Generalization bounds of ERM algorithm with V-geometrically Ergodic Markov chains

Abstract

The previous results describing the generalization ability of Empirical Risk Minimization (ERM) algorithm are usually based on the assumption of independent and identically distributed (i.i.d.) samples. In this paper we go far beyond this classical framework by establishing the first exponential bound on the rate of uniform convergence of the ERM algorithm with Vgeometrically ergodic Markov chain samples, as the application of the bound on the rate of uniform convergence, we also obtain the generalization bounds of the ERM algorithm with V-geometrically ergodic Markov chain samples and prove that the ERM algorithm with V-geometrically ergodic Markov chain samples is consistent. The main results obtained in this paper extend the previously known results of i.i.d. observations to the case of V-geometrically ergodic Markov chain samples.

Generalization and Robustness of Batched Weighted Average Algorithm with V-Geometrically Ergodic Markov Data

“…The proofs in our paper use a previous result on the uniform convergence rate of the empirical loss for V-geometrically ergodic Markov chains [6]. Convergence of the empirical loss is a fundamental problem in statistics and statistical learning theory, and it has been studied for other types of Markov chains such as α-mixing [4,14,15], β-mixing [16,17], φ-mixing [16], and uniformly ergodic [5] chains.…”

“…We now introduce the V-geometrically ergodic Markov chains and the settings for our analyses. We will follow the definitions in [6]. We also review a result on the uniform convergence rate of the empirical loss for V-geometrically ergodic Markov data [6] which will be used in the subsequent sections.…”

“…Let H be the set of all hypotheses, where a hypothesis h is a function from X to Y. Throughout this paper, we make the following assumption: H is contained in a ball B(C q (X )) of a Hölder space C q (X ) for some q > 0, which is similar to the assumption in [6]. The Hölder space C q (X ) is the space of all continuous functions on X with the following norm [6,23]:…”

“…We review a previous result [6] which gives a PAC-style bound on the training set size for the empirical loss to converge uniformly to the expected loss when training data are V-geometrically ergodic Markov chains. This result will be used to prove the generalization and robustness bounds for the BWA algorithm in subsequent sections.…”

“…Secondly, the fact that these chains can be defined on a general, possibly uncountable, state space makes their learning models more general than previous models which learn from finite or countable state space Markov chains [11]. Thirdly, the V-geometrically ergodic assumption is not too restrictive since it includes all uniformly ergodic chains as well as all ergodic chains on a finite state space [6,12]. Nevertheless, we emphasize that our proof idea can be applied to other types of mixing Markov chains if we have the uniform convergence rate of the empirical loss for these chains.…”