arXiv subjects

Qiyang Han

Publications and source records attributed to Qiyang Han.

At least 19 recordsLinked to original sources

arXiv2026

Empirical Bayes for correlated Gaussian sequence model

Empirical Bayes methods are among the most widely used statistical methods for large-scale inference. A central paradigm is the NPMLE, whose theoretical guarantees are by now well understood for the independent Gaussian sequence model. In this paper, we study empirical Bayes estimation from dependent observations in the Gaussian sequence model. We show that the maximum Composite Marginal Likelihood (CML) estimator, which ignores all correlations in the likelihood, converges in weighted Hellinger distance at the rate $n_*^{-1/2}$, where $n_*=n/\kappa_0$ is the `effective sample size' determined solely by the number of observations $n$ and the spectral radius $\kappa_0$ of the correlation matrix of the Gaussian observations. A complementary minimax lower bound shows that $n_*$ indeed serves as the right complexity measure, and that the CML estimator is nearly rate optimal under general dependence. We consider two concrete applications. In the first, we consider Bayesian linear regression, where the signal prior is estimated via CML applied to the least squares estimator. In the second, we consider the more challenging Bayesian nonlinear single-index model, where prior is estimated by CML applied to a one-step debiased gradient descent. In both applications, although the full likelihood landscape can be arbitrarily complicated and intractable, our CML method is facilitated by exploiting the high-dimensional distribution of the auxiliary statistics through a correlated Gaussian sequence model. The key ingredient in the proof of our results is a sharp local maximal inequality for the log composite marginal likelihood process under dependent Gaussian observations. In contrast to standard empirical process methods, we prove this inequality by leveraging a recent geometric Brascamp-Lieb inequality for Gaussian measures.

math.ST↗

arXiv2026

Thompson sampling: Precise arm-pull dynamics and adaptive inference

Adaptive sampling schemes are well known to create complex dependence that may invalidate conventional inference methods. A recent line of work shows that this need not be the case for UCB-type algorithms in multi-armed bandits. A central emerging theme is a `stability' property with asymptotically deterministic arm-pull counts in these algorithms, making inference as easy as in the i.i.d. setting. In this paper, we study the precise arm-pull dynamics in another canonical class of Thompson-sampling type algorithms. We show that the phenomenology is qualitatively different: the arm-pull count is asymptotically deterministic if and only if the arm is suboptimal or is the unique optimal arm; otherwise it converges in distribution to the unique invariant law of an SDE. This dichotomy uncovers a unifying principle behind many existing (in)stability results: an arm is stable if and only if its interaction with statistical noise is asymptotically negligible. As an application, we show that normalized arm means obey the same dichotomy, with Gaussian limits for stable arms and a semi-universal, non-Gaussian limit for unstable arms. This not only enables the construction of confidence intervals for the unknown mean rewards despite non-normality, but also reveals the potential of developing tractable inference procedures beyond the stable regime. The proofs rely on two new approaches. For suboptimal arms, we develop an `inverse process' approach that characterizes the inverse of the arm-pull count process via a Stieltjes integral. For optimal arms, we adopt a reparametrization of the arm-pull and noise processes that reduces the singularity in the natural SDE to proving the uniqueness of the invariant law of another SDE. We prove the latter by a set of analytic tools, including the parabolic H\"ormander condition and the Stroock-Varadhan support theorem.

math.ST↗

arXiv2025

Long-time dynamics and universality of nonconvex gradient descent

This paper develops a general approach to characterize the long-time trajectory behavior of nonconvex gradient descent in generalized single-index models in the large aspect ratio regime. In this regime, we show that for each iteration the gradient descent iterate concentrates around a deterministic vector called the `Gaussian theoretical gradient descent', whose dynamics can be tracked by a state evolution system of two recursive equations for two scalars. Our concentration guarantees hold universally for a broad class of design matrices and remain valid over long time horizons until algorithmic convergence or divergence occurs. Moreover, our approach reveals that gradient descent iterates are in general approximately independent of the data and strongly incoherent with the feature vectors, a phenomenon previously known as the `implicit regularization' effect of gradient descent in specific models under Gaussian data. As an illustration of the utility of our general theory, we present two applications of different natures in the regression setting. In the first, we prove global convergence of nonconvex gradient descent with general independent initialization for a broad class of structured link functions, and establish universality of randomly initialized gradient descent in phase retrieval for large aspect ratios. In the second, we develop a data-free iterative algorithm for estimating state evolution parameters along the entire gradient descent trajectory, thereby providing a low-cost yet statistically valid tool for practical tasks such as hyperparameter tuning and runtime determination. As a by-product of our analysis, we show that in the large aspect ratio regime, the Gaussian theoretical gradient descent coincides with a recent line of dynamical mean-field theory for gradient descent over the constant-time horizon.

