How to Filter an "Individual Sequence With Feedback"

We consider causally estimating (filtering) the components of a noise-corrupted sequence relative to a reference class of filters. The noiseless sequence to be filtered is designed by a “well-informed antagonist,” meaning it may evolve according to an arbitrary law, unknown to the filter, based on past noiseless and noisy sequence components. We show that this setting is more challenging than that of an individual noiseless sequence (a.k.a. the “semi-stochastic” setting) in the sense that any deterministic filter, even one guaranteed to do well on every noiseless individual sequence, fails under some well-informed antagonist. On the other hand, we constructively establish the existence of a randomized filter which successfully competes with an arbitrary given finite reference class of filters, under every antagonist. Thus, unlike in the semi-stochastic setting, randomization is crucial in the antagonist framework. Our noise model allows for channels whose noisy output depends on the l past channel outputs (in addition to the noiseless channel input symbol). Memoryless channels are obtained as a special case of our model by taking l = 0. In this case, our scheme coincides with one that was recently shown to compete with an arbitrary reference class when the underlying noiseless sequence is an individual sequence. Hence, our results show that the latter scheme is universal not only for the semi-stochastic setting in which it was originally proposed, but also under the well-informed antagonist.