Julia & Lux for the Uninitiated

This is a quick intro to Lux loosely based on:

PyTorch's tutorial.
Flux's tutorial (the link for which has now been lost to abyss).
Jax's tutorial.

It introduces basic Julia programming, as well Zygote, a source-to-source automatic differentiation (AD) framework in Julia. We'll use these tools to build a very simple neural network. Let's start with importing Lux.jl

julia

using Lux, Random

Precompiling Lux...
    571.7 ms  ✓ ConcreteStructs
    507.4 ms  ✓ IfElse
    513.1 ms  ✓ Reexport
    512.2 ms  ✓ SIMDTypes
    544.1 ms  ✓ CommonWorldInvalidations
    543.5 ms  ✓ FastClosures
    599.8 ms  ✓ ArgCheck
    631.4 ms  ✓ OpenLibm_jll
    702.1 ms  ✓ ConstructionBase
    761.3 ms  ✓ CompilerSupportLibraries_jll
    812.2 ms  ✓ DocStringExtensions
    835.5 ms  ✓ Statistics
   1029.8 ms  ✓ EnzymeCore
    533.0 ms  ✓ StaticArraysCore
    537.9 ms  ✓ ManualMemory
    543.7 ms  ✓ Future
    547.0 ms  ✓ HashArrayMappedTries
    681.0 ms  ✓ Requires
    908.2 ms  ✓ ADTypes
    816.3 ms  ✓ Compat
    723.8 ms  ✓ JLLWrappers
    664.9 ms  ✓ NaNMath
    919.7 ms  ✓ CpuId
   1572.8 ms  ✓ IrrationalConstants
    745.4 ms  ✓ ConstructionBase → ConstructionBaseLinearAlgebraExt
    555.0 ms  ✓ DiffResults
    534.0 ms  ✓ ScopedValues
    621.3 ms  ✓ Adapt
    494.8 ms  ✓ ADTypes → ADTypesEnzymeCoreExt
    526.4 ms  ✓ Compat → CompatLinearAlgebraExt
   1199.8 ms  ✓ Static
    599.5 ms  ✓ ADTypes → ADTypesConstructionBaseExt
    470.4 ms  ✓ EnzymeCore → AdaptExt
   1291.5 ms  ✓ ThreadingUtilities
    830.4 ms  ✓ LogExpFunctions
    989.7 ms  ✓ OpenSpecFun_jll
   2505.0 ms  ✓ UnsafeAtomics
    690.2 ms  ✓ GPUArraysCore
    623.9 ms  ✓ BitTwiddlingConvenienceFunctions
    766.4 ms  ✓ ArrayInterface
    892.3 ms  ✓ Functors
   2890.7 ms  ✓ MacroTools
    486.2 ms  ✓ ArrayInterface → ArrayInterfaceStaticArraysCoreExt
    680.7 ms  ✓ Atomix
    602.5 ms  ✓ ArrayInterface → ArrayInterfaceGPUArraysCoreExt
   1583.4 ms  ✓ CPUSummary
   1701.1 ms  ✓ ChainRulesCore
    938.5 ms  ✓ CommonSubexpressions
   1042.9 ms  ✓ MLDataDevices
