Beyond the Mean: Fisher-Orthogonal Projection for Natural Gradient Descent in Large Batch Training

Modern GPUs are equipped with large amounts of high-bandwidth memory, enabling them to support mini-batch sizes of up to tens of thousands of training samples. However, most existing optimizers struggle to perform effectively at such a large batch size. As batch size increases, gradient noise decreases due to averaging over many samples, limiting the ability of first-order methods to escape sharp or suboptimal minima and reach the global minimum. Meanwhile, second-order methods like the natural gradient with Kronecker-Factored Approximate Curvature (KFAC) often require excessively high damping to remain stable at large batch sizes. This high damping effectively “washes out” the curvature information that gives these methods their advantage, reducing their performance to that of simple gradient descent. In this paper, we introduce Fisher-Orthogonal Projection (FOP), a novel technique that restores the effectiveness of the second-order method at very large batch sizes, enabling scalable training with improved generalization and faster convergence. FOP constructs a variance-aware update direction by leveraging gradients from two sub-batches, enhancing the average gradient with a component of the gradient difference that is orthogonal to the average under the Fisher-metric. Through extensive benchmarks, we show that FOP accelerates convergence by ×1.2–1.3 over KFAC and ×1.5–1.7 over SGD/AdamW at the same moderate batch sizes, while at extreme scales it achieves up to a ×7.5 speedup. Unlike other methods, FOP maintains small-batch accuracy when scaling to extremely large batch sizes. Moreover, it reduces Top-1 error by 2.3–3.3% on long-tailed CI-FAR benchmarks, demonstrating robust generalization under severe class imbalance. Our lightweight, geometry-aware use of intra-batch variance makes natural-gradient optimization practical on modern data-centre GPUs. FOP is open-source and pip-installable, which can be integrated into existing training code with a single line and no extra configuration.