Publication

The Conditional Regret-Capacity Theorem for Batch Universal Prediction

Michael Christoph Gastpar
2025

Abstract

We derive a conditional version of the classical regret-capacity theorem. This result can be used in universal prediction to find lower bounds on the minimal batch regret, which is a recently introduced generalization of the average regret, when batches of training data are available to the predictor. As an example, we apply this result to the class of binary memoryless sources. Finally, we generalize the theorem to Rényi information measures, revealing a deep connection between the conditional Rényi divergence and the conditional Sibson’s mutual information.

Official source

https://infoscience.epfl.ch/entities/publication/dd2bedab-2a2c-4f6c-874e-e7817b17eff6

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Michael Christoph Gastpar
2025

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/dd2bedab-2a2c-4f6c-874e-e7817b17eff6

About this result

Related concepts (32)

Information theory

Information theory is the mathematical study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field, in applied mathematics, is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure in information theory is entropy.

Rényi entropy

In information theory, the Rényi entropy is a quantity that generalizes various notions of entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi, who looked for the most general way to quantify information while preserving additivity for independent events. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions. The Rényi entropy is important in ecology and statistics as index of diversity.

Entropy (information theory)

In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable , which takes values in the alphabet and is distributed according to : where denotes the sum over the variable's possible values. The choice of base for , the logarithm, varies for different applications. Base 2 gives the unit of bits (or "shannons"), while base e gives "natural units" nat, and base 10 gives units of "dits", "bans", or "hartleys".

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