arXiv subjects
Publications and source records attributed to Arun Kumar Kuchibhotla.
In this note, we improve some concentration inequalities for martingales with bounded increments. These results recover the missing factor in Freedman-style inequalities and are near optimal. We also provide minor refinements of concentration inequalities for functions of independent random variables. These proofs use techniques from the works of Bentkus and Pinelis.
This paper provides some foundations for valid forecasting of rare and extreme heat waves through a better understanding of the similarities and differences between several consecutive hot days under normal circumstances and rare, extreme heat waves. We analyze AIRS data from the American Pacific Northwest and AIRS data from the Phoenix, Arizona region. A genetic algorithm is used to help determine the most promising predictors. Classification accuracy with supervised learning is excellent for the Pacific Northwest and is replicated for Phoenix. Conformal prediction sets are considered as a way to represent forecasting uncertainty. Complications caused by endogenous sampling are discussed.
Many functionals of interest in statistics and machine learning can be written as minimizers of expected loss functions. Such functionals are called $M$-estimands, and can be estimated by $M$-estimators -- minimizers of empirical average losses. Traditionally, statistical inference (e.g., hypothesis tests and confidence sets) for $M$-estimands is obtained by proving asymptotic normality of $M$-estimators centered at the target. However, asymptotic normality is only one of several possible limiting distributions and (asymptotically) valid inference becomes significantly difficult with non-normal limits. In this paper, we provide conditions for the symmetry of three general classes of limiting distributions, enabling inference using HulC (Kuchibhotla et al. (2024)).
In a decision-theoretic framework, the minimax lower bound provides the worst-case performance of estimators relative to a given class of statistical models. For parametric and semiparametric models, the Hájek--Le Cam local asymptotic minimax (LAM) theorem provides the sharp local asymptotic lower bound. Despite its relative generality, this result comes with limitations as it only applies to the estimation of differentiable functionals under regular statistical models. On the other hand, minimax lower bound techniques such as Fano's or Assoud's are applicable in more general settings but are not sharp enough to imply the LAM theorem. To address this gap, we provide new non-asymptotic minimax lower bounds under minimal regularity assumptions, which imply sharp asymptotic constants. The proposed lower bounds do not require the differentiability of functionals or regularity of statistical models, extending the efficiency theory to broader situations where standard results fail. The use of the new lower bounds is illustrated through the local minimax lower bound constants for estimating the density at a point and directionally differentiable parameters.
Conformal prediction is a generic methodology for finite-sample valid distribution-free prediction. This technique has garnered a lot of attention in the literature partly because it can be applied with any machine learning algorithm that provides point predictions to yield valid prediction regions. Of course, the efficiency (width/volume) of the resulting prediction region depends on the performance of the machine learning algorithm. In the context of point prediction, several techniques (such as cross-validation) exist to select one of many machine learning algorithms for better performance. In contrast, such selection techniques are seldom discussed in the context of set prediction (or prediction regions). In this paper, we consider the problem of obtaining the smallest conformal prediction region given a family of machine learning algorithms. We provide two general-purpose selection algorithms and consider coverage as well as width properties of the final prediction region. The first selection method yields the smallest width prediction region among the family of conformal prediction regions for all sample sizes but only has an approximate coverage guarantee. The second selection method has a finite sample coverage guarantee but only attains close to the smallest width. The approximate optimal width property of the second method is quantified via an oracle inequality. As an illustration, we consider the use of aggregation of non-parametric regression estimators in the split conformal method with the absolute residual conformal score.