cs.LG↗

arXiv2025

Precise gradient descent training dynamics for finite-width multi-layer neural networks

In this paper, we provide the first precise distributional characterization of gradient descent iterates for general multi-layer neural networks under the canonical single-index regression model, in the `finite-width proportional regime' where the sample size and feature dimension grow proportionally while the network width and depth remain bounded. Our non-asymptotic state evolution theory captures Gaussian fluctuations in first-layer weights and concentration in deeper-layer weights, and remains valid for non-Gaussian features. Our theory differs from existing neural tangent kernel (NTK), mean-field (MF) theories and tensor program (TP) in several key aspects. First, our theory operates in the finite-width regime whereas these existing theories are fundamentally infinite-width. Second, our theory allows weights to evolve from individual initializations beyond the lazy training regime, whereas NTK and MF are either frozen at or only weakly sensitive to initialization, and TP relies on special initialization schemes. Third, our theory characterizes both training and generalization errors for general multi-layer neural networks beyond the uniform convergence regime, whereas existing theories study generalization almost exclusively in two-layer settings. As a statistical application, we show that vanilla gradient descent can be augmented to yield consistent estimates of the generalization error at each iteration, which can be used to guide early stopping and hyperparameter tuning. As a further theoretical implication, we show that despite model misspecification, the model learned by gradient descent retains the structure of a single-index function with an effective signal determined by a linear combination of the true signal and the initialization.

cs.LG↗

arXiv2024

Gradient descent inference in empirical risk minimization

Gradient descent is one of the most widely used iterative algorithms in modern statistical learning. However, its precise algorithmic dynamics in high-dimensional settings remain only partially understood, which has limited its broader potential for statistical inference applications. This paper provides a precise, non-asymptotic joint distributional characterization of gradient descent iterates and their debiased statistics in a broad class of empirical risk minimization problems, in the so-called mean-field regime where the sample size is proportional to the signal dimension. Our non-asymptotic state evolution theory holds for both general non-convex loss functions and non-Gaussian data, and reveals the central role of two Onsager correction matrices that precisely characterize the non-trivial dependence among all gradient descent iterates in the mean-field regime. Leveraging the joint state evolution characterization, we show that the gradient descent iterate retrieves approximate normality after a debiasing correction via a linear combination of all past iterates, where the debiasing coefficients can be estimated by the proposed gradient descent inference algorithm. This leads to a new algorithmic statistical inference framework based on debiased gradient descent, which (i) applies to a broad class of models with both convex and non-convex losses, (ii) remains valid at each iteration without requiring algorithmic convergence, and (iii) exhibits a certain robustness to possible model misspecification. As a by-product, our framework also provides algorithmic estimates of the generalization error at each iteration. As canonical examples, we demonstrate our theory and inference methods in the single-index regression model and a generalized logistic regression model, where the natural loss functions may exhibit arbitrarily non-convex landscapes.

math.ST↗

arXiv2024

UCB algorithms for multi-armed bandits: Precise regret and adaptive inference

Upper Confidence Bound (UCB) algorithms are a widely-used class of sequential algorithms for the $K$-armed bandit problem. Despite extensive research over the past decades aimed at understanding their asymptotic and (near) minimax optimality properties, a precise understanding of their regret behavior remains elusive. This gap has not only hindered the evaluation of their actual algorithmic efficiency, but also limited further developments in statistical inference in sequential data collection. This paper bridges these two fundamental aspects--precise regret analysis and adaptive statistical inference--through a deterministic characterization of the number of arm pulls for an UCB index algorithm [Lai87, Agr95, ACBF02]. Our resulting precise regret formula not only accurately captures the actual behavior of the UCB algorithm for finite time horizons and individual problem instances, but also provides significant new insights into the regimes in which the existing theory remains informative. In particular, we show that the classical Lai-Robbins regret formula is exact if and only if the sub-optimality gaps exceed the order $\sigma\sqrt{K\log T/T}$. We also show that its maximal regret deviates from the minimax regret by a logarithmic factor, and therefore settling its strict minimax optimality in the negative. The deterministic characterization of the number of arm pulls for the UCB algorithm also has major implications in adaptive statistical inference. Building on the seminal work of [Lai82], we show that the UCB algorithm satisfies certain stability properties that lead to quantitative central limit theorems in two settings including the empirical means of unknown rewards in the bandit setting. These results have an important practical implication: conventional confidence sets designed for i.i.d. data remain valid even when data are collected sequentially.