    589.9 ms  ✓ ArrayInterface → ArrayInterfaceChainRulesCoreExt
    611.9 ms  ✓ ADTypes → ADTypesChainRulesCoreExt
    766.8 ms  ✓ PolyesterWeave
   1780.6 ms  ✓ StaticArrayInterface
    827.2 ms  ✓ MLDataDevices → MLDataDevicesChainRulesCoreExt
   1879.3 ms  ✓ DispatchDoctor
   1990.3 ms  ✓ Setfield
    680.5 ms  ✓ CloseOpenIntervals
   1547.4 ms  ✓ Optimisers
    805.8 ms  ✓ LayoutPointers
   1601.5 ms  ✓ LogExpFunctions → LogExpFunctionsChainRulesCoreExt
    483.0 ms  ✓ DispatchDoctor → DispatchDoctorEnzymeCoreExt
    710.6 ms  ✓ DispatchDoctor → DispatchDoctorChainRulesCoreExt
   3256.3 ms  ✓ SpecialFunctions
    471.0 ms  ✓ Optimisers → OptimisersAdaptExt
    474.0 ms  ✓ Optimisers → OptimisersEnzymeCoreExt
   1371.6 ms  ✓ LuxCore
   1011.2 ms  ✓ StrideArraysCore
    701.2 ms  ✓ DiffRules
    509.9 ms  ✓ LuxCore → LuxCoreEnzymeCoreExt
    642.0 ms  ✓ LuxCore → LuxCoreSetfieldExt
    655.2 ms  ✓ LuxCore → LuxCoreMLDataDevicesExt
    679.9 ms  ✓ LuxCore → LuxCoreFunctorsExt
    706.5 ms  ✓ LuxCore → LuxCoreChainRulesCoreExt
    865.3 ms  ✓ Polyester
   2042.2 ms  ✓ SpecialFunctions → SpecialFunctionsChainRulesCoreExt
   3030.3 ms  ✓ WeightInitializers
   7948.7 ms  ✓ StaticArrays
    671.0 ms  ✓ Adapt → AdaptStaticArraysExt
    695.1 ms  ✓ StaticArrays → StaticArraysStatisticsExt
    700.6 ms  ✓ ConstructionBase → ConstructionBaseStaticArraysExt
    714.5 ms  ✓ StaticArrays → StaticArraysChainRulesCoreExt
    730.9 ms  ✓ StaticArrayInterface → StaticArrayInterfaceStaticArraysExt
   1027.7 ms  ✓ WeightInitializers → WeightInitializersChainRulesCoreExt
   3814.0 ms  ✓ ForwardDiff
    831.5 ms  ✓ ForwardDiff → ForwardDiffStaticArraysExt
   3580.9 ms  ✓ KernelAbstractions
    692.9 ms  ✓ KernelAbstractions → LinearAlgebraExt
    769.5 ms  ✓ KernelAbstractions → EnzymeExt
   5209.2 ms  ✓ NNlib
    816.3 ms  ✓ NNlib → NNlibEnzymeCoreExt
    817.0 ms  ✓ NNlib → NNlibSpecialFunctionsExt
    861.7 ms  ✓ NNlib → NNlibForwardDiffExt
   7688.7 ms  ✓ LuxLib
   8928.9 ms  ✓ Lux
  94 dependencies successfully precompiled in 37 seconds. 11 already precompiled.

