Still no free lunches: the price to pay for tighter PAC-Bayes bounds

Benjamin Guedj (1, 2, 3, 4, 5) , Louis Pujol (6)

1 UCL - University College of London [London]
2 UCL-CS - Department of Computer science [University College of London]
3 Inria-CWI - Inria-CWI
4 MODAL - MOdel for Data Analysis and Learning
5 Inria-London - The Inria London Programme
6 Université Paris-Saclay

"No free lunch" results state the impossibility of obtaining meaningful bounds on the error of a learning algorithm without prior assumptions and modelling. Some models are expensive (strong assumptions, such as as subgaussian tails), others are cheap (simply finite variance). As it is well known, the more you pay, the more you get: in other words, the most expensive models yield the more interesting bounds. Recent advances in robust statistics have investigated procedures to obtain tight bounds while keeping the cost minimal. The present paper explores and exhibits what the limits are for obtaining tight PAC-Bayes bounds in a robust setting for cheap models, addressing the question: is PAC-Bayes good value for money?

Domaines: Machine Learning [stat.ML]; Apprentissage [cs.LG]; Théorie [stat.TH]

Benjamin Guedj, Louis Pujol. Still no free lunches: the price to pay for tighter PAC-Bayes bounds. Entropy, 2021, ⟨10.3390/e23111529⟩. ⟨hal-02401286⟩