# Backward-State Policy Is Part of the Learning Algorithm

arXiv: https://arxiv.org/abs/2609.39813 · HTML: https://mlsanity.com/paper/2609.39813/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 39 of its 52 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 30 of 39)

- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [GQA: Training Generalized Multi-Query Transformer Models from Multi-Head Checkpoints](https://mlsanity.com/paper/2305.13245/) (2023)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [MiniCPM: Unveiling the Potential of Small Language Models with Scalable Training Strategies](https://mlsanity.com/paper/2404.06395/) (2024)
- [Deep Learning with Limited Numerical Precision](https://mlsanity.com/paper/1502.02551/) (2015)
- [Understanding Straight-Through Estimator in Training Activation Quantized Neural Nets](https://mlsanity.com/paper/1903.05662/) (2019)
- [Small-scale proxies for large-scale Transformer training instabilities](https://mlsanity.com/paper/2309.14322/) (2023)
- [Scaling Laws and Compute-Optimal Training Beyond Fixed Training Durations](https://mlsanity.com/paper/2405.18392/) (2024)
- [Flipout: Efficient Pseudo-Independent Weight Perturbations on Mini-Batches](https://mlsanity.com/paper/1803.04386/) (2018)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [Scaling Laws for Precision](https://mlsanity.com/paper/2411.04330/) (2024)
- [FP8-LM: Training FP8 Large Language Models](https://mlsanity.com/paper/2310.18313/) (2023)
- [Stable and low-precision training for large-scale vision-language models](https://mlsanity.com/paper/2304.13013/) (2023)
- [SageAttention3: Microscaling FP4 Attention for Inference and An Exploration of 8-Bit Training](https://mlsanity.com/paper/2505.11594/) (2025)
- [Scaling FP8 training to trillion-token LLMs](https://mlsanity.com/paper/2409.12517/) (2024)
- [Inverse Path Tracing for Joint Material and Lighting Estimation](https://mlsanity.com/paper/1903.07145/) (2019)
- [ActNN: Reducing Training Memory Footprint via 2-Bit Activation Compressed Training](https://mlsanity.com/paper/2104.14129/) (2021)
- [FP4 All the Way: Fully Quantized Training of LLMs](https://mlsanity.com/paper/2505.19115/) (2025)
- [Quartet: Native FP4 Training Can Be Optimal for Large Language Models](https://mlsanity.com/paper/2505.14669/) (2025)
- [COAT: Compressing Optimizer states and Activation for Memory-Efficient FP8 Training](https://mlsanity.com/paper/2410.19313/) (2024)
- [GACT: Activation Compressed Training for Generic Network Architectures](https://mlsanity.com/paper/2206.11357/) (2022)
- [Recipes for Pre-training LLMs with MXFP8](https://mlsanity.com/paper/2506.08027/) (2025)
- [Quartet II: Accurate LLM Pre-Training in NVFP4 by Improved Unbiased Gradient Estimation](https://mlsanity.com/paper/2601.22813/) (2026)
- [Unit Scaling: Out-of-the-Box Low-Precision Training](https://mlsanity.com/paper/2303.11257/) (2023)

## Abstract (arXiv)

Low-precision training rounds tensors that the backward pass reads again, often for several gradients; each use can read the forward's rounded value, the original, or a new random rounding. This backward-state policy looks like a memory and precision detail, settled by copy accuracy and final loss. We argue that it is part of the learning algorithm, and that neither check shows whether it is right. Copy accuracy does not decide the outcome: in three pairs of 390M runs with an emulated FP8 backward, training fails when attention's backward reuses the forward's rounded output and succeeds with a new rounding from the same distribution. Even the most accurate copy, the original itself, can be wrong by our reference: the gradient of the forward pass as it actually ran, with gradients passed through rounding unchanged. For example, a normalization output stored in low precision feeds two gradients: the gain's gradient needs the original, but the next layer's weight gradient needs the rounded value that layer multiplied. Final loss, the other check, does not rule out the error of reading the original for both: it persists in models trained with such a store, while planned loss comparisons stay within a margin fixed in advance. We therefore derive from this reference which value each use must read, or which substitute gives the same gradient on average with the forward held fixed, and check these per-use requirements on single operators, without training. In three tests using PyTorch and Transformer Engine, the requirements predicted beforehand whether reuse changes what the backward computes on average relative to an independent copy, and every prediction held. Backward-state policy is thus part of the learning algorithm: it should be specified and checked use by use, not settled by copy accuracy and final loss.