A learning problem whose consistency is equivalent to the non-existence of real-valued measurable cardinals
We show that the $k$-nearest neighbour learning rule is universally consistent in a metric space $X$ if and only if it is universally consistent in every separable subspace of $X$ and the density of $X$ is less than every real-measurable cardinal. In particular, the $k$-NN classifier is universally consistent in every metric space whose separable subspaces are sigma-finite dimensional in the sense of Nagata and Preiss if and only if there are no real-valued measurable cardinals. The latter assumption is relatively consistent with ZFC, however the consistency of the existence of such cardinals cannot be proved within ZFC. Our results were inspired by an example sketched by C\'erou and Guyader in 2006 at an intuitive level of rigour.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Detection of arbitrage opportunities in multi-asset derivatives markets
We are interested in the existence of equivalent martingale measures and the detection of arbitrage opportunities in markets where several multi-asset derivatives are traded simultaneously. More specifically, we consider…
On the Time-Inconsistent Deterministic Linear-Quadratic Control
A fundamental theory of deterministic linear-quadratic (LQ) control is the equivalent relationship between control problems, two-point boundary value problems and Riccati equations. In this paper, we extend the equivalen…
Kernel Bayesian Inference with Posterior Regularization
We propose a vector-valued regression problem whose solution is equivalent to the reproducing kernel Hilbert space (RKHS) embedding of the Bayesian posterior distribution. This equivalence provides a new understanding of…
Bayesian InferenceregressionNonuniqueness and Convergence to Equivalent Solutions in Observer-based Inverse Reinforcement Learning
A key challenge in solving the deterministic inverse reinforcement learning (IRL) problem online and in real-time is the existence of multiple solutions. Nonuniqueness necessitates the study of the notion of equivalent s…
reinforcement-learningReinforcement Learning (RL)Dynamic portfolio selection under generalized disappointment aversion
This paper addresses the continuous-time portfolio selection problem under generalized disappointment aversion (GDA). The implicit definition of the certainty equivalent within GDA preferences introduces time inconsisten…