arXiv subjects

Christopher Musco

Publications and source records attributed to Christopher Musco.

At least 37 records · Page 2Linked to original sources

arXiv2025

Kernel Banzhaf: A Fast and Robust Estimator for Banzhaf Values

Banzhaf values provide a popular, interpretable alternative to the widely-used Shapley values for quantifying the importance of features in machine learning models. Like Shapley values, computing Banzhaf values exactly requires time exponential in the number of features, necessitating the use of efficient estimators. Existing estimators, however, are limited to Monte Carlo sampling methods. In this work, we introduce Kernel Banzhaf, the first regression-based estimator for Banzhaf values. Our approach leverages a novel regression formulation, whose exact solution corresponds to the exact Banzhaf values. Inspired by the success of Kernel SHAP for Shapley values, Kernel Banzhaf efficiently solves a sampled instance of this regression problem. Through empirical evaluations across eight datasets, we find that Kernel Banzhaf significantly outperforms existing Monte Carlo methods in terms of accuracy, sample efficiency, robustness to noise, and feature ranking recovery. Finally, we complement our experimental evaluation with strong theoretical guarantees on Kernel Banzhaf's performance.

cs.LG↗

arXiv2025

Matrix Product Sketching via Coordinated Sampling

We revisit the well-studied problem of approximating a matrix product, $\mathbf{A}^T\mathbf{B}$, based on small space sketches $\mathcal{S}(\mathbf{A})$ and $\mathcal{S}(\mathbf{B})$ of $\mathbf{A} \in \R^{n \times d}$ and $\mathbf{B}\in \R^{n \times m}$. We are interested in the setting where the sketches must be computed independently of each other, except for the use of a shared random seed. We prove that, when $\mathbf{A}$ and $\mathbf{B}$ are sparse, methods based on \emph{coordinated random sampling} can outperform classical linear sketching approaches, like Johnson-Lindenstrauss Projection or CountSketch. For example, to obtain Frobenius norm error $ε\|\mathbf{A}\|_F\|\mathbf{B}\|_F$, coordinated sampling requires sketches of size $O(s/ε^2)$ when $\mathbf{A}$ and $\mathbf{B}$ have at most $s \leq d,m$ non-zeros per row. In contrast, linear sketching leads to sketches of size $O(d/ε^2)$ and $O(m/ε^2)$ for $\mathbf{A}$ and $\mathbf{B}$. We empirically evaluate our approach on two applications: 1) distributed linear regression in databases, a problem motivated by tasks like dataset discovery and augmentation, and 2) approximating attention matrices in transformer-based language models. In both cases, our sampling algorithms yield an order of magnitude improvement over linear sketching.

cs.DS↗

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.

cs.DS↗

arXiv2024

Stability of the Lanczos Method for Matrix Function Approximation

The ubiquitous Lanczos method can approximate $f(A)x$ for any symmetric $n \times n$ matrix $A$, vector $x$, and function $f$. In exact arithmetic, the method's error after $k$ iterations is bounded by the error of the best degree-$k$ polynomial uniformly approximating $f(x)$ on the range $[λ_{min}(A), λ_{max}(A)]$. However, despite decades of work, it has been unclear if this powerful guarantee holds in finite precision. We resolve this problem, proving that when $\max_{x \in [λ_{min}, λ_{max}]}|f(x)| \le C$, Lanczos essentially matches the exact arithmetic guarantee if computations use roughly $\log(nC\|A\|)$ bits of precision. Our proof extends work of Druskin and Knizhnerman [DK91], leveraging the stability of the classic Chebyshev recurrence to bound the stability of any polynomial approximating $f(x)$. We also study the special case of $f(A) = A^{-1}$, where stronger guarantees hold. In exact arithmetic Lanczos performs as well as the best polynomial approximating $1/x$ at each of $A$'s eigenvalues, rather than on the full eigenvalue range. In seminal work, Greenbaum gives an approach to extending this bound to finite precision: she proves that finite precision Lanczos and the related CG method match any polynomial approximating $1/x$ in a tiny range around each eigenvalue [Gre89]. For $A^{-1}$, this bound appears stronger than ours. However, we exhibit matrices with condition number $κ$ where exact arithmetic Lanczos converges in $polylog(κ)$ iterations, but Greenbaum's bound predicts $Ω(κ^{1/5})$ iterations. It thus cannot offer significant improvement over the $O(κ^{1/2})$ bound achievable via our result. Our analysis raises the question of if convergence in less than $poly(κ)$ iterations can be expected in finite precision, even for matrices with clustered, skewed, or otherwise favorable eigenvalue distributions.

