Finite Sample Analysis Research Articles

False discovery rate control for high-dimensional Cox model with uneven data splitting

ABSTRACT Statistical inference for high-dimensional survival data is important for obtaining valid scientific results in many research areas, including biomedical studies and financial risk management. In this paper, a novel framework for feature selection in Cox model is proposed, which achieves false discovery rate (FDR) control asymptotically. The key step is to construct a sequence of ranking statistics based on two independent estimators of the regression coefficients. The FDR control is reached by choosing a data-driven threshold along the ranking of symmetric-based statistics. The de-sparsified estimator and uneven data splitting strategy are employed to improve the robustness of variable selection results and the power in finite sample analysis. We establish the asymptotic FDR control property for the proposed approach at any designated level. Extensive simulation studies and an empirical application on a P2P loan dataset confirm the robustness of the proposed method in FDR control, and show that it often leads to higher power among competitors.

Finite-Sample Analysis of Deep CCA-Based Unsupervised Post-Nonlinear Multimodal Learning.

Canonical correlation analysis (CCA) has been essential in unsupervised multimodal/multiview latent representation learning and data fusion. Classic CCA extracts shared information from multiple modalities of data using linear transformations. In recent years, deep neural networks-based nonlinear feature extractors were combined with CCA to come up with new variants, namely the ``DeepCCA'' line of work. These approaches were shown to have enhanced performance in many applications. However, theoretical supports of DeepCCA are often lacking. To address this challenge, the recent work of Lyu and Fu (2020) showed that, under a reasonable postnonlinear generative model, a carefully designed DeepCCA criterion provably removes unknown distortions in data generation and identifies the shared information across modalities. Nonetheless, a critical assumption used by Lyu and Fu (2020) for identifiability analysis was that unlimited data is available, which is unrealistic. This brief paper puts forth a finite-sample analysis of the DeepCCA method by Lyu and Fu (2020). The main result is that the finite-sample version of the method can still estimate the shared information with a guaranteed accuracy when the number of samples is sufficiently large. Our analytical approach is a nontrivial integration of statistical learning, numerical differentiation, and robust system identification, which may be of interest beyond the scope of DeepCCA and benefit other unsupervised learning paradigms.

A Lyapunov Theory for Finite-Sample Guarantees of Markovian Stochastic Approximation

The stochastic approximation (SA) method stands as the foundational mathematical tool for modern large-scale optimization and machine learning. Therefore, gaining a theoretical understanding of SA algorithms is of fundamental interest. In their paper titled “A Lyapunov Theory for Finite-Sample Guarantees of Markovian Stochastic Approximation,” Chen et al. present a unified Lyapunov framework for the finite-sample analysis of a Markovian SA algorithm under a contractive operator with respect to an arbitrary norm. The key novelty lies in the construction of a smooth Lyapunov function called the generalized Moreau envelope. The authors demonstrate the effectiveness of their SA results in the context of reinforcement learning (RL), specifically through popular algorithms such as variants of temporal difference (TD) learning and Q-learning. As byproducts, the results provide theoretical insights into the efficiency of bootstrapping in TD learning with eligibility traces and the bias-variance tradeoff in off-policy learning.

Treatment choice, mean square regret and partial identification

We consider a decision maker who faces a binary treatment choice when their welfare is only partially identified from data. We contribute to the literature by anchoring our finite-sample analysis on mean square regret, a decision criterion advocated by Kitagawa et al. in (2022) "Treatment Choice with Nonlinear Regret" . We find that optimal rules are always fractional, irrespective of the width of the identified set and precision of its estimate. The optimal treatment fraction is a simple logistic transformation of the commonly used t-statistic multiplied by a factor calculated by a simple constrained optimization. This treatment fraction gets closer to 0.5 as the width of the identified set becomes wider, implying the decision maker becomes more cautious against the adversarial Nature.

