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Ryan Tibshirani

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7 papers

1 author row

NeurIPS Conference 2019 Conference Paper

Conformal Prediction Under Covariate Shift

Ryan Tibshirani
Rina Foygel Barber
Emmanuel Candes
Aaditya Ramdas

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 the two distributions is known---or, in practice, can be estimated accurately from a set of unlabeled data (test covariate points). Our weighted extension of conformal prediction also applies more broadly, 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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NeurIPS Conference 2019 Conference Paper

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

Maria Jahja
David Farrow
Roni Rosenfeld
Ryan Tibshirani

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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NeurIPS Conference 2017 Conference Paper

A Sharp Error Analysis for the Fused Lasso, with Application to Approximate Changepoint Screening

Kevin Lin
James Sharpnack
Alessandro Rinaldo
Ryan Tibshirani

In the 1-dimensional multiple changepoint detection problem, we derive a new fast error rate for the fused lasso estimator, under the assumption that the mean vector has a sparse number of changepoints. This rate is seen to be suboptimal (compared to the minimax rate) by only a factor of $\log\log{n}$. Our proof technique is centered around a novel construction that we call a lower interpolant. We extend our results to misspecified models and exponential family distributions. We also describe the implications of our error analysis for the approximate screening of changepoints.

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NeurIPS Conference 2017 Conference Paper

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

Ryan Tibshirani

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 penalty is a separable sum of support functions, 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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NeurIPS Conference 2017 Conference Paper

Higher-Order Total Variation Classes on Grids: Minimax Theory and Trend Filtering Methods

Veeranjaneyulu Sadhanala
Yu-Xiang Wang
James Sharpnack
Ryan Tibshirani

We consider the problem of estimating the values of a function over $n$ nodes of a $d$-dimensional grid graph (having equal side lengths $n^{1/d}$) from noisy observations. The function is assumed to be smooth, but is allowed to exhibit different amounts of smoothness at different regions in the grid. Such heterogeneity eludes classical measures of smoothness from nonparametric statistics, such as Holder smoothness. Meanwhile, total variation (TV) smoothness classes allow for heterogeneity, but are restrictive in another sense: only constant functions count as perfectly smooth (achieve zero TV). To move past this, we define two new higher-order TV classes, based on two ways of compiling the discrete derivatives of a parameter across the nodes. We relate these two new classes to Holder classes, and derive lower bounds on their minimax errors. We also analyze two naturally associated trend filtering methods; when $d=2$, each is seen to be rate optimal over the appropriate class.

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NeurIPS Conference 2016 Conference Paper

The Multiple Quantile Graphical Model

Alnur Ali
J. Zico Kolter
Ryan Tibshirani

We introduce the Multiple Quantile Graphical Model (MQGM), which extends the neighborhood selection approach of Meinshausen and Buhlmann for learning sparse graphical models. The latter is defined by the basic subproblem of modeling the conditional mean of one variable as a sparse function of all others. Our approach models a set of conditional quantiles of one variable as a sparse function of all others, and hence offers a much richer, more expressive class of conditional distribution estimates. We establish that, under suitable regularity conditions, the MQGM identifies the exact conditional independencies with probability tending to one as the problem size grows, even outside of the usual homoskedastic Gaussian data model. We develop an efficient algorithm for fitting the MQGM using the alternating direction method of multipliers. We also describe a strategy for sampling from the joint distribution that underlies the MQGM estimate. Lastly, we present detailed experiments that demonstrate the flexibility and effectiveness of the MQGM in modeling hetereoskedastic non-Gaussian data.

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NeurIPS Conference 2016 Conference Paper

Total Variation Classes Beyond 1d: Minimax Rates, and the Limitations of Linear Smoothers

Veeranjaneyulu Sadhanala
Yu-Xiang Wang
Ryan Tibshirani

We consider the problem of estimating a function defined over $n$ locations on a $d$-dimensional grid (having all side lengths equal to $n^{1/d}$). When the function is constrained to have discrete total variation bounded by $C_n$, we derive the minimax optimal (squared) $\ell_2$ estimation error rate, parametrized by $n, C_n$. Total variation denoising, also known as the fused lasso, is seen to be rate optimal. Several simpler estimators exist, such as Laplacian smoothing and Laplacian eigenmaps. A natural question is: can these simpler estimators perform just as well? We prove that these estimators, and more broadly all estimators given by linear transformations of the input data, are suboptimal over the class of functions with bounded variation. This extends fundamental findings of Donoho and Johnstone (1998) on 1-dimensional total variation spaces to higher dimensions. The implication is that the computationally simpler methods cannot be used for such sophisticated denoising tasks, without sacrificing statistical accuracy. We also derive minimax rates for discrete Sobolev spaces over $d$-dimensional grids, which are, in some sense, smaller than the total variation function spaces. Indeed, these are small enough spaces that linear estimators can be optimal---and a few well-known ones are, such as Laplacian smoothing and Laplacian eigenmaps, as we show. Lastly, we investigate the adaptivity of the total variation denoiser to these smaller Sobolev function spaces.

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