cs.DS↗

arXiv2024

Near-optimal hierarchical matrix approximation from matrix-vector products

We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an $n\times n$ matrix $\mathbf{A}$, accessible only though matrix-vector products with $\mathbf{A}$ and $\mathbf{A}^{\mathsf{T}}$. We prove that, for the rank-$k$ HODLR approximation problem, our method achieves a $(1+β)^{\log(n)}$-optimal approximation in expected Frobenius norm using $O(k\log(n)/β^3)$ matrix-vector products. In particular, the algorithm obtains a $(1+\varepsilon)$-optimal approximation with $O(k\log^4(n)/\varepsilon^3)$ matrix-vector products, and for any constant $c$, an $n^c$-optimal approximation with $O(k \log(n))$ matrix-vector products. Apart from matrix-vector products, the additional computational cost of our method is just $O(n \operatorname{poly}(\log(n), k, β))$. We complement the upper bound with a lower bound, which shows that any matrix-vector query algorithm requires at least $Ω(k\log(n) + k/\varepsilon)$ queries to obtain a $(1+\varepsilon)$-optimal approximation. Our algorithm can be viewed as a robust version of widely used "peeling" methods for recovering HODLR matrices and is, to the best of our knowledge, the first matrix-vector query algorithm to enjoy theoretical worst-case guarantees for approximation by any hierarchical matrix class. To control the propagation of error between levels of hierarchical approximation, we introduce a new perturbation bound for low-rank approximation, which shows that the widely used Generalized Nyström method enjoys inherent stability when implemented with noisy matrix-vector products. We also introduce a novel randomly perforated matrix sketching method to further control the error in the peeling algorithm.

cs.DS↗

arXiv2024

Benchmarking Estimators for Natural Experiments: A Novel Dataset and a Doubly Robust Algorithm

Estimating the effect of treatments from natural experiments, where treatments are pre-assigned, is an important and well-studied problem. We introduce a novel natural experiment dataset obtained from an early childhood literacy nonprofit. Surprisingly, applying over 20 established estimators to the dataset produces inconsistent results in evaluating the nonprofit's efficacy. To address this, we create a benchmark to evaluate estimator accuracy using synthetic outcomes, whose design was guided by domain experts. The benchmark extensively explores performance as real world conditions like sample size, treatment correlation, and propensity score accuracy vary. Based on our benchmark, we observe that the class of doubly robust treatment effect estimators, which are based on simple and intuitive regression adjustment, generally outperform other more complicated estimators by orders of magnitude. To better support our theoretical understanding of doubly robust estimators, we derive a closed form expression for the variance of any such estimator that uses dataset splitting to obtain an unbiased estimate. This expression motivates the design of a new doubly robust estimator that uses a novel loss function when fitting functions for regression adjustment. We release the dataset and benchmark in a Python package; the package is built in a modular way to facilitate new datasets and estimators.

stat.ML↗

arXiv2024

Sampling Methods for Inner Product Sketching

Recently, Bessa et al. (PODS 2023) showed that sketches based on coordinated weighted sampling theoretically and empirically outperform popular linear sketching methods like Johnson-Lindentrauss projection and CountSketch for the ubiquitous problem of inner product estimation. We further develop this finding by introducing and analyzing two alternative sampling-based methods. In contrast to the computationally expensive algorithm in Bessa et al., our methods run in linear time (to compute the sketch) and perform better in practice, significantly beating linear sketching on a variety of tasks. For example, they provide state-of-the-art results for estimating the correlation between columns in unjoined tables, a problem that we show how to reduce to inner product estimation in a black-box way. While based on known sampling techniques (threshold and priority sampling) we introduce significant new theoretical analysis to prove approximation guarantees for our methods.