Finite Sample Analysis for Structured Discrete System Identification

We consider a discrete-time dynamical system over a discrete state-space, which evolves according to a structured Markov model called Bernoulli Autoregressive (BAR) model. Our goal is to obtain sample complexity bounds for the problem of estimating the parameters of this model using an indirect Maximum Likelihood Estimator. Our sample complexity bounds exploit the structure of the BAR model and are established using concentration inequalities for random matrices and Lipschitz functions.

Near-Optimal Multi-Robot Motion Planning with Finite Sampling

An underlying structure in several sampling-based methods for continuous multirobot motion planning (MRMP) is the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tensor roadmap</i> , which emerges from combining multiple probabilistic roadmap (PRM) graphs constructed for the individual robots via a tensor product. We study the conditions under which the tensor roadmap encodes a near-optimal solution for MRMP—satisfying these conditions implies near optimality for a variety of popular planners, including dRRT*, and the discrete methods M* and conflict-based search, when applied to the continuous domain. We develop the first finite-sample analysis of this kind, which specifies the number of samples, their deterministic distribution, and magnitude of the connection radii that should be used by each individual PRM graph, to guarantee near-optimality using the tensor roadmap. This significantly improves upon a previous asymptotic analysis, wherein the number of samples tends to infinity. Our new finite sample-size analysis supports guaranteed high-quality solutions in practice within finite time. To achieve our new result, we first develop a sampling scheme, which we call the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">staggered grid</i> , for finite-sample motion planning for individual robots, which requires significantly fewer samples than previous work. We then extend it to the much more involved MRMP setting, which requires to account for interactions among multiple robots. Finally, we report on a few experiments that serve as a verification of our theoretical findings and raise interesting questions for further investigation.

Finite-Sample Analysis of Two-Time-Scale Natural Actor–Critic Algorithm

Actor-critic style two-time-scale algorithms are one of the most popular methods in reinforcement learning, and have seen great empirical success. However, their performance is not completely understood theoretically. In this paper, we characterize the <i>global</i> convergence of an online natural actor-critic algorithm in the tabular setting using a single trajectory of samples. Our analysis applies to very general settings, as we only assume ergodicity of the underlying Markov decision process. In order to ensure enough exploration, we employ an <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula>-greedy sampling of the trajectory. For a fixed and small enough exploration parameter <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula>, we show that the two-time-scale natural actor-critic algorithm has a rate of convergence of <inline-formula><tex-math notation="LaTeX">$\tilde{\mathcal {O}}(1/T^{1/4})$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> is the number of samples, and this leads to a sample complexity of <inline-formula><tex-math notation="LaTeX">$\tilde{\mathcal {O}}(1/\delta ^{8})$</tex-math></inline-formula> samples to find a policy that is within an error of <inline-formula><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> from the <i>global optimum</i>. Moreover, by carefully decreasing the exploration parameter <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> as the iterations proceed, we present an improved sample complexity of <inline-formula><tex-math notation="LaTeX">$\tilde{\mathcal {O}}(1/\delta ^{6})$</tex-math></inline-formula> for convergence to the global optimum.

Online Learning of the Kalman Filter With Logarithmic Regret

In this paper, we consider the problem of predicting observations generated online by an unknown, partially observable linear system, which is driven by Gaussian noise. In the linear Gaussian setting, the optimal predictor in the mean square error sense is the celebrated Kalman filter, which can be explicitly computed when the system model is known. When the system model is unknown, we have to learn how to predict observations online based on finite data, suffering possibly a non-zero regret with respect to the Kalman filter's prediction. We show that it is possible to achieve a regret of the order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{poly}\log (\mathsf {N})$</tex-math></inline-formula> with high probability, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> is the number of observations collected. This is achieved using an online least-squares algorithm, which exploits the approximately linear relation between future observations and past observations. The regret analysis is based on the stability properties of the Kalman filter, recent statistical tools for finite sample analysis of system identification, and classical results for the analysis of least-squares algorithms for time series. Our regret analysis can also be applied to other predictors, e.g. multiple step-ahead prediction, or prediction under exogenous inputs including closed-loop prediction. A fundamental technical contribution is that our bounds hold even for the class of non-explosive systems (including marginally stable systems), which was not addressed before in the case of online prediction.

