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Researcher profile

Arya Mazumdar

Arya Mazumdar contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Information TheoryMachine Learningmath.ITData Structures and AlgorithmsDiscrete MathematicsDistributed, Parallel, and Cluster Computingeess.SPCryptography and SecurityDatabasesInformation Retrievalmath.PRmath.STNetworking and Internet ArchitectureStatistics Theory

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Published work

12 published item(s)

preprint2026arXiv

Approximate Distributed Coded Computing: Polynomial Codes and Randomized Sketching

Coded computing is a distributed paradigm that uses coding theory to introduce \textit{redundancy} and overcome bottlenecks in large-scale systems. In the same vein, randomized numerical linear algebra employs probabilistic methods to \textit{compress} and accelerate linear algebraic operations, addressing challenges in high-dimensional data analysis. This article reviews the foundations of both fields and presents distributed schemes that combine techniques from both to speed up optimization and machine learning algorithms, in the presence of slow or non-responsive servers. Along the way, we touch on various related topics and mathematical concepts.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Binary Iterative Hard Thresholding Converges with Optimal Number of Measurements for 1-Bit Compressed Sensing

Compressed sensing has been a very successful high-dimensional signal acquisition and recovery technique that relies on linear operations. However, the actual measurements of signals have to be quantized before storing or processing. 1(One)-bit compressed sensing is a heavily quantized version of compressed sensing, where each linear measurement of a signal is reduced to just one bit: the sign of the measurement. Once enough of such measurements are collected, the recovery problem in 1-bit compressed sensing aims to find the original signal with as much accuracy as possible. The recovery problem is related to the traditional "halfspace-learning" problem in learning theory. For recovery of sparse vectors, a popular reconstruction method from 1-bit measurements is the binary iterative hard thresholding (BIHT) algorithm. The algorithm is a simple projected sub-gradient descent method, and is known to converge well empirically, despite the nonconvexity of the problem. The convergence property of BIHT was not theoretically justified, except with an exorbitantly large number of measurements (i.e., a number of measurement greater than $\max\{k^{10}, 24^{48}, k^{3.5}/ε\}$, where $k$ is the sparsity, $ε$ denotes the approximation error, and even this expression hides other factors). In this paper we show that the BIHT algorithm converges with only $\tilde{O}(\frac{k}ε)$ measurements. Note that, this dependence on $k$ and $ε$ is optimal for any recovery method in 1-bit compressed sensing. With this result, to the best of our knowledge, BIHT is the only practical and efficient (polynomial time) algorithm that requires the optimal number of measurements in all parameters (both $k$ and $ε$). This is also an example of a gradient descent algorithm converging to the correct solution for a nonconvex problem, under suitable structural conditions.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Decentralized Competing Bandits in Non-Stationary Matching Markets

Understanding complex dynamics of two-sided online matching markets, where the demand-side agents compete to match with the supply-side (arms), has recently received substantial interest. To that end, in this paper, we introduce the framework of decentralized two-sided matching market under non stationary (dynamic) environments. We adhere to the serial dictatorship setting, where the demand-side agents have unknown and different preferences over the supply-side (arms), but the arms have fixed and known preference over the agents. We propose and analyze a decentralized and asynchronous learning algorithm, namely Decentralized Non-stationary Competing Bandits (\texttt{DNCB}), where the agents play (restrictive) successive elimination type learning algorithms to learn their preference over the arms. The complexity in understanding such a system stems from the fact that the competing bandits choose their actions in an asynchronous fashion, and the lower ranked agents only get to learn from a set of arms, not \emph{dominated} by the higher ranked agents, which leads to \emph{forced exploration}. With carefully defined complexity parameters, we characterize this \emph{forced exploration} and obtain sub-linear (logarithmic) regret of \texttt{DNCB}. Furthermore, we validate our theoretical findings via experiments.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Lower Bounds on the Total Variation Distance Between Mixtures of Two Gaussians

Mixtures of high dimensional Gaussian distributions have been studied extensively in statistics and learning theory. While the total variation distance appears naturally in the sample complexity of distribution learning, it is analytically difficult to obtain tight lower bounds for mixtures. Exploiting a connection between total variation distance and the characteristic function of the mixture, we provide fairly tight functional approximations. This enables us to derive new lower bounds on the total variation distance between pairs of two-component Gaussian mixtures that have a shared covariance matrix.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

On Learning Mixture of Linear Regressions in the Non-Realizable Setting

While mixture of linear regressions (MLR) is a well-studied topic, prior works usually do not analyze such models for prediction error. In fact, {\em prediction} and {\em loss} are not well-defined in the context of mixtures. In this paper, first we show that MLR can be used for prediction where instead of predicting a label, the model predicts a list of values (also known as {\em list-decoding}). The list size is equal to the number of components in the mixture, and the loss function is defined to be minimum among the losses resulted by all the component models. We show that with this definition, a solution of the empirical risk minimization (ERM) achieves small probability of prediction error. This begs for an algorithm to minimize the empirical risk for MLR, which is known to be computationally hard. Prior algorithmic works in MLR focus on the {\em realizable} setting, i.e., recovery of parameters when data is probabilistically generated by a mixed linear (noisy) model. In this paper we show that a version of the popular alternating minimization (AM) algorithm finds the best fit lines in a dataset even when a realizable model is not assumed, under some regularity conditions on the dataset and the initial points, and thereby provides a solution for the ERM. We further provide an algorithm that runs in polynomial time in the number of datapoints, and recovers a good approximation of the best fit lines. The two algorithms are experimentally compared.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Support Recovery in Mixture Models with Sparse Parameters

