  Differential machine learning combines automatic adjoint differentiation
(AAD) with modern machine learning (ML) in the context of risk management of
financial Derivatives. We introduce novel algorithms for training fast,
accurate pricing and risk approximations, online, in real-time, with
convergence guarantees. Our machinery is applicable to arbitrary Derivatives
instruments or trading books, under arbitrary stochastic models of the
underlying market variables. It effectively resolves computational bottlenecks
of Derivatives risk reports and capital calculations.
  Differential ML is a general extension of supervised learning, where ML
models are trained on examples of not only inputs and labels but also
differentials of labels wrt inputs. It is also applicable in many situations
outside finance, where high quality first-order derivatives wrt training inputs
are available. Applications in Physics, for example, may leverage differentials
known from first principles to learn function approximations more effectively.
  In finance, AAD computes pathwise differentials with remarkable efficacy so
differential ML algorithms provide extremely effective pricing and risk
approximations. We can produce fast 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.
  TensorFlow implementation is available on
https://github.com/differential-machine-learning
