### Non-linear Inequalities between Predictive and Kolmogorov Complexities

Authors: Michael V. Vyugin and Vladimir V. V'yugin.

Source: Lecture Notes in Artificial Intelligence Vol. 2225, 2001, 190 - 204.

Abstract. Predictive complexity is a generalization of Kolmogorov complexity. It corresponds to an "optimal" prediction strategy which gives a lower bound to ability of any algorithm to predict elements of a sequence of outcomes. A variety of types of loss functions makes it interesting to study relations between corresponding predictive complexities.

Non-linear inequalities (with variable coefficients) between predictive complexity of non-logarithmic type and Kolmogorov complexity (which is close to predictive complexity for logarithmic loss function) are the main subject of consideration in this paper. We deduce from these inequalities an asymptotic relation , when , where a is a constant and is the length of a sequence . An analogous asymptotic result holds for relative complexities and . To obtain these inequalities we present estimates of the cardinality of all sequences of given predictive complexity.