Mixture models are widely used to fit complex and multimodal datasets. In this paper we study mixtures with high dimensional sparse latent parameter vectors and consider the problem of support recovery of those vectors. While parameter learning in mixture models is well-studied, the sparsity constraint remains relatively unexplored. Sparsity of parameter vectors is a natural constraint in variety of settings, and support recovery is a major step towards parameter estimation. We provide efficient algorithms for support recovery that have a logarithmic sample complexity dependence on the dimensionality of the latent space. Our algorithms are quite general, namely they are applicable to 1) mixtures of many different canonical distributions including Uniform, Poisson, Laplace, Gaussians, etc. 2) Mixtures of linear regressions and linear classifiers with Gaussian covariates under different assumptions on the unknown parameters. In most of these settings, our results are the first guarantees on the problem while in the rest, our results provide improvements on existing works.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A workload-adaptive mechanism for linear queries under local differential privacy

We propose a new mechanism to accurately answer a user-provided set of linear counting queries under local differential privacy (LDP). Given a set of linear counting queries (the workload) our mechanism automatically adapts to provide accuracy on the workload queries. We define a parametric class of mechanisms that produce unbiased estimates of the workload, and formulate a constrained optimization problem to select a mechanism from this class that minimizes expected total squared error. We solve this optimization problem numerically using projected gradient descent and provide an efficient implementation that scales to large workloads. We demonstrate the effectiveness of our optimization-based approach in a wide variety of settings, showing that it outperforms many competitors, even outperforming existing mechanisms on the workloads for which they were intended.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Algebraic and Analytic Approaches for Parameter Learning in Mixture Models

We present two different approaches for parameter learning in several mixture models in one dimension. Our first approach uses complex-analytic methods and applies to Gaussian mixtures with shared variance, binomial mixtures with shared success probability, and Poisson mixtures, among others. An example result is that $\exp(O(N^{1/3}))$ samples suffice to exactly learn a mixture of $k<N$ Poisson distributions, each with integral rate parameters bounded by $N$. Our second approach uses algebraic and combinatorial tools and applies to binomial mixtures with shared trial parameter $N$ and differing success parameters, as well as to mixtures of geometric distributions. Again, as an example, for binomial mixtures with $k$ components and success parameters discretized to resolution $ε$, $O(k^2(N/ε)^{8/\sqrtε})$ samples suffice to exactly recover the parameters. For some of these distributions, our results represent the first guarantees for parameter estimation.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Connectivity in Random Annulus Graphs and the Geometric Block Model

We provide new connectivity results for {\em vertex-random graphs} or {\em random annulus graphs} which are significant generalizations of random geometric graphs. Random geometric graphs (RGG) are one of the most basic models of random graphs for spatial networks proposed by Gilbert in 1961, shortly after the introduction of the Erdős-R\&#39;{en}yi random graphs. They resemble social networks in many ways (e.g. by spontaneously creating cluster of nodes with high modularity). The connectivity properties of RGG have been studied since its introduction, and analyzing them has been significantly harder than their Erdős-R\&#39;{en}yi counterparts due to correlated edge formation. Our next contribution is in using the connectivity of random annulus graphs to provide necessary and sufficient conditions for efficient recovery of communities for {\em the geometric block model} (GBM). The GBM is a probabilistic model for community detection defined over an RGG in a similar spirit as the popular {\em stochastic block model}, which is defined over an Erdős-R\&#39;{en}yi random graph. The geometric block model inherits the transitivity properties of RGGs and thus models communities better than a stochastic block model. However, analyzing them requires fresh perspectives as all prior tools fail due to correlation in edge formation. We provide a simple and efficient algorithm that can recover communities in GBM exactly with high probability in the regime of connectivity.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Recovery of Sparse Signals from a Mixture of Linear Samples

Mixture of linear regressions is a popular learning theoretic model that is used widely to represent heterogeneous data. In the simplest form, this model assumes that the labels are generated from either of two different linear models and mixed together. Recent works of Yin et al. and Krishnamurthy et al., 2019, focus on an experimental design setting of model recovery for this problem. It is assumed that the features can be designed and queried with to obtain their label. When queried, an oracle randomly selects one of the two different sparse linear models and generates a label accordingly. How many such oracle queries are needed to recover both of the models simultaneously? This question can also be thought of as a generalization of the well-known compressed sensing problem (Candès and Tao, 2005, Donoho, 2006). In this work, we address this query complexity problem and provide efficient algorithms that improves on the previously best known results.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Reliable Distributed Clustering with Redundant Data Assignment

In this paper, we present distributed generalized clustering algorithms that can handle large scale data across multiple machines in spite of straggling or unreliable machines. We propose a novel data assignment scheme that enables us to obtain global information about the entire data even when some machines fail to respond with the results of the assigned local computations. The assignment scheme leads to distributed algorithms with good approximation guarantees for a variety of clustering and dimensionality reduction problems.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Storage Capacity of Repairable Networks

In this paper, we introduce a model of a distributed storage system that is locally recoverable from any single server failure. Unlike the usual local recovery model of codes for distributed storage, this model accounts for the fact that each server or storage node in a network is connectible to only some, and not all other, nodes. This may happen for reasons such as physical separation, inhomogeneity in storage platforms etc. We estimate the storage capacity of both undirected and directed networks under this model and propose some constructive schemes. From a coding theory point of view, we show that this model is approximately dual of the well-studied index coding problem. Further in this paper, we extend the above model to handle multiple server failures. Among other results, we provide an upper bound on the minimum pairwise distance of a set of words that can be stored in a graph with the local repair guarantee. The well-known impossibility bounds on the distance of locally recoverable codes follow from our result.

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