Elsevier

Theoretical Computer Science

Volume 313, Issue 2, 17 February 2004, Pages 195-207
Theoretical Computer Science

Loss functions, complexities, and the Legendre transformation

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Abstract

The paper introduces a way of re-constructing a loss function from predictive complexity. We show that a loss function and expectations of the corresponding predictive complexity w.r.t. the Bernoulli distribution are related through the Legendre transformation. It is shown that if two loss functions specify the same complexity then they are equivalent in a strong sense. The expectations are also related to the so-called generalized entropy.

Keywords

On-line prediction
Predictive complexity
Generalized entropy

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A previous version of this paper appeared in proceedings of The Twelfth International Conference on Algorithmic Learning Theory, Lecture Notes in Artificial Intelligence, Vol. 2225, Springer, Berlin, Heidelberg, 2001.