math.ST↗

arXiv2024

Entrywise dynamics and universality of general first order methods

General first order methods (GFOMs), including various gradient descent and AMP algorithms, constitute a broad class of iterative algorithms in modern statistical learning problems. Some GFOMs also serve as constructive proof devices, iteratively characterizing the empirical distributions of statistical estimators in the large system limits for any fixed number of iterations. This paper develops a non-asymptotic, entrywise characterization for a general class of GFOMs. Our characterizations capture the precise entrywise behavior of the GFOMs, and hold universally across a broad class of heterogeneous random matrix models. As a corollary, we provide the first non-asymptotic description of the empirical distributions of the GFOMs beyond Gaussian ensembles. We demonstrate the utility of these general results in two applications. In the first application, we prove entrywise universality for regularized least squares estimators in the linear model, by controlling the entrywise error relative to a suitably constructed GFOM. This algorithmic proof method also leads to systematically improved averaged universality results for regularized regression estimators in the linear model, and resolves the universality conjecture for (regularized) MLEs in logistic regression. In the second application, we obtain entrywise Gaussian approximations for a class of gradient descent algorithms. Our approach provides non-asymptotic state evolution for the bias and variance of the algorithm along the iteration path, applicable for non-convex loss functions. The proof relies on a new recursive leave-k-out method that provides almost delocalization for the GFOMs and their derivatives. Crucially, our method ensures entrywise universality for up to poly-logarithmic many iterations, which facilitates effective $\ell_2/\ell_\infty$ control between certain GFOMs and statistical estimators in applications.

math.ST↗

arXiv2023

A leave-one-out approach to approximate message passing

Approximate message passing (AMP) has emerged both as a popular class of iterative algorithms and as a powerful analytic tool in a wide range of statistical estimation problems and statistical physics models. A well established line of AMP theory proves Gaussian approximations for the empirical distributions of the AMP iterate in the high dimensional limit, under the GOE random matrix model and its variants. This paper provides a non-asymptotic, leave-one-out representation for the AMP iterate that holds under a broad class of Gaussian random matrix models with general variance profiles. In contrast to the typical AMP theory that describes the empirical distributions of the AMP iterate via a low dimensional state evolution, our leave-one-out representation yields an intrinsically high dimensional state evolution formula which provides non-asymptotic characterizations for the possibly heterogeneous, entrywise behavior of the AMP iterate under the prescribed random matrix models. To exemplify some distinct features of our AMP theory in applications, we analyze, in the context of regularized linear estimation, the precise stochastic behavior of the Ridge estimator for independent and non-identically distributed observations whose covariates exhibit general variance profiles. We find that its finite-sample distribution is characterized via a weighted Ridge estimator in a heterogeneous Gaussian sequence model. Notably, in contrast to the i.i.d. sampling scenario, the effective noise and regularization are now full dimensional vectors determined via a high dimensional system of equations. Our leave-one-out method of proof differs significantly from the widely adopted conditioning approach for rotational invariant ensembles, and relies instead on an inductive method that utilizes almost solely integration-by-parts and concentration techniques.

