A Universal Probability Assignment for Prediction of Individual Sequences

TL;DR

This paper introduces a universal probability assignment method for predicting individual sequences, which uses randomized empirical frequencies to create a simple, effective predictor applicable to various loss functions, supporting probabilistic assumptions.

Contribution

It proposes a novel universal sequential probability assignment based on randomized empirical frequencies, enabling effective prediction across diverse loss functions.

Findings

The method provides a good universal predictor for large classes of loss functions.

Randomization of empirical frequencies is essential for the predictor's effectiveness.

The approach offers a partial justification for the use of probabilistic models in sequence prediction.

Abstract

Is it a good idea to use the frequency of events in the past, as a guide to their frequency in the future (as we all do anyway)? In this paper the question is attacked from the perspective of universal prediction of individual sequences. It is shown that there is a universal sequential probability assignment, such that for a large class loss functions (optimization goals), the predictor minimizing the expected loss under this probability, is a good universal predictor. The proposed probability assignment is based on randomly dithering the empirical frequencies of states in the past, and it is easy to show that randomization is essential. This yields a very simple universal prediction scheme which is similar to Follow-the-Perturbed-Leader (FPL) and works for a large class of loss functions, as well as a partial justification for using probabilistic assumptions.