MicroGrad.jl

Minimal automatic differentiation in Julia
Author LiorSinai
Github
Popularity
5 Stars
Updated Last
20 Days Ago
Started In
June 2024

MicroGrad.jl: Julia Automatic Differentiation

A minimal implementation of automatic differentation (AD) based on Zygote.jl and ChainRules.jl. See limitations below.

This package is built on 3 of Julia's main principles: functional programming, multiple dispatch and metaprogramming.

It is inspired by Andrej Karpathy's micrograd package. However the internal workings of this package are very different. Micrograd creates an object (Value) which implements custom versions of mathematical operations that also calculate the derivative. All operations are only done with those objects. In this package existing functions and their arguments are dispatched to a derivative function (rrule). Metaprogamming is used to generate code and dispatch it following the rules of differentiation.

The two functions exposed are rrule and pullback.

Example using rrule:

x = 0.6
y1, back1 = rrule(cos, x) # (0.82533, cos_back(...))
y2, back2 = rrule(sin, y1) # (0.73477, sin_back(...))
Δy2 = 1.0
Δsin, Δy1 = back2(Δy2) # (nothing,  0.67831)
Δcos, Δx = back1(Δy1) # (nothing, -0.38300)

The pullback function will automatically transform the forward pass and return a Pullback struct that can generate the backward pass:

foo(x) = sin(cos(x))
z, back = pullback(foo, 0.6) # (0.73477, ∂(foo))
Δfoo, Δx = back(1.0) # (nothing, -0.38300)

Note that pullback returns derivatives for both the function and the arguments and so it is a mimic of Zygote._pullback and not Zygote.pullback.

Decision boundary for moon dataset.

This code can be used to train a simple model. The above shows the boundary region after training a small multi-layer perceptron model to differentiate between to two half-circles.

The model is defined as:

model = Chain(
    Dense(2 => 16, activation=relu),
    Dense(16 => 16, activation=relu),
    Dense(16 => 2, activation=relu),
)

Loss and activation functions

Because this AD is limited (especially with regards to control flow and broadcasting), explicit activation and loss functions and their rrules are also provided. They are based on functions in NNlib.jl.

These are:

  • sigmoid
  • tanh_act
  • relu
  • softmax
  • logsoftmax
  • mse
  • cross_entropy
  • logit_cross_entropy
  • hinge_loss (max-margin loss)

Debug

Try a pullback:

z, back = pullback(f, args...)
Δ = similar(z) # Flux.jl calls this sensitivity
grads = back(Δ)

Debug the IR pullback code:

using IRTools: @code_ir
using MicroGrad: instrument, primal, reverse_differentiate
foo(x) = sin(cos(x))
ir = @code_ir foo(1.0)
ir = instrument(ir)
T = Tuple{typeof(foo), Float64}
pr, calls = primal(ir, T)
back_ir = reverse_differentiate(ir)

Alternatively, the ir struct can be created with:

using IRTools: IR, meta
m = meta(T, world=Base.get_world_counter())
ir = IR(m)

When using the expression pullback code, do the following:

using MicroGrad: primal, reverse_differentiate
ci = @code_lowered foo(1.0)
T = Tuple{typeof(foo), Float64}
pr, calls = primal(ci, T)
back_ci = reverse_differentiate(ci, :methodinstance, :Δ)

Alternatively, the ci struct can be created with:

m = meta(T; world=Base.get_world_counter())
type_signature, sps, method_ = m
ci = Base.uncompressed_ast(method_)

The pullback function is recursive and so pullback calls in the above IR/CodeInfo blocks should be inspected as well. The type T is the first parameter in back::Pullback{T, S...}.

Also inspect the output of:

world = Base.get_world_counter()
MicroGrad._generate_pullback(world, typeof(f), typeof.(args)...)
MicroGrad._generate_callable_pullback(typeof(back), world, typeof(Δ))

Limitations

This package has the following limitations.

  • Keyword arguments are not supported.
  • Control flow is not supported.
  • Like with Zygote.jl, mutating operations and try-catch statements are not supported.
  • Minimal set of rrules are implemented. Many core functions from Base are not implemented e.g. generic broadcasting, Base.iterate etc.
  • The generated reverse code is only tested in limited scenarios and is not guaranteed to be correct in general.
  • Cannot handle code with superfluous calls. This will result in nothing being passed to callbacks. Zygote overcomes this issue by wrapping the pullbacks with ZBack which returns nothing if the input is nothing. See the "confused" test.

Further, the reverse_expr code cannot handle:

  • Creation of new variables e.g. with map.
  • Generated code e.g. in the Chain layer in the examples.

Install

Download the GitHub repository (it is not registered). Then in the Julia REPL:

julia> ] # enter package mode
(@v1.x) pkg> dev path\\to\\MicroGrad.jl
julia> using Revise # for dynamic editing of code. Note: will not always recompile generated functions.
julia> using MicroGrad

Resources

Differentiation:

Code inspection and modificaiton:

Struct inspection: