Finite-Memory Prediction as Well as the Empirical Mean

The problem of universally predicting an individual continuous sequence using a deterministic finite-state machine (FSM) is considered. The empirical mean is used as a reference as it is the constant that fits a given sequence within a minimal square error. A reasonable prediction performance is the regret, namely the excess square-error over the reference loss. This paper analyzes the tradeoff between the number of states of the universal FSM and the attainable regret. This paper first studies the case of a small number of states. A class of machines, termed degenerated tracking memory (DTM), is defined and shown to be optimal for small enough number of states. Unfortunately, DTM machines become suboptimal and their regret does not vanish as the number of available states increases. Next, the exponential decaying memory (EDM) machine, previously used for predicting binary sequences, is considered. While the EDM machine has poorer performance for small number of states, it achieves a vanishing regret for large number of states. Following that, an asymptotic lower bound of O(k^-2/3) on the achievable regret of any k-state machine is derived. This bound is attained asymptotically by the EDM machine. Finally, the enhanced exponential decaying memory machine is presented and shown to outperform the EDM machine for any number of states.