arXiv subjects

Rajarshi Bhattacharjee

Publications and source records attributed to Rajarshi Bhattacharjee.

13 recordsLinked to original sources

arXiv2026

Sublinear Time Eigenvector Approximation via Column Sampling

We study sublinear time sampling methods for approximating the outlying eigenvectors of large matrices. Our main result is an algorithm that uniformly samples just $\tilde{O}(\log n/ε^4)$ columns of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries bounded in magnitude by $1$, and, for any eigenvalue $λ$ of $A$ with $|λ| \ge εn$, outputs an approximate eigenvector $v$ satisfying $\|Av - λv\|_2 \le εn$. For approximating just the eigenvector of the largest magnitude eigenvalue, our algorithm samples only $\tilde{O}(\log n/ε^2)$ columns. Given the ability to sample rows and columns of $A$ proportional to their squared norms, we give a similar result with an improved error bound of $ε\|A\|_F$. For top eigenvector approximation, we show our bound is tight up to logarithmic terms. A key feature of our algorithms is that the output eigenvectors are spanned by a small number of $A$'s columns, and individual entries can be computed rapidly, in poly(log n, 1/epsilon) time per entry. This makes them applicable in the quantum-inspired algorithms framework of [Tang, STOC 2019], where we give the first sublinear time classical algorithms for eigenvector approximation with additive error $ε\|A\|_F$. Finally, we present an alternative approach, based on a truncated Nystrom method, that, while not allowing poly(log n, 1/epsilon) time entrywise computation of the approximate eigenvectors, achieves near optimal sample complexity for general symmetric matrices, and improved bounds for positive semidefinite matrices. Technically, our bounds build on recent work on approximating the outlying eigenvalues of symmetric matrices via random sampling in [Bhattacharjee et al. '22] and [Swartworth and Woodruff '25]. We demonstrate for the first time that these approaches extend to the problem of eigenvector estimation.

cs.DS↗

arXiv2026

Fast Length-Squared Sampling for Positive-Semidefinite Matrices

We describe a simple rejection-sampling-based algorithm to perform length-squared sampling on an $n \times n$ positive-semidefinite (psd) matrix: that is, to sample a column with probability proportional to its squared $\ell_2$-norm. The algorithm runs in just $O(n)$ expected time, which is significantly sublinear in the input matrix size. The runtime is optimal, even when the input is assumed to be diagonal. Our result has several applications. Length-squared sampling is used by a number of sublinear time algorithms for matrix problems, like low-rank approximation and eigenvalue approximation. Often, it is assumed that the algorithm is given access to the matrix column norms, and thus can perform length-squared sampling efficiently. Our result shows that, at least for psd matrices, we can remove this assumption. We also discuss an application to an asymptotically optimal algorithm for estimating the Frobenius norm of a psd matrix to relative error. Finally, we show that our sampling algorithm yields a very simple sublinear time algorithm for the robust psd low-rank approximation problem introduced by Bakshi et al. (FOCS, 2020), which nearly matches the more complex method developed there.

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arXiv2026

Safe RLHF Beyond Expectation: Stochastic Dominance for Universal Spectral Risk Control

Safe Reinforcement Learning from Human Feedback (RLHF) typically enforces safety through expected cost constraints, but the expectation captures only a single statistic of the cost distribution and fails to account for distributional uncertainty, particularly under heavy tails or rare catastrophic events. This limitation is problematic when robustness and risk sensitivity are critical. Stochastic dominance offers a principled alternative by comparing entire cost distributions rather than just their averages, enabling direct control over tail risks and potential out-of-distribution failures that expectation-based constraints may overlook. In this work, we propose Risk-sensitive Alignment via Dominance (RAD), a novel alignment framework that replaces scalar expected cost constraints with First-Order Stochastic Dominance (FSD) constraints. We operationalize this constraint by comparing the target policy's cost distribution to that of a reference policy within an Optimal Transport (OT) framework, using entropic regularization and Sinkhorn iterations to obtain a differentiable and computationally efficient objective for stable end-to-end optimization. Furthermore, we introduce quantile-weighted FSD constraints and show that weighted FSD universally controls a broad class of Spectral Risk Measures (SRMs), so that improvements under weighted dominance imply guaranteed improvements in the corresponding spectral risk. This provides a principled mechanism for tuning a model's risk profile via the quantile weighting function. Empirical results demonstrate that RAD improves harmlessness over baselines while remaining competitive in helpfulness, and exhibits greater robustness on out-of-distribution harmlessness evaluations.

