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Source author record

Garvesh Raskutti

Garvesh Raskutti appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Machine Learningmath.STStatistics TheoryMethodologyInformation Theorymath.ITApplicationsmath.NANumerical Analysis

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Published work

16 published item(s)

preprint2026arXiv

Online Localized Conformal Prediction

Conformal prediction is a framework that provides valid uncertainty quantification for general models with exchangeable data. However, in the online learning and time-series settings, exchangeability is not satisfied. Existing online conformal methods, such as adaptive conformal inference (ACI), can achieve long-run validity, yet they remain inefficient under covariate heterogeneity because they rely on global calibration. We propose \emph{Online Localized Conformal Prediction (OLCP)}, which combines online adaptation with covariate-dependent localization to better reflect heterogeneity. To reduce sensitivity to the localization bandwidth, we further develop \emph{OLCP-Hedge}, which performs bandwidth selection as an online expert aggregation problem using a constrained online convex optimization framework. Importantly, we provide coverage guarantees for both algorithms and demonstrate through simulations and real-data experiments that the proposed methods attain valid long-run coverage with narrower prediction sets than existing baselines.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Lazy Estimation of Variable Importance for Large Neural Networks

As opaque predictive models increasingly impact many areas of modern life, interest in quantifying the importance of a given input variable for making a specific prediction has grown. Recently, there has been a proliferation of model-agnostic methods to measure variable importance (VI) that analyze the difference in predictive power between a full model trained on all variables and a reduced model that excludes the variable(s) of interest. A bottleneck common to these methods is the estimation of the reduced model for each variable (or subset of variables), which is an expensive process that often does not come with theoretical guarantees. In this work, we propose a fast and flexible method for approximating the reduced model with important inferential guarantees. We replace the need for fully retraining a wide neural network by a linearization initialized at the full model parameters. By adding a ridge-like penalty to make the problem convex, we prove that when the ridge penalty parameter is sufficiently large, our method estimates the variable importance measure with an error rate of $O(\frac{1}{\sqrt{n}})$ where $n$ is the number of training samples. We also show that our estimator is asymptotically normal, enabling us to provide confidence bounds for the VI estimates. We demonstrate through simulations that our method is fast and accurate under several data-generating regimes, and we demonstrate its real-world applicability on a seasonal climate forecasting example.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Context-dependent self-exciting point processes: models, methods, and risk bounds in high dimensions

High-dimensional autoregressive point processes model how current events trigger or inhibit future events, such as activity by one member of a social network can affect the future activity of his or her neighbors. While past work has focused on estimating the underlying network structure based solely on the times at which events occur on each node of the network, this paper examines the more nuanced problem of estimating context-dependent networks that reflect how features associated with an event (such as the content of a social media post) modulate the strength of influences among nodes. Specifically, we leverage ideas from compositional time series and regularization methods in machine learning to conduct network estimation for high-dimensional marked point processes. Two models and corresponding estimators are considered in detail: an autoregressive multinomial model suited to categorical marks and a logistic-normal model suited to marks with mixed membership in different categories. Importantly, the logistic-normal model leads to a convex negative log-likelihood objective and captures dependence across categories. We provide theoretical guarantees for both estimators, which we validate by simulations and a synthetic data-generating model. We further validate our methods through two real data examples and demonstrate the advantages and disadvantages of both approaches.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Convex and Non-convex Approaches for Statistical Inference with Class-Conditional Noisy Labels

We study the problem of estimation and testing in logistic regression with class-conditional noise in the observed labels, which has an important implication in the Positive-Unlabeled (PU) learning setting. With the key observation that the label noise problem belongs to a special sub-class of generalized linear models (GLM), we discuss convex and non-convex approaches that address this problem. A non-convex approach based on the maximum likelihood estimation produces an estimator with several optimal properties, but a convex approach has an obvious advantage in optimization. We demonstrate that in the low-dimensional setting, both estimators are consistent and asymptotically normal, where the asymptotic variance of the non-convex estimator is smaller than the convex counterpart. We also quantify the efficiency gap which provides insight into when the two methods are comparable. In the high-dimensional setting, we show that both estimation procedures achieve $\ell_2$-consistency at the minimax optimal $\sqrt{s\log p/n}$ rates under mild conditions. Finally, we propose an inference procedure using a de-biasing approach. We validate our theoretical findings through simulations and a real-data example.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

