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Numerical calculations carrying along derivatives
In mathematics and computer algebra, automatic differentiation (auto differentiation, autodiff, or AD), also called algorithmic differentiation, computational differentiation, is a set of techniques to evaluate the partial derivative of a function specified by a computer program. Automatic differentiation exploits the fact that every computer calculation, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, partial derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor of more arithmetic operations than the original program.

Automatic differentiation is distinct from symbolic differentiation and numerical differentiation. Symbolic differentiation faces the difficulty of converting a computer program into a single mathematical expression and can lead to inefficient code. Numerical differentiation (the method of finite differences) can introduce round off errors in the discretization process and cancellation. Both of these classical methods have problems with calculating higher derivatives, where complexity and errors increase. Finally, both of these classical methods are slow at computing partial derivatives of a function with respect to many inputs, as is needed for gradient based optimization algorithms. Automatic differentiation solves all of these problems.

Automatic differentiation is particularly important in the field of machine learning. For example, it allows one to implement backpropagation in a neural network without a manually computed derivative.

Forward and reverse accumulation are just two (extreme) ways of traversing the chain rule. The problem of computing a full Jacobian of f : 'R'^(n) → 'R'^(m) with a minimum number of arithmetic operations is known as the optimal Jacobian accumulation (OJA) problem, which is NP complete. Central to this proof is the idea that algebraic dependencies may exist between the local partials that label the edges of the graph. In particular, two or more edge labels may be recognized as equal. The complexity of the problem is still open if it is assumed that all edge labels are unique and algebraically independent.

An example implementation based on the dual number approach follows.

Multivariate functions can be handled with the same efficiency and mechanisms as univariate functions by adopting a directional derivative operator. That is, if it is sufficient to compute , the directional derivative of at in the direction may be calculated as using the same arithmetic as above. If all the elements of are desired, then function evaluations are required. Note that in many optimization applications, the directional derivative is indeed sufficient.

The above arithmetic can be generalized to calculate second order and higher derivatives of multivariate functions. However, the arithmetic rules quickly grow complicated: complexity is quadratic in the highest derivative degree. Instead, truncated Taylor polynomial algebra can be used. The resulting arithmetic, defined on generalized dual numbers, allows efficient computation using functions as if they were a data type. Once the Taylor polynomial of a function is known, the derivatives are easily extracted.

Forward mode AD is implemented by a nonstandard interpretation of the program in which real numbers are replaced by dual numbers, constants are lifted to dual numbers with a zero epsilon coefficient, and the numeric primitives are lifted to operate on dual numbers. This nonstandard interpretation is generally implemented using one of two strategies: source code transformation or operator overloading.

The source code for a function is replaced by an automatically generated source code that includes statements for calculating the derivatives interleaved with the original instructions. Source code transformation can be implemented for all programming languages, and it is also easier for the compiler to do compile time optimizations. However, the implementation of the AD tool itself is more difficult and the build system is more complex.

Operator overloading is a possibility for source code written in a language supporting it. Objects for real numbers and elementary mathematical operations must be overloaded to cater for the augmented arithmetic depicted above. This requires no change in the form or sequence of operations in the original source code for the function to be differentiated, but often requires changes in basic data types for numbers and vectors to support overloading and often also involves the insertion of special flagging operations. Due to the inherent operator overloading overhead on each loop, this approach usually demonstrates weaker speed performance.

Overloaded Operators can be used to extract the valuation graph, followed by automatic generation of the AD version of the primal function at run time. Unlike the classic OO AAD, such AD function does not change from one iteration to the next one. Hence there is any OO or tape interpretation run time overhead per Xi sample. With the AD function being generated at runtime, it can be optimised to take into account the current state of the program and precompute certain values. In addition, it can be generated in a way to consistently utilize native CPU vectorization to process 4(8) double chunks of user data (AVX2\AVX512 speed up x4 x8). With multithreading added into account, such approach can lead to a final acceleration of order 8 × #Cores compared to the traditional AAD tools. A reference implementation is available on GitHub.