cs.DB↗

arXiv2024

Simple Analysis of Priority Sampling

We prove a tight upper bound on the variance of the priority sampling method (aka sequential Poisson sampling). Our proof is significantly shorter and simpler than the original proof given by Mario Szegedy at STOC 2006, which resolved a conjecture by Duffield, Lund, and Thorup.

cs.DS↗

arXiv2024

Agnostic Active Learning of Single Index Models with Linear Sample Complexity

We study active learning methods for single index models of the form $F({\mathbf x}) = f(\langle {\mathbf w}, {\mathbf x}\rangle)$, where $f:\mathbb{R} \to \mathbb{R}$ and ${\mathbf x,\mathbf w} \in \mathbb{R}^d$. In addition to their theoretical interest as simple examples of non-linear neural networks, single index models have received significant recent attention due to applications in scientific machine learning like surrogate modeling for partial differential equations (PDEs). Such applications require sample-efficient active learning methods that are robust to adversarial noise. I.e., that work even in the challenging agnostic learning setting. We provide two main results on agnostic active learning of single index models. First, when $f$ is known and Lipschitz, we show that $\tilde{O}(d)$ samples collected via {statistical leverage score sampling} are sufficient to learn a near-optimal single index model. Leverage score sampling is simple to implement, efficient, and already widely used for actively learning linear models. Our result requires no assumptions on the data distribution, is optimal up to log factors, and improves quadratically on a recent ${O}(d^{2})$ bound of \cite{gajjar2023active}. Second, we show that $\tilde{O}(d)$ samples suffice even in the more difficult setting when $f$ is \emph{unknown}. Our results leverage tools from high dimensional probability, including Dudley's inequality and dual Sudakov minoration, as well as a novel, distribution-aware discretization of the class of Lipschitz functions.

cs.LG↗

arXiv2024

Algorithm-agnostic low-rank approximation of operator monotone matrix functions

Low-rank approximation of a matrix function, $f(A)$, is an important task in computational mathematics. Most methods require direct access to $f(A)$, which is often considerably more expensive than accessing $A$. Persson and Kressner (SIMAX 2023) avoid this issue for symmetric positive semidefinite matrices by proposing funNyström, which first constructs a Nyström approximation to $A$ using subspace iteration, and then uses the approximation to directly obtain a low-rank approximation for $f(A)$. They prove that the method yields a near-optimal approximation whenever $f$ is a continuous operator monotone function with $f(0) = 0$. We significantly generalize the results of Persson and Kressner beyond subspace iteration. We show that if $\widehat{A}$ is a near-optimal low-rank Nyström approximation to $A$ then $f(\widehat{A})$ is a near-optimal low-rank approximation to $f(A)$, independently of how $\widehat{A}$ is computed. Further, we show sufficient conditions for a basis $Q$ to produce a near-optimal Nyström approximation $\widehat{A} = AQ(Q^T AQ)^{\dagger} Q^T A$. We use these results to establish that many common low-rank approximation methods produce near-optimal Nyström approximations to $A$ and therefore to $f(A)$.

math.NA↗

arXiv2024

Faster Spectral Density Estimation and Sparsification in the Nuclear Norm

We consider the problem of estimating the spectral density of the normalized adjacency matrix of an $n$-node undirected graph. We provide a randomized algorithm that, with $O(nε^{-2})$ queries to a degree and neighbor oracle and in $O(nε^{-3})$ time, estimates the spectrum up to $ε$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $O(nε^{-7})$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $ε$, a $2^{O(ε^{-1})}$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call nuclear sparsification. We provide an $O(nε^{-2})$-query and $O(nε^{-2})$-time algorithm for computing $O(nε^{-2})$-sparse nuclear sparsifiers. We show that this bound is optimal in both its sparsity and query complexity, and we separate our results from the related notion of additive spectral sparsification. Of independent interest, we show that our sparsification method also yields the first deterministic algorithm for spectral density estimation that scales linearly with $n$ (sublinear in the representation size of the graph).

