Information-Theoretic Analysis of Minimax Excess Risk
This work provides foundational insights into excess risk analysis, which is incremental but addresses a gap in theoretical machine learning for researchers in the field.
The paper tackles the problem of understanding the information-theoretic nature of minimax excess risk in machine learning theory by framing it as a zero-sum game and proving that, under regularity conditions, the duality gap is zero, allowing order of play to be swapped and enabling bounds via Bayesian learning results.
Two main concepts studied in machine learning theory are generalization gap (difference between train and test error) and excess risk (difference between test error and the minimum possible error). While information-theoretic tools have been used extensively to study the generalization gap of learning algorithms, the information-theoretic nature of excess risk has not yet been fully investigated. In this paper, some steps are taken toward this goal. We consider the frequentist problem of minimax excess risk as a zero-sum game between the algorithm designer and the world. Then, we argue that it is desirable to modify this game in a way that the order of play can be swapped. We then prove that, under some regularity conditions, if the world and designer can play randomly the duality gap is zero and the order of play can be changed. In this case, a Bayesian problem surfaces in the dual representation. This makes it possible to utilize recent information-theoretic results on minimum excess risk in Bayesian learning to provide bounds on the minimax excess risk. We demonstrate the applicability of the results by providing information theoretic insight on two important classes of problems: classification when the hypothesis space has finite VC-dimension, and regularized least squares.