Adversarial Prediction: Lossless Predictors and Fractal Like Adversaries

Abstract

In this talk we will look at the classical prediction game where the adversary (or nature) is producing a sequence of bits and a prediction algorithm is trying to predict the future bit(s) from the past bits. This is like gambling on the future bits which involves the risk of making mistakes while shooting for profit from right predictions. Say the algorithm gets a payoff of 1 on a right prediction and − 1 on wrong predictions (and is also make fractional bets c ≤ 1 in which case its payoff is + c or − c). We will see an algorithm [1] that has a good performance while almost never taking a risk of having a net loss where loss is said to happen when the number of wrong predictions exceeds the number of right predictions. Our algorithm gets no more than an exponentially small loss $e^{-\Omega(\epsilon^2 T)}$ over T bits on any sequence (where ε is a constant parameter). Further as compared to the payoff that would have been achieved by predicting the majority bit (in hindsight) our algorithms payoff is not lower by more than O(εT) (which is commonly known as regret). We will also see experimental results on how these algorithms perform on stock data. Our algorithms build upon several classical works on the experts problem [2-4]

We will also see what kind of sequences are best from the adversary’s perspective. We will show that under a certain formulation of predictive payoff it is best for the adversary to generate a “fractal like” sequence [5].