ISLET: Fast and Optimal Low-rank Tensor Regression via Importance Sketching

In this paper, we develop a novel procedure for low-rank tensor regression, namely \emph{\underline{I}mportance \underline{S}ketching \underline{L}ow-rank \underline{E}stimation for \underline{T}ensors} (ISLET). The central idea behind ISLET is \emph{importance sketching}, i.e., carefully designed sketches based on both the responses and low-dimensional structure of the parameter of interest. We show that the proposed method is sharply minimax optimal in terms of the mean-squared error under low-rank Tucker assumptions and under randomized Gaussian ensemble design. In addition, if a tensor is low-rank with group sparsity, our procedure also achieves minimax optimality. Further, we show through numerical study that ISLET achieves comparable or better mean-squared error performance to existing state-of-the-art methods while having substantial storage and run-time advantages including capabilities for parallel and distributed computing. In particular, our procedure performs reliable estimation with tensors of dimension $p = O(10^8)$ and is $1$ or $2$ orders of magnitude faster than baseline methods.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

The bias of isotonic regression

We study the bias of the isotonic regression estimator. While there is extensive work characterizing the mean squared error of the isotonic regression estimator, relatively little is known about the bias. In this paper, we provide a sharp characterization, proving that the bias scales as $O(n^{-β/3})$ up to log factors, where $1 \leq β\leq 2$ is the exponent corresponding to H{ö}lder smoothness of the underlying mean. Importantly, this result only requires a strictly monotone mean and that the noise distribution has subexponential tails, without relying on symmetric noise or other restrictive assumptions.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Identifiability Assumptions and Algorithm for Directed Graphical Models with Feedback

Directed graphical models provide a useful framework for modeling causal or directional relationships for multivariate data. Prior work has largely focused on identifiability and search algorithms for directed acyclic graphical (DAG) models. In many applications, feedback naturally arises and directed graphical models that permit cycles occur. In this paper we address the issue of identifiability for general directed cyclic graphical (DCG) models satisfying the Markov assumption. In particular, in addition to the faithfulness assumption which has already been introduced for cyclic models, we introduce two new identifiability assumptions, one based on selecting the model with the fewest edges and the other based on selecting the DCG model that entails the maximum number of d-separation rules. We provide theoretical results comparing these assumptions which show that: (1) selecting models with the largest number of d-separation rules is strictly weaker than the faithfulness assumption; (2) unlike for DAG models, selecting models with the fewest edges does not necessarily result in a milder assumption than the faithfulness assumption. We also provide connections between our two new principles and minimality assumptions. We use our identifiability assumptions to develop search algorithms for small-scale DCG models. Our simulation study supports our theoretical results, showing that the algorithms based on our two new principles generally out-perform algorithms based on the faithfulness assumption in terms of selecting the true skeleton for DCG models.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Non-Convex Projected Gradient Descent for Generalized Low-Rank Tensor Regression

In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying optimization problems are often non-convex. While convex relaxations could lead to polynomial-time algorithms they are often slow in practice. On the other hand, limited theoretical guarantees exist for non-convex methods. In this paper we provide a general framework that provides theoretical guarantees for learning high-dimensional tensor regression models under different low-rank structural assumptions using the projected gradient descent algorithm applied to a potentially non-convex constraint set $Θ$ in terms of its \emph{localized Gaussian width}. We juxtapose our theoretical results for non-convex projected gradient descent algorithms with previous results on regularized convex approaches. The two main differences between the convex and non-convex approach are: (i) from a computational perspective whether the non-convex projection operator is computable and whether the projection has desirable contraction properties and (ii) from a statistical upper bound perspective, the non-convex approach has a superior rate for a number of examples. We provide three concrete examples of low-dimensional structure which address these issues and explain the pros and cons for the non-convex and convex approaches. We supplement our theoretical results with simulations which show that, under several common settings of generalized low rank tensor regression, the projected gradient descent approach is superior both in terms of statistical error and run-time provided the step-sizes of the projected descent algorithm are suitably chosen.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

A Statistical Perspective on Randomized Sketching for Ordinary Least-Squares

We consider statistical as well as algorithmic aspects of solving large-scale least-squares (LS) problems using randomized sketching algorithms. For a LS problem with input data $(X, Y) \in \mathbb{R}^{n \times p} \times \mathbb{R}^n$, sketching algorithms use a sketching matrix, $S\in\mathbb{R}^{r \times n}$ with $r \ll n$. Then, rather than solving the LS problem using the full data $(X,Y)$, sketching algorithms solve the LS problem using only the sketched data $(SX, SY)$. Prior work has typically adopted an algorithmic perspective, in that it has made no statistical assumptions on the input $X$ and $Y$, and instead it has been assumed that the data $(X,Y)$ are fixed and worst-case (WC). Prior results show that, when using sketching matrices such as random projections and leverage-score sampling algorithms, with $p < r \ll n$, the WC error is the same as solving the original problem, up to a small constant. From a statistical perspective, we typically consider the mean-squared error performance of randomized sketching algorithms, when data $(X, Y)$ are generated according to a statistical model $Y = X β+ ε$, where $ε$ is a noise process. We provide a rigorous comparison of both perspectives leading to insights on how they differ. To do this, we first develop a framework for assessing algorithmic and statistical aspects of randomized sketching methods. We then consider the statistical prediction efficiency (PE) and the statistical residual efficiency (RE) of the sketched LS estimator; and we use our framework to provide upper bounds for several types of random projection and random sampling sketching algorithms. Among other results, we show that the RE can be upper bounded when $p < r \ll n$ while the PE typically requires the sample size $r$ to be substantially larger. Lower bounds developed in subsequent results show that our upper bounds on PE can not be improved.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Statistical and Algorithmic Perspectives on Randomized Sketching for Ordinary Least-Squares -- ICML

