Still no free lunches: the price to pay for tighter PAC-Bayes bounds
This addresses the problem of balancing cost and performance in statistical learning theory for researchers, but it is incremental as it builds on existing robust statistics and PAC-Bayes frameworks.
The paper investigates the trade-off between model assumptions and tightness in PAC-Bayes bounds, showing that cheaper models with weaker assumptions yield less tight bounds, with specific numerical limits derived for robust settings.
"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?