arXiv subjects

Ryan J. Tibshirani

Publications and source records attributed to Ryan J. Tibshirani.

At least 55 records · Page 3Linked to original sources

arXiv2021

Kalman Filter, Sensor Fusion, and Constrained Regression: Equivalences and Insights

The Kalman filter (KF) is one of the most widely used tools for data assimilation and sequential estimation. In this work, we show that the state estimates from the KF in a standard linear dynamical system setting are equivalent to those given by the KF in a transformed system, with infinite process noise (i.e., a "flat prior") and an augmented measurement space. This reformulation -- which we refer to as augmented measurement sensor fusion (SF) -- is conceptually interesting, because the transformed system here is seemingly static (as there is effectively no process model), but we can still capture the state dynamics inherent to the KF by folding the process model into the measurement space. Further, this reformulation of the KF turns out to be useful in settings in which past states are observed eventually (at some lag). Here, when the measurement noise covariance is estimated by the empirical covariance, we show that the state predictions from SF are equivalent to those from a regression of past states on past measurements, subject to particular linear constraints (reflecting the relationships encoded in the measurement map). This allows us to port standard ideas (say, regularization methods) in regression over to dynamical systems. For example, we can posit multiple candidate process models, fold all of them into the measurement model, transform to the regression perspective, and apply $\ell_1$ penalization to perform process model selection. We give various empirical demonstrations, and focus on an application to nowcasting the weekly incidence of influenza in the US.

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arXiv2021

Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs

In this paper we study the statistical properties of Laplacian smoothing, a graph-based approach to nonparametric regression. Under standard regularity conditions, we establish upper bounds on the error of the Laplacian smoothing estimator $\widehat{f}$, and a goodness-of-fit test also based on $\widehat{f}$. These upper bounds match the minimax optimal estimation and testing rates of convergence over the first-order Sobolev class $H^1(\mathcal{X})$, for $\mathcal{X}\subseteq \mathbb{R}^d$ and $1 \leq d < 4$; in the estimation problem, for $d = 4$, they are optimal modulo a $\log n$ factor. Additionally, we prove that Laplacian smoothing is manifold-adaptive: if $\mathcal{X} \subseteq \mathbb{R}^d$ is an $m$-dimensional manifold with $m < d$, then the error rate of Laplacian smoothing (in either estimation or testing) depends only on $m$, in the same way it would if $\mathcal{X}$ were a full-dimensional set in $\mathbb{R}^d$.

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arXiv2020

Surprises in High-Dimensional Ridgeless Least Squares Interpolation

Interpolators -- estimators that achieve zero training error -- have attracted growing attention in machine learning, mainly because state-of-the art neural networks appear to be models of this type. In this paper, we study minimum $\ell_2$ norm ("ridgeless") interpolation in high-dimensional least squares regression. We consider two different models for the feature distribution: a linear model, where the feature vectors $x_i \in {\mathbb R}^p$ are obtained by applying a linear transform to a vector of i.i.d. entries, $x_i = Σ^{1/2} z_i$ (with $z_i \in {\mathbb R}^p$); and a nonlinear model, where the feature vectors are obtained by passing the input through a random one-layer neural network, $x_i = φ(W z_i)$ (with $z_i \in {\mathbb R}^d$, $W \in {\mathbb R}^{p \times d}$ a matrix of i.i.d. entries, and $φ$ an activation function acting componentwise on $W z_i$). We recover -- in a precise quantitative way -- several phenomena that have been observed in large-scale neural networks and kernel machines, including the "double descent" behavior of the prediction risk, and the potential benefits of overparametrization.

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arXiv2020

Conformal Prediction Under Covariate Shift

We extend conformal prediction methodology beyond the case of exchangeable data. In particular, we show that a weighted version of conformal prediction can be used to compute distribution-free prediction intervals for problems in which the test and training covariate distributions differ, but the likelihood ratio between these two distributions is known---or, in practice, can be estimated accurately with access to a large set of unlabeled data (test covariate points). Our weighted extension of conformal prediction also applies more generally, to settings in which the data satisfies a certain weighted notion of exchangeability. We discuss other potential applications of our new conformal methodology, including latent variable and missing data problems.

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arXiv2020

The Implicit Regularization of Stochastic Gradient Flow for Least Squares

We study the implicit regularization of mini-batch stochastic gradient descent, when applied to the fundamental problem of least squares regression. We leverage a continuous-time stochastic differential equation having the same moments as stochastic gradient descent, which we call stochastic gradient flow. We give a bound on the excess risk of stochastic gradient flow at time $t$, over ridge regression with tuning parameter $λ= 1/t$. The bound may be computed from explicit constants (e.g., the mini-batch size, step size, number of iterations), revealing precisely how these quantities drive the excess risk. Numerical examples show the bound can be small, indicating a tight relationship between the two estimators. We give a similar result relating the coefficients of stochastic gradient flow and ridge. These results hold under no conditions on the data matrix $X$, and across the entire optimization path (not just at convergence).

