Finite memory universal portfolios

We consider the memory requirements of stock-market investment algorithms through their finite state machine (FSM) implementations. The regret of an online algorithm is the limit difference between its capital growth rate and that of the optimal (in hindsight) constant rebalanced portfolio. Let lscr, isin, and m be the number of states, the regret, and the number of stocks, respectively. We consider the relationships between mnplus and isin for large m. For individual markets (with no underlying distributions) and deterministic FSMs, we show that any isin-regret FSM must have Omega ((1/isin)^m-1/m-1/2) states, and also show an isin-regret FSMs with O ((1/isin)^4m) states. These space-complexity questions are especially pertinent to state portfolio algorithms, where both market history and side-information are taken into account.