Adjoint sensitivity methods

函数的导数测量目标输出变量对输入参数变化的灵敏度。一些应用需要计算大量输入参数的导数。例如，金融市场由许多因素（例如利率）驱动，了解每个因素的变化如何影响金融产品的价格是关键。如果我们单独计算每个输入参数的衍生品，计算成本与输入参数的数量呈线性关系。线性成本对于大量/大量的输入参数来说往往不够好。相反，使用伴随公式产生了一个令人惊讶的结果：“给定一个计算标量输出的计算机程序，我们可以获得该输出对所有程序输入的灵敏度，其成本不超过原始评估成本的4倍。（预印本的书“吸烟的邻接”教授迈克贾尔斯）。这些伴随灵敏度方法在计算机科学中称为自动（或算法）微分，在机器学习中称为反向传播，在金融中称为AAD。在金融学中，资产价格动态通常被假设遵循一个（随机）模型。一个好的模型的必要条件是，它能够尽可能准确地匹配市场上观察到的价格。在实践中，金融模型通常具有自由参数，这些参数以这样的方式进行校准，即最小化模型得出的价格与市场上观察到的价格之间的差异。为了找到自由参数的最佳选择，人们可以迭代地进行搜索方向，该搜索方向由价格差异对自由参数变化的敏感度给出（基于梯度的优化方法）。因此，需要计算每次迭代的导数。伴随灵敏度方法保证了一个金融产品的衍生产品的有效计算，每个金融产品需要一个新的灵敏度计算。因此，计算成本随着金融产品数量的增加而增加。本研究项目开发了一种技术，使计算成本与校准过程中使用的金融产品数量无关。通过降低相关程度，本项目福尔斯以下EPSRC研究领域：（1）数值分析，（2）运筹学，（3）人工智能技术。该研究项目与汇丰银行合作。我们合作的目的是计算伴随灵敏度的财务工作流程与多个组件，近似神经网络。

The derivative of a function measures the sensitivity of the target output variable to changes in the input parameters. Several applications require the calculation of derivatives with respect to a large number of input parameters. The financial market, for example, is driven by many factors (e.g. interest rates) and it is key to understand how changes in each factor can affect the price of a financial product.If we individually compute derivatives for each input parameter, computational cost scales linearly with the number of input parameters. Linear cost is often not good enough for a large/vast amount number of input parameters. Instead, using an adjoint formulation yields a surprising result: "Given a computer program to compute a scalar output, we can obtain the sensitivity of that output to all the program's inputs for a cost which is no more than a factor 4 greater than the original evaluation cost." (Preprint of the book "Smoking Adjoints" by Prof. Mike Giles). These adjoint sensitivity methods go under the name of automatic (or algorithmic) differentiation in computer science, backpropagation in machine learning, and AAD in finance. In finance, asset price dynamics are often assumed to follow a (random) model. A necessary requirement of a good model is that it is able to match prices observed in the market as accurately as possible. In practice, a financial model often has free parameters that are calibrated in such a way that minimizes the difference between prices resulting from the model and prices observed in the market. In order to find the best choice for the free parameters, one might proceed iteratively with the search direction given by the sensitivity of the price difference to changes in the free parameters (gradient-based optimization approach). Thus, one needs to compute derivatives for each iteration. While adjoint sensitivity methods guarantee the efficient computation of derivatives of one financial product, each financial product requires a new sensitivity computation. As a consequence, computational cost increases as the number of financial products increases. This research project develops a technique to render computational cost independent of the number of financial products used in the calibration procedure.By decreasing degree of relevance, this project falls within the following EPSRC research areas: (1) Numerical Analysis, (2) Operational Research, (3) Artificial Intelligence Technologies. The research project is in collaboration with HSBC. The aim of our collaboration is to compute adjoint sensitivities of a financial workflow with multiple components that are approximated by neural networks.