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 $O(n\epsilon^{-2})$ queries to a degree and neighbor oracle and in $O(n\epsilon^{-3})$ time, estimates the spectrum up to $\epsilon$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $O(n\epsilon^{-7})$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $\epsilon$, a $2^{O(\epsilon^{-1})}$ 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 $O(n\epsilon^{-2})$-query and $O(n\epsilon^{-2})$-time algorithm for computing $O(n\epsilon^{-2})$-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).

Navigable Graphs for High-Dimensional Nearest Neighbor Search: Constructions and Limits

There has been significant recent interest in graph-based nearest neighbor search methods, many of which are centered on the construction of navigable graphs over high-dimensional point sets. A graph is navigable if we can successfully move from any starting node to any target node using a greedy routing strategy where we always move to the neighbor that is closest to the destination according to a given distance function. The complete graph is navigable for any point set, but the important question for applications is if sparser graphs can be constructed. While this question is fairly well understood in low-dimensions, we establish some of the first upper and lower bounds for high-dimensional point sets. First, we give a simple and efficient way to construct a navigable graph with average degree $O(\sqrt{n \log n })$ for any set of $n$ points, in any dimension, for any distance function. We compliment this result with a nearly matching lower bound: even under the Euclidean metric in $O(\log n)$ dimensions, a random point set has no navigable graph with average degree $O(n^{\alpha})$ for any $\alpha < 1/2$. Our lower bound relies on sharp anti-concentration bounds for binomial random variables, which we use to show that the near-neighborhoods of a set of random points do not overlap significantly, forcing any navigable graph to have many edges.

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.

Faster Linear Systems and Matrix Norm Approximation via Multi-level Sketched Preconditioning

We present a new class of preconditioned iterative methods for solving linear systems of the form $Ax = b$. Our methods are based on constructing a low-rank Nystr\"om approximation to $A$ using sparse random sketching. This approximation is used to construct a preconditioner, which itself is inverted quickly using additional levels of random sketching and preconditioning. We prove that the convergence of our methods depends on a natural average condition number of $A$, which improves as the rank of the Nystr\"om approximation increases. Concretely, this allows us to obtain faster runtimes for a number of fundamental linear algebraic problems: 1. We show how to solve any $n\times n$ linear system that is well-conditioned except for $k$ outlying large singular values in $\tilde{O}(n^{2.065} + k^\omega)$ time, improving on a recent result of [Derezi\'nski, Yang, STOC 2024] for all $k \gtrsim n^{0.78}$. 2. We give the first $\tilde{O}(n^2 + {d_\lambda}^{\omega}$) time algorithm for solving a regularized linear system $(A + \lambda I)x = b$, where $A$ is positive semidefinite with effective dimension $d_\lambda$. This problem arises in applications like Gaussian process regression. 3. We give faster algorithms for approximating Schatten $p$-norms and other matrix norms. For example, for the Schatten 1 (nuclear) norm, we give an algorithm that runs in $\tilde{O}(n^{2.11})$ time, improving on an $\tilde{O}(n^{2.18})$ method of [Musco et al., ITCS 2018]. Interestingly, previous state-of-the-art algorithms for most of the problems above relied on stochastic iterative methods, like stochastic coordinate and gradient descent. Our work takes a completely different approach, instead leveraging tools from matrix sketching.

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.

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 $\epsilon$-optimal diagonal preconditioner in time $\widetilde{O}(\mathrm{nnz}(\mathbf{K}) \cdot \mathrm{poly}(\kappa^\star,\epsilon^{-1}))$, where $\kappa^\star$ 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 $\Omega(d^{3.5})$ attained by general-purpose semidefinite programming, and our solvers improve state-of-the-art runtimes of $\Omega(d^{\omega})$ where $\omega > 2.3$ 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.

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 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 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 of $O(d)$ samples.

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 $\epsilon$ in Wasserstein-1 distance even if we know \emph{all} of their moments to multiplicative accuracy $(1\pm2^{-\Omega(1/\epsilon)})$; 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 $2^{O(1/\epsilon)}$ random walks initiated at uniformly random nodes in the graph. As a strengthening of our main result, we show that improving the dependence on $1/\epsilon$ in this result would require a new algorithmic approach. Specifically, no algorithm can compute an $\epsilon$-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability, even when given the transcript of $2^{\Omega(1/\epsilon)}$ random walks of length $2^{\Omega(1/\epsilon)}$ started at random nodes.

Dimensionality Reduction for General KDE Mode Finding

Finding the mode of a high dimensional probability distribution $D$ is a fundamental algorithmic problem in statistics and data analysis. There has been particular interest in efficient methods for solving the problem when $D$ is represented as a mixture model or kernel density estimate, although few algorithmic results with worst-case approximation and runtime guarantees are known. In this work, we significantly generalize a result of (LeeLiMusco:2021) on mode approximation for Gaussian mixture models. We develop randomized dimensionality reduction methods for mixtures involving a broader class of kernels, including the popular logistic, sigmoid, and generalized Gaussian kernels. As in Lee et al.'s work, our dimensionality reduction results yield quasi-polynomial algorithms for mode finding with multiplicative accuracy $(1-\epsilon)$ for any $\epsilon > 0$. Moreover, when combined with gradient descent, they yield efficient practical heuristics for the problem. In addition to our positive results, we prove a hardness result for box kernels, showing that there is no polynomial time algorithm for finding the mode of a kernel density estimate, unless $\mathit{P} = \mathit{NP}$. Obtaining similar hardness results for kernels used in practice (like Gaussian or logistic kernels) is an interesting future direction.

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.