Automatic differentiation

Fundamental to automatic differentiation is the decomposition of differentials provided by the chain rule of partial derivatives of composite functions. For the simple composition

$${\displaystyle {\begin{aligned}y&=f(g(h(x)))=f(g(h(w_{0})))=f(g(w_{1}))=f(w_{2})=w_{3}\\w_{0}&=x\\w_{1}&=h(w_{0})\\w_{2}&=g(w_{1})\\w_{3}&=f(w_{2})=y\end{aligned}}}$$

the chain rule gives

$${\displaystyle {\frac {\partial y}{\partial x}}={\frac {\partial y}{\partial w_{2}}}{\frac {\partial w_{2}}{\partial w_{1}}}{\frac {\partial w_{1}}{\partial x}}={\frac {\partial f(w_{2})}{\partial w_{2}}}{\frac {\partial g(w_{1})}{\partial w_{1}}}{\frac {\partial h(w_{0})}{\partial x}}}$$

Usually, two distinct modes of automatic differentiation are presented.

• forward accumulation (also called bottom-up, forward mode, or tangent mode)
• reverse accumulation (also called top-down, reverse mode, or adjoint mode)

Forward accumulation specifies that one traverses the chain rule from inside to outside (that is, first compute ${\displaystyle \partial w_{1}/\partial x}$ and then ${\displaystyle \partial w_{2}/\partial w_{1}}$ and at last ${\displaystyle \partial y/\partial w_{2}}$), while reverse accumulation has the traversal from outside to inside (first compute ${\displaystyle \partial y/\partial w_{2}}$ and then ${\displaystyle \partial w_{2}/\partial w_{1}}$ and at last ${\displaystyle \partial w_{1}/\partial x}$). More succinctly,

• Forward accumulation computes the recursive relation: ${\displaystyle {\frac {\partial w_{i}}{\partial x}}={\frac {\partial w_{i}}{\partial w_{i-1}}}{\frac {\partial w_{i-1}}{\partial x}}}$ with ${\displaystyle w_{3}=y}$, and,
• Reverse accumulation computes the recursive relation: ${\displaystyle {\frac {\partial y}{\partial w_{i}}}={\frac {\partial y}{\partial w_{i+1}}}{\frac {\partial w_{i+1}}{\partial w_{i}}}}$ with ${\displaystyle w_{0}=x}$.

The value of the partial derivative, called seed, is propagated forward or backward and is initially ${\displaystyle {\frac {\partial x}{\partial x}}=1}$ or ${\displaystyle {\frac {\partial y}{\partial y}}=1}$. Forward accumulation evaluates the function and calculates the derivative with respect to one independent variable in one pass. For each independent variable ${\displaystyle x_{1},x_{2},\dots ,x_{n}}$ a separate pass is therefore necessary in which the derivative with respect to that independent variable is set to one (${\displaystyle {\frac {\partial x_{1}}{\partial x_{1}}}=1}$) and of all others to zero (${\displaystyle {\frac {\partial x_{2}}{\partial x_{1}}}=\dots ={\frac {\partial x_{n}}{\partial x_{1}}}=0}$). In contrast, reverse accumulation requires the evaluated partial functions for the partial derivatives. Reverse accumulation therefore evaluates the function first and calculates the derivatives with respect to all independent variables in an additional pass.

Which of these two types should be used depends on the sweep count. The computational complexity of one sweep is proportional to the complexity of the original code.

• Forward accumulation is more efficient than reverse accumulation for functions f : Rⁿ → R^m with n ≪ m as only n sweeps are necessary, compared to m sweeps for reverse accumulation.
• Reverse accumulation is more efficient than forward accumulation for functions f : Rⁿ → R^m with n ≫ m as only m sweeps are necessary, compared to n sweeps for forward accumulation.

Backpropagation of errors in multilayer perceptrons, a technique used in machine learning, is a special case of reverse accumulation.[2]

Forward accumulation was introduced by R.E. Wengert in 1964.[3] According to Andreas Griewank, reverse accumulation has been suggested since the late 1960s, but the inventor is unknown.[4] Seppo Linnainmaa published reverse accumulation in 1976.[5]

Forward accumulation

In forward accumulation AD, one first fixes the independent variable with respect to which differentiation is performed and computes the derivative of each sub-expression recursively. In a pen-and-paper calculation, this involves repeatedly substituting the derivative of the inner functions in the chain rule:

$${\displaystyle {\begin{aligned}{\frac {\partial y}{\partial x}}&={\frac {\partial y}{\partial w_{n-1}}}{\frac {\partial w_{n-1}}{\partial x}}\\[6pt]&={\frac {\partial y}{\partial w_{n-1}}}\left({\frac {\partial w_{n-1}}{\partial w_{n-2}}}{\frac {\partial w_{n-2}}{\partial x}}\right)\\[6pt]&={\frac {\partial y}{\partial w_{n-1}}}\left({\frac {\partial w_{n-1}}{\partial w_{n-2}}}\left({\frac {\partial w_{n-2}}{\partial w_{n-3}}}{\frac {\partial w_{n-3}}{\partial x}}\right)\right)\\[6pt]&=\cdots \end{aligned}}}$$

This can be generalized to multiple variables as a matrix product of Jacobians.

