Patterns and Probabilities: A Study in Algorithmic Randomness and Computable Learning
Francesca Zaffora Blando
Abstract:
This dissertation bridges the theory of algorithmic randomness - a branch of computability theory - and the foundations of inductive learning. Algorithmic randomness provides a mathematical analysis of the notion of an individual object (such as a string of bytes representing a computer file or a sequence of experimental outcomes) displaying no effective patterns or regularities. Here, we investigate the role that algorithmic randomness plays in inductive learning when randomness is taken to be a property of sequences of observations, or data streams, and the learners are computationally limited. Our results constitute a first step towards a systematic classification and analysis of the learning scenarios where algorithmic randomness is beneficial for inductive learning and the scenarios where it is instead detrimental for the learning process.
We focus on three main themes. Firstly, we explore the connections between algorithmic randomness and Bayesian merging-of-opinions theorems. In particular, we show that algorithmic randomness leads to merging of opinions in the following sense. When two computable Bayesian agents perform the same experiment, agreeing on which data streams are algorithmically random suffices to guarantee that they will eventually reach a consensus with probability one even if, at the beginning of the learning process, their beliefs are otherwise dissimilar. Secondly, we consider the role of algorithmic randomness in Bayesian convergence-to-the-truth results. More precisely, we show that there is a robust correspondence between algorithmic randomness and the collection of truth-conducive data streams. When a computable Bayesian agent is faced with an effective inductive problem, the algorithmically random data streams are in fact exactly the ones that ensure that their beliefs will asymptotically align with the truth. Lastly, we investigate a learning-theoretic approach - in the spirit of formal learning theory - for modelling algorithmic randomness itself. Building on the local irregularity and unruliness that is the hallmark of algorithmic randomness, we show that the algorithmically random data streams are, systematically, unlearnable: i.e., they coincide with the data streams from which no computable qualitative learning method can extrapolate any patterns.
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