Now let us control the randomness in our code using proper Pseudo Random Number Generator (PRNG)

julia

rng = Random.default_rng()
Random.seed!(rng, 0)

Random.TaskLocalRNG()

Arrays

The starting point for all of our models is the Array (sometimes referred to as a Tensor in other frameworks). This is really just a list of numbers, which might be arranged into a shape like a square. Let's write down an array with three elements.

julia

x = [1, 2, 3]

3-element Vector{Int64}:
 1
 2
 3

Here's a matrix – a square array with four elements.

julia

x = [1 2; 3 4]

2×2 Matrix{Int64}:
 1  2
 3  4

We often work with arrays of thousands of elements, and don't usually write them down by hand. Here's how we can create an array of 5×3 = 15 elements, each a random number from zero to one.

julia

x = rand(rng, 5, 3)

5×3 Matrix{Float64}:
 0.455238   0.746943   0.193291
 0.547642   0.746801   0.116989
 0.773354   0.97667    0.899766
 0.940585   0.0869468  0.422918
 0.0296477  0.351491   0.707534

There's a few functions like this; try replacing rand with ones, zeros, or randn.

By default, Julia works stores numbers is a high-precision format called Float64. In ML we often don't need all those digits, and can ask Julia to work with Float32 instead. We can even ask for more digits using BigFloat.

julia

x = rand(BigFloat, 5, 3)

5×3 Matrix{BigFloat}:
 0.981339    0.793159  0.459019
 0.043883    0.624384  0.56055
 0.164786    0.524008  0.0355555
 0.414769    0.577181  0.621958
 0.00823197  0.30215   0.655881

julia

x = rand(Float32, 5, 3)

5×3 Matrix{Float32}:
 0.567794   0.369178   0.342539
 0.0985227  0.201145   0.587206
 0.776598   0.148248   0.0851708
 0.723731   0.0770206  0.839303
 0.404728   0.230954   0.679087

We can ask the array how many elements it has.

julia

length(x)

Or, more specifically, what size it has.

julia

size(x)

(5, 3)

We sometimes want to see some elements of the array on their own.

julia

5×3 Matrix{Float32}:
 0.567794   0.369178   0.342539
 0.0985227  0.201145   0.587206
 0.776598   0.148248   0.0851708
 0.723731   0.0770206  0.839303
 0.404728   0.230954   0.679087

julia

x[2, 3]

0.58720636f0

This means get the second row and the third column. We can also get every row of the third column.

julia

x[:, 3]

5-element Vector{Float32}:
 0.34253937
 0.58720636
 0.085170805
 0.8393034
 0.67908657

We can add arrays, and subtract them, which adds or subtracts each element of the array.

julia

x + x

5×3 Matrix{Float32}:
 1.13559   0.738356  0.685079
 0.197045  0.40229   1.17441
 1.5532    0.296496  0.170342
 1.44746   0.154041  1.67861
 0.809456  0.461908  1.35817

julia

x - x

5×3 Matrix{Float32}:
 0.0  0.0  0.0
 0.0  0.0  0.0
 0.0  0.0  0.0
 0.0  0.0  0.0
 0.0  0.0  0.0

Julia supports a feature called broadcasting, using the . syntax. This tiles small arrays (or single numbers) to fill bigger ones.

julia

x .+ 1

5×3 Matrix{Float32}:
 1.56779  1.36918  1.34254
 1.09852  1.20114  1.58721
 1.7766   1.14825  1.08517
 1.72373  1.07702  1.8393
 1.40473  1.23095  1.67909

We can see Julia tile the column vector 1:5 across all rows of the larger array.

julia

zeros(5, 5) .+ (1:5)

5×5 Matrix{Float64}:
 1.0  1.0  1.0  1.0  1.0
 2.0  2.0  2.0  2.0  2.0
 3.0  3.0  3.0  3.0  3.0
 4.0  4.0  4.0  4.0  4.0
 5.0  5.0  5.0  5.0  5.0

The x' syntax is used to transpose a column 1:5 into an equivalent row, and Julia will tile that across columns.

julia

zeros(5, 5) .+ (1:5)'

5×5 Matrix{Float64}:
 1.0  2.0  3.0  4.0  5.0
 1.0  2.0  3.0  4.0  5.0
 1.0  2.0  3.0  4.0  5.0
 1.0  2.0  3.0  4.0  5.0
 1.0  2.0  3.0  4.0  5.0

We can use this to make a times table.

julia

(1:5) .* (1:5)'

5×5 Matrix{Int64}:
 1   2   3   4   5
 2   4   6   8  10
 3   6   9  12  15
 4   8  12  16  20
 5  10  15  20  25

Finally, and importantly for machine learning, we can conveniently do things like matrix multiply.

julia

W = randn(5, 10)
x = rand(10)
W * x

5-element Vector{Float64}:
  1.2197981041108443
 -2.62625877100596
 -2.8573820474674845
 -2.4319346874291314
  1.0108668577150213

Julia's arrays are very powerful, and you can learn more about what they can do here.

CUDA Arrays

CUDA functionality is provided separately by the CUDA.jl package. If you have a GPU and LuxCUDA is installed, Lux will provide CUDA capabilities. For additional details on backends see the manual section.

You can manually add CUDA. Once CUDA is loaded you can move any array to the GPU with the cu function (or the gpu function exported by `Lux``), and it supports all of the above operations with the same syntax.

julia

using LuxCUDA

if LuxCUDA.functional()
    x_cu = cu(rand(5, 3))
    @show x_cu
end

(Im)mutability

Lux as you might have read is Immutable by convention which means that the core library is built without any form of mutation and all functions are pure. However, we don't enforce it in any form. We do strongly recommend that users extending this framework for their respective applications don't mutate their arrays.

julia

x = reshape(1:8, 2, 4)

