# Loss-Shift Transfer via Bayes Quotients

arXiv: https://arxiv.org/abs/2606.13178 · HTML: https://mlsanity.com/paper/2606.13178/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 48 references that are in the map.
- First posted on arXiv 11 Jun 2026.

## Builds on (most cited first, 12 of 12)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [A Comprehensive Survey on Transfer Learning](https://mlsanity.com/paper/1911.02685/) (2019)
- [Human uncertainty makes classification more robust](https://mlsanity.com/paper/1908.07086/) (2019)
- [Emergence of Invariance and Disentanglement in Deep Representations](https://mlsanity.com/paper/1706.01350/) (2017)
- [Fundamental Limits and Tradeoffs in Invariant Representation Learning](https://mlsanity.com/paper/2012.10713/) (2020)
- [Minimum Excess Risk in Bayesian Learning](https://mlsanity.com/paper/2012.14868/) (2020)
- [From Classification Accuracy to Proper Scoring Rules: Elicitability of Probabilistic Top List Predictions](https://mlsanity.com/paper/2301.11797/) (2023)
- [A Fiber Criterion for Representation Identifiability in Supervised Learning](https://mlsanity.com/paper/2606.01092/) (2026)

## Abstract (arXiv)

Transfer learning is usually studied as a consequence of distribution shift. This paper identifies an orthogonal failure mode in which the data distribution is fixed and the loss changes. This setting is called loss shift. A loss determines which information in (X) is Bayes-relevant, and two losses may therefore require different representations even under the same joint law (P(X,Y)). The idea is formalized using Bayes quotients, which allow losses to be ordered by refinement. In the Bayes-quotient formulation, strict refinement gives an immediate qualitative obstruction. A source-minimal representation for a coarser loss is insufficient for a strictly finer target loss. For finite-output log loss, this obstruction becomes an exact quantitative identity. The excess risk is the conditional information about (Y) discarded by the representation. Experiments in controlled, learned, synthetic-image, and real-image settings show the predicted effect, i.e., classification-equivalent representations can have different optimal log-loss performance under a fixed data distribution.