Learning, complexity and information density

Abstract

What is the relationship between the complexity of a learner and the randomness of his mistakes? This question was posed in \cite{rat0903} who showed that the more complex the learner the higher the possibility that his mistakes deviate from a true random sequence. In the current paper we report on an empirical investigation of this problem. We investigate two characteristics of randomness, the stochastic and algorithmic complexity of the binary sequence of mistakes. A learner with a Markov model of order $k$ is trained on a finite binary sequence produced by a Markov source of order $k^{*}$ and is tested on a different random sequence. As a measure of learner's complexity we define a quantity called the \emph{sysRatio}, denoted by $\rho$, which is the ratio between the compressed and uncompressed lengths of the binary string whose $i^{th}$ bit represents the maximum \emph{a posteriori} decision made at state $i$ of the learner's model. The quantity $\rho$ is a measure of information density. The main result of the paper shows that this ratio is crucial in answering the above posed question. The result indicates that there is a critical threshold $\rho^{*}$ such that when $\rho\leq\rho^{*}$ the sequence of mistakes possesses the following features: (1)\emph{}low divergence $\Delta$ from a random sequence, (2) low variance in algorithmic complexity. When $\rho>\rho^{*}$, the characteristics of the mistake sequence changes sharply towards a\emph{}high\emph{$\Delta$} and high variance in algorithmic complexity.