2×4 reshape(::UnitRange{Int64}, 2, 4) with eltype Int64:
 1  3  5  7
 2  4  6  8

To update this array, we should first copy the array.

julia

x_copy = copy(x)
view(x_copy, :, 1) .= 0

println("Original Array ", x)
println("Mutated Array ", x_copy)

Original Array [1 3 5 7; 2 4 6 8]
Mutated Array [0 3 5 7; 0 4 6 8]

Note that our current default AD engine (Zygote) is unable to differentiate through this mutation, however, for these specialized cases it is quite trivial to write custom backward passes. (This problem will be fixed once we move towards Enzyme.jl)

Managing Randomness

We rely on the Julia StdLib Random for managing the randomness in our execution. First, we create an PRNG (pseudorandom number generator) and seed it.

julia

rng = Xoshiro(0)     # Creates a Xoshiro PRNG with seed 0

Random.Xoshiro(0xdb2fa90498613fdf, 0x48d73dc42d195740, 0x8c49bc52dc8a77ea, 0x1911b814c02405e8, 0x22a21880af5dc689)

If we call any function that relies on rng and uses it via randn, rand, etc. rng will be mutated. As we have already established we care a lot about immutability, hence we should use Lux.replicate on PRNGs before using them.

First, let us run a random number generator 3 times with the replicated rng.

julia

random_vectors = Vector{Vector{Float64}}(undef, 3)
for i in 1:3
    random_vectors[i] = rand(Lux.replicate(rng), 10)
    println("Iteration $i ", random_vectors[i])
end
@assert random_vectors[1] ≈ random_vectors[2] ≈ random_vectors[3]

Iteration 1 [0.4552384158732863, 0.5476424498276177, 0.7733535276924052, 0.9405848223512736, 0.02964765308691042, 0.74694291453392, 0.7468008914093891, 0.9766699015845924, 0.08694684883050086, 0.35149138733595564]
Iteration 2 [0.4552384158732863, 0.5476424498276177, 0.7733535276924052, 0.9405848223512736, 0.02964765308691042, 0.74694291453392, 0.7468008914093891, 0.9766699015845924, 0.08694684883050086, 0.35149138733595564]
Iteration 3 [0.4552384158732863, 0.5476424498276177, 0.7733535276924052, 0.9405848223512736, 0.02964765308691042, 0.74694291453392, 0.7468008914093891, 0.9766699015845924, 0.08694684883050086, 0.35149138733595564]

As expected we get the same output. We can remove the replicate call and we will get different outputs.

julia

for i in 1:3
    println("Iteration $i ", rand(rng, 10))
end

Iteration 1 [0.4552384158732863, 0.5476424498276177, 0.7733535276924052, 0.9405848223512736, 0.02964765308691042, 0.74694291453392, 0.7468008914093891, 0.9766699015845924, 0.08694684883050086, 0.35149138733595564]
Iteration 2 [0.018743665453639813, 0.8601828553599953, 0.6556360448565952, 0.7746656838366666, 0.7817315740767116, 0.5553797706980106, 0.1261990389976131, 0.4488101521328277, 0.624383955429775, 0.05657739601024536]
Iteration 3 [0.19597391412112541, 0.6830945313415872, 0.6776220912718907, 0.6456416023530093, 0.6340362477836592, 0.5595843665394066, 0.5675557670686644, 0.34351700231383653, 0.7237308297251812, 0.3691778381831775]

Automatic Differentiation

Julia has quite a few (maybe too many) AD tools. For the purpose of this tutorial, we will use:

ForwardDiff.jl – For Jacobian-Vector Product (JVP)
Zygote.jl – For Vector-Jacobian Product (VJP)

Slight Detour: We have had several questions regarding if we will be considering any other AD system for the reverse-diff backend. For now we will stick to Zygote.jl, however once we have tested Lux extensively with Enzyme.jl, we will make the switch.

