Reverse-mode automatic differentiation in Julia



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Reverse-mode automatic differentiation for static and dynamic graphs.


mutable struct Linear{T}

forward(m::Linear, X) = m.W * X

loss(m::Linear, X) = sum(forward(m, X))

m = Linear(rand(3,4), rand(3))
X = rand(4,5)

val, g = grad(loss, m, X)

g is an object of type GradientResult holding gradients w.r.t. input variables. For scalars and tensors it returns gradient value, for structs it returns dictionary of (field path → gradient) pairs:

julia> g[1]
Dict{Tuple{Symbol},Array{Float64,2}} with 1 entry:
  (:W,) => [3.38128 2.97142 2.39706 1.55525; 3.38128 2.97142 2.39706 1.55525; 3.38128 2.97142 2.39706 1.55525]   # gradient w.r.t. m.W

julia> g[2]  # gradient w.r.t. X
4×5 Array{Float64,2}:
 0.910691  0.910691  0.910691  0.910691  0.910691
 1.64994   1.64994   1.64994   1.64994   1.64994
 1.81215   1.81215   1.81215   1.81215   1.81215
 2.31594   2.31594   2.31594   2.31594   2.31594

GradientResult can be used in conjunction with update!() function to modify tensors and fields of (mutable) structs. To continue out previous example:

for i=1:100
    val, g = grad(loss, m, X)
    println("Loss value in $(i)th epoch: $val")
    update!(m, g[1], (x, gx) -> x .- 0.01gx)

(Note that our simplified loss function doesn't actually represent an error to be minimized, so loss value quickly goes below zero. For more realistic and much more complex examples see vae.jl.)

Custom derivatives

You can add custom derivatives using @diffrule macro (see list of allowed variable names below).

logistic(x) = 1 / (1 + exp(-x))
# for an expression like `y = logistic(x)` where x is a Number
# gradient w.r.t. x
# is `(logistic(x) * (1 - logistic(x)) * dy)` where "dy" stands for derivative "dL/dy"
@diffrule logistic(x::Number) x (logistic(x) * (1 - logistic(x)) * dy)

L(x) = sum(logistic.(x))
val, g = grad(L, rand(5))

For functions accepting keyword arguments use @diffrule_kw instead:

import NNlib: conv, ∇conv_data, ∇conv_filter

@diffrule_kw conv(x, w) x ∇conv_data(dy, w)
@diffrule_kw conv(x, w) w ∇conv_filter(dy, x)

During reverse pass Yota will generate call to derivative function with the same keyword arguments that were passed to the original one. For example, if you have:

conv(A, W; pad=1)

corresponding derivative will be:

∇conv_data(dy, w; pad=1)

There's also @nodiff(call_pattern, variable) macro which stops Yota from backpropagating through that variable.

Allowed variable names

To distinguish between variable names that can be matched to (a.k.a. placeholders) and fixed symbols (e.g. function names), @diffrule uses several rules:

  • y means return value of a primal expression, e.g. y = f(x)
  • dy means derivative of a loss function w.r.t. to y
  • t, u, v, w, x, as well as i, j, k (all listed in Yota.DIFF_PHS) are "placeholders" and can be used as names of variables, e.g. @diffrule foo(u, v) u ∇foo(dy, u, v)
  • anything starting with _ is also considered a placeholder, e.g. @diffrule bar(u, _state) _state ∇bar(dy, u, _state)

Tracer and the Tape

Being a reverse-mode AD package, Yota works in 2 steps:

  1. Record all primitive operations onto a "tape".
  2. Go back trough the tape, recording derivatives for each operation.

"Tape" here is simply a list of operations. You can get the tape as a .tape field of GradientResult or construct it directly using trace function:

import Yota: trace

val, tape = trace(L, rand(5))

# Tape
#   inp %1::Array{Float64,1}
#   const %2 = logistic::typeof(logistic)
#   %3 = broadcast(%2, %1)::Array{Float64,1}
#   %4 = sum(%3)::Float64

trace uses IRTools.jl to collect function calls during execution. Functions are divided into 2 groups:

  • primitive, which are recorded to the tape;
  • non-primitive, which are traced-through down to primitive ones.

By default, set of primitive functions is defined in Yota.PRIMITIVES and includes such beasts as *, broadcast, getproperty as well as all functions for which @diffrule is defined. You can also specify custom primitives using primitive=Set([...]) keyword to trace().

Also note that broadcast's first argument is always considered a primitive and recorded to the tape as is, so backpropagation will only work if there's a @diffrule for it.

Tape can also be executed and compiled:

using BenchmarkTools
import Yota: play!, compile!

x = rand(5)

@btime play!(tape, x)
# 3.526 μs (13 allocations: 640 bytes)

@btime play!(tape, x)
# 492.063 ns (2 allocations: 144 bytes)

Note that trace() is an alias to irtrace() - IRTools-based tracer. As of Yota 0.4.0, two other tracers are available:

These tracers can be used for experimental purposes, but their reliability or even existence is not guaranteed in future. For any long-term code please use alias trace() which always points to the most recent and well-tested implementation.

CUDA support

CuArray is fully supported. If you encounter an issue with CUDA arrays which you don't have with ordinary arrays, please file a bug.

Static vs. dynamic (experimental)

Tracer records operations as they are executed the first time with given arguments. For example, for a loop like this:

function iterative(x, n)
    for i=1:n
        x = 2 .* x
    return sum(x)

exactly n iterations will be recorded to the tape and replaying tape with any other values of n will make no effect. This also applies to a standard grad():

x = rand(4)
_, g = grad(iterative, x, 1)   # g[1] == [2.0, 2.0, 2.0, 2.0]
_, g = grad(iterative, x, 2)   # g[1] == [2.0, 2.0, 2.0, 2.0]
_, g = grad(iterative, x, 3)   # g[1] == [2.0, 2.0, 2.0, 2.0]

Nevertheless, Yota provides pseudo-dynamic capabilities by caching gradient results for all ever generated tapes. This doesn't eliminate cost of re-tracing, but avoids repeated backpropagation and tape optimization. You can tell grad() to use dynamic caching using dynamic=true keyword argument:

x = rand(4)
_, g = grad(iterative, x, 1; dynamic=true)   # g[1] == [2.0, 2.0, 2.0, 2.0]
_, g = grad(iterative, x, 2; dynamic=true)   # g[1] == [4.0, 4.0, 4.0, 4.0]
_, g = grad(iterative, x, 3; dynamic=true)   # g[1] == [8.0, 8.0, 8.0, 8.0]

Note that this feature is experimental and may be removed in future versions.

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