Compared to reverse accumulation, forward accumulation is natural and easy to implement as the flow of derivative information coincides with the order of evaluation. Each variable ${\displaystyle w_{i}}$ is augmented with its derivative ${\displaystyle {\dot {w}}_{i}}$ (stored as a numerical value, not a symbolic expression),

$${\displaystyle {\dot {w}}_{i}={\frac {\partial w_{i}}{\partial x}}}$$

as denoted by the dot. The derivatives are then computed in sync with the evaluation steps and combined with other derivatives via the chain rule.

Using the chain rule, if ${\displaystyle w_{i}}$ has predecessors in the computational graph:

${\displaystyle {\dot {w}}_{i}=\sum _{j\in \{{\text{predecessors of i}}\}}{\frac {\partial w_{i}}{\partial w_{j}}}{\dot {w}}_{j}}$

Figure 2: Example of forward accumulation with computational graph

As an example, consider the function:

$${\displaystyle {\begin{aligned}y&=f(x_{1},x_{2})\\&=x_{1}x_{2}+\sin x_{1}\\&=w_{1}w_{2}+\sin w_{1}\\&=w_{3}+w_{4}\\&=w_{5}\end{aligned}}}$$

For clarity, the individual sub-expressions have been labeled with the variables ${\displaystyle w_{i}}$.

The choice of the independent variable to which differentiation is performed affects the seed values ẇ₁ and ẇ₂. Given interest in the derivative of this function with respect to x₁, the seed values should be set to:

$${\displaystyle {\begin{aligned}{\dot {w}}_{1}={\frac {\partial w_{1}}{\partial x_{1}}}={\frac {\partial x_{1}}{\partial x_{1}}}=1\\{\dot {w}}_{2}={\frac {\partial w_{2}}{\partial x_{1}}}={\frac {\partial x_{2}}{\partial x_{1}}}=0\end{aligned}}}$$

With the seed values set, the values propagate using the chain rule as shown. Figure 2 shows a pictorial depiction of this process as a computational graph.

Operations to compute value	Operations to compute derivative
${\displaystyle w_{1}=x_{1}}$	${\displaystyle {\dot {w}}_{1}=1}$ (seed)
${\displaystyle w_{2}=x_{2}}$	${\displaystyle {\dot {w}}_{2}=0}$ (seed)
${\displaystyle w_{3}=w_{1}\cdot w_{2}}$	${\displaystyle {\dot {w}}_{3}=w_{2}\cdot {\dot {w}}_{1}+w_{1}\cdot {\dot {w}}_{2}}$
${\displaystyle w_{4}=\sin w_{1}}$	${\displaystyle {\dot {w}}_{4}=\cos w_{1}\cdot {\dot {w}}_{1}}$
${\displaystyle w_{5}=w_{3}+w_{4}}$	${\displaystyle {\dot {w}}_{5}={\dot {w}}_{3}+{\dot {w}}_{4}}$

To compute the gradient of this example function, which requires not only ${\displaystyle {\tfrac {\partial y}{\partial x_{1}}}}$ but also ${\displaystyle {\tfrac {\partial y}{\partial x_{2}}}}$, an additional sweep is performed over the computational graph using the seed values ${\displaystyle {\dot {w}}_{1}=0;{\dot {w}}_{2}=1}$.

Forward accumulation calculates the function and the derivative (but only for one independent variable each) in one pass. The associated method call expects the expression Z to be derived with regard to a variable V. The method returns a pair of the evaluated function and its derivation. The method traverses the expression tree recursively until a variable is reached. If the derivative with respect to this variable is requested, its derivative is 1, 0 otherwise. Then the partial function as well as the partial derivative are evaluated.[6]

tuple<float,float> evaluateAndDerive(Expression Z, Variable V) {
   if isVariable(Z)
      if (Z = V) return {valueOf(Z), 1};
      else return {valueOf(Z), 0};
   else if (Z = A + B)
      {a, a'} = evaluateAndDerive(A, V);
      {b, b'} = evaluateAndDerive(B, V);
      return {a + b, a' + b'};
   else if (Z = A - B)
      {a, a'} = evaluateAndDerive(A, V);
      {b, b'} = evaluateAndDerive(B, V);
      return {a - b, a' - b'};
   else if (Z = A * B)
      {a, a'} = evaluateAndDerive(A, V);
      {b, b'} = evaluateAndDerive(B, V);
      return {a * b, b * a' + a * b'};
}

