Adaptive Minimax Estimation over Sparse lq-Hulls

Given a dictionary of Mn initial estimates of the unknown true regression function, we aim to construct linearly aggregated estimators that target the best performance among all the linear combinations under a sparse q-norm (0 ≤ q ≤ 1) constraint on the linear coefficients. Besides identifying the optimal rates of aggregation for these `q-aggregation problems, our multi-directional (or universal) aggregation strategies by model mixing or model selection achieve the optimal rates simultaneously over the full range of 0 ≤ q ≤ 1 for general Mn and upper bound tn of the q-norm. Both random and fixed designs, with known or unknown error variance, are handled, and the `q-aggregations examined in this work cover major types of aggregation problems previously studied in the literature. Consequences on minimax-rate adaptive regression under `q-constrained true coefficients (0 ≤ q ≤ 1) are also provided

Sharp Oracle Inequalities for Aggregation of Affine Estimators

We consider the problem of combining a (possibly uncountably infinite) set of affine estimators in non-parametric regression model with heteroscedastic Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a PAC-Bayesian type inequality that leads to sharp oracle inequalities in discrete but also in continuous settings. The framework is general enough to cover the combinations of various procedures such as least square regression, kernel ridge regression, shrinking estimators and many other estimators used in the literature on statistical inverse problems. As a consequence, we show that the proposed aggregate provides an adaptive estimator in the exact minimax sense without neither discretizing the range of tuning parameters nor splitting the set of observations. We also illustrate numerically the good performance achieved by the exponentially weighted aggregate

Kernel Estimation and Model Combination in a Bandit Problem with Covariates

Multi-armed bandit problem is an important optimization game that requires an exploration-exploitation tradeoff to achieve optimal total reward. Motivated from industrial applications such as online advertising and clinical research, we consider a setting where the rewards of bandit machines are associated with covariates, and the accurate estimation of the corresponding mean reward functions plays an important role in the performance of allocation rules. Under a flexible problem setup, we establish asymptotic strong consistency and perform a finite-time regret analysis for a sequential randomized allocation strategy based on kernel estimation. In addition, since many nonparametric and parametric methods in supervised learning may be applied to estimating the mean reward functions but guidance on how to choose among them is generally unavailable, we propose a model combining allocation strategy for adaptive performance. Simulations and a real data evaluation are conducted to illustrate the performance of the proposed allocation strategy

Minimax rates of convergence for nonparametric location-scale models

This paper studies minimax rates of convergence for nonparametric location-scale models, which include mean, quantile and expectile regression settings. Under Hellinger differentiability on the error distribution and other mild conditions, we show that the minimax rate of convergence for estimating the regression function under the squared $L_2$ loss is determined by the metric entropy of the nonparametric function class. Different error distributions, including asymmetric Laplace distribution, asymmetric connected double truncated gamma distribution, connected normal-Laplace distribution, Cauchy distribution and asymmetric normal distribution are studied as examples. Applications on low order interaction models and multiple index models are also given