math.ST↗

arXiv2023

The distribution of Ridgeless least squares interpolators

The Ridgeless minimum $\ell_2$-norm interpolator in overparametrized linear regression has attracted considerable attention in recent years in both machine learning and statistics communities. While it seems to defy conventional wisdom that overfitting leads to poor prediction, recent theoretical research on its $\ell_2$-type risks reveals that its norm minimizing property induces an `implicit regularization' that helps prediction in spite of interpolation. This paper takes a further step that aims at understanding its precise stochastic behavior as a statistical estimator. Specifically, we characterize the distribution of the Ridgeless interpolator in high dimensions, in terms of a Ridge estimator in an associated Gaussian sequence model with positive regularization, which provides a precise quantification of the prescribed implicit regularization in the most general distributional sense. Our distributional characterizations hold for general non-Gaussian random designs and extend uniformly to positively regularized Ridge estimators. As a direct application, we obtain a complete characterization for a general class of weighted $\ell_q$ risks of the Ridge(less) estimators that are previously only known for $q=2$ by random matrix methods. These weighted $\ell_q$ risks not only include the standard prediction and estimation errors, but also include the non-standard covariate shift settings. Our uniform characterizations further reveal a surprising feature of the commonly used generalized and $k$-fold cross-validation schemes: tuning the estimated $\ell_2$ prediction risk by these methods alone lead to simultaneous optimal $\ell_2$ in-sample, prediction and estimation risks, as well as the optimal length of debiased confidence intervals.

math.ST↗

arXiv2022

Gaussian random projections of convex cones: approximate kinematic formulae and applications

Understanding the stochastic behavior of random projections of geometric sets constitutes a fundamental problem in high dimension probability that finds wide applications in diverse fields. This paper provides a kinematic description for the behavior of Gaussian random projections of closed convex cones, in analogy to that of randomly rotated cones studied in [ALMT14]. Formally, let $K$ be a closed convex cone in $\mathbb{R}^n$, and $G\in \mathbb{R}^{m\times n}$ be a Gaussian matrix with i.i.d. $\mathcal{N}(0,1)$ entries. We show that $GK\equiv \{G\mu: \mu \in K\}$ behaves like a randomly rotated cone in $\mathbb{R}^m$ with statistical dimension $\min\{\delta(K),m\}$, in the following kinematic sense: for any fixed closed convex cone $L$ in $\mathbb{R}^m$, \begin{align*} &\delta(L)+\delta(K)\ll m\, \Rightarrow\, L\cap GK = \{0\} \hbox{ with high probability},\\ &\delta(L)+\delta(K)\gg m\, \Rightarrow\, L\cap GK \neq \{0\} \hbox{ with high probability}. \end{align*} A similar kinematic description is obtained for $G^{-1}L\equiv \{\mu \in \mathbb{R}^n: G\mu \in L\}$. The practical utility and broad applicability of the prescribed approximate kinematic formulae are demonstrated in a number of distinct problems arising from statistical learning, mathematical programming and asymptotic geometric analysis. In particular, we prove (i) new phase transitions of the existence of cone constrained maximum likelihood estimators in logistic regression, (ii) new phase transitions of the cost optimum of deterministic conic programs with random constraints, and (iii) a local version of the Gaussian Dvoretzky-Milman theorem that describes almost deterministic, low-dimensional behaviors of subspace sections of randomly projected convex sets.

math.PR↗

arXiv2022

Exact bounds for some quadratic empirical processes with applications

Let $Z_1,\ldots,Z_n$ be i.i.d. isotropic random vectors in $\mathbb{R}^p$, and $T \subset \mathbb{R}^p$ be a compact set. A classical line of empirical process theory characterizes the size of the suprema of the quadratic process \begin{align*} \sup_{t \in T} \bigg| \frac{1}{n}\sum_{i=1}^n \langle Z_i,t \rangle^2-\lVert t \rVert^2 \bigg|, \end{align*} via a single parameter known as the Gaussian width of $T$. This paper introduces an improved bound for the suprema of this quadratic process for standard Gaussian vectors $\{Z_i\}$ that can be exactly attained for certain choices of $T$, and is thus referred to as an exact bound. Our exact bound is expressed via a collection of (stochastic) Gaussian widths over spherical sections of $T$ that serves as a natural multi-scale analogue to the Gaussian width of $T$. Compared to the classical bounds for the quadratic process, our new bounds not only determine the optimal constants in the classical bounds that can be attained for some $T$, but also precisely capture certain subtle phase transitional behavior of the quadratic process beyond the reach of the classical bounds. To illustrate the utility of our results, we obtain tight versions of the Gaussian Dvoretzky-Milman theorem for random projection, and the Koltchinskii-Lounici theorem for covariance estimation, both with optimal constants. Moreover, our bounds recover the celebrated BBP phase transitional behavior of the top eigenvalue of the sample covariance and its generalization to the sample covariance error. The proof of our results exploits recently sharpened Gaussian comparison inequalities. The technical scope of our method of proof is further demonstrated in obtaining an exact bound for a two-sided Chevet inequality.