cs.LG↗

arXiv2024

Improved Spectral Density Estimation via Explicit and Implicit Deflation

We study algorithms for approximating the spectral density of a symmetric matrix $A$ that is accessed through matrix-vector product queries. By combining a previously studied Chebyshev polynomial moment matching method with a deflation step that approximately projects off the largest magnitude eigendirections of $A$ before estimating the spectral density, we give an $ε\cdotσ_\ell(A)$ error approximation to the spectral density in the Wasserstein-$1$ metric using $O(\ell\log n+ 1/ε)$ matrix-vector products, where $σ_\ell(A)$ is the $\ell^{th}$ largest singular value of $A$. In the common case when $A$ exhibits fast singular value decay, our bound can be much stronger than prior work, which gives an error bound of $ε\cdot ||A||_2$ using $O(1/ε)$ matrix-vector products. We also show that it is nearly tight: any algorithm giving error $ε\cdot σ_\ell(A)$ must use $Ω(\ell+1/ε)$ matrix-vector products. We further show that the popular Stochastic Lanczos Quadrature (SLQ) method matches the above bound, even though SLQ itself is parameter-free and performs no explicit deflation. This bound explains the strong practical performance of SLQ, and motivates a simple variant of SLQ that achieves an even tighter error bound. Our error bound for SLQ leverages an analysis that views it as an implicit polynomial moment matching method, along with recent results on low-rank approximation with single-vector Krylov methods. We use these results to show that the method can perform implicit deflation as part of moment matching.

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arXiv2024

Universal Matrix Sparsifiers and Fast Deterministic Algorithms for Linear Algebra

Let $\mathbf S \in \mathbb R^{n \times n}$ satisfy $\|\mathbf 1-\mathbf S\|_2\leεn$, where $\mathbf 1$ is the all ones matrix and $\|\cdot\|_2$ is the spectral norm. It is well-known that there exists such an $\mathbf S$ with just $O(n/ε^2)$ non-zero entries: we can let $\mathbf S$ be the scaled adjacency matrix of a Ramanujan expander graph. We show that such an $\mathbf S$ yields a $universal$ $sparsifier$ for any positive semidefinite (PSD) matrix. In particular, for any PSD $\mathbf A \in \mathbb{R}^{n\times n}$ with entries bounded in magnitude by $1$, $\|\mathbf A - \mathbf A\circ\mathbf S\|_2 \le εn$, where $\circ$ denotes the entrywise (Hadamard) product. Our techniques also give universal sparsifiers for non-PSD matrices. In this case, letting $\mathbf S$ be the scaled adjacency matrix of a Ramanujan graph with $\tilde O(n/ε^4)$ edges, we have $\|\mathbf A - \mathbf A \circ \mathbf S \|_2 \le ε\cdot \max(n,\|\mathbf A\|_1)$, where $\|\mathbf A\|_1$ is the nuclear norm. We show that the above bounds for both PSD and non-PSD matrices are tight up to log factors. Since $\mathbf A \circ \mathbf S$ can be constructed deterministically, our result for PSD matrices derandomizes and improves upon known results for randomized matrix sparsification, which require randomly sampling ${O}(\frac{n \log n}{ε^2})$ entries. We also leverage our results to give the first deterministic algorithms for several problems related to singular value approximation that run in faster than matrix multiplication time. Finally, if $\mathbf A \in \{-1,0,1\}^{n \times n}$ is PSD, we show that $\mathbf{\tilde A}$ with $\|\mathbf A - \mathbf{\tilde A}\|_2 \le εn$ can be obtained by deterministically reading $\tilde O(n/ε)$ entries of $\mathbf A$. This improves the $1/ε$ dependence on our result for general PSD matrices and is near-optimal.