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arXiv2020

Predictive inference with the jackknife+

This paper introduces the jackknife+, which is a novel method for constructing predictive confidence intervals. Whereas the jackknife outputs an interval centered at the predicted response of a test point, with the width of the interval determined by the quantiles of leave-one-out residuals, the jackknife+ also uses the leave-one-out predictions at the test point to account for the variability in the fitted regression function. Assuming exchangeable training samples, we prove that this crucial modification permits rigorous coverage guarantees regardless of the distribution of the data points, for any algorithm that treats the training points symmetrically. Such guarantees are not possible for the original jackknife and we demonstrate examples where the coverage rate may actually vanish. Our theoretical and empirical analysis reveals that the jackknife and the jackknife+ intervals achieve nearly exact coverage and have similar lengths whenever the fitting algorithm obeys some form of stability. Further, we extend the jackknife+ to K-fold cross validation and similarly establish rigorous coverage properties. Our methods are related to cross-conformal prediction proposed by Vovk [2015] and we discuss connections.

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arXiv2020

The limits of distribution-free conditional predictive inference

We consider the problem of distribution-free predictive inference, with the goal of producing predictive coverage guarantees that hold conditionally rather than marginally. Existing methods such as conformal prediction offer marginal coverage guarantees, where predictive coverage holds on average over all possible test points, but this is not sufficient for many practical applications where we would like to know that our predictions are valid for a given individual, not merely on average over a population. On the other hand, exact conditional inference guarantees are known to be impossible without imposing assumptions on the underlying distribution. In this work we aim to explore the space in between these two, and examine what types of relaxations of the conditional coverage property would alleviate some of the practical concerns with marginal coverage guarantees while still being possible to achieve in a distribution-free setting.

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arXiv2019

The Generalized Lasso Problem and Uniqueness

We study uniqueness in the generalized lasso problem, where the penalty is the $\ell_1$ norm of a matrix $D$ times the coefficient vector. We derive a broad result on uniqueness that places weak assumptions on the predictor matrix $X$ and penalty matrix $D$; the implication is that, if $D$ is fixed and its null space is not too large (the dimension of its null space is at most the number of samples), and $X$ and response vector $y$ jointly follow an absolutely continuous distribution, then the generalized lasso problem has a unique solution almost surely, regardless of the number of predictors relative to the number of samples. This effectively generalizes previous uniqueness results for the lasso problem (which corresponds to the special case $D=I$). Further, we extend our study to the case in which the loss is given by the negative log-likelihood from a generalized linear model. In addition to uniqueness results, we derive results on the local stability of generalized lasso solutions that might be of interest in their own right.

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arXiv2019

A Higher-Order Kolmogorov-Smirnov Test

We present an extension of the Kolmogorov-Smirnov (KS) two-sample test, which can be more sensitive to differences in the tails. Our test statistic is an integral probability metric (IPM) defined over a higher-order total variation ball, recovering the original KS test as its simplest case. We give an exact representer result for our IPM, which generalizes the fact that the original KS test statistic can be expressed in equivalent variational and CDF forms. For small enough orders ($k \leq 5$), we develop a linear-time algorithm for computing our higher-order KS test statistic; for all others ($k \geq 6$), we give a nearly linear-time approximation. We derive the asymptotic null distribution for our test, and show that our nearly linear-time approximation shares the same asymptotic null. Lastly, we complement our theory with numerical studies.

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arXiv2019

A Continuous-Time View of Early Stopping for Least Squares

We study the statistical properties of the iterates generated by gradient descent, applied to the fundamental problem of least squares regression. We take a continuous-time view, i.e., consider infinitesimal step sizes in gradient descent, in which case the iterates form a trajectory called gradient flow. Our primary focus is to compare the risk of gradient flow to that of ridge regression. Under the calibration $t=1/λ$---where $t$ is the time parameter in gradient flow, and $λ$ the tuning parameter in ridge regression---we prove that the risk of gradient flow is no less than 1.69 times that of ridge, along the entire path (for all $t \geq 0$). This holds in finite samples with very weak assumptions on the data model (in particular, with no assumptions on the features $X$). We prove that the same relative risk bound holds for prediction risk, in an average sense over the underlying signal $β_0$. Finally, we examine limiting risk expressions (under standard Marchenko-Pastur asymptotics), and give supporting numerical experiments.

