arXiv subjects

Aaditya Ramdas

Publications and source records attributed to Aaditya Ramdas.

At least 199 records · Page 11Linked to original sources

arXiv2021

Distribution-free uncertainty quantification for classification under label shift

Trustworthy deployment of ML models requires a proper measure of uncertainty, especially in safety-critical applications. We focus on uncertainty quantification (UQ) for classification problems via two avenues -- prediction sets using conformal prediction and calibration of probabilistic predictors by post-hoc binning -- since these possess distribution-free guarantees for i.i.d. data. Two common ways of generalizing beyond the i.i.d. setting include handling covariate and label shift. Within the context of distribution-free UQ, the former has already received attention, but not the latter. It is known that label shift hurts prediction, and we first argue that it also hurts UQ, by showing degradation in coverage and calibration. Piggybacking on recent progress in addressing label shift (for better prediction), we examine the right way to achieve UQ by reweighting the aforementioned conformal and calibration procedures whenever some unlabeled data from the target distribution is available. We examine these techniques theoretically in a distribution-free framework and demonstrate their excellent practical performance.

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arXiv2021

Fast and Powerful Conditional Randomization Testing via Distillation

We consider the problem of conditional independence testing: given a response Y and covariates (X,Z), we test the null hypothesis that Y is independent of X given Z. The conditional randomization test (CRT) was recently proposed as a way to use distributional information about X|Z to exactly (non-asymptotically) control Type-I error using any test statistic in any dimensionality without assuming anything about Y|(X,Z). This flexibility in principle allows one to derive powerful test statistics from complex prediction algorithms while maintaining statistical validity. Yet the direct use of such advanced test statistics in the CRT is prohibitively computationally expensive, especially with multiple testing, due to the CRT's requirement to recompute the test statistic many times on resampled data. We propose the distilled CRT, a novel approach to using state-of-the-art machine learning algorithms in the CRT while drastically reducing the number of times those algorithms need to be run, thereby taking advantage of their power and the CRT's statistical guarantees without suffering the usual computational expense. In addition to distillation, we propose a number of other tricks like screening and recycling computations to further speed up the CRT without sacrificing its high power and exact validity. Indeed, we show in simulations that all our proposals combined lead to a test that has similar power to the most powerful existing CRT implementations but requires orders of magnitude less computation, making it a practical tool even for large data sets. We demonstrate these benefits on a breast cancer dataset by identifying biomarkers related to cancer stage.

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arXiv2021

Dynamic Algorithms for Online Multiple Testing

We derive new algorithms for online multiple testing that provably control false discovery exceedance (FDX) while achieving orders of magnitude more power than previous methods. This statistical advance is enabled by the development of new algorithmic ideas: earlier algorithms are more "static" while our new ones allow for the dynamical adjustment of testing levels based on the amount of wealth the algorithm has accumulated. We demonstrate that our algorithms achieve higher power in a variety of synthetic experiments. We also prove that SupLORD can provide error control for both FDR and FDX, and controls FDR at stopping times. Stopping times are particularly important as they permit the experimenter to end the experiment arbitrarily early while maintaining desired control of the FDR. SupLORD is the first non-trivial algorithm, to our knowledge, that can control FDR at stopping times in the online setting.

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arXiv2021

Nonparametric iterated-logarithm extensions of the sequential generalized likelihood ratio test

We develop a nonparametric extension of the sequential generalized likelihood ratio (GLR) test and corresponding time-uniform confidence sequences for the mean of a univariate distribution. By utilizing a geometric interpretation of the GLR statistic, we derive a simple analytic upper bound on the probability that it exceeds any prespecified boundary; these are intractable to approximate via simulations due to infinite horizon of the tests and the composite nonparametric nulls under consideration. Using time-uniform boundary-crossing inequalities, we carry out a unified nonasymptotic analysis of expected sample sizes of one-sided and open-ended tests over nonparametric classes of distributions (including sub-Gaussian, sub-exponential, sub-gamma, and exponential families). Finally, we present a flexible and practical method to construct time-uniform confidence sequences that are easily tunable to be uniformly close to the pointwise Chernoff bound over any target time interval.

