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。一些应用需要计算对大量输入参数的导数。例如，金融市场是由许多因素（如利率）驱动的，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导数，计算成本与输入参数的数量成线性关系。对于大量的输入参数，线性成本通常不够好。相反，使用伴随公式会产生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敏感性，其成本不超过原始评估成本的4倍。”（Mike Giles教授的《Smoking adjoint》一书的预印本）。这些伴随灵敏度方法在计算机科学中被称为自动（或算法）微分，在机器学习中被称为反向传播，在金融中被称为AAD。在金融领域，资产价格动态通常被认为遵循一种（随机）模型。一个好的模型的一个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，这些参数以这样一种方式进行校准，以最小化模型得出的价格与市场观察到的价格之间的差异。为了找到自由参数的最佳选择，可以根据价差对自由参数变化的敏感性给出的搜索方向进行迭代（基于梯度的优化方法）。因此，需要为每次迭代计算导数。伴随敏感性方法保证了单个金融产品衍生品的高效计算，但每个金融产品都需要进行新的敏感性计算。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，使计算成本独立于校准过程中使用的金融产品的数量。按关联度递减，本项目属于以下EPSRC研究领域：(1)数值分析，(2)运筹学，(3)人工智能技术。该研究项目是与汇丰银行合作进行的。我们合作的目的是通过神经网络近似计算具有多个组件的财务工作流的伴随灵敏度。

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.