batches. We show that BatchAvgLeastSquares also has near-optimal sample complexity. - CauchyEst takes the median of solutions to several batches of linear systems at each node. We show that the algorithm specialized to polytrees, CauchyEstTree, has near-optimal sample complexity. Experimentally, we show that for uncontaminated, realizable data, the LeastSquares algorithm performs best, but in the presence of contamination or DAG misspecification, CauchyEst/CauchyEstTree and BatchAvgLeastSquares respectively perform better.

cs.DS↗

arXiv2022

Verification and search algorithms for causal DAGs

We study two problems related to recovering causal graphs from interventional data: (i) $\textit{verification}$, where the task is to check if a purported causal graph is correct, and (ii) $\textit{search}$, where the task is to recover the correct causal graph. For both, we wish to minimize the number of interventions performed. For the first problem, we give a characterization of a minimal sized set of atomic interventions that is necessary and sufficient to check the correctness of a claimed causal graph. Our characterization uses the notion of $\textit{covered edges}$, which enables us to obtain simple proofs and also easily reason about earlier known results. We also generalize our results to the settings of bounded size interventions and node-dependent interventional costs. For all the above settings, we provide the first known provable algorithms for efficiently computing (near)-optimal verifying sets on general graphs. For the second problem, we give a simple adaptive algorithm based on graph separators that produces an atomic intervention set which fully orients any essential graph while using $\mathcal{O}(\log n)$ times the optimal number of interventions needed to $\textit{verify}$ (verifying size) the underlying DAG on $n$ vertices. This approximation is tight as $\textit{any}$ search algorithm on an essential line graph has worst case approximation ratio of $Ω(\log n)$ with respect to the verifying size. With bounded size interventions, each of size $\leq k$, our algorithm gives an $\mathcal{O}(\log n \cdot \log k)$ factor approximation. Our result is the first known algorithm that gives a non-trivial approximation guarantee to the verifying size on general unweighted graphs and with bounded size interventions.

cs.LG↗

arXiv2022

Universal 1-Bit Compressive Sensing for Bounded Dynamic Range Signals

A {\em universal 1-bit compressive sensing (CS)} scheme consists of a measurement matrix $A$ such that all signals $x$ belonging to a particular class can be approximately recovered from $\textrm{sign}(Ax)$. 1-bit CS models extreme quantization effects where only one bit of information is revealed per measurement. We focus on universal support recovery for 1-bit CS in the case of {\em sparse} signals with bounded {\em dynamic range}. Specifically, a vector $x \in \mathbb{R}^n$ is said to have sparsity $k$ if it has at most $k$ nonzero entries, and dynamic range $R$ if the ratio between its largest and smallest nonzero entries is at most $R$ in magnitude. Our main result shows that if the entries of the measurement matrix $A$ are i.i.d.~Gaussians, then under mild assumptions on the scaling of $k$ and $R$, the number of measurements needs to be $\tildeΩ(Rk^{3/2})$ to recover the support of $k$-sparse signals with dynamic range $R$ using $1$-bit CS. In addition, we show that a near-matching $O(R k^{3/2} \log n)$ upper bound follows as a simple corollary of known results. The $k^{3/2}$ scaling contrasts with the known lower bound of $\tildeΩ(k^2 \log n)$ for the number of measurements to recover the support of arbitrary $k$-sparse signals.

cs.IT↗

arXiv2022

Low Degree Testing over the Reals

We study the problem of testing whether a function $f: \mathbb{R}^n \to \mathbb{R}$ is a polynomial of degree at most $d$ in the \emph{distribution-free} testing model. Here, the distance between functions is measured with respect to an unknown distribution $\mathcal{D}$ over $\mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we do not assume that $\mathcal{D}$ has finite support. We design a tester that given query access to $f$, and sample access to $\mathcal{D}$, makes $(d/\varepsilon)^{O(1)}$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $\varepsilon$ with respect to $\mathcal{D}$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest.

