Related papers: Estimation bounds and sharp oracle inequalities of…

Based on the ideas of arXiv:1710.06612, we consider the problem of minimization of the Holder-continuous non-smooth functional $f$ with non-positive convex (generally, non-smooth) Lipschitz-continuous functional constraint. We propose some…

We study Empirical Risk Minimizers (ERM) and Regularized Empirical Risk Minimizers (RERM) for regression problems with convex and $L$-Lipschitz loss functions. We consider a setting where $|\cO|$ malicious outliers contaminate the labels.…

The Lasso has attracted the attention of many authors these last years. While many efforts have been made to prove that the Lasso behaves like a variable selection procedure at the price of strong (though unavoidable) assumptions on the…

We characterize the complexity of minimizing $\max_{i\in[N]} f_i(x)$ for convex, Lipschitz functions $f_1,\ldots, f_N$. For non-smooth functions, existing methods require $O(N\epsilon^{-2})$ queries to a first-order oracle to compute an…

In binary classification and regression problems, it is well understood that Lipschitz continuity and smoothness of the loss function play key roles in governing generalization error bounds for empirical risk minimization algorithms. In…

We develop a minimax theory for operator learning, where the goal is to estimate an unknown operator between separable Hilbert spaces from finitely many noisy input-output samples. For uniformly bounded Lipschitz operators, we prove…

This paper consider penalized empirical loss minimization of convex loss functions with unknown non-linear target functions. Using the elastic net penalty we establish a finite sample oracle inequality which bounds the loss of our estimator…

Although kernel methods are widely used in many learning problems, they have poor scalability to large datasets. To address this problem, sketching and stochastic gradient methods are the most commonly used techniques to derive efficient…

We investigate an extension of classical empirical risk minimization, where the hypothesis space consists of a random subspace within a given Hilbert space. Specifically, we examine the Nystr\"om method where the subspaces are defined by a…

We derive oracle inequalities for the problems of isotonic and convex regression using the combination of $Q$-aggregation procedure and sparsity pattern aggregation. This improves upon the previous results including the oracle inequalities…

Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method…

Many practical optimization problems lack strong convexity. Fortunately, recent studies have revealed that first-order algorithms also enjoy linear convergences under various weaker regularity conditions. While the relationship among…

We provide theoretical analysis of the statistical and computational properties of penalized $M$-estimators that can be formulated as the solution to a possibly nonconvex optimization problem. Many important estimators fall in this…

We prove local Lipschitz regularity for local minimiser of \[ W^{1,1}(\Omega)\ni v\mapsto \int_\Omega F(Dv)\, dx \] where $\Omega\subseteq {\mathbb R}^N$, $N\ge 2$ and $F:{\mathbb R}^N\to {\mathbb R}$ is a quasiuniformly convex integrand in…

Error bounds, which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function, have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for…

We consider a statistical inverse learning problem, where we observe the image of a function $f$ through a linear operator $A$ at i.i.d. random design points $X_i$, superposed with an additive noise. The distribution of the design points is…

In this paper, we propose some accelerated methods for solving optimization problems under the condition of relatively smooth and relatively Lipschitz continuous functions with an inexact oracle. We consider the problem of minimizing the…

We study the oracle complexity of nonsmooth nonconvex optimization, with the algorithm assumed to have access only to local function information. It has been shown by Davis, Drusvyatskiy, and Jiang (2023) that for nonsmooth Lipschitz…

A general lower bound is developed for the minimax risk when estimating an arbitrary functional. The bound is based on testing two composite hypotheses and is shown to be effective in estimating the nonsmooth functional…

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…