math.PR↗

arXiv2022

Universality of regularized regression estimators in high dimensions

The Convex Gaussian Min-Max Theorem (CGMT) has emerged as a prominent theoretical tool for analyzing the precise stochastic behavior of various statistical estimators in the so-called high dimensional proportional regime, where the sample size and the signal dimension are of the same order. However, a well recognized limitation of the existing CGMT machinery rests in its stringent requirement on the exact Gaussianity of the design matrix, therefore rendering the obtained precise high dimensional asymptotics largely a specific Gaussian theory in various important statistical models. This paper provides a structural universality framework for a broad class of regularized regression estimators that is particularly compatible with the CGMT machinery. In particular, we show that with a good enough $\ell_\infty$ bound for the regression estimator $\hat{\mu}_A$, any `structural property' that can be detected via the CGMT for $\hat{\mu}_G$ (under a standard Gaussian design $G$) also holds for $\hat{\mu}_A$ under a general design $A$ with independent entries. As a proof of concept, we demonstrate our new universality framework in three key examples of regularized regression estimators: the Ridge, Lasso and regularized robust regression estimators, where new universality properties of risk asymptotics and/or distributions of regression estimators and other related quantities are proved. As a major statistical implication of the Lasso universality results, we validate inference procedures using the degrees-of-freedom adjusted debiased Lasso under general design and error distributions. We also provide a counterexample, showing that universality properties for regularized regression estimators do not extend to general isotropic designs.

math.ST↗

arXiv2022

Noisy linear inverse problems under convex constraints: Exact risk asymptotics in high dimensions

In the standard Gaussian linear measurement model $Y=Xμ_0+ξ\in \mathbb{R}^m$ with a fixed noise level $σ>0$, we consider the problem of estimating the unknown signal $μ_0$ under a convex constraint $μ_0 \in K$, where $K$ is a closed convex set in $\mathbb{R}^n$. We show that the risk of the natural convex constrained least squares estimator (LSE) $\hatμ(σ)$ can be characterized exactly in high dimensional limits, by that of the convex constrained LSE $\hatμ_K^{\mathsf{seq}}$ in the corresponding Gaussian sequence model at a different noise level. The characterization holds (uniformly) for risks in the maximal regime that ranges from constant order all the way down to essentially the parametric rate, as long as certain necessary non-degeneracy condition is satisfied for $\hatμ(σ)$. The precise risk characterization reveals a fundamental difference between noiseless (or low noise limit) and noisy linear inverse problems in terms of the sample complexity for signal recovery. A concrete example is given by the isotonic regression problem: While exact recovery of a general monotone signal requires $m\gg n^{1/3}$ samples in the noiseless setting, consistent signal recovery in the noisy setting requires as few as $m\gg \log n$ samples. Such a discrepancy occurs when the low and high noise risk behavior of $\hatμ_K^{\mathsf{seq}}$ differ significantly. In statistical languages, this occurs when $\hatμ_K^{\mathsf{seq}}$ estimates $0$ at a faster `adaptation rate' than the slower `worst-case rate' for general signals. Several other examples, including non-negative least squares and generalized Lasso (in constrained forms), are also worked out to demonstrate the concrete applicability of the theory in problems of different types.

math.ST↗

arXiv2021

Nonparametric, tuning-free estimation of S-shaped functions

We consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimisation problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst-case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm the desirable finite-sample properties of the estimator, and our algorithm is implemented in the R package Sshaped.

stat.ME↗

arXiv2021

Berry-Esseen bounds for Chernoff-type non-standard asymptotics in isotonic regression

A Chernoff-type distribution is a nonnormal distribution defined by the slope at zero of the greatest convex minorant of a two-sided Brownian motion with a polynomial drift. While a Chernoff-type distribution is known to appear as the distributional limit in many non-regular statistical estimation problems, the accuracy of Chernoff-type approximations has remained largely unknown. In the present paper, we tackle this problem and derive Berry-Esseen bounds for Chernoff-type limit distributions in the canonical non-regular statistical estimation problem of isotonic (or monotone) regression. The derived Berry-Esseen bounds match those of the oracle local average estimator with optimal bandwidth in each scenario of possibly different Chernoff-type asymptotics, up to multiplicative logarithmic factors. Our method of proof differs from standard techniques on Berry-Esseen bounds, and relies on new localization techniques in isotonic regression and an anti-concentration inequality for the supremum of a Brownian motion with a Lipschitz drift.