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arXiv2023

No-regret Algorithms for Fair Resource Allocation

We consider a fair resource allocation problem in the no-regret setting against an unrestricted adversary. The objective is to allocate resources equitably among several agents in an online fashion so that the difference of the aggregate $α$-fair utilities of the agents between an optimal static clairvoyant allocation and that of the online policy grows sub-linearly with time. The problem is challenging due to the non-additive nature of the $α$-fairness function. Previously, it was shown that no online policy can exist for this problem with a sublinear standard regret. In this paper, we propose an efficient online resource allocation policy, called Online Proportional Fair (OPF), that achieves $c_α$-approximate sublinear regret with the approximation factor $c_α=(1-α)^{-(1-α)}\leq 1.445,$ for $0\leq α< 1$. The upper bound to the $c_α$-regret for this problem exhibits a surprising phase transition phenomenon. The regret bound changes from a power-law to a constant at the critical exponent $α=\frac{1}{2}.$ As a corollary, our result also resolves an open problem raised by Even-Dar et al. [2009] on designing an efficient no-regret policy for the online job scheduling problem in certain parameter regimes. The proof of our results introduces new algorithmic and analytical techniques, including greedy estimation of the future gradients for non-additive global reward functions and bootstrapping adaptive regret bounds, which may be of independent interest.

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arXiv2022

Sublinear Time Eigenvalue Approximation via Random Sampling

We study the problem of approximating the eigenspectrum of a symmetric matrix $\mathbf A \in \mathbb{R}^{n \times n}$ with bounded entries (i.e., $\|\mathbf A\|_{\infty} \leq 1$). We present a simple sublinear time algorithm that approximates all eigenvalues of $\mathbf{A}$ up to additive error $\pm εn$ using those of a randomly sampled $\tilde {O}\left (\frac{\log^3 n}{ε^3}\right ) \times \tilde O\left (\frac{\log^3 n}{ε^3}\right )$ principal submatrix. Our result can be viewed as a concentration bound on the complete eigenspectrum of a random submatrix, significantly extending known bounds on just the singular values (the magnitudes of the eigenvalues). We give improved error bounds of $\pm ε\sqrt{\text{nnz}(\mathbf{A})}$ and $\pm ε\|\mathbf A\|_F$ when the rows of $\mathbf A$ can be sampled with probabilities proportional to their sparsities or their squared $\ell_2$ norms respectively. Here $\text{nnz}(\mathbf{A})$ is the number of non-zero entries in $\mathbf{A}$ and $\|\mathbf A\|_F$ is its Frobenius norm. Even for the strictly easier problems of approximating the singular values or testing the existence of large negative eigenvalues (Bakshi, Chepurko, and Jayaram, FOCS '20), our results are the first that take advantage of non-uniform sampling to give improved error bounds. From a technical perspective, our results require several new eigenvalue concentration and perturbation bounds for matrices with bounded entries. Our non-uniform sampling bounds require a new algorithmic approach, which judiciously zeroes out entries of a randomly sampled submatrix to reduce variance, before computing the eigenvalues of that submatrix as estimates for those of $\mathbf A$. We complement our theoretical results with numerical simulations, which demonstrate the effectiveness of our algorithms in practice.

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arXiv2022

Optimizing Age-of-Information in Adversarial and Stochastic Environments

We design efficient online scheduling policies to maximize the freshness of information delivered to the users in a cellular network under both adversarial and stochastic channel and mobility assumptions. The information freshness achieved by a policy is investigated through the lens of a recently proposed metric - Age-of-Information (AoI). We show that a natural greedy scheduling policy is competitive against any optimal offline policy in minimizing the AoI in the adversarial setting. We also derive universal lower bounds to the competitive ratio achievable by any online policy in the adversarial framework. In the stochastic setting, we show that a simple index policy is near-optimal for minimizing the average AoI in two different mobility scenarios. Further, we prove that the greedy scheduling policy minimizes the peak AoI for static users in the stochastic setting. Simulation results show that the proposed policies perform well under realistic conditions.

cs.IT↗

arXiv2021

Optimizing Age-of-Information in Adversarial Environments with Channel State Information

