On prediction of individual sequences relative to a set of experts in the presence of noise

Abstract

The problem of predicting the next outcome of an individual binary sequence, based on noisy observations of the past is considered. The goal of the predictor is to perform, for each individual clean sequence (almost) as well as the best 'expert' in a finite set, where performance is evaluated using a general loss function. This setup is a generalization of that considered extensively by researchers from various disciplines of universal prediction and forecasting for the case where the observation of the past data is corrupted by noise. In this work, we restrict ourselves to the case where the noise is an additive, i.i.d Bernoulli(p) process and where its parameter p is assumed known to the predictor. In the expected loss regime, for a given sequence, we define the regret as the difference between the total expected loss of the predictor on the entire sequence and that of the best expert in the comparison class. We generalize and improve the recent order results of Haussler et al. to this noisy setup for a large class of loss functions and show that under certain conditions on the loss functions, the minimax regret is θ(log N) while under other conditions it is θ(√n log N), where n is the sequence length and N is the cardinality of the expert class. It is also seen that this new setup, which involves a stochastic noise process, gives rise to additional performance criteria by which our predictors can be shown to do well.