cs.DS↗

arXiv2021

Efficient inference of interventional distributions

We consider the problem of efficiently inferring interventional distributions in a causal Bayesian network from a finite number of observations. Let $\mathcal{P}$ be a causal model on a set $\mathbf{V}$ of observable variables on a given causal graph $G$. For sets $\mathbf{X},\mathbf{Y}\subseteq \mathbf{V}$, and setting ${\bf x}$ to $\mathbf{X}$, let $P_{\bf x}(\mathbf{Y})$ denote the interventional distribution on $\mathbf{Y}$ with respect to an intervention ${\bf x}$ to variables ${\bf x}$. Shpitser and Pearl (AAAI 2006), building on the work of Tian and Pearl (AAAI 2001), gave an exact characterization of the class of causal graphs for which the interventional distribution $P_{\bf x}({\mathbf{Y}})$ can be uniquely determined. We give the first efficient version of the Shpitser-Pearl algorithm. In particular, under natural assumptions, we give a polynomial-time algorithm that on input a causal graph $G$ on observable variables $\mathbf{V}$, a setting ${\bf x}$ of a set $\mathbf{X} \subseteq \mathbf{V}$ of bounded size, outputs succinct descriptions of both an evaluator and a generator for a distribution $\hat{P}$ that is $\varepsilon$-close (in total variation distance) to $P_{\bf x}({\mathbf{Y}})$ where $Y=\mathbf{V}\setminus \mathbf{X}$, if $P_{\bf x}(\mathbf{Y})$ is identifiable. We also show that when $\mathbf{Y}$ is an arbitrary set, there is no efficient algorithm that outputs an evaluator of a distribution that is $\varepsilon$-close to $P_{\bf x}({\mathbf{Y}})$ unless all problems that have statistical zero-knowledge proofs, including the Graph Isomorphism problem, have efficient randomized algorithms.

cs.DS↗

arXiv2021

Near-Optimal Learning of Tree-Structured Distributions by Chow-Liu

We provide finite sample guarantees for the classical Chow-Liu algorithm (IEEE Trans.~Inform.~Theory, 1968) to learn a tree-structured graphical model of a distribution. For a distribution $P$ on $Σ^n$ and a tree $T$ on $n$ nodes, we say $T$ is an $\varepsilon$-approximate tree for $P$ if there is a $T$-structured distribution $Q$ such that $D(P\;||\;Q)$ is at most $\varepsilon$ more than the best possible tree-structured distribution for $P$. We show that if $P$ itself is tree-structured, then the Chow-Liu algorithm with the plug-in estimator for mutual information with $\widetilde{O}(|Σ|^3 n\varepsilon^{-1})$ i.i.d.~samples outputs an $\varepsilon$-approximate tree for $P$ with constant probability. In contrast, for a general $P$ (which may not be tree-structured), $Ω(n^2\varepsilon^{-2})$ samples are necessary to find an $\varepsilon$-approximate tree. Our upper bound is based on a new conditional independence tester that addresses an open problem posed by Canonne, Diakonikolas, Kane, and Stewart~(STOC, 2018): we prove that for three random variables $X,Y,Z$ each over $Σ$, testing if $I(X; Y \mid Z)$ is $0$ or $\geq \varepsilon$ is possible with $\widetilde{O}(|Σ|^3/\varepsilon)$ samples. Finally, we show that for a specific tree $T$, with $\widetilde{O} (|Σ|^2n\varepsilon^{-1})$ samples from a distribution $P$ over $Σ^n$, one can efficiently learn the closest $T$-structured distribution in KL divergence by applying the add-1 estimator at each node.

cs.DS↗

arXiv2021

Identifiability of AMP chain graph models

We study identifiability of Andersson-Madigan-Perlman (AMP) chain graph models, which are a common generalization of linear structural equation models and Gaussian graphical models. AMP models are described by DAGs on chain components which themselves are undirected graphs. For a known chain component decomposition, we show that the DAG on the chain components is identifiable if the determinants of the residual covariance matrices of the chain components are monotone non-decreasing in topological order. This condition extends the equal variance identifiability criterion for Bayes nets, and it can be generalized from determinants to any super-additive function on positive semidefinite matrices. When the component decomposition is unknown, we describe conditions that allow recovery of the full structure using a polynomial time algorithm based on submodular function minimization. We also conduct experiments comparing our algorithm's performance against existing baselines.