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arXiv2018

Post-Selection Inference for Changepoint Detection Algorithms with Application to Copy Number Variation Data

Changepoint detection methods are used in many areas of science and engineering, e.g., in the analysis of copy number variation data, to detect abnormalities in copy numbers along the genome. Despite the broad array of available tools, methodology for quantifying our uncertainty in the strength (or presence) of given changepoints, post-detection, are lacking. Post-selection inference offers a framework to fill this gap, but the most straightforward application of these methods results in low-powered tests and leaves open several important questions about practical usability. In this work, we carefully tailor post-selection inference methods towards changepoint detection, focusing as our main scientific application on copy number variation data. As for changepoint algorithms, we study binary segmentation, and two of its most popular variants, wild and circular, and the fused lasso. We implement some of the latest developments in post-selection inference theory: we use auxiliary randomization to improve power, which requires implementations of MCMC algorithms (importance sampling and hit-and-run sampling) to carry out our tests. We also provide recommendations for improving practical useability, detailed simulations, and an example analysis on array comparative genomic hybridization (CGH) data.

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arXiv2018

Additive Models with Trend Filtering

We study additive models built with trend filtering, i.e., additive models whose components are each regularized by the (discrete) total variation of their $k$th (discrete) derivative, for a chosen integer $k \geq 0$. This results in $k$th degree piecewise polynomial components, (e.g., $k=0$ gives piecewise constant components, $k=1$ gives piecewise linear, $k=2$ gives piecewise quadratic, etc.). Analogous to its advantages in the univariate case, additive trend filtering has favorable theoretical and computational properties, thanks in large part to the localized nature of the (discrete) total variation regularizer that it uses. On the theory side, we derive fast error rates for additive trend filtering estimates, and show these rates are minimax optimal when the underlying function is additive and has component functions whose derivatives are of bounded variation. We also show that these rates are unattainable by additive smoothing splines (and by additive models built from linear smoothers, in general). On the computational side, as per the standard in additive models, backfitting is an appealing method for optimization, but it is particularly appealing for additive trend filtering because we can leverage a few highly efficient univariate trend filtering solvers. Going one step further, we describe a new backfitting algorithm whose iterations can be run in parallel, which (as far as we know) is the first of its kind. Lastly, we present experiments to examine the empirical performance of additive trend filtering.

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arXiv2017

Uniform Asymptotic Inference and the Bootstrap After Model Selection

Recently, Tibshirani et al. (2016) proposed a method for making inferences about parameters defined by model selection, in a typical regression setting with normally distributed errors. Here, we study the large sample properties of this method, without assuming normality. We prove that the test statistic of Tibshirani et al. (2016) is asymptotically valid, as the number of samples n grows and the dimension d of the regression problem stays fixed. Our asymptotic result holds uniformly over a wide class of nonnormal error distributions. We also propose an efficient bootstrap version of this test that is provably (asymptotically) conservative, and in practice, often delivers shorter intervals than those from the original normality-based approach. Finally, we prove that the test statistic of Tibshirani et al. (2016) does not enjoy uniform validity in a high-dimensional setting, when the dimension d is allowed grow.

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arXiv2017

Extended Comparisons of Best Subset Selection, Forward Stepwise Selection, and the Lasso

In exciting new work, Bertsimas et al. (2016) showed that the classical best subset selection problem in regression modeling can be formulated as a mixed integer optimization (MIO) problem. Using recent advances in MIO algorithms, they demonstrated that best subset selection can now be solved at much larger problem sizes that what was thought possible in the statistics community. They presented empirical comparisons of best subset selection with other popular variable selection procedures, in particular, the lasso and forward stepwise selection. Surprisingly (to us), their simulations suggested that best subset selection consistently outperformed both methods in terms of prediction accuracy. Here we present an expanded set of simulations to shed more light on these comparisons. The summary is roughly as follows: (a) neither best subset selection nor the lasso uniformly dominate the other, with best subset selection generally performing better in high signal-to-noise (SNR) ratio regimes, and the lasso better in low SNR regimes; (b) best subset selection and forward stepwise perform quite similarly throughout; (c) the relaxed lasso (actually, a simplified version of the original relaxed estimator defined in Meinshausen, 2007) is the overall winner, performing just about as well as the lasso in low SNR scenarios, and as well as best subset selection in high SNR scenarios.

