Abstract:
Predictive complexity is a generalization of Kolmogorov complexity 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 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 prove that asymptotically they differ on sequences of length $n$ in the worst case by a factor equal to $\log n$. These estimates cannot be improved. To obtain these inequalities we present estimates of the cardinality of all sequences of given predictive complexity.