arXiv subjects
Publications and source records attributed to Sivaraman Balakrishnan.
We introduce a new notion of regularity of an estimator called median regularity. We prove that uniformly valid (honest) inference for a functional is possible if and only if there exists a median regular estimator of that functional. To our knowledge, such a notion of regularity that is necessary for uniformly valid inference is unavailable in the literature.
Permutation tests are widely used in statistics, providing a finite-sample guarantee on the type I error rate whenever the distribution of the samples under the null hypothesis is invariant to some rearrangement. Despite its increasing popularity and empirical success, theoretical properties of the permutation test, especially its power, have not been fully explored beyond simple cases. In this paper, we attempt to partly fill this gap by presenting a general non-asymptotic framework for analyzing the minimax power of the permutation test. The utility of our proposed framework is illustrated in the context of two-sample and independence testing under both discrete and continuous settings. In each setting, we introduce permutation tests based on U-statistics and study their minimax performance. We also develop exponential concentration bounds for permuted U-statistics based on a novel coupling idea, which may be of independent interest. Building on these exponential bounds, we introduce permutation tests which are adaptive to unknown smoothness parameters without losing much power. The proposed framework is further illustrated using more sophisticated test statistics including weighted U-statistics for multinomial testing and Gaussian kernel-based statistics for density testing. Finally, we provide some simulation results that further justify the permutation approach.
Motivated by the growing popularity of variants of the Wasserstein distance in statistics and machine learning, we study statistical inference for the Sliced Wasserstein distance--an easily computable variant of the Wasserstein distance. Specifically, we construct confidence intervals for the Sliced Wasserstein distance which have finite-sample validity under no assumptions or under mild moment assumptions. These intervals are adaptive in length to the regularity of the underlying distributions. We also bound the minimax risk of estimating the Sliced Wasserstein distance, and as a consequence establish that the lengths of our proposed confidence intervals are minimax optimal over appropriate distribution classes. To motivate the choice of these classes, we also study minimax rates of estimating a distribution under the Sliced Wasserstein distance. These theoretical findings are complemented with a simulation study demonstrating the deficiencies of the classical bootstrap, and the advantages of our proposed methods. We also show strong correspondences between our theoretical predictions and the adaptivity of our confidence interval lengths in simulations. We conclude by demonstrating the use of our confidence intervals in the setting of simulator-based likelihood-free inference. In this setting, contrasting popular approximate Bayesian computation methods, we develop uncertainty quantification methods with rigorous frequentist coverage guarantees.
We consider the task of heavy-tailed statistical estimation given streaming $p$-dimensional samples. This could also be viewed as stochastic optimization under heavy-tailed distributions, with an additional $O(p)$ space complexity constraint. We design a clipped stochastic gradient descent algorithm and provide an improved analysis, under a more nuanced condition on the noise of the stochastic gradients, which we show is critical when analyzing stochastic optimization problems arising from general statistical estimation problems. Our results guarantee convergence not just in expectation but with exponential concentration, and moreover does so using $O(1)$ batch size. We provide consequences of our results for mean estimation and linear regression. Finally, we provide empirical corroboration of our results and algorithms via synthetic experiments for mean estimation and linear regression.
In this paper, we investigate local permutation tests for testing conditional independence between two random vectors $X$ and $Y$ given $Z$. The local permutation test determines the significance of a test statistic by locally shuffling samples which share similar values of the conditioning variables $Z$, and it forms a natural extension of the usual permutation approach for unconditional independence testing. Despite its simplicity and empirical support, the theoretical underpinnings of the local permutation test remain unclear. Motivated by this gap, this paper aims to establish theoretical foundations of local permutation tests with a particular focus on binning-based statistics. We start by revisiting the hardness of conditional independence testing and provide an upper bound for the power of any valid conditional independence test, which holds when the probability of observing collisions in $Z$ is small. This negative result naturally motivates us to impose additional restrictions on the possible distributions under the null and alternate. To this end, we focus our attention on certain classes of smooth distributions and identify provably tight conditions under which the local permutation method is universally valid, i.e. it is valid when applied to any (binning-based) test statistic. To complement this result on type I error control, we also show that in some cases, a binning-based statistic calibrated via the local permutation method can achieve minimax optimal power. We also introduce a double-binning permutation strategy, which yields a valid test over less smooth null distributions than the typical single-binning method without compromising much power. Finally, we present simulation results to support our theoretical findings.
We study the Personalized PageRank (PPR) algorithm, a local spectral method for clustering, which extracts clusters using locally-biased random walks around a given seed node. In contrast to previous work, we adopt a classical statistical learning setup, where we obtain samples from an unknown nonparametric distribution, and aim to identify sufficiently salient clusters. We introduce a trio of population-level functionals -- the normalized cut, conductance, and local spread, analogous to graph-based functionals of the same name -- and prove that PPR, run on a neighborhood graph, recovers clusters with small population normalized cut and large conductance and local spread. We apply our general theory to establish that PPR identifies connected regions of high density (density clusters) that satisfy a set of natural geometric conditions. We also show a converse result, that PPR can fail to recover geometrically poorly-conditioned density clusters, even asymptotically. Finally, we provide empirical support for our theory.