cs.DS↗

arXiv2024

Improved Active Learning via Dependent Leverage Score Sampling

We show how to obtain improved active learning methods in the agnostic (adversarial noise) setting by combining marginal leverage score sampling with non-independent sampling strategies that promote spatial coverage. In particular, we propose an easily implemented method based on the \emph{pivotal sampling algorithm}, which we test on problems motivated by learning-based methods for parametric PDEs and uncertainty quantification. In comparison to independent sampling, our method reduces the number of samples needed to reach a given target accuracy by up to $50\%$. We support our findings with two theoretical results. First, we show that any non-independent leverage score sampling method that obeys a weak \emph{one-sided $\ell_{\infty}$ independence condition} (which includes pivotal sampling) can actively learn $d$ dimensional linear functions with $O(d\log d)$ samples, matching independent sampling. This result extends recent work on matrix Chernoff bounds under $\ell_{\infty}$ independence, and may be of interest for analyzing other sampling strategies beyond pivotal sampling. Second, we show that, for the important case of polynomial regression, our pivotal method obtains an improved bound on $O(d)$ samples.

cs.LG↗

arXiv2024

Fixed-sparsity matrix approximation from matrix-vector products

We study the problem of approximating a matrix $\mathbf{A}$ with a matrix that has a fixed sparsity pattern (e.g., diagonal, banded, etc.), when $\mathbf{A}$ is accessed only by matrix-vector products. We describe a simple randomized algorithm that returns an approximation with the given sparsity pattern with Frobenius-norm error at most $(1+\varepsilon)$ times the best possible error. When each row of the desired sparsity pattern has at most $s$ nonzero entries, this algorithm requires $O(s/\varepsilon)$ non-adaptive matrix-vector products with $\mathbf{A}$. We also prove a matching lower-bound, showing that, for any sparsity pattern with $Θ(s)$ nonzeros per row and column, any algorithm achieving $(1+ε)$ approximation requires $Ω(s/\varepsilon)$ matrix-vector products in the worst case. We thus resolve the matrix-vector product query complexity of the problem up to constant factors, even for the well-studied case of diagonal approximation, for which no previous lower bounds were known.

cs.DS↗

arXiv2023

A Simple and Practical Method for Reducing the Disparate Impact of Differential Privacy

Differentially private (DP) mechanisms have been deployed in a variety of high-impact social settings (perhaps most notably by the U.S. Census). Since all DP mechanisms involve adding noise to results of statistical queries, they are expected to impact our ability to accurately analyze and learn from data, in effect trading off privacy with utility. Alarmingly, the impact of DP on utility can vary significantly among different sub-populations. A simple way to reduce this disparity is with stratification. First compute an independent private estimate for each group in the data set (which may be the intersection of several protected classes), then, to compute estimates of global statistics, appropriately recombine these group estimates. Our main observation is that naive stratification often yields high-accuracy estimates of population-level statistics, without the need for additional privacy budget. We support this observation theoretically and empirically. Our theoretical results center on the private mean estimation problem, while our empirical results center on extensive experiments on private data synthesis to demonstrate the effectiveness of stratification on a variety of private mechanisms. Overall, we argue that this straightforward approach provides a strong baseline against which future work on reducing utility disparities of DP mechanisms should be compared.

cs.CR↗

arXiv2023

On the Unreasonable Effectiveness of Single Vector Krylov Methods for Low-Rank Approximation

Krylov subspace methods are a ubiquitous tool for computing near-optimal rank $k$ approximations of large matrices. While "large block" Krylov methods with block size at least $k$ give the best known theoretical guarantees, block size one (a single vector) or a small constant is often preferred in practice. Despite their popularity, we lack theoretical bounds on the performance of such "small block" Krylov methods for low-rank approximation. We address this gap between theory and practice by proving that small block Krylov methods essentially match all known low-rank approximation guarantees for large block methods. Via a black-box reduction we show, for example, that the standard single vector Krylov method run for $t$ iterations obtains the same spectral norm and Frobenius norm error bounds as a Krylov method with block size $\ell \geq k$ run for $O(t/\ell)$ iterations, up to a logarithmic dependence on the smallest gap between sequential singular values. That is, for a given number of matrix-vector products, single vector methods are essentially as effective as any choice of large block size. By combining our result with tail-bounds on eigenvalue gaps in random matrices, we prove that the dependence on the smallest singular value gap can be eliminated if the input matrix is perturbed by a small random matrix. Further, we show that single vector methods match the more complex algorithm of [Bakshi et al. `22], which combines the results of multiple block sizes to achieve an improved algorithm for Schatten $p$-norm low-rank approximation.

