Fractal structures in Adversarial Prediction

TL;DR

This paper demonstrates that in a prediction game with an adversary, generating fractal-like sequences is optimal for minimizing predictability, revealing a natural emergence of fractals in adversarial settings and time series data.

Contribution

It formally proves the emergence of fractal-like sequences as optimal strategies for adversaries in prediction games and provides trade-offs between predictability and deviation.

Findings

Fractal-like sequences are optimal for adversaries under certain payoff models.

Optimal trade-offs between predictability and deviation are characterized.

Time series with higher deviations can be explained by fractal-like adversarial strategies.

Abstract

Fractals are self-similar recursive structures that have been used in modeling several real world processes. In this work we study how "fractal-like" processes arise in a prediction game where an adversary is generating a sequence of bits and an algorithm is trying to predict them. We will see that under a certain formalization of the predictive payoff for the algorithm it is most optimal for the adversary to produce a fractal-like sequence to minimize the algorithm's ability to predict. Indeed it has been suggested before that financial markets exhibit a fractal-like behavior. We prove that a fractal-like distribution arises naturally out of an optimization from the adversary's perspective. In addition, we give optimal trade-offs between predictability and expected deviation (i.e. sum of bits) for our formalization of predictive payoff. This result is motivated by the observation that several time series data exhibit higher deviations than expected for a completely random walk.