# FORGE: Fused On-Register Gradient Elimination for Memory-Efficient LLM Training

arXiv: https://arxiv.org/abs/2606.22932 · HTML: https://mlsanity.com/paper/2606.22932/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 54 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Builds on (most cited first, 30 of 41)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Megatron-LM: Training Multi-Billion Parameter Language Models Using Model Parallelism](https://mlsanity.com/paper/1909.08053/) (2019)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [ZeRO: Memory Optimizations Toward Training Trillion Parameter Models](https://mlsanity.com/paper/1910.02054/) (2019)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [Mixed Precision Training](https://mlsanity.com/paper/1710.03740/) (2017)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [PyTorch FSDP: Experiences on Scaling Fully Sharded Data Parallel](https://mlsanity.com/paper/2304.11277/) (2023)
- [Large Batch Training of Convolutional Networks](https://mlsanity.com/paper/1708.03888/) (2017)
- [FlashAttention-3: Fast and Accurate Attention with Asynchrony and Low-precision](https://mlsanity.com/paper/2407.08608/) (2024)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [8-bit Optimizers via Block-wise Quantization](https://mlsanity.com/paper/2110.02861/) (2021)
- [Deep Learning with Limited Numerical Precision](https://mlsanity.com/paper/1502.02551/) (2015)
- [Reducing Activation Recomputation in Large Transformer Models](https://mlsanity.com/paper/2205.05198/) (2022)
- [DeepSpeed Ulysses: System Optimizations for Enabling Training of Extreme Long Sequence Transformer Models](https://mlsanity.com/paper/2309.14509/) (2023)
- [OpenMathInstruct-2: Accelerating AI for Math with Massive Open-Source Instruction Data](https://mlsanity.com/paper/2410.01560/) (2024)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [Adam-mini: Use Fewer Learning Rates To Gain More](https://mlsanity.com/paper/2406.16793/) (2024)
- [Full Parameter Fine-tuning for Large Language Models with Limited Resources](https://mlsanity.com/paper/2306.09782/) (2023)
- [Liger Kernel: Efficient Triton Kernels for LLM Training](https://mlsanity.com/paper/2410.10989/) (2024)
- [FlashAttention-4: Algorithm and Kernel Pipelining Co-Design for Asymmetric Hardware Scaling](https://mlsanity.com/paper/2603.05451/) (2026)
- [APOLLO: SGD-like Memory, AdamW-level Performance](https://mlsanity.com/paper/2412.05270/) (2024)

## Built on by (most cited first, 2 of 2)

- [Auditing Invisible Weight Updates with Reference Traces](https://mlsanity.com/paper/2607.09800/) (2026)
- [MMOE: Modernizing Diffusion Transformers with Efficient Expert Design](https://mlsanity.com/paper/2607.24665/) (2026)

## Abstract (arXiv)

Reverse-mode differentiation computes every weight gradient, writes it to memory, and only then lets the optimizer read it back. This two-phase schedule sets the memory ceiling of modern training: at the seam between the phases, every layer's gradient is live at once. We argue that this materialized gradient is an artifact of how differentiation is staged, not a quantity that learning requires -- and we eliminate it. FORGE folds the optimizer step into the backward pass and applies it one tile at a time, entirely in registers, so each gradient tile is consumed the instant it is produced and never becomes a tensor. The fusion changes only when the update happens, not what it computes: in full precision the fused step is provably exact -- the identical optimizer update, for every element-wise rule -- and that exactness survives tensor- and sequence-parallel sharding; in the bf16 and 8-bit regimes used in practice it is faithful rather than bit-identical, its deviation bounded and, for the weight store, rendered unbiased by stochastic rounding. Because each gradient tile is born and consumed in the same registers, it is never converted down to bf16 to be stored and read back; FORGE thus preserves the full-precision fidelity that both bf16 and 8-bit optimizers lose to that conversion. Nor is the method tied to one architecture or one optimizer: linear layers are ubiquitous, and FORGE reclaims the gradient memory of any of them under any element-wise rule. Empirically FORGE more than halves the memory of an optimizer step and, at the small batch sizes typical of fine-tuning and continued pretraining, runs about 1.5x faster; integrated into tensor-parallel Megatron-LM it fits 8B training at four times the micro-batch a standard optimizer allows on the same GPUs.