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The problem of universally predicting an individual continuous sequence usinga deterministic finite-state machine (FSM) is considered. The empirical mean isused as a reference as it is the constant that fits a given sequence within aminimal square error. With this reference, a reasonable prediction performanceis the regret, namely the excess square-error over the reference loss, theempirical variance. The paper analyzes the tradeoff between the number ofstates of the universal FSM and the attainable regret. It first studies thecase of a small number of states. A class of machines, denoted DegeneratedTracking Memory (DTM), is defined and the optimal machine in this class isshown to be the optimal among all machines for small enough number of states.Unfortunately, DTM machines become suboptimal as the number of available statesincreases. Next, the Exponential Decaying Memory (EDM) machine, previously usedfor predicting binary sequences, is considered. While this machine has poorerperformance for small number of states, it achieves a vanishing regret forlarge number of states. Following that, an asymptotic lower bound ofO(k^{-2/3}) on the achievable regret of any k-state machine is derived. Thisbound is attained asymptotically by the EDM machine. Furthermore, a newmachine, denoted the Enhanced Exponential Decaying Memory machine, is shown tooutperform the EDM machine for any number of states.

Finite-Memory Prediction as Well as the Empirical Mean