cs.DS↗

arXiv2021

Testing Product Distributions: A Closer Look

We study the problems of identity and closeness testing of $n$-dimensional product distributions. Prior works by Canonne, Diakonikolas, Kane and Stewart (COLT 2017) and Daskalakis and Pan (COLT 2017) have established tight sample complexity bounds for non-tolerant testing over a binary alphabet: given two product distributions $P$ and $Q$ over a binary alphabet, distinguish between the cases $P = Q$ and $d_{\mathrm{TV}}(P, Q) > ε$. We build on this prior work to give a more comprehensive map of the complexity of testing of product distributions by investigating tolerant testing with respect to several natural distance measures and over an arbitrary alphabet. Our study gives a fine-grained understanding of how the sample complexity of tolerant testing varies with the distance measures for product distributions. In addition, we also extend one of our upper bounds on product distributions to bounded-degree Bayes nets.

cs.DS↗

arXiv2021

Model Counting meets F0 Estimation

Constraint satisfaction problems (CSP's) and data stream models are two powerful abstractions to capture a wide variety of problems arising in different domains of computer science. Developments in the two communities have mostly occurred independently and with little interaction between them. In this work, we seek to investigate whether bridging the seeming communication gap between the two communities may pave the way to richer fundamental insights. To this end, we focus on two foundational problems: model counting for CSP's and computation of zeroth frequency moments ($F_0$) for data streams. Our investigations lead us to observe striking similarity in the core techniques employed in the algorithmic frameworks that have evolved separately for model counting and $F_0$ computation. We design a recipe for translation of algorithms developed for $F_0$ estimation to that of model counting, resulting in new algorithms for model counting. We then observe that algorithms in the context of distributed streaming can be transformed to distributed algorithms for model counting. We next turn our attention to viewing streaming from the lens of counting and show that framing $F_0$ estimation as a special case of #DNF counting allows us to obtain a general recipe for a rich class of streaming problems, which had been subjected to case-specific analysis in prior works. In particular, our view yields a state-of-the art algorithm for multidimensional range efficient $F_0$ estimation with a simpler analysis.

cs.DS↗

arXiv2020

Eco-Routing Using Open Street Maps

A vehicle's fuel consumption depends on its type, the speed, the condition, and the gradients of the road on which it is moving. We developed a Routing Engine for finding an eco-route (one with low fuel consumption) between a source and a destination. Open Street Maps has data on road conditions. We used CGIAR-CSI road elevation data 16[4] to integrate the road gradients into the proposed route-finding algorithm that modifies Open Street Routing Machine (OSRM). It allowed us to dynamically predict a vehicle's velocity, considering both the conditions and the road segment's slope. Using Highway EneRgy Assessment (HERA) methodology, we calculated the fuel consumed by a vehicle given its type and velocity. We have created both web and mobile interfaces through which users can specify Geo coordinates or human-readable addresses of a source and a destination. The user interface graphically displays the route obtained from the proposed Routing Engine with a detailed travel itinerary.

cs.NI↗

arXiv2020

Learning and Sampling of Atomic Interventions from Observations

We study the problem of efficiently estimating the effect of an intervention on a single variable (atomic interventions) using observational samples in a causal Bayesian network. Our goal is to give algorithms that are efficient in both time and sample complexity in a non-parametric setting. Tian and Pearl (AAAI `02) have exactly characterized the class of causal graphs for which causal effects of atomic interventions can be identified from observational data. We make their result quantitative. Suppose P is a causal model on a set $\vec{V}$ of n observable variables with respect to a given causal graph G with observable distribution $P$. Let $P_x$ denote the interventional distribution over the observables with respect to an intervention of a designated variable X with x. Assuming that $G$ has bounded in-degree, bounded c-components ($k$), and that the observational distribution is identifiable and satisfies certain strong positivity condition, we give an algorithm that takes $m=\tilde{O}(nε^{-2})$ samples from $P$ and $O(mn)$ time, and outputs with high probability a description of a distribution $\hat{P}$ such that $d_{\mathrm{TV}}(P_x, \hat{P}) \leq ε$, and: 1. [Evaluation] the description can return in $O(n)$ time the probability $\hat{P}(\vec{v})$ for any assignment $\vec{v}$ to $\vec{V}$ 2. [Generation] the description can return an iid sample from $\hat{P}$ in $O(n)$ time. We also show lower bounds for the sample complexity showing that our sample complexity has an optimal dependence on the parameters $n$ and $ε$, as well as if $k=1$ on the strong positivity parameter.

cs.LG↗