cs.DS↗

arXiv2023

Structured Semidefinite Programming for Recovering Structured Preconditioners

We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including the following. We give an algorithm which, given positive definite $\mathbf{K} \in \mathbb{R}^{d \times d}$ with $\mathrm{nnz}(\mathbf{K})$ nonzero entries, computes an $ε$-optimal diagonal preconditioner in time $\widetilde{O}(\mathrm{nnz}(\mathbf{K}) \cdot \mathrm{poly}(κ^\star,ε^{-1}))$, where $κ^\star$ is the optimal condition number of the rescaled matrix. We give an algorithm which, given $\mathbf{M} \in \mathbb{R}^{d \times d}$ that is either the pseudoinverse of a graph Laplacian matrix or a constant spectral approximation of one, solves linear systems in $\mathbf{M}$ in $\widetilde{O}(d^2)$ time. Our diagonal preconditioning results improve state-of-the-art runtimes of $Ω(d^{3.5})$ attained by general-purpose semidefinite programming, and our solvers improve state-of-the-art runtimes of $Ω(d^ω)$ where $ω> 2.3$ is the current matrix multiplication constant. We attain our results via new algorithms for a class of semidefinite programs (SDPs) we call matrix-dictionary approximation SDPs, which we leverage to solve an associated problem we call matrix-dictionary recovery.

cs.DS↗

arXiv2023

Active Learning for Single Neuron Models with Lipschitz Non-Linearities

We consider the problem of active learning for single neuron models, also sometimes called ``ridge functions'', in the agnostic setting (under adversarial label noise). Such models have been shown to be broadly effective in modeling physical phenomena, and for constructing surrogate data-driven models for partial differential equations. Surprisingly, we show that for a single neuron model with any Lipschitz non-linearity (such as the ReLU, sigmoid, absolute value, low-degree polynomial, among others), strong provable approximation guarantees can be obtained using a well-known active learning strategy for fitting \emph{linear functions} in the agnostic setting. % -- i.e. for the case when there is no non-linearity. Namely, we can collect samples via statistical \emph{leverage score sampling}, which has been shown to be near-optimal in other active learning scenarios. We support our theoretical results with empirical simulations showing that our proposed active learning strategy based on leverage score sampling outperforms (ordinary) uniform sampling when fitting single neuron models.

cs.LG↗

arXiv2023

Moments, Random Walks, and Limits for Spectrum Approximation

We study lower bounds for the problem of approximating a one dimensional distribution given (noisy) measurements of its moments. We show that there are distributions on $[-1,1]$ that cannot be approximated to accuracy $ε$ in Wasserstein-1 distance even if we know \emph{all} of their moments to multiplicative accuracy $(1\pm2^{-Ω(1/ε)})$; this result matches an upper bound of Kong and Valiant [Annals of Statistics, 2017]. To obtain our result, we provide a hard instance involving distributions induced by the eigenvalue spectra of carefully constructed graph adjacency matrices. Efficiently approximating such spectra in Wasserstein-1 distance is a well-studied algorithmic problem, and a recent result of Cohen-Steiner et al. [KDD 2018] gives a method based on accurately approximating spectral moments using $2^{O(1/ε)}$ random walks initiated at uniformly random nodes in the graph. As a strengthening of our main result, we show that improving the dependence on $1/ε$ in this result would require a new algorithmic approach. Specifically, no algorithm can compute an $ε$-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability, even when given the transcript of $2^{Ω(1/ε)}$ random walks of length $2^{Ω(1/ε)}$ started at random nodes.

cs.DS↗