This paper considers a multi-user downlink scheduling problem with access to the channel state information at the transmitter (CSIT) to minimize the Age-of-Information (AoI) in a non-stationary environment. The non-stationary environment is modelled using a novel adversarial framework. In this setting, we propose a greedy scheduling policy, called MA-CSIT, that takes into account the current channel state information. We establish a finite upper bound on the competitive ratio achieved by the MA-CSIT policy for a small number of users and show that the proposed policy has a better performance guarantee than a recently proposed greedy scheduler that operates without CSIT. In particular, we show that access to the additional channel state information improves the competitive ratio from 8 to 2 in the two-user case and from 18 to 8/3 in the three-user case. Finally, we carry out extensive numerical simulations to quantify the advantage of knowing CSIT in order to minimize the Age-of-Information for an arbitrary number of users.

cs.IT↗

arXiv2020

Competitive Algorithms for Minimizing the Maximum Age-of-Information

In this short paper, we consider the problem of designing a near-optimal competitive scheduling policy for $N$ mobile users, to maximize the freshness of available information uniformly across all users. Prompted by the unreliability and non-stationarity of the emerging 5G-mmWave channels for high-speed users, we forego of any statistical assumptions of the wireless channels and user-mobility. Instead, we allow the channel states and the mobility patterns to be dictated by an omniscient adversary. It is not difficult to see that no competitive scheduling policy can exist for the corresponding throughput-maximization problem in this adversarial model. Surprisingly, we show that there exists a simple online distributed scheduling policy with a finite competitive ratio for maximizing the freshness of information in this adversarial model. Moreover, we also prove that the proposed policy is competitively optimal up to an $O(\ln N)$ factor.

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arXiv2020

Fundamental Limits of Online Network-Caching

Optimal caching of files in a content distribution network (CDN) is a problem of fundamental and growing commercial interest. Although many different caching algorithms are in use today, the fundamental performance limits of network caching algorithms from an online learning point-of-view remain poorly understood to date. In this paper, we resolve this question in the following two settings: (1) a single user connected to a single cache, and (2) a set of users and a set of caches interconnected through a bipartite network. Recently, an online gradient-based coded caching policy was shown to enjoy sub-linear regret. However, due to the lack of known regret lower bounds, the question of the optimality of the proposed policy was left open. In this paper, we settle this question by deriving tight non-asymptotic regret lower bounds in both of the above settings. In addition to that, we propose a new Follow-the-Perturbed-Leader-based uncoded caching policy with near-optimal regret. Technically, the lower-bounds are obtained by relating the online caching problem to the classic probabilistic paradigm of balls-into-bins. Our proofs make extensive use of a new result on the expected load in the most populated half of the bins, which might also be of independent interest. We evaluate the performance of the caching policies by experimenting with the popular MovieLens dataset and conclude the paper with design recommendations and a list of open problems.

cs.IT↗

arXiv2020

Fundamental Limits of Age-of-Information in Stationary and Non-stationary Environments

We study the multi-user scheduling problem for minimizing the Age of Information (AoI) in cellular wireless networks under stationary and non-stationary regimes. We derive fundamental lower bounds for the scheduling problem and design efficient online policies with provable performance guarantees. In the stationary setting, we consider the AoI optimization problem for a set of mobile users travelling around multiple cells. In this setting, we propose a scheduling policy and show that it is $2$-optimal. Next, we propose a new adversarial channel model for studying the scheduling problem in non-stationary environments. For $N$ users, we show that the competitive ratio of any online scheduling policy in this setting is at least $Ω(N)$. We then propose an online policy and show that it achieves a competitive ratio of $O(N^2)$. Finally, we introduce a relaxed adversarial model with channel state estimations for the immediate future. We propose a heuristic model predictive control policy that exploits this feature and compare its performance through numerical simulations.

cs.IT↗

arXiv2019

Online Algorithms for Multiclass Classification using Partial Labels

In this paper, we propose online algorithms for multiclass classification using partial labels. We propose two variants of Perceptron called Avg Perceptron and Max Perceptron to deal with the partial labeled data. We also propose Avg Pegasos and Max Pegasos, which are extensions of Pegasos algorithm. We also provide mistake bounds for Avg Perceptron and regret bound for Avg Pegasos. We show the effectiveness of the proposed approaches by experimenting on various datasets and comparing them with the standard Perceptron and Pegasos.

cs.LG↗