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arXiv2021

On the bias, risk and consistency of sample means in multi-armed bandits

The sample mean is among the most well studied estimators in statistics, having many desirable properties such as unbiasedness and consistency. However, when analyzing data collected using a multi-armed bandit (MAB) experiment, the sample mean is biased and much remains to be understood about its properties. For example, when is it consistent, how large is its bias, and can we bound its mean squared error? This paper delivers a thorough and systematic treatment of the bias, risk and consistency of MAB sample means. Specifically, we identify four distinct sources of selection bias (sampling, stopping, choosing and rewinding) and analyze them both separately and together. We further demonstrate that a new notion of \emph{effective sample size} can be used to bound the risk of the sample mean under suitable loss functions. We present several carefully designed examples to provide intuition on the different sources of selection bias we study. Our treatment is nonparametric and algorithm-agnostic, meaning that it is not tied to a specific algorithm or goal. In a nutshell, our proofs combine variational representations of information-theoretic divergences with new martingale concentration inequalities.

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arXiv2021

Familywise Error Rate Control by Interactive Unmasking

We propose a method for multiple hypothesis testing with familywise error rate (FWER) control, called the i-FWER test. Most testing methods are predefined algorithms that do not allow modifications after observing the data. However, in practice, analysts tend to choose a promising algorithm after observing the data; unfortunately, this violates the validity of the conclusion. The i-FWER test allows much flexibility: a human (or a computer program acting on the human's behalf) may adaptively guide the algorithm in a data-dependent manner. We prove that our test controls FWER if the analysts adhere to a particular protocol of "masking" and "unmasking". We demonstrate via numerical experiments the power of our test under structured non-nulls, and then explore new forms of masking.

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arXiv2021

Path Length Bounds for Gradient Descent and Flow

We derive bounds on the path length $ζ$ of gradient descent (GD) and gradient flow (GF) curves for various classes of smooth convex and nonconvex functions. Among other results, we prove that: (a) if the iterates are linearly convergent with factor $(1-c)$, then $ζ$ is at most $\mathcal{O}(1/c)$; (b) under the Polyak-Kurdyka-Lojasiewicz (PKL) condition, $ζ$ is at most $\mathcal{O}(\sqrtκ)$, where $κ$ is the condition number, and at least $\widetildeΩ(\sqrt{d} \wedge κ^{1/4})$; (c) for quadratics, $ζ$ is $Θ(\min\{\sqrt{d},\sqrt{\log κ}\})$ and in some cases can be independent of $κ$; (d) assuming just convexity, $ζ$ can be at most $2^{4d\log d}$; (e) for separable quasiconvex functions, $ζ$ is $Θ(\sqrt{d})$. Thus, we advance current understanding of the properties of GD and GF curves beyond rates of convergence. We expect our techniques to facilitate future studies for other algorithms.

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arXiv2021

The Power of Batching in Multiple Hypothesis Testing

One important partition of algorithms for controlling the false discovery rate (FDR) in multiple testing is into offline and online algorithms. The first generally achieve significantly higher power of discovery, while the latter allow making decisions sequentially as well as adaptively formulating hypotheses based on past observations. Using existing methodology, it is unclear how one could trade off the benefits of these two broad families of algorithms, all the while preserving their formal FDR guarantees. To this end, we introduce $\text{Batch}_{\text{BH}}$ and $\text{Batch}_{\text{St-BH}}$, algorithms for controlling the FDR when a possibly infinite sequence of batches of hypotheses is tested by repeated application of one of the most widely used offline algorithms, the Benjamini-Hochberg (BH) method or Storey's improvement of the BH method. We show that our algorithms interpolate between existing online and offline methodology, thus trading off the best of both worlds.

