Differential machine learning (DML) is a recently proposed technique that uses samplewise state derivatives to regularize least square fits to learn conditional expectations of functionals of stochastic processes as functions of state variables. Exploiting the derivative information leads to fewer samples than a vanilla ML approach for the same level of precision. This paper extends the methodology to parametric problems where the processes and functionals also depend on model and contract parameters, respectively. In addition, we propose adaptive parameter sampling to improve relative accuracy when the functionals have different magnitudes for different parameter sets. For calibration, we construct pricing surrogates for calibration instruments and optimize over them globally. We discuss strategies for robust calibration. We demonstrate the usefulness of our methodology on one-factor Cheyette models with benchmark rate volatility specification with an extra stochastic volatility factor on (two-curve) caplet prices at different strikes and maturities, first for parametric pricing, and then by calibrating to a given caplet volatility surface. To allow convenient and efficient simulation of processes and functionals and in particular the corresponding computation of samplewise derivatives, we propose to specify the processes and functionals in a low-code way close to mathematical notation which is then used to generate efficient computation of the functionals and derivatives in TensorFlow.

翻译：微分机器学习（DML）是一种近期提出的技术，它利用逐样本状态导数对最小二乘拟合进行正则化，以学习随机过程泛函的条件期望作为状态变量的函数。利用导数信息使得在达到相同精度水平时，所需样本量少于普通机器学习方法。本文将该方法扩展至参数化问题，其中随机过程和泛函分别依赖于模型参数和合约参数。此外，我们提出自适应参数采样方法，以在泛函对于不同参数集具有不同量级时提高相对精度。在校准方面，我们为校准工具构建定价代理函数，并在全局范围内对其进行优化。我们讨论了稳健校准的策略。我们通过带基准利率波动率规范的单因子Cheyette模型（含额外随机波动率因子）对（双曲线）上限期权价格在不同行权价和期限下的表现，首先用于参数化定价，然后通过校准至给定上限期权波动率曲面，验证了该方法的有效性。为便于高效便捷地模拟过程与泛函（特别是对应的逐样本导数计算），我们建议以接近数学符号的低代码方式指定过程与泛函，进而生成高效的计算代码用于TensorFlow中的泛函与导数运算。