Series or orthogonal basis regression is one of the most popular non-parametric regression techniques in practice, obtained by regressing the response on features generated by evaluating the basis functions at observed covariate values. The most routinely used series estimator is based on ordinary least squares fitting, which is known to be minimax rate optimal in various settings, albeit under stringent restrictions on the basis functions and the distribution of covariates. In this work, inspired by the recently developed Forster-Warmuth (FW) learner, we propose an alternative series regression estimator that can attain the minimax estimation rate under strictly weaker conditions imposed on the basis functions and the joint law of covariates, than existing series estimators in the literature. Moreover, a key contribution of this work generalizes the FW-learner to a so-called counterfactual regression problem, in which the response variable of interest may not be directly observed (hence, the name ``counterfactual'') on all sampled units, and therefore needs to be inferred in order to identify and estimate the regression in view from the observed data. Although counterfactual regression is not entirely a new area of inquiry, we propose the first-ever systematic study of this challenging problem from a unified pseudo-outcome perspective. In fact, we provide what appears to be the first generic and constructive approach for generating the pseudo-outcome (to substitute for the unobserved response) which leads to the estimation of the counterfactual regression curve of interest with small bias, namely bias of second order. Several applications are used to illustrate the resulting FW-learner including many nonparametric regression problems in missing data and causal inference literature, for which we establish high-level conditions for minimax rate optimality of the proposed FW-learner.
We consider the problem of inference for projection parameters in linear regression with increasing dimensions. This problem has been studied under a variety of assumptions in the literature. The classical asymptotic normality result for the least squares estimator of the projection parameter only holds when the dimension $d$ of the covariates is of a smaller order than $n^{1/2}$, where $n$ is the sample size. Traditional sandwich estimator-based Wald intervals are asymptotically valid in this regime. In this work, we propose a bias correction for the least squares estimator and prove the asymptotic normality of the resulting debiased estimator. Precisely, we provide an explicit finite sample Berry Esseen bound on the Normal approximation to the law of the linear contrasts of the proposed estimator normalized by the sandwich standard error estimate. Our bound, under only finite moment conditions on covariates and errors, tends to 0 as long as $d = o(n^{2/3})$ up to the polylogarithmic factors. Furthermore, we leverage recent methods of statistical inference that do not require an estimator of the variance to perform asymptotically valid statistical inference and that leads to a sharper miscoverage control compared to Wald's. We provide a discussion of how our techniques can be generalized to increase the allowable range of $d$ even further.
Ensemble methods such as bagging and random forests are ubiquitous in various fields, from finance to genomics. Despite their prevalence, the question of the efficient tuning of ensemble parameters has received relatively little attention. This paper introduces a cross-validation method, ECV (Extrapolated Cross-Validation), for tuning the ensemble and subsample sizes in randomized ensembles. Our method builds on two primary ingredients: initial estimators for small ensemble sizes using out-of-bag errors and a novel risk extrapolation technique that leverages the structure of prediction risk decomposition. By establishing uniform consistency of our risk extrapolation technique over ensemble and subsample sizes, we show that ECV yields $δ$-optimal (with respect to the oracle-tuned risk) ensembles for squared prediction risk. Our theory accommodates general ensemble predictors, only requires mild moment assumptions, and allows for high-dimensional regimes where the feature dimension grows with the sample size. As a practical case study, we employ ECV to predict surface protein abundances from gene expressions in single-cell multiomics using random forests. In comparison to sample-split cross-validation and $K$-fold cross-validation, ECV achieves higher accuracy avoiding sample splitting. At the same time, its computational cost is considerably lower owing to the use of the risk extrapolation technique. Additional numerical results validate the finite-sample accuracy of ECV for several common ensemble predictors under a computational constraint on the maximum ensemble size.
Bagging is a commonly used ensemble technique in statistics and machine learning to improve the performance of prediction procedures. In this paper, we study the prediction risk of variants of bagged predictors under the proportional asymptotics regime, in which the ratio of the number of features to the number of observations converges to a constant. Specifically, we propose a general strategy to analyze the prediction risk under squared error loss of bagged predictors using classical results on simple random sampling. Specializing the strategy, we derive the exact asymptotic risk of the bagged ridge and ridgeless predictors with an arbitrary number of bags under a well-specified linear model with arbitrary feature covariance matrices and signal vectors. Furthermore, we prescribe a generic cross-validation procedure to select the optimal subsample size for bagging and discuss its utility to eliminate the non-monotonic behavior of the limiting risk in the sample size (i.e., double or multiple descents). In demonstrating the proposed procedure for bagged ridge and ridgeless predictors, we thoroughly investigate the oracle properties of the optimal subsample size and provide an in-depth comparison between different bagging variants.