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arXiv2021

Interactive Martingale Tests for the Global Null

Global null testing is a classical problem going back about a century to Fisher's and Stouffer's combination tests. In this work, we present simple martingale analogs of these classical tests, which are applicable in two distinct settings: (a) the online setting in which there is a possibly infinite sequence of $p$-values, and (b) the batch setting, where one uses prior knowledge to preorder the hypotheses. Through theory and simulations, we demonstrate that our martingale variants have higher power than their classical counterparts even when the preordering is only weakly informative. Finally, using a recent idea of "masking" $p$-values, we develop a novel interactive test for the global null that can take advantage of covariates and repeated user guidance to create a data-adaptive ordering that achieves higher detection power against structured alternatives.

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arXiv2021

Uncertainty quantification using martingales for misspecified Gaussian processes

We address uncertainty quantification for Gaussian processes (GPs) under misspecified priors, with an eye towards Bayesian Optimization (BO). GPs are widely used in BO because they easily enable exploration based on posterior uncertainty bands. However, this convenience comes at the cost of robustness: a typical function encountered in practice is unlikely to have been drawn from the data scientist's prior, in which case uncertainty estimates can be misleading, and the resulting exploration can be suboptimal. We present a frequentist approach to GP/BO uncertainty quantification. We utilize the GP framework as a working model, but do not assume correctness of the prior. We instead construct a confidence sequence (CS) for the unknown function using martingale techniques. There is a necessary cost to achieving robustness: if the prior was correct, posterior GP bands are narrower than our CS. Nevertheless, when the prior is wrong, our CS is statistically valid and empirically outperforms standard GP methods, in terms of both coverage and utility for BO. Additionally, we demonstrate that powered likelihoods provide robustness against model misspecification.

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arXiv2021

On conditional versus marginal bias in multi-armed bandits

The bias of the sample means of the arms in multi-armed bandits is an important issue in adaptive data analysis that has recently received considerable attention in the literature. Existing results relate in precise ways the sign and magnitude of the bias to various sources of data adaptivity, but do not apply to the conditional inference setting in which the sample means are computed only if some specific conditions are satisfied. In this paper, we characterize the sign of the conditional bias of monotone functions of the rewards, including the sample mean. Our results hold for arbitrary conditioning events and leverage natural monotonicity properties of the data collection policy. We further demonstrate, through several examples from sequential testing and best arm identification, that the sign of the conditional and marginal bias of the sample mean of an arm can be different, depending on the conditioning event. Our analysis offers new and interesting perspectives on the subtleties of assessing the bias in data adaptive settings.

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arXiv2021

Off-policy Confidence Sequences

We develop confidence bounds that hold uniformly over time for off-policy evaluation in the contextual bandit setting. These confidence sequences are based on recent ideas from martingale analysis and are non-asymptotic, non-parametric, and valid at arbitrary stopping times. We provide algorithms for computing these confidence sequences that strike a good balance between computational and statistical efficiency. We empirically demonstrate the tightness of our approach in terms of failure probability and width and apply it to the "gated deployment" problem of safely upgrading a production contextual bandit system.

cs.LG↗

arXiv2021

Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy

We propose a method to optimize the representation and distinguishability of samples from two probability distributions, by maximizing the estimated power of a statistical test based on the maximum mean discrepancy (MMD). This optimized MMD is applied to the setting of unsupervised learning by generative adversarial networks (GAN), in which a model attempts to generate realistic samples, and a discriminator attempts to tell these apart from data samples. In this context, the MMD may be used in two roles: first, as a discriminator, either directly on the samples, or on features of the samples. Second, the MMD can be used to evaluate the performance of a generative model, by testing the model's samples against a reference data set. In the latter role, the optimized MMD is particularly helpful, as it gives an interpretable indication of how the model and data distributions differ, even in cases where individual model samples are not easily distinguished either by eye or by classifier.

