We present EGN, a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges to an $\epsilon$-stationary point at a linear rate. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, and SGN optimizers across various supervised and reinforcement learning tasks.

翻译：本文提出EGN——一种随机二阶优化算法，该算法将广义高斯-牛顿（GN）海森矩阵近似与低秩线性代数相结合来计算下降方向。通过利用邓肯-古特曼矩阵恒等式，参数更新可通过分解一个尺寸与小批量数据规模相当的矩阵获得。这对于神经网络参数向量维度比批量大小高出数个数量级的大规模机器学习问题尤为有利。此外，我们展示了如何将线性搜索、自适应正则化和动量等改进措施无缝集成到EGN中，以进一步加速算法。在温和假设条件下，我们证明了该算法能以线性速率收敛至$\epsilon$-稳定点。最终，数值实验表明，在各种监督学习和强化学习任务中，EGN始终优于或至少匹配经过充分调优的SGD、Adam和SGN优化器的泛化性能。