Many important policy decisions concerning policing hinge on our understanding of how likely various criminal offenses are to result in arrests. Since many crimes are never reported to law enforcement, estimates based on police records alone must be adjusted to account for the likelihood that each crime would have been reported to the police. In this paper, we present a methodological framework for estimating the likelihood of arrest from police data that incorporates estimates of crime reporting rates computed from a victimization survey. We propose a parametric regression-based two-step estimator that (i) estimates the likelihood of crime reporting using logistic regression with survey weights; and then (ii) applies a second regression step to model the likelihood of arrest. Our empirical analysis focuses on racial disparities in arrests for violent crimes (sex offenses, robbery, aggravated and simple assaults) from 2006--2015 police records from the National Incident Based Reporting System (NIBRS), with estimates of crime reporting obtained using 2003--2020 data from the National Crime Victimization Survey (NCVS). We find that, after adjusting for unreported crimes, the likelihood of arrest computed from police records decreases significantly. We also find that, while incidents with white offenders on average result in arrests more often than those with black offenders, the disparities tend to be small after accounting for crime characteristics and unreported crimes.
We develop and analyze the HulC, an intuitive and general method for constructing confidence sets using the convex hull of estimates constructed from subsets of the data. Unlike classical methods which are based on estimating the (limiting) distribution of an estimator, the HulC is often simpler to use and effectively bypasses this step. In comparison to the bootstrap, the HulC requires fewer regularity conditions and succeeds in many examples where the bootstrap provably fails. Unlike subsampling, the HulC does not require knowledge of the rate of convergence of the estimators on which it is based. The validity of the HulC requires knowledge of the (asymptotic) median-bias of the estimators. We further analyze a variant of our basic method, called the Adaptive HulC, which is fully data-driven and estimates the median-bias using subsampling. We show that the Adaptive HulC retains the aforementioned strengths of the HulC. In certain cases where the underlying estimators are pathologically asymmetric the HulC and Adaptive HulC can fail to provide useful confidence sets. We propose a final variant, the Unimodal HulC, which can salvage the situation in cases where the distribution of the underlying estimator is (asymptotically) unimodal. We discuss these methods in the context of several challenging inferential problems which arise in parametric, semi-parametric, and non-parametric inference. Although our focus is on validity under weak regularity conditions, we also provide some general results on the width of the HulC confidence sets, showing that in many cases the HulC confidence sets have near-optimal width.
We study subsampling-based ridge ensembles in the proportional asymptotics regime, where the feature size grows proportionally with the sample size such that their ratio converges to a constant. By analyzing the squared prediction risk of ridge ensembles as a function of the explicit penalty $λ$ and the limiting subsample aspect ratio $ϕ_s$ (the ratio of the feature size to the subsample size), we characterize contours in the $(λ, ϕ_s)$-plane at any achievable risk. As a consequence, we prove that the risk of the optimal full ridgeless ensemble (fitted on all possible subsamples) matches that of the optimal ridge predictor. In addition, we prove strong uniform consistency of generalized cross-validation (GCV) over the subsample sizes for estimating the prediction risk of ridge ensembles. This allows for GCV-based tuning of full ridgeless ensembles without sample splitting and yields a predictor whose risk matches optimal ridge risk.
Conformal inference has played a pivotal role in providing uncertainty quantification for black-box ML prediction algorithms with finite sample guarantees. Traditionally, conformal prediction inference requires a data-independent specification of miscoverage level. In practical applications, one might want to update the miscoverage level after computing the prediction set. For example, in the context of binary classification, the analyst might start with a 95$\%$ prediction sets and see that most prediction sets contain all outcome classes. Prediction sets with both classes being undesirable, the analyst might desire to consider, say 80$\%$ prediction set. Construction of prediction sets that guarantee coverage with data-dependent miscoverage level can be considered as a post-selection inference problem. In this work, we develop simultaneous conformal inference to account for data-dependent miscoverage levels. Under the assumption of independent and identically distributed observations, our proposed methods have a finite sample simultaneous guarantee over all miscoverage levels. This allows practitioners to trade freely coverage probability for the quality of the prediction set by any criterion of their choice (say size of prediction set) while maintaining the finite sample guarantees similar to traditional conformal inference.