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arXiv2021

Confidence sequences for sampling without replacement

Many practical tasks involve sampling sequentially without replacement (WoR) from a finite population of size $N$, in an attempt to estimate some parameter $θ^\star$. Accurately quantifying uncertainty throughout this process is a nontrivial task, but is necessary because it often determines when we stop collecting samples and confidently report a result. We present a suite of tools for designing confidence sequences (CS) for $θ^\star$. A CS is a sequence of confidence sets $(C_n)_{n=1}^N$, that shrink in size, and all contain $θ^\star$ simultaneously with high probability. We present a generic approach to constructing a frequentist CS using Bayesian tools, based on the fact that the ratio of a prior to the posterior at the ground truth is a martingale. We then present Hoeffding- and empirical-Bernstein-type time-uniform CSs and fixed-time confidence intervals for sampling WoR, which improve on previous bounds in the literature and explicitly quantify the benefit of WoR sampling.

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arXiv2020

STAR: A general interactive framework for FDR control under structural constraints

We propose a general framework based on selectively traversed accumulation rules (STAR) for interactive multiple testing with generic structural constraints on the rejection set. It combines accumulation tests from ordered multiple testing with data-carving ideas from post-selection inference, allowing for highly flexible adaptation to generic structural information. Our procedure defines an interactive protocol for gradually pruning a candidate rejection set, beginning with the set of all hypotheses and shrinking with each step. By restricting the information at each step via a technique we call masking, our protocol enables interaction while controlling the false discovery rate (FDR) in finite samples for any data-adaptive update rule that the analyst may choose. We suggest update rules for a variety of applications with complex structural constraints, show that STAR performs well for problems ranging from convex region detection to FDR control on directed acyclic graphs, and show how to extend it to regression problems where knockoff statistics are available in lieu of $p$-values.

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arXiv2020

Asynchronous Online Testing of Multiple Hypotheses

We consider the problem of asynchronous online testing, aimed at providing control of the false discovery rate (FDR) during a continual stream of data collection and testing, where each test may be a sequential test that can start and stop at arbitrary times. This setting increasingly characterizes real-world applications in science and industry, where teams of researchers across large organizations may conduct tests of hypotheses in a decentralized manner. The overlap in time and space also tends to induce dependencies among test statistics, a challenge for classical methodology, which either assumes (overly optimistically) independence or (overly pessimistically) arbitrary dependence between test statistics. We present a general framework that addresses both of these issues via a unified computational abstraction that we refer to as "conflict sets." We show how this framework yields algorithms with formal FDR guarantees under a more intermediate, local notion of dependence. We illustrate our algorithms in simulations by comparing to existing algorithms for online FDR control.

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arXiv2020

The leave-one-covariate-out conditional randomization test

Conditional independence testing is an important problem, yet provably hard without assumptions. One of the assumptions that has become popular of late is called "model-X", where we assume we know the joint distribution of the covariates, but assume nothing about the conditional distribution of the outcome given the covariates. Knockoffs is a popular methodology associated with this framework, but it suffers from two main drawbacks: only one-bit $p$-values are available for inference on each variable, and the method is randomized with significant variability across runs in practice. The conditional randomization test (CRT) is thought to be the "right" solution under model-X, but usually viewed as computationally inefficient. This paper proposes a computationally efficient leave-one-covariate-out (LOCO) CRT that addresses both drawbacks of knockoffs. LOCO CRT produces valid $p$-values that can be used to control the familywise error rate, and has nearly zero algorithmic variability. For L1 regularized M-estimators, we develop an even faster variant called L1ME CRT, which reuses computation by leveraging a novel observation about the stability of the cross-validated lasso to removing inactive variables. Last, for multivariate Gaussian covariates, we present a closed form expression for the LOCO CRT $p$-value, thus completely eliminating resampling in this important special case.

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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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