Differential Machine Learning
16 Pages Posted: 4 Jun 2020
Date Written: January 3, 2020
Differential machine learning (ML) is an extension of supervised learning, where ML models are trained on examples of not only inputs and labels but also differentials of labels to inputs.
Differential ML is applicable in situations where high quality first order derivatives wrt training inputs are available. In the context of financial Derivatives and risk management, pathwise differentials are efficiently computed with automatic adjoint differentiation (AAD). Differential machine learning, combined with AAD, provides extremely effective pricing and risk approximations. We can produce fast pricing analytics in models too complex for closed form solutions, extract the risk factors of complex transactions and trading books, and effectively compute risk management metrics like reports across a large number of scenarios, backtesting and simulation of hedge strategies, or regulations like XVA, CCR, FRTB or SIMM-MVA.
The article focuses on differential deep learning (DL), arguably the strongest application. Standard DL trains neural networks (NN) on punctual examples, whereas differential DL teaches them the shape of the target function, resulting in vastly improved performance, illustrated with a number of numerical examples, both idealized and real world. In the online appendices, we apply differential learning to other ML models, like classic regression or principal component analysis (PCA), with equally remarkable results.
This paper is meant to be read in conjunction with its companion GitHub repo https://github.com/differential-machine-learning, where we posted a TensorFlow implementation, tested on Google Colab, along with examples from the article and additional ones. We also posted appendices covering many practical implementation details not covered in the paper, mathematical proofs, application to ML models besides neural networks and extensions necessary for a reliable implementation in production.
Keywords: automatic differentiation, machine learning, deep learning, neural networks, quantitative finance, risk management, derivatives, trading systems, financial regulation, options, pricing, monte carlo, least square method, regression, PCA
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