Bayesian Robustness: A Nonasymptotic Viewpoint

We study the problem of robustly estimating the posterior distribution for the setting where observed data can be contaminated with potentially adversarial outliers. We propose Rob-ULA, a robust variant of the Unadjusted Langevin Algorithm (ULA), and provide a finite-sample analysis of its sampling distribution. In particular, we show that after T = O ˜ ( d / ε acc ) iterations, we can sample from pT such that dist ( p T , p * ) ≤ ε acc + O ˜ ( ϵ ) , where ϵ is the fraction of corruptions and dist represents the squared 2-Wasserstein distance metric. Our results for the class of posteriors p * which satisfy log-concavity and smoothness assumptions. We corroborate our theoretical analysis with experiments on both synthetic and real-world datasets for mean estimation, regression and binary classification. Supplementary materials for this article are available online.

A Novel EM-Type Algorithm to Estimate Semi-Parametric Mixtures of Partially Linear Models

Semi- and non-parametric mixture of normal regression models are a flexible class of mixture of regression models. These models assume that the component mixing proportions, regression functions and/or variances are non-parametric functions of the covariates. Among this class of models, the semi-parametric mixture of partially linear models (SPMPLMs) combine the desirable interpretability of a parametric model and the flexibility of a non-parametric model. However, local-likelihood estimation of the non-parametric term poses a computational challenge. Traditional EM optimisation of the local-likelihood functions is not appropriate due to the label-switching problem. Separately applying the EM algorithm on each local-likelihood function will likely result in non-smooth function estimates. This is because the local responsibilities calculated at the E-step of each local EM are not guaranteed to be aligned. To prevent this, the EM algorithm must be modified so that the same (global) responsibilities are used at each local M-step. In this paper, we propose a one-step backfitting EM-type algorithm to estimate the SPMPLMs and effectively address the label-switching problem. The proposed algorithm estimates the non-parametric term using each set of local responsibilities in turn and then incorporates a smoothing step to obtain the smoothest estimate. In addition, to reduce the computational burden imposed by the use of the partial-residuals estimator of the parametric term, we propose a plug-in estimator. The performance and practical usefulness of the proposed methods was tested using a simulated dataset and two real datasets, respectively. Our finite sample analysis revealed that the proposed methods are effective at solving the label-switching problem and producing reasonable and interpretable results in a reasonable amount of time.

Conditional predictive inference for stable algorithms

We investigate generically applicable and intuitively appealing prediction intervals based on k-fold cross-validation. We focus on the conditional coverage probability of the proposed intervals, given the observations in the training sample (hence, training conditional validity), and show that it is close to the nominal level, in an appropriate sense, provided that the underlying algorithm used for computing point predictions is sufficiently stable when feature-response pairs are omitted. Our results are based on a finite sample analysis of the empirical distribution function of k-fold cross-validation residuals and hold in nonparametric settings with only minimal assumptions on the error distribution. To illustrate our results, we also apply them to high-dimensional linear predictors, where we obtain uniform asymptotic training conditional validity as both sample size and dimension tend to infinity at the same rate and consistent parameter estimation typically fails. These results show that despite the serious problems of resampling procedures for inference on the unknown parameters (cf. in A Festschrift for Erich L. Lehmann (1983) 28–48 Wadsworth; Ann. Statist. 24 (1996) 307–335; J. Mach. Learn. Res. 19 (2018) 5), cross-validation methods can be successfully applied to obtain reliable predictive inference even in high dimensions and conditionally on the training data.