Reverse accumulation

In reverse accumulation AD, the dependent variable to be differentiated is fixed and the derivative is computed with respect to each sub-expression recursively. In a pen-and-paper calculation, the derivative of the outer functions is repeatedly substituted in the chain rule:

$${\displaystyle {\begin{aligned}{\frac {\partial y}{\partial x}}&={\frac {\partial y}{\partial w_{1}}}{\frac {\partial w_{1}}{\partial x}}\\&=\left({\frac {\partial y}{\partial w_{2}}}{\frac {\partial w_{2}}{\partial w_{1}}}\right){\frac {\partial w_{1}}{\partial x}}\\&=\left(\left({\frac {\partial y}{\partial w_{3}}}{\frac {\partial w_{3}}{\partial w_{2}}}\right){\frac {\partial w_{2}}{\partial w_{1}}}\right){\frac {\partial w_{1}}{\partial x}}\\&=\cdots \end{aligned}}}$$

In reverse accumulation, the quantity of interest is the adjoint, denoted with a bar ${\displaystyle {\bar {w}}_{i}}$; it is a derivative of a chosen dependent variable with respect to a subexpression ${\displaystyle w_{i}}$:

$${\displaystyle {\bar {w}}_{i}={\frac {\partial y}{\partial w_{i}}}}$$

Using the chain rule, if ${\displaystyle w_{i}}$ has successors in the computational graph:

${\displaystyle {\bar {w}}_{i}=\sum _{j\in \{{\text{successors of i}}\}}{\bar {w}}_{j}{\frac {\partial w_{j}}{\partial w_{i}}}}$

Reverse accumulation traverses the chain rule from outside to inside, or in the case of the computational graph in Figure 3, from top to bottom. The example function is scalar-valued, and thus there is only one seed for the derivative computation, and only one sweep of the computational graph is needed to calculate the (two-component) gradient. This is only half the work when compared to forward accumulation, but reverse accumulation requires the storage of the intermediate variables w_i as well as the instructions that produced them in a data structure known as a "tape" or a Wengert list[7] (however, Wengert published forward accumulation, not reverse accumulation[3]), which may consume significant memory if the computational graph is large. This can be mitigated to some extent by storing only a subset of the intermediate variables and then reconstructing the necessary work variables by repeating the evaluations, a technique known as rematerialization. Checkpointing is also used to save intermediary states.

Figure 3: Example of reverse accumulation with computational graph

The operations to compute the derivative using reverse accumulation are shown in the table below (note the reversed order):

The data flow graph of a computation can be manipulated to calculate the gradient of its original calculation. This is done by adding an adjoint node for each primal node, connected by adjoint edges which parallel the primal edges but flow in the opposite direction. The nodes in the adjoint graph represent multiplication by the derivatives of the functions calculated by the nodes in the primal. For instance, addition in the primal causes fanout in the adjoint; fanout in the primal causes addition in the adjoint;[lower-alpha 1] a unary function y = f(x) in the primal causes x̄ = ȳ f′(x) in the adjoint; etc.

Reverse accumulation requires two passes: In the forward pass, the function is evaluated first and the partial results are cached. In the reverse pass, the partial derivatives are calculated and the previously derived value is backpropagated. The corresponding method call expects the expression Z to be derived and seed with the derived value of the parent expression. For the top expression, Z derived with regard to Z, this is 1. The method traverses the expression tree recursively until a variable is reached and adds the current seed value to the derivative expression.[8][9]

void derive(Expression Z, float seed) {
   if isVariable(Z)
      partialDerivativeOf(Z) += seed;
   else if (Z = A + B)
      derive(A, seed);
      derive(B, seed);
   else if (Z = A - B)
      derive(A, seed);
      derive(B, -seed);
   else if (Z = A * B)
      derive(A, valueOf(B) * seed);
      derive(B, valueOf(A) * seed);
}

Forward and reverse accumulation are just two (extreme) ways of traversing the chain rule. The problem of computing a full Jacobian of f : Rⁿ → R^m with a minimum number of arithmetic operations is known as the optimal Jacobian accumulation (OJA) problem, which is NP-complete.[10] 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 ${\displaystyle y'=\nabla f(x)\cdot x'}$, the directional derivative ${\displaystyle y'\in \mathbb {R} ^{m}}$ of ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ at ${\displaystyle x\in \mathbb {R} ^{n}}$ in the direction ${\displaystyle x'\in \mathbb {R} ^{n}}$ may be calculated as ${\displaystyle (\langle y_{1},y'_{1}\rangle ,\ldots ,\langle y_{m},y'_{m}\rangle )=f(\langle x_{1},x'_{1}\rangle ,\ldots ,\langle x_{n},x'_{n}\rangle )}$ using the same arithmetic as above. If all the elements of ${\displaystyle \nabla f}$ are desired, then ${\displaystyle n}$ 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.

Figure 4: Example of how source code transformation could work

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.

Figure 5: Example of how operator overloading could work

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.