Fast Rates for Structured Prediction

Vivien Cabannes (1) , Francis Bach (1) , Alessandro Rudi (1)

1 SIERRA - Statistical Machine Learning and Parsimony

Discrete supervised learning problems such as classification are often tackled by introducing a continuous surrogate problem akin to regression. Bounding the original error, between estimate and solution, by the surrogate error endows discrete problems with convergence rates already shown for continuous instances. Yet, current approaches do not leverage the fact that discrete problems are essentially predicting a discrete output when continuous problems are predicting a continuous value. In this paper, we tackle this issue for general structured prediction problems, opening the way to "super fast" rates, that is, convergence rates for the excess risk faster than n −1 , where n is the number of observations, with even exponential rates with the strongest assumptions. We first illustrate it for predictors based on nearest neighbors, generalizing rates known for binary classification to any discrete problem within the framework of structured prediction. We then consider kernel ridge regression where we improve known rates in n −1/4 to arbitrarily fast rates, depending on a parameter characterizing the hardness of the problem, thus allowing, under smoothness assumptions, to bypass the curse of dimensionality.

Mots clés: Structured prediction; fast convergence rates; generalization bounds; low-density separation; margin condition; local averaging method; nearest neighbors; kernel methods; kernel ridge regression

Domaines: Machine Learning [stat.ML]

Vivien Cabannes, Francis Bach, Alessandro Rudi. Fast Rates for Structured Prediction. COLT 2021 - 34th Annual Conference on Learning Theory, Aug 2021, Boulder, Colorado, United States. ⟨hal-03384304⟩