math.ST↗

arXiv2021

High dimensional asymptotics of likelihood ratio tests in the Gaussian sequence model under convex constraints

In the Gaussian sequence model $Y=μ+ξ$, we study the likelihood ratio test (LRT) for testing $H_0: μ=μ_0$ versus $H_1: μ\in K$, where $μ_0 \in K$, and $K$ is a closed convex set in $\mathbb{R}^n$. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair $(μ_0,K)$, in the high dimensional regime where the estimation error of the associated least squares estimator diverges in an appropriate sense. The normal approximation further leads to a precise characterization of the power behavior of the LRT in the high dimensional regime. These characterizations show that the power behavior of the LRT is in general non-uniform with respect to the Euclidean metric, and illustrate the conservative nature of existing minimax optimality and sub-optimality results for the LRT. A variety of examples, including testing in the orthant/circular cone, isotonic regression, Lasso, and testing parametric assumptions versus shape-constrained alternatives, are worked out to demonstrate the versatility of the developed theory.

math.ST↗

arXiv2021

Generalized kernel distance covariance in high dimensions: non-null CLTs and power universality

Distance covariance is a popular dependence measure for two random vectors $X$ and $Y$ of possibly different dimensions and types. Recent years have witnessed concentrated efforts in the literature to understand the distributional properties of the sample distance covariance in a high-dimensional setting, with an exclusive emphasis on the null case that $X$ and $Y$ are independent. This paper derives the first non-null central limit theorem for the sample distance covariance, and the more general sample (Hilbert-Schmidt) kernel distance covariance in high dimensions, primarily in the Gaussian case. The new non-null central limit theorem yields an asymptotically exact first-order power formula for the widely used generalized kernel distance correlation test of independence between $X$ and $Y$. The power formula in particular unveils an interesting universality phenomenon: the power of the generalized kernel distance correlation test is completely determined by $n\cdot \text{dcor}^2(X,Y)/\sqrt{2}$ in the high dimensional limit, regardless of a wide range of choices of the kernels and bandwidth parameters. Furthermore, this separation rate is also shown to be optimal in a minimax sense. The key step in the proof of the non-null central limit theorem is a precise expansion of the mean and variance of the sample distance covariance in high dimensions, which shows, among other things, that the non-null Gaussian approximation of the sample distance covariance involves a rather subtle interplay between the dimension-to-sample ratio and the dependence between $X$ and $Y$.

math.ST↗

arXiv2021

Oracle posterior contraction rates under hierarchical priors

We offer a general Bayes theoretic framework to derive posterior contraction rates under a hierarchical prior design: the first-step prior serves to assess the model selection uncertainty, and the second-step prior quantifies the prior belief on the strength of the signals within the model chosen from the first step. In particular, we establish non-asymptotic oracle posterior contraction rates under (i) a local Gaussianity condition on the log likelihood ratio of the statistical experiment, (ii) a local entropy condition on the dimensionality of the models, and (iii) a sufficient mass condition on the second-step prior near the best approximating signal for each model. The first-step prior can be designed generically. The posterior distribution enjoys Gaussian tail behavior and therefore the resulting posterior mean also satisfies an oracle inequality, automatically serving as an adaptive point estimator in a frequentist sense. Model mis-specification is allowed in these oracle rates. The local Gaussianity condition serves as a unified attempt of non-asymptotic Gaussian quantification of the experiments, and can be easily verified in various experiments considered in [GvdV07a] and beyond. The general results are applied in various problems including: (i) trace regression, (ii) shape-restricted isotonic/convex regression, (iii) high-dimensional partially linear regression, (iv) covariance matrix estimation in the sparse factor model, (v) detection of non-smooth polytopal image boundary, and (vi) intensity estimation in a Poisson point process model. These new results serve either as theoretical justification of practical prior proposals in the literature, or as an illustration of the generic construction scheme of a (nearly) minimax adaptive estimator for a complicated experiment.

math.ST↗