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We show how any PAC learning algorithm that works under the uniform distribution can be transformed, in a blackbox fashion, into one that works under an arbitrary and unknown distribution D. The efficiency of our transformation scales with the inherent complexity of D, running in (n, (md)d) time for distributions over n whose pmfs are computed by depth-d decision trees, where m is the sample complexity of the original algorithm. For monotone distributions our transformation uses only samples from D, and for general ones it uses subcube conditioning samples. A key technical ingredient is an algorithm which, given the aforementioned access to D, produces an optimal decision tree decomposition of D: an approximation of D as a mixture of uniform distributions over disjoint subcubes. With this decomposition in hand, we run the uniform-distribution learner on each subcube and combine the hypotheses using the decision tree. This algorithmic decomposition lemma also yields new algorithms for learning decision tree distributions with runtimes that exponentially improve on the prior state of the art—results of independent interest in distribution learning.
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Misspecification in Prediction Problems and Robustness via Improper Learning
We study probabilistic prediction games when the underlying model is misspecified, investigating the consequences of predicting using an incorrect parametric model. We show that for a broad class of loss functions and parametric families of distributions, the regret of playing a ``proper'' predictor--one from the putative model class--relative to the best predictor in the same model class has lower bound scaling at least as sqrt(c n), where c is a measure of the model misspecification to the true distribution in terms of total variation distance. In contrast, using an aggregation-based (improper) learner, one can obtain regret d log n for any underlying generating distribution, where d is the dimension of the parameter; we exhibit instances in which this is unimprovable even over the family of all learners that may play distributions in the convex hull of the parametric family. These results suggest that simple strategies for aggregating multiple learners together should be more robust, and several experiments conform to this hypothesis.
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- PAR ID:
- 10275097
- Date Published:
- Journal Name:
- AISTATS
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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