Affine invariant integrated rank-weighted statistical depth: properties and finite sample analysis

Affine invariant integrated rank-weighted statistical depth: properties and finite sample analysis

Finite-Sample Analysis of Identification of Switched Linear Systems With Arbitrary or Restricted Switching

For the identification of switched systems with measured states and a measured switching signal, this letter aims to analyze the effect of switching strategies on the estimation error. The data is assumed to be collected from globally asymptotically or marginally stable switched systems under switches that are arbitrary or subject to an average dwell time constraint. Then the switched system is estimated by the least-squares (LS) estimator. To capture the effect of the parameters of the switching strategies on the LS estimation error, finite-sample error bounds are developed in this letter. The obtained error bounds show that the estimation error is logarithmic of the switching parameters when there are only stable modes; however, when there are unstable modes, the estimation error bound can increase linearly as the switching parameter changes. This suggests that in the presence of unstable modes, the switching strategy should be properly designed to avoid the significant increase of the estimation error.

A Unified Lyapunov Framework for Finite-Sample Analysis of Reinforcement Learning Algorithms

Reinforcement learning (RL) is a paradigm where an agent learns to accomplish tasks by interacting with the environment, similar to how humans learn. RL is therefore viewed as a promising approach to achieve artificial intelligence, as evidenced by the remarkable empirical successes. However, many RL algorithms are theoretically not well-understood, especially in the setting where function approximation and off-policy sampling are employed. My thesis [1] aims at developing thorough theoretical understanding to the performance of various RL algorithms through finite-sample analysis.Since most of the RL algorithms are essentially stochastic approximation (SA) algorithms for solving variants of the Bellman equation, the first part of thesis is dedicated to the analysis of general SA involving a contraction operator, and under Markovian noise. We develop a Lyapunov approach where we construct a novel Lyapunov function called the generaled Moreau envelope. The results on SA enable us to establish finite-sample bounds of various RL algorithms in the tabular setting (cf. Part II of the thesis) and when using function approximation (cf. Part III of the thesis), which in turn provide theoretical insights to several important problems in the RL community, such as the efficiency of bootstrapping, the bias-variance trade-off in off-policy learning, and the stability of off-policy control.The main body of this document provides an overview of the contributions of my thesis.

Asymptotic and finite sample properties of Hill-type estimators in the presence of errors in observations

We establish asymptotic and finite sample properties of the Hill and Harmonic Moment estimators applied to heavy-tailed data contaminated by errors. We formulate conditions on the errors and the number of upper order statistics under which these estimators continue to be asymptotically normal. We specify analogous conditions which must hold in finite samples for the confidence intervals derived from the asymptotic normal distribution to be reliable. In the large sample analysis, we specify conditions related to second-order regular variation and divergence rates for the number of upper order statistics, k, used to compute the estimators. In the finite sample analysis, we examine several data-driven methods of selecting k, and determine which of them are most suitable for confidence interval inference. The results of these investigations are applied to interarrival times of internet traffic anomalies, which are available only with a round-off error.

Finite-sample analysis of nonlinear stochastic approximation with applications in reinforcement learning

Motivated by applications in reinforcement learning (RL), we study a nonlinear stochastic approximation (SA) algorithm under Markovian noise, and establish its finite-sample convergence bounds under various stepsizes. Specifically, we show that when using constant stepsize (i.e., αk≡α), the algorithm achieves exponential fast convergence to a neighborhood (with radius O(αlog(1/α))) around the desired limit point. When using diminishing stepsizes with appropriate decay rate, the algorithm converges with rate O(log(k)/k). Our proof is based on Lyapunov drift arguments, and to handle the Markovian noise, we exploit the fast mixing of the underlying Markov chain. To demonstrate the generality of our theoretical results on Markovian SA, we use it to derive the finite-sample bounds of the popular Q-learning algorithm with linear function approximation, under a condition on the behavior policy. Importantly, we do not need to make the assumption that the samples are i.i.d., and do not require an artificial projection step in the algorithm. Numerical simulations corroborate our theoretical results.