In this paper we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for nonparametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed responses ${\bf Y} = (Y_1,\ldots,Y_n)$ onto a subspace spanned by certain eigenvectors of a neighborhood graph Laplacian. We show that PCR-LE achieves minimax rates of convergence for random design regression over Sobolev spaces. Under sufficient smoothness conditions on the design density $p$, PCR-LE achieves the optimal rates for both estimation (where the optimal rate in squared $L^2$ norm is known to be $n^{-2s/(2s + d)}$) and goodness-of-fit testing ($n^{-4s/(4s + d)}$). We also show that PCR-LE is \emph{manifold adaptive}: that is, we consider the situation where the design is supported on a manifold of small intrinsic dimension $m$, and give upper bounds establishing that PCR-LE achieves the faster minimax estimation ($n^{-2s/(2s + m)}$) and testing ($n^{-4s/(4s + m)}$) rates of convergence. Interestingly, these rates are almost always much faster than the known rates of convergence of graph Laplacian eigenvectors to their population-level limits; in other words, for this problem regression with estimated features appears to be much easier, statistically speaking, than estimating the features itself. We support these theoretical results with empirical evidence.
To assess generalization, machine learning scientists typically either (i) bound the generalization gap and then (after training) plug in the empirical risk to obtain a bound on the true risk; or (ii) validate empirically on holdout data. However, (i) typically yields vacuous guarantees for overparameterized models. Furthermore, (ii) shrinks the training set and its guarantee erodes with each re-use of the holdout set. In this paper, we introduce a method that leverages unlabeled data to produce generalization bounds. After augmenting our (labeled) training set with randomly labeled fresh examples, we train in the standard fashion. Whenever classifiers achieve low error on clean data and high error on noisy data, our bound provides a tight upper bound on the true risk. We prove that our bound is valid for 0-1 empirical risk minimization and with linear classifiers trained by gradient descent. Our approach is especially useful in conjunction with deep learning due to the early learning phenomenon whereby networks fit true labels before noisy labels but requires one intuitive assumption. Empirically, on canonical computer vision and NLP tasks, our bound provides non-vacuous generalization guarantees that track actual performance closely. This work provides practitioners with an option for certifying the generalization of deep nets even when unseen labeled data is unavailable and provides theoretical insights into the relationship between random label noise and generalization.
Given only positive examples and unlabeled examples (from both positive and negative classes), we might hope nevertheless to estimate an accurate positive-versus-negative classifier. Formally, this task is broken down into two subtasks: (i) Mixture Proportion Estimation (MPE) -- determining the fraction of positive examples in the unlabeled data; and (ii) PU-learning -- given such an estimate, learning the desired positive-versus-negative classifier. Unfortunately, classical methods for both problems break down in high-dimensional settings. Meanwhile, recently proposed heuristics lack theoretical coherence and depend precariously on hyperparameter tuning. In this paper, we propose two simple techniques: Best Bin Estimation (BBE) (for MPE); and Conditional Value Ignoring Risk (CVIR), a simple objective for PU-learning. Both methods dominate previous approaches empirically, and for BBE, we establish formal guarantees that hold whenever we can train a model to cleanly separate out a small subset of positive examples. Our final algorithm (TED)$^n$, alternates between the two procedures, significantly improving both our mixture proportion estimator and classifier
Modern policy gradient algorithms such as Proximal Policy Optimization (PPO) rely on an arsenal of heuristics, including loss clipping and gradient clipping, to ensure successful learning. These heuristics are reminiscent of techniques from robust statistics, commonly used for estimation in outlier-rich (``heavy-tailed'') regimes. In this paper, we present a detailed empirical study to characterize the heavy-tailed nature of the gradients of the PPO surrogate reward function. We demonstrate that the gradients, especially for the actor network, exhibit pronounced heavy-tailedness and that it increases as the agent's policy diverges from the behavioral policy (i.e., as the agent goes further off policy). Further examination implicates the likelihood ratios and advantages in the surrogate reward as the main sources of the observed heavy-tailedness. We then highlight issues arising due to the heavy-tailed nature of the gradients. In this light, we study the effects of the standard PPO clipping heuristics, demonstrating that these tricks primarily serve to offset heavy-tailedness in gradients. Thus motivated, we propose incorporating GMOM, a high-dimensional robust estimator, into PPO as a substitute for three clipping tricks. Despite requiring less hyperparameter tuning, our method matches the performance of PPO (with all heuristics enabled) on a battery of MuJoCo continuous control tasks.
