Given the massive cost of language model pre-training,
a non-trivial improvement of the optimization algorithm would lead to a material reduction on the time and cost of training.
Adam and its variants have been state-of-the-art for years, and more sophisticated second-order (Hessian-based)  optimizers often incur too much per-step overhead. In this paper, we propose Sophia,Second-order Clipped Stochastic Optimization, a simple scalable second-order optimizer that uses a light-weight estimate of the diagonal Hessian as the pre-conditioner. The update is the moving average of the gradients divided by the moving average of the estimated Hessian, followed by element-wise clipping. The clipping controls the worst-case update size and tames the negative impact of non-convexity and rapid change of Hessian along the trajectory.
Sophia only estimates the diagonal Hessian every handful of iterations, which has negligible average per-step time and memory overhead. On language modeling with GPT models of sizes ranging from 125M to 1.5B, Sophia achieves a 2x speed-up compared to Adam in the number of steps, total compute, and wall-clock time, achieving the same perplexity with 50fewer steps, less total compute, and reduced wall-clock time.
Theoretically, we show that Sophia, in a much simplified setting, adapts to the heterogeneous curvatures in different parameter dimensions, and thus has a run-time bound that does not depend on the condition number of the loss.