Several machine learning applications involve the optimization of higher-order derivatives (e.g., gradients of gradients) during training, which can be expensive in respect to memory and computation even with automatic differentiation. As a typical example in generative modeling, score matching (SM) involves the optimization of the trace of a Hessian. To improve computing efficiency, we rewrite the SM objective and its variants in terms of directional derivatives, and present a generic strategy to efficiently approximate any-order directional derivative with finite difference (FD). Our approximation only involves function evaluations, which can be executed in parallel, and no gradient computations. Thus, it reduces the total computational cost while also improving numerical stability. We provide two instantiations by reformulating variants of SM objectives into the FD forms. Empirically, we demonstrate that our methods produce results comparable to the gradient-based counterparts while being much more computationally efficient.

通过使用有限差分法（FD）对一些机器学习应用中的高阶导数进行近似计算，例如生成模型中的SM（Score Matching），可以大大降低计算成本，并提高数值稳定性。

通过有限差分得分匹配高效学习生成模型