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arXiv2017

From Fixed-X to Random-X Regression: Bias-Variance Decompositions, Covariance Penalties, and Prediction Error Estimation

In statistical prediction, classical approaches for model selection and model evaluation based on covariance penalties are still widely used. Most of the literature on this topic is based on what we call the "Fixed-X" assumption, where covariate values are assumed to be nonrandom. By contrast, it is often more reasonable to take a "Random-X" view, where the covariate values are independently drawn for both training and prediction. To study the applicability of covariance penalties in this setting, we propose a decomposition of Random-X prediction error in which the randomness in the covariates contributes to both the bias and variance components. This decomposition is general, but we concentrate on the fundamental case of least squares regression. We prove that in this setting the move from Fixed-X to Random-X prediction results in an increase in both bias and variance. When the covariates are normally distributed and the linear model is unbiased, all terms in this decomposition are explicitly computable, which yields an extension of Mallows' Cp that we call $RCp$. $RCp$ also holds asymptotically for certain classes of nonnormal covariates. When the noise variance is unknown, plugging in the usual unbiased estimate leads to an approach that we call $\hat{RCp}$, which is closely related to Sp (Tukey 1967), and GCV (Craven and Wahba 1978). For excess bias, we propose an estimate based on the "shortcut-formula" for ordinary cross-validation (OCV), resulting in an approach we call $RCp^+$. Theoretical arguments and numerical simulations suggest that $RCP^+$ is typically superior to OCV, though the difference is small. We further examine the Random-X error of other popular estimators. The surprising result we get for ridge regression is that, in the heavily-regularized regime, Random-X variance is smaller than Fixed-X variance, which can lead to smaller overall Random-X error.

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arXiv2017

Dykstra's Algorithm, ADMM, and Coordinate Descent: Connections, Insights, and Extensions

We study connections between Dykstra's algorithm for projecting onto an intersection of convex sets, the augmented Lagrangian method of multipliers or ADMM, and block coordinate descent. We prove that coordinate descent for a regularized regression problem, in which the (separable) penalty functions are seminorms, is exactly equivalent to Dykstra's algorithm applied to the dual problem. ADMM on the dual problem is also seen to be equivalent, in the special case of two sets, with one being a linear subspace. These connections, aside from being interesting in their own right, suggest new ways of analyzing and extending coordinate descent. For example, from existing convergence theory on Dykstra's algorithm over polyhedra, we discern that coordinate descent for the lasso problem converges at an (asymptotically) linear rate. We also develop two parallel versions of coordinate descent, based on the Dykstra and ADMM connections.

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arXiv2017

Distribution-Free Predictive Inference For Regression

We develop a general framework for distribution-free predictive inference in regression, using conformal inference. The proposed methodology allows for the construction of a prediction band for the response variable using any estimator of the regression function. The resulting prediction band preserves the consistency properties of the original estimator under standard assumptions, while guaranteeing finite-sample marginal coverage even when these assumptions do not hold. We analyze and compare, both empirically and theoretically, the two major variants of our conformal framework: full conformal inference and split conformal inference, along with a related jackknife method. These methods offer different tradeoffs between statistical accuracy (length of resulting prediction intervals) and computational efficiency. As extensions, we develop a method for constructing valid in-sample prediction intervals called {\it rank-one-out} conformal inference, which has essentially the same computational efficiency as split conformal inference. We also describe an extension of our procedures for producing prediction bands with locally varying length, in order to adapt to heteroskedascity in the data. Finally, we propose a model-free notion of variable importance, called {\it leave-one-covariate-out} or LOCO inference. Accompanying this paper is an R package {\tt conformalInference} that implements all of the proposals we have introduced. In the spirit of reproducibility, all of our empirical results can also be easily (re)generated using this package.

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arXiv2017

The DFS Fused Lasso: Linear-Time Denoising over General Graphs

The fused lasso, also known as (anisotropic) total variation denoising, is widely used for piecewise constant signal estimation with respect to a given undirected graph. The fused lasso estimate is highly nontrivial to compute when the underlying graph is large and has an arbitrary structure. But for a special graph structure, namely, the chain graph, the fused lasso---or simply, 1d fused lasso---can be computed in linear time. In this paper, we establish a surprising connection between the total variation of a generic signal defined over an arbitrary graph, and the total variation of this signal over a chain graph induced by running depth-first search (DFS) over the nodes of the graph. Specifically, we prove that for any signal, its total variation over the induced chain graph is no more than twice its total variation over the original graph. This connection leads to several interesting theoretical and computational conclusions. Denoting by $m$ and $n$ the number of edges and nodes, respectively, of the graph in question, our result implies that for an underlying signal with total variation $t$ over the graph, the fused lasso achieves a mean squared error rate of \smash{$t^{2/3} n^{-2/3}$}. Moreover, precisely the same mean squared error rate is achieved by running the 1d fused lasso on the induced chain graph from running DFS. Importantly, the latter estimator is simple and computationally cheap, requiring only $O(m)$ operations for constructing the DFS-induced chain and $O(n)$ operations for computing the 1d fused lasso solution over this chain. Further, for trees that have bounded max degree, the error rate of \smash{$t^{2/3} n^{-2/3}$} cannot be improved, in the sense that it is the minimax rate for signals that have total variation $t$ over the tree.

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