Z-Loss Backward Geometry in Dense Output Heads and Sparse Routers
Z-loss has been widely applied to the logits of language-model output heads and sparse mixture-of-experts routers. Z-loss constrains the softmax log-normalizers of these output heads and routers, thereby limiting large-logit excursions, reducing finite-precision roundoff exposure, and avoiding training-loss divergence. These use cases arise in modern Transformer settings where large-vocabulary softmax heads, top-$k$ routing, fused losses, and mixed-precision optimizers interact. Z-loss has typically been understood only as a scalar penalty on the log-normalizer. This paper instead analyzes Z-loss from a backward-pass perspective, focusing on the gradients produced by the Z-loss penalty. T...
| Authors | Bum Jun Kim |
|---|---|
| Primary category | cs.LG |
| Published | 31 July 2026 |
| Last updated at source | 31 July 2026 |
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