We consider statistical and algorithmic aspects of solving large-scale least-squares (LS) problems using randomized sketching algorithms. Prior results show that, from an \emph{algorithmic perspective}, when using sketching matrices constructed from random projections and leverage-score sampling, if the number of samples $r$ much smaller than the original sample size $n$, then the worst-case (WC) error is the same as solving the original problem, up to a very small relative error. From a \emph{statistical perspective}, one typically considers the mean-squared error performance of randomized sketching algorithms, when data are generated according to a statistical linear model. In this paper, we provide a rigorous comparison of both perspectives leading to insights on how they differ. To do this, we first develop a framework for assessing, in a unified manner, algorithmic and statistical aspects of randomized sketching methods. We then consider the statistical prediction efficiency (PE) and the statistical residual efficiency (RE) of the sketched LS estimator; and we use our framework to provide upper bounds for several types of random projection and random sampling algorithms. Among other results, we show that the RE can be upper bounded when $r$ is much smaller than $n$, while the PE typically requires the number of samples $r$ to be substantially larger. Lower bounds developed in subsequent work show that our upper bounds on PE can not be improved.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Minimax Optimal Rates for Poisson Inverse Problems with Physical Constraints

This paper considers fundamental limits for solving sparse inverse problems in the presence of Poisson noise with physical constraints. Such problems arise in a variety of applications, including photon-limited imaging systems based on compressed sensing. Most prior theoretical results in compressed sensing and related inverse problems apply to idealized settings where the noise is i.i.d., and do not account for signal-dependent noise and physical sensing constraints. Prior results on Poisson compressed sensing with signal-dependent noise and physical constraints provided upper bounds on mean squared error performance for a specific class of estimators. However, it was unknown whether those bounds were tight or if other estimators could achieve significantly better performance. This work provides minimax lower bounds on mean-squared error for sparse Poisson inverse problems under physical constraints. Our lower bounds are complemented by minimax upper bounds. Our upper and lower bounds reveal that due to the interplay between the Poisson noise model, the sparsity constraint and the physical constraints: (i) the mean-squared error does not depend on the sample size $n$ other than to ensure the sensing matrix satisfies RIP-like conditions and the intensity $T$ of the input signal plays a critical role; and (ii) the mean-squared error has two distinct regimes, a low-intensity and a high-intensity regime and the transition point from the low-intensity to high-intensity regime depends on the input signal $f^*$. In the low-intensity regime the mean-squared error is independent of $T$ while in the high-intensity regime, the mean-squared error scales as $\frac{s \log p}{T}$, where $s$ is the sparsity level, $p$ is the number of pixels or parameters and $T$ is the signal intensity.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

The Information Geometry of Mirror Descent

Information geometry applies concepts in differential geometry to probability and statistics and is especially useful for parameter estimation in exponential families where parameters are known to lie on a Riemannian manifold. Connections between the geometric properties of the induced manifold and statistical properties of the estimation problem are well-established. However developing first-order methods that scale to larger problems has been less of a focus in the information geometry community. The best known algorithm that incorporates manifold structure is the second-order natural gradient descent algorithm introduced by Amari. On the other hand, stochastic approximation methods have led to the development of first-order methods for optimizing noisy objective functions. A recent generalization of the Robbins-Monro algorithm known as mirror descent, developed by Nemirovski and Yudin is a first order method that induces non-Euclidean geometries. However current analysis of mirror descent does not precisely characterize the induced non-Euclidean geometry nor does it consider performance in terms of statistical relative efficiency. In this paper, we prove that mirror descent induced by Bregman divergences is equivalent to the natural gradient descent algorithm on the dual Riemannian manifold. Using this equivalence, it follows that (1) mirror descent is the steepest descent direction along the Riemannian manifold of the exponential family; (2) mirror descent with log-likelihood loss applied to parameter estimation in exponential families asymptotically achieves the classical Cramér-Rao lower bound and (3) natural gradient descent for manifolds corresponding to exponential families can be implemented as a first-order method through mirror descent.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Early stopping and non-parametric regression: An optimal data-dependent stopping rule

