Non-parametric search for significant inputs of unknown system

Suppose that we can assign arbitrary t-tuple of binary inputs x:=(x(i),i∈[t]),[t]:={1,...,t} and measure the output Z which depends only on unknown s-tuple x(A) of significant inputs (SIs). A priori, A is equally likely to be any s-tuple of different elements from [t]. The successive measurements Z_i,i=1,...,N, are independent random variables (RVs) given an N-sequence (N-design) of input t-tuples (memoryless channel). For a sequence D of N designs, N=1,..., we introduce the asymptotic rate AR_T(D)=lim sup_t→∞log t/N(γ) of a test T. Here N(γ) is the minimal sample (block) size such that the probability of T to miss s-tuple A (mean error probability) is less than γ,0<γ<1. It is shown for a general class of tests and designs that AR does not depend on γ,0<γ<1, and remains the same for slowly decreasing γ(t):|log γ(t)|=o(log t) as t→∞. This definition is well-applicable to the combinatorial problems (when noise is absent) whereas in the case of sequential design N(γ) must be replaced with the mean duration of sequential strategy guaranteeing the same upper bound for the error probability. It is well-known that the maximum likelihood (ML)-decision minimizes the error probability for any design. This implies that AR_ML is optimal among all tests if applicable. We outline the progress in three directions