We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) We show that a real number z ∈ [0, 1] is computably random if and only if every nondecreasing computable function [0, 1] → R is differentiable at z. (2) A real number z ∈ [0, 1] is weakly 2-random if and only if every almost everywhere differentiable computable function [0, 1] → R is diff...

We consider the learning task consisting in predicting as well as the best function in a finite reference set G up to the smallest possible additive term. If R(g) denotes the generalization error of a prediction function g, under reasonable assumptions on the loss function (typically satisfied by the least square loss when the output is bounded), it is known that the progressive mixture rule ĝ ...

We investigate statistical properties for a broad class of modern kernel-based regression (KBR) methods. These kernel methods were developed during the last decade and are inspired by convex risk minimization in infinite-dimensional Hilbert spaces. One leading example is support vector regression. We first describe the relationship between the loss function L of the KBR method and the tail of t...

We consider the learning task consisting in predicting as well as the best function in a finite reference set G up to the smallest possible additive term. IfR(g) denotes the generalization error of a prediction function g, under reasonable assumptions on the loss function (typically satisfied by the least square loss when the output is bounded), it is known that the progressive mixture rule ĝ s...

Nonasymptotic risk bounds are provided for maximum likelihood-type isotonic estimators of an unknown nondecreasing regression function, with general average loss at design points. These bounds are optimal up to scale constants, and they imply uniform n−1/3-consistency of the p risk for unknown regression functions of uniformly bounded variation, under mild assumptions on the joint probability d...

The goal of this note is to construct a uniformly antisymmetric function f : R → R with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson [6]. (See also the list of problems in Thomson [9] and Problem 2(b) from Ciesielski’s survey [5].) A problem of existence of uniformly antisymmetric function f : R → R with finite range remains open. A function f : R → R is said to...

There are several methods for solving fuzzy linear programming (FLP)problems. When the constraints and/or the objective function are fuzzy, the methodsproposed by Zimmermann, Verdegay, Chanas and Werners are used more often thanthe others. In the Zimmerman method (ZM) the main objective function cx is addedto the constraints as a fuzzy goal and the corresponding linear programming (LP)problem w...

We prove a function field version of the Bounded Height Conjecture formulated by Chatzidakis, Ghioca, Masser and Maurin in 2013.