Approximate Natural Gradient Descent (NGD) methods are an important family of optimisers for deep learning models, which use approximate Fisher information matrices to pre-condition gradients during training. The empirical Fisher (EF) method approximates the Fisher information matrix empirically by reusing the per-sample gradients collected during back-propagation. Despite its ease of implementation, the EF approximation has its theoretical and practical limitations. This paper first investigates the inversely-scaled projection issue of EF, which is shown to be a major cause of the poor empirical approximation quality. An improved empirical Fisher (iEF) method, motivated as a generalised NGD method from a loss reduction perspective, is proposed to address this issue, meanwhile retaining the practical convenience of EF. The exact iEF and EF methods are experimentally evaluated using practical deep learning setups, including widely-used setups for parameter-efficient fine-tuning of pre-trained models (T5-base with LoRA and Prompt-Tuning on GLUE tasks, and ViT with LoRA for CIFAR100). Optimisation experiments show that applying exact iEF as an optimiser provides strong convergence and generalisation. It achieves the best test performance and the lowest training loss for majority of the tasks, even when compared with well-tuned AdamW/Adafactor baselines. Additionally, under a novel empirical evaluation framework, the proposed iEF method shows consistently better approximation quality to the exact Natural Gradient updates than both EF and the more expensive sampled Fisher (SF). Further investigation also shows that the superior approximation quality of iEF is robust to damping across tasks and training stages. Improving existing approximate NGD optimisers with iEF is expected to lead to better convergence ability and stronger robustness to choice of damping.

通过提出改进的经验Fisher（iEF）方法，该论文研究了近似自然梯度下降（NGD）方法中经验Fisher信息矩阵的逆比例缩放问题，并在实验中评估了该方法的性能，在参数高效微调、深度学习优化等方面取得了较好的收敛性和拟合能力。

一种改进的经验费歇近似方法用于自然梯度下降