Learning Effectiveness and Memory Size

We study learning effectiveness as a function of memory size. We quantify the maximal level that a bounded memory machine (or agent) can match (or reproduce) a long string of inputs as a function of the input length k and the memory size n. The input string is an element of Ik and the output string is an element of Jk and the loss of the agent when matching an input coordinate i ∈ I with an output coordinate j ∈ J is g(i, j). This level is expressed by a function v(p, θ) of two variables: a probability p on I and a nonnegative θ ≥ 0. The function v(p, θ) is defined as a function of the triple G = 〈I, J, g〉. It equals the minimum of EQg(i, j), where the minimization is over all distributions Q on action pairs with marginal p on I, denoted QI , and the mutual information IQ(i; j) = H(QI) +H(QJ) −H(Q) ≤ θ, where H is the entropy function. If i1, . . . , ik are iid I-valued random variables with distribution p, then for T ⊂ Jk we have Emin(j1,...,jk)∈T 1 k ∑k t=1 g(it, jt) ≥ v(p, log |T | k ). Moreover, for every finite set T of functions τ from the finite strings I∗ of I-elements to J we have Eminτ∈T 1 k ∑k t=1 g(it, τ(i1, . . . , it−1)) ≥ v(p, log |T | k ). It follows that if σ is the mixed strategy of player 1 in the infinite repetition of the stage game G that plays a k-periodic sequence i1, i2, . . ., where i1, . . . , ik are iid random variables with distribution p, then for every strategy τ of player 2 that is defined by an automaton ∗Institute of Mathematics and Center for the Study of Rationality, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel (e-mail: aneyman@math.huji.ac.il). This research was supported in part by Israel Science Foundation grants 263/03 and 1123/06.