In lab meeting this week we discussed Automatic Differentiation in Machine Learning: a Survey, which addresses the general technique of autodifferentiation for functions expressed in computer programs.
The paper initially outlines three main approaches to the computation of derivatives in computer programs: 1) manually working out derivates by hand and coding the result. 2) using numerical approximations to derivates in the form of assessing function values at small discrete steps, and 3) computer manipulation of symbolic mathematic expressions that automatically generates differential expressions via standard calculus rules. The limitations of approach 1 are that derivatives can involve the calculation of a large number of terms (expression swell) that are both manually cumbersome to deal with and lend themselves to easy algebraic mistakes. The disadvantage of approach 2 is a finite difference approach to numerical differentiation inevitably involves round-off errors and is sensitive to step-size. The disadvantage to 3 is that this approach is computationally cumbersome and often standard calculus rules involve forms of derivates that still need to be manually reduced. In contrast to these methods, the paper introduces autodifferentiation as a set of techniques that operates at the elementary computation level via step-wise implementation of the chain rule. The assessment of a function in a computer program involves an execution of procedures that can be represented via a computational graph where each node in the graph is an intermediary temporary variable. Computers execute this trace of elementary operations via intermediary variables and this procedure can be utilized efficiently to additionally concurrently calculate a derivative. The paper outlines two general approaches to autodifferentiation: forward mode and backward mode. In forward mode, a derivative trace is evaluated alongside the computers execution of the functional trace, which intermediary values from the functional evaluation trace are used in concert with the derivative trace calculation. This execution can be conveniently realized by extending the representation of the real numbers to include a dual component. Analogous to imaginary numbers, this dual component is denoted with an , where . That is, each number in the execution of the function on a computer is now extended to include tuple which includes this additional dual component whose expressions evaluate using intermediate variables alongside the evaluation of the function. Using this simple rule of the dual number (), evaluations implement the notion of the chain rule in differentiation to automatically calculate derivatives at every computational step along the evaluation trace. In executing this forward mode autodifferentiation, a derivative trace is initially ‘seeded’ with an input derivative vector as a function is evaluated with an initial input. Whereas symbolic differentiation can be thought of as the mapping of to , forward autodifferentiation is the mapping of to . As such, considering a functional mapping from dimension to , forward mode requires n trace evaluations to calculate the entire Jacobian, each time the seeded derivative vector is an index for a particular dimension in n. Thus, the calculation of the Jacobian can be completed in approximately n times the total number of operations in an evaluation of . Backward mode, by contrast, computes to . That is, one evaluation of the derivative trace in backward mode can similarly compute a Jacobian vector dot product, in this case extracting a row of the Jacobian (as opposed to a column as in forward mode). In this case, the calculation of the full Jacobian can be completed in approximately m times the total number of operations in an evaluation of . A disadvantage to backward mode autodifferentiation is that each intermediate variable along a function evaluation trace must be stored in memory for use during the backward execution of the derivative trace. |