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The oracle property of model selection procedures has attracted a large volume of favorable publications in the literature, but also faced criticisms of being ineffective and misleading in applications. In this paper, we introduce a class…
This paper studies the numerical solution of strictly convex unconstrained optimization problems by linesearch Newton-CG methods. We focus on methods employing inexact evaluations of the objective function and inexact and possibly random…
We study the problem of minimizing a convex function on a nonempty, finite subset of the integer lattice when the function cannot be evaluated at noninteger points. We propose a new underestimator that does not require access to…
We study a high-dimensional generalized linear model and penalized empirical risk minimization with $\ell_1$ penalty. Our aim is to provide a non-trivial illustration that non-asymptotic bounds for the estimator can be obtained without…
Under the linear regression framework, we study the variable selection problem when the underlying model is assumed to have a small number of nonzero coefficients (i.e., the underlying linear model is sparse). Non-convex penalties in…
High-dimensional data analysis has motivated a spectrum of regularization methods for variable selection and sparse modeling, with two popular classes of convex ones and concave ones. A long debate has been on whether one class dominates…
Penalized likelihood methods are fundamental to ultra-high dimensional variable selection. How high dimensionality such methods can handle remains largely unknown. In this paper, we show that in the context of generalized linear models,…
This paper is concerned with goal-oriented a posteriori error estimation for nonlinear functionals in the context of nonlinear variational problems solved with continuous Galerkin finite element discretizations. A two-level, or discrete,…
We consider the finite sample properties of the regularized high-dimensional Cox regression via lasso. Existing literature focuses on linear models or generalized linear models with Lipschitz loss functions, where the empirical risk…
We consider the estimation of two-sample integral functionals, of the type that occur naturally, for example, when the object of interest is a divergence between unknown probability densities. Our first main result is that, in wide…
In the convex optimization approach to online regret minimization, many methods have been developed to guarantee a $O(\sqrt{T})$ bound on regret for subdifferentiable convex loss functions with bounded subgradients, by using a reduction to…
We use the fitted Pareto law to construct an accompanying approximation of the excess distribution function. A selection rule of the location of the excess distribution function is proposed based on a stagewise lack-of-fit testing…
In this paper we analyze the necessary number of samples to estimate the gradient of any multidimensional smooth (possibly non-convex) function in a zero-order stochastic oracle model. In this model, an estimator has access to noisy values…
We consider the high-dimensional heteroscedastic regression model, where the mean and the log variance are modeled as a linear combination of input variables. Existing literature on high-dimensional linear regres- sion models has largely…
It is well-known that given a smooth, bounded-from-below, and possibly nonconvex function, standard gradient-based methods can find $\epsilon$-stationary points (with gradient norm less than $\epsilon$) in $\mathcal{O}(1/\epsilon^2)$…
Analysis of non-asymptotic estimation error and structured statistical recovery based on norm regularized regression, such as Lasso, needs to consider four aspects: the norm, the loss function, the design matrix, and the noise model. This…
We extend recent computer-assisted design and analysis techniques for first-order optimization over structured functions--known as performance estimation--to apply to structured sets. We prove "interpolation theorems" for smooth and…
Penalized estimation can conduct variable selection and parameter estimation simultaneously. The general framework is to minimize a loss function subject to a penalty designed to generate sparse variable selection. The…
We consider the problem of estimating a function $f\_{0}$ in logistic regression model. We propose to estimate this function $f\_{0}$ by a sparse approximation build as a linear combination of elements of a given dictionary of $p$…
We study unconstrained optimization problems with nonsmooth and convex objective function in the form of a mathematical expectation. The proposed method approximates the expected objective function with a sample average function using…