Non-negative Bregman Divergence Minimization for Deep Direct Density Ratio Estimation
International Conference on Machine Learning (ICML), 2021.
- density ratio estimation
- Bregman divergence
- overfitting
- outlier detection
In one sentence. Bregman-divergence density ratio estimation overfits badly with deep models because the empirical objective can be driven down without learning the ratio; a non-negative correction removes that failure mode, with a generalization error bound and gains in inlier-based outlier detection.
Train-loss hacking
Density ratio estimation underlies anomaly detection, domain adaptation, and the weighting used in causal inference. The dominant approach minimizes a Bregman divergence between a model and the true ratio, which works well for restricted model classes.
With a highly flexible model such as a deep network, it fails in a specific way the paper calls train-loss hacking: a characteristic of the empirical Bregman divergence estimator allows the training objective to be pushed arbitrarily low by exploiting the finite sample rather than by fitting the ratio. The result is overfitting that the training loss does not reveal.
The correction
The paper proposes a non-negative correction to the empirical Bregman divergence estimator that removes the region in which the objective can be exploited. The soundness of the correction is supported by a generalization error bound, and experiments show favourable performance on inlier-based outlier detection.
The construction is in the same spirit as non-negative risk correction in PU learning: the fix is to respect a constraint that the population objective satisfies but the empirical one does not.