The strategy of early stopping is a regularization technique based on choosing a stopping time for an iterative algorithm. Focusing on non-parametric regression in a reproducing kernel Hilbert space, we analyze the early stopping strategy for a form of gradient-descent applied to the least-squares loss function. We propose a data-dependent stopping rule that does not involve hold-out or cross-validation data, and we prove upper bounds on the squared error of the resulting function estimate, measured in either the $L^2(P)$ and $L^2(P_n)$ norm. These upper bounds lead to minimax-optimal rates for various kernel classes, including Sobolev smoothness classes and other forms of reproducing kernel Hilbert spaces. We show through simulation that our stopping rule compares favorably to two other stopping rules, one based on hold-out data and the other based on Stein's unbiased risk estimate. We also establish a tight connection between our early stopping strategy and the solution path of a kernel ridge regression estimator.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Geometry of the faithfulness assumption in causal inference

Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient for statistical estimation, and strong-faithfulness has been proposed and assumed to achieve uniform or high-dimensional consistency. In contrast to the plain faithfulness assumption, the set of distributions that is not strong-faithful has nonzero Lebesgue measure and in fact, can be surprisingly large as we show in this paper. We study the strong-faithfulness condition from a geometric and combinatorial point of view and give upper and lower bounds on the Lebesgue measure of strong-faithful distributions for various classes of directed acyclic graphs. Our results imply fundamental limitations for the PC-algorithm and potentially also for other algorithms based on partial correlation testing in the Gaussian case.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Minimax-optimal rates for sparse additive models over kernel classes via convex programming

Sparse additive models are families of $d$-variate functions that have the additive decomposition $f^* = \sum_{j \in S} f^*_j$, where $S$ is an unknown subset of cardinality $s \ll d$. In this paper, we consider the case where each univariate component function $f^*_j$ lies in a reproducing kernel Hilbert space (RKHS), and analyze a method for estimating the unknown function $f^*$ based on kernels combined with $\ell_1$-type convex regularization. Working within a high-dimensional framework that allows both the dimension $d$ and sparsity $s$ to increase with $n$, we derive convergence rates (upper bounds) in the $L^2(\mathbb{P})$ and $L^2(\mathbb{P}_n)$ norms over the class $\MyBigClass$ of sparse additive models with each univariate function $f^*_j$ in the unit ball of a univariate RKHS with bounded kernel function. We complement our upper bounds by deriving minimax lower bounds on the $L^2(\mathbb{P})$ error, thereby showing the optimality of our method. Thus, we obtain optimal minimax rates for many interesting classes of sparse additive models, including polynomials, splines, and Sobolev classes. We also show that if, in contrast to our univariate conditions, the multivariate function class is assumed to be globally bounded, then much faster estimation rates are possible for any sparsity $s = Ω(\sqrt{n})$, showing that global boundedness is a significant restriction in the high-dimensional setting.

0 citations0 reviewsNo code linkedOpen access
preprint2009arXiv

Minimax rates of estimation for high-dimensional linear regression over $\ell_q$-balls

Consider the standard linear regression model $\y = \Xmat \betastar + w$, where $\y \in \real^\numobs$ is an observation vector, $\Xmat \in \real^{\numobs \times \pdim}$ is a design matrix, $\betastar \in \real^\pdim$ is the unknown regression vector, and $w \sim \mathcal{N}(0, σ^2 I)$ is additive Gaussian noise. This paper studies the minimax rates of convergence for estimation of $\betastar$ for $\ell_\rpar$-losses and in the $\ell_2$-prediction loss, assuming that $\betastar$ belongs to an $\ell_{\qpar}$-ball $\Ballq(\myrad)$ for some $\qpar \in [0,1]$. We show that under suitable regularity conditions on the design matrix $\Xmat$, the minimax error in $\ell_2$-loss and $\ell_2$-prediction loss scales as $\Rq \big(\frac{\log \pdim}{n}\big)^{1-\frac{\qpar}{2}}$. In addition, we provide lower bounds on minimax risks in $\ell_{\rpar}$-norms, for all $\rpar \in [1, +\infty], \rpar \neq \qpar$. Our proofs of the lower bounds are information-theoretic in nature, based on Fano's inequality and results on the metric entropy of the balls $\Ballq(\myrad)$, whereas our proofs of the upper bounds are direct and constructive, involving direct analysis of least-squares over $\ell_{\qpar}$-balls. For the special case $q = 0$, a comparison with $\ell_2$-risks achieved by computationally efficient $\ell_1$-relaxations reveals that although such methods can achieve the minimax rates up to constant factors, they require slightly stronger assumptions on the design matrix $\Xmat$ than algorithms involving least-squares over the $\ell_0$-ball.

0 citations0 reviewsNo code linkedOpen access