In Part I of this article (Banerjee and Kuchibhotla (2023)), we have introduced a new method to bound the difference in expectations of an average of independent random vector and the limiting Gaussian random vector using level sets. In the current article, we further explore this idea using finite sample Edgeworth expansions and also established integral representation theorems.
Recent development in high-dimensional statistical inference has necessitated concentration inequalities for a broader range of random variables. We focus on sub-Weibull random variables, which extend sub-Gaussian or sub-exponential random variables to allow heavy-tailed distributions. This paper presents concentration inequalities for independent sub-Weibull random variables with finite Generalized Bernstein-Orlicz norms, providing generalized Bernstein's inequalities and Rosenthal-type moment bounds. The tightness of the proposed bounds is shown through lower bounds of the concentration inequalities obtained via the Paley-Zygmund inequality. The results are applied to a graphical model inference problem, improving previous sample complexity bounds.
Conformal prediction has received tremendous attention in recent years and has offered new solutions to problems in missing data and causal inference; yet these advances have not leveraged modern semiparametric efficiency theory for more robust and efficient uncertainty quantification. In this paper, we consider the problem of obtaining distribution-free prediction regions accounting for a shift in the distribution of the covariates between the training and test data. Under an explainable covariate shift assumption analogous to the standard missing at random assumption, we propose three variants of a general framework to construct well-calibrated prediction regions for the unobserved outcome in the test sample. Our approach is based on the efficient influence function for the quantile of the unobserved outcome in the test population combined with an arbitrary machine learning prediction algorithm, without compromising asymptotic coverage. Next, we extend our approach to account for departure from the explainable covariate shift assumption in a semiparametric sensitivity analysis for potential latent covariate shift. In all cases, we establish that the resulting prediction sets eventually attain nominal average coverage in large samples. This guarantee is a consequence of the product bias form of our proposal which implies correct coverage if either the propensity score or the conditional distribution of the response is estimated sufficiently well. Our results also provide a framework for construction of doubly robust prediction sets of individual treatment effects, under both unconfoundedness and allowing for some degree of unmeasured confounding. Finally, we discuss aggregation of prediction sets from different machine learning algorithms for optimal prediction and illustrate the performance of our methods in both synthetic and real data.
In this work, we provide a $(n/m)^{-1/2}$-rate finite sample Berry-Esseen bound for $m$-dependent high-dimensional random vectors over the class of hyper-rectangles. This bound imposes minimal assumptions on the random vectors such as nondegenerate covariances and finite third moments. The proof uses inductive relationships between anti-concentration inequalities and Berry--Esseen bounds, which are inspired by the telescoping method of Chen and Shao (2004) and the recursion method of Kuchibhotla and Rinaldo (2020). Performing a dual induction based on the relationships, we obtain tight Berry-Esseen bounds for dependent samples.
In the United States and elsewhere, risk assessment algorithms are being used to help inform criminal justice decision-makers. A common intent is to forecast an offender's ``future dangerousness.'' Such algorithms have been correctly criticized for potential unfairness, and there is an active cottage industry trying to make repairs. In this paper, we use counterfactual reasoning to consider the prospects for improved fairness when members of a less privileged group are treated by a risk algorithm as if they are members of a more privileged group. We combine a machine learning classifier trained in a novel manner with an optimal transport adjustment for the relevant joint probability distributions, which together provide a constructive response to claims of bias-in-bias-out. A key distinction is between fairness claims that are empirically testable and fairness claims that are not. We then use confusion tables and conformal prediction sets to evaluate achieved fairness for projected risk. Our data are a random sample of 300,000 offenders at their arraignments for a large metropolitan area in the United States during which decisions to release or detain are made. We show that substantial improvement in fairness can be achieved consistent with a Pareto improvement for protected groups.