We consider the problem of conditional independence testing of $X$ and $Y$ given $Z$ where $X,Y$ and $Z$ are three real random variables and $Z$ is continuous. We focus on two main cases - when $X$ and $Y$ are both discrete, and when $X$ and $Y$ are both continuous. In view of recent results on conditional independence testing (Shah and Peters, 2018), one cannot hope to design non-trivial tests, which control the type I error for all absolutely continuous conditionally independent distributions, while still ensuring power against interesting alternatives. Consequently, we identify various, natural smoothness assumptions on the conditional distributions of $X,Y|Z=z$ as $z$ varies in the support of $Z$, and study the hardness of conditional independence testing under these smoothness assumptions. We derive matching lower and upper bounds on the critical radius of separation between the null and alternative hypotheses in the total variation metric. The tests we consider are easily implementable and rely on binning the support of the continuous variable $Z$. To complement these results, we provide a new proof of the hardness result of Shah and Peters.
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 $\mathbb{R}^d$.
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.
This paper studies the minimax rate of nonparametric conditional density estimation under a weighted absolute value loss function in a multivariate setting. We first demonstrate that conditional density estimation is impossible if one only requires that $p_{X|Z}$ is smooth in $x$ for all values of $z$. This motivates us to consider a sub-class of absolutely continuous distributions, restricting the conditional density $p_{X|Z}(x|z)$ to not only be Hölder smooth in $x$, but also be total variation smooth in $z$. We propose a corresponding kernel-based estimator and prove that it achieves the minimax rate. We give some simple examples of densities satisfying our assumptions which imply that our results are not vacuous. Finally, we propose an estimator which achieves the minimax optimal rate adaptively, i.e., without the need to know the smoothness parameter values in advance. Crucially, both of our estimators (the adaptive and non-adaptive ones) impose no assumptions on the marginal density $p_Z$, and are not obtained as a ratio between two kernel smoothing estimators which may sound like a go to approach in this problem.
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.
Causal effects are often characterized with averages, which can give an incomplete picture of the underlying counterfactual distributions. Here we consider estimating the entire counterfactual density and generic functionals thereof. We focus on two kinds of target parameters. The first is a density approximation, defined by a projection onto a finite-dimensional model using a generalized distance metric, which includes f-divergences as well as $L_p$ norms. The second is the distance between counterfactual densities, which can be used as a more nuanced effect measure than the mean difference, and as a tool for model selection. We study nonparametric efficiency bounds for these targets, giving results for smooth but otherwise generic models and distances. Importantly, we show how these bounds connect to means of particular non-trivial functions of counterfactuals, linking the problems of density and mean estimation. We go on to propose doubly robust-style estimators for the density approximations and distances, and study their rates of convergence, showing they can be optimally efficient in large nonparametric models. We also give analogous methods for model selection and aggregation, when many models may be available and of interest. Our results all hold for generic models and distances, but throughout we highlight what happens for particular choices, such as $L_2$ projections on linear models, and KL projections on exponential families. Finally we illustrate by estimating the density of CD4 count among patients with HIV, had all been treated with combination therapy versus zidovudine alone, as well as a density effect. Our results suggest combination therapy may have increased CD4 count most for high-risk patients. Our methods are implemented in the freely available R package npcausal on GitHub.
The task of aggregating and denoising crowd-labeled data has gained increased significance with the advent of crowdsourcing platforms and massive datasets. We propose a permutation-based model for crowd labeled data that is a significant generalization of the classical Dawid-Skene model, and introduce a new error metric by which to compare different estimators. We derive global minimax rates for the permutation-based model that are sharp up to logarithmic factors, and match the minimax lower bounds derived under the simpler Dawid-Skene model. We then design two computationally-efficient estimators: the WAN estimator for the setting where the ordering of workers in terms of their abilities is approximately known, and the OBI-WAN estimator where that is not known. For each of these estimators, we provide non-asymptotic bounds on their performance. We conduct synthetic simulations and experiments on real-world crowdsourcing data, and the experimental results corroborate our theoretical findings.
A number of applications require two-sample testing on ranked preference data. For instance, in crowdsourcing, there is a long-standing question of whether pairwise comparison data provided by people is distributed similar to ratings-converted-to-comparisons. Other examples include sports data analysis and peer grading. In this paper, we design two-sample tests for pairwise comparison data and ranking data. For our two-sample test for pairwise comparison data, we establish an upper bound on the sample complexity required to correctly distinguish between the distributions of the two sets of samples. Our test requires essentially no assumptions on the distributions. We then prove complementary lower bounds showing that our results are tight (in the minimax sense) up to constant factors. We investigate the role of modeling assumptions by proving lower bounds for a range of pairwise comparison models (WST, MST,SST, parameter-based such as BTL and Thurstone). We also provide testing algorithms and associated sample complexity bounds for the problem of two-sample testing with partial (or total) ranking data.Furthermore, we empirically evaluate our results via extensive simulations as well as two real-world datasets consisting of pairwise comparisons. By applying our two-sample test on real-world pairwise comparison data, we conclude that ratings and rankings provided by people are indeed distributed differently. On the other hand, our test recognizes no significant difference in the relative performance of European football teams across two seasons. Finally, we apply our two-sample test on a real-world partial and total ranking dataset and find a statistically significant difference in Sushi preferences across demographic divisions based on gender, age and region of residence.