(Almost) Provable Error Bounds Under Distribution Shift via Disagreement Discrepancy

We derive a new, (almost) guaranteed upper bound on the error of deep neural networks under distribution shift using unlabeled test data. Prior methods are either vacuous in practice or accurate on average but heavily underestimate error for a sizeable fraction of shifts. In particular, the latter only give guarantees based on complex continuous measures such as test calibration, which cannot be identified without labels, and are therefore unreliable. Instead, our bound requires a simple, intuitive condition which is well justified by prior empirical works and holds in practice effectively 100\% of the time. The bound is inspired by $\mathcal{H}\Delta\mathcal{H}$-divergence but is easier to evaluate and substantially tighter, consistently providing non-vacuous test error upper bounds. Estimating the bound requires optimizing one multiclass classifier to disagree with another, for which some prior works have used sub-optimal proxy losses; we devise a "disagreement loss" which is theoretically justified and performs better in practice. We expect this loss can serve as a drop-in replacement for future methods which require maximizing multiclass disagreement. Across a wide range of natural and synthetic distribution shift benchmarks, our method gives valid error bounds while achieving average accuracy comparable to—though not better than—competitive estimation baselines.