Computationally efficient sparse clustering

Abstract We study statistical and computational limits of clustering when the means of the centres are sparse and their dimension is possibly much larger than the sample size. Our theoretical analysis focuses on the model $X_i=z_i \theta +\varepsilon _{i}, \ z_i \in \{-1,1\}, \ \varepsilon _i \thicksim \mathcal{N}(0, I)$, which has two clusters with centres $\theta $ and $-\theta $. We provide a finite sample analysis of a new sparse clustering algorithm based on sparse Principal Component Analysis (PCA) and show that it achieves the minimax optimal misclustering rate in the regime $\|\theta \| \rightarrow \infty $. Our results require the sparsity to grow slower than the square root of the sample size. Using a recent framework for computational lower bounds—the low-degree likelihood ratio—we give evidence that this condition is necessary for any polynomial-time clustering algorithm to succeed below the Baik-Ben Arous-Péché (BBP) threshold. This complements the existing evidence based on reductions and statistical query lower bounds. Compared with these existing results, we cover a wider set of parameter regimes and give a more precise understanding of the runtime required and the misclustering error achievable. Our results imply that a large class of tests based on low-degree polynomials fail to solve even the weak testing task.

Learning Topic Models: Identifiability and Finite-Sample Analysis

Topic models provide a useful text-mining tool for learning, extracting, and discovering latent structures in large text corpora. Although a plethora of methods have been proposed for topic modeling, lacking in the literature is a formal theoretical investigation of the statistical identifiability and accuracy of latent topic estimation. In this article, we propose a maximum likelihood estimator (MLE) of latent topics based on a specific integrated likelihood that is naturally connected to the concept, in computational geometry, of volume minimization. Our theory introduces a new set of geometric conditions for topic model identifiability, conditions that are weaker than conventional separability conditions, which typically rely on the existence of pure topic documents or of anchor words. Weaker conditions allow a wider and thus potentially more fruitful investigation. We conduct finite-sample error analysis for the proposed estimator and discuss connections between our results and those of previous investigations. We conclude with empirical studies employing both simulated and real datasets. Supplementary materials for this article are available online.

Edgeworth expansions for network moments

Network method of moments (Ann. Statist. 39 (2011) 2280–2301) is an important tool for nonparametric network inference. However, there has been little investigation on accurate descriptions of the sampling distributions of network moment statistics. In this paper, we present the first higher-order accurate approximation to the sampling CDF of a studentized network moment by Edgeworth expansion. In sharp contrast to classical literature on noiseless U-statistics, we show that the Edgeworth expansion of a network moment statistic as a noisy U-statistic can achieve higher-order accuracy without nonlattice or smoothness assumptions but just requiring weak regularity conditions. Behind this result is our surprising discovery that the two typically-hated factors in network analysis, namely, sparsity and edgewise observational errors, jointly play a blessing role, contributing a crucial self-smoothing effect in the network moment statistic and making it analytically tractable. Our assumptions match the minimum requirements in related literature. For sparse networks, our theory shows that our empirical Edgeworth expansion and a simple normal approximation both achieve the same gradually depreciating Berry–Esseen-type bound as the network becomes sparser. This result also significantly refines the best previous theoretical result. For practitioners, our empirical Edgeworth expansion is highly accurate and computationally efficient. It is also easy to implement and convenient for parallel computing. We demonstrate the clear advantage of our method by several comprehensive simulation studies. As a byproduct, we also provide a finite-sample analysis of the network jackknife. We showcase three applications of our results in network inference. We prove, to our knowledge, the first theoretical guarantee of higher-order accuracy for some network bootstrap schemes, and moreover, the first theoretical guidance for selecting the subsample size for network subsampling. We also derive a one-sample test and the Cornish–Fisher confidence interval for a given moment with higher-order accurate controls of confidence level and type I error, respectively.

The Fragility of Robust Causality Analysis: Finite-Sample Analysis and the Inflation-Economic Growth Relationship

The Fragility of Robust Causality Analysis: Finite-Sample Analysis and the Inflation-Economic Growth Relationship