Even though, theoretically, a VJP (Vector-Jacobian product - reverse autodiff) and a JVP (Jacobian-Vector product - forward-mode autodiff) are similar—they compute a product of a Jacobian and a vector—they differ by the computational complexity of the operation. In short, when you have a large number of parameters (hence a wide matrix), a JVP is less efficient computationally than a VJP, and, conversely, a JVP is more efficient when the Jacobian matrix is a tall matrix.

julia

using ComponentArrays, ForwardDiff, Zygote

Precompiling ComponentArrays...
    835.1 ms  ✓ ComponentArrays
  1 dependency successfully precompiled in 1 seconds. 23 already precompiled.
Precompiling MLDataDevicesComponentArraysExt...
    460.0 ms  ✓ MLDataDevices → MLDataDevicesComponentArraysExt
  1 dependency successfully precompiled in 0 seconds. 26 already precompiled.
Precompiling LuxComponentArraysExt...
    464.4 ms  ✓ ComponentArrays → ComponentArraysOptimisersExt
   1318.6 ms  ✓ Lux → LuxComponentArraysExt
   1903.8 ms  ✓ ComponentArrays → ComponentArraysKernelAbstractionsExt
  3 dependencies successfully precompiled in 2 seconds. 107 already precompiled.
Precompiling Zygote...
    349.7 ms  ✓ DataValueInterfaces
    343.9 ms  ✓ IteratorInterfaceExtensions
    397.0 ms  ✓ DataAPI
    403.3 ms  ✓ RealDot
    584.6 ms  ✓ OrderedCollections
    587.8 ms  ✓ SuiteSparse_jll
    650.5 ms  ✓ AbstractFFTs
    686.4 ms  ✓ Serialization
    353.5 ms  ✓ TableTraits
   1059.7 ms  ✓ FillArrays
    421.5 ms  ✓ AbstractFFTs → AbstractFFTsChainRulesCoreExt
   1138.8 ms  ✓ ZygoteRules
    394.9 ms  ✓ FillArrays → FillArraysStatisticsExt
    793.1 ms  ✓ Tables
   1991.8 ms  ✓ IRTools
    710.4 ms  ✓ StructArrays
   1712.8 ms  ✓ Distributed
    399.7 ms  ✓ StructArrays → StructArraysAdaptExt
    400.3 ms  ✓ StructArrays → StructArraysLinearAlgebraExt
   3696.5 ms  ✓ SparseArrays
    614.9 ms  ✓ SuiteSparse
    628.7 ms  ✓ Adapt → AdaptSparseArraysExt
    638.3 ms  ✓ Statistics → SparseArraysExt
    653.1 ms  ✓ StructArrays → StructArraysSparseArraysExt
    656.0 ms  ✓ ChainRulesCore → ChainRulesCoreSparseArraysExt
    674.7 ms  ✓ FillArrays → FillArraysSparseArraysExt
    581.4 ms  ✓ SparseInverseSubset
   5204.9 ms  ✓ ChainRules
  25063.7 ms  ✓ Zygote
  29 dependencies successfully precompiled in 36 seconds. 36 already precompiled.
Precompiling ArrayInterfaceSparseArraysExt...
    583.5 ms  ✓ ArrayInterface → ArrayInterfaceSparseArraysExt
  1 dependency successfully precompiled in 1 seconds. 8 already precompiled.
Precompiling SparseArraysExt...
    899.2 ms  ✓ KernelAbstractions → SparseArraysExt
  1 dependency successfully precompiled in 1 seconds. 27 already precompiled.
Precompiling MLDataDevicesSparseArraysExt...
    615.4 ms  ✓ MLDataDevices → MLDataDevicesSparseArraysExt
  1 dependency successfully precompiled in 1 seconds. 18 already precompiled.
Precompiling StructArraysStaticArraysExt...
    635.7 ms  ✓ StructArrays → StructArraysStaticArraysExt
  1 dependency successfully precompiled in 1 seconds. 19 already precompiled.
Precompiling StructArraysGPUArraysCoreExt...
    664.6 ms  ✓ StructArrays → StructArraysGPUArraysCoreExt
  1 dependency successfully precompiled in 1 seconds. 34 already precompiled.
Precompiling ArrayInterfaceChainRulesExt...
    720.2 ms  ✓ ArrayInterface → ArrayInterfaceChainRulesExt
  1 dependency successfully precompiled in 1 seconds. 40 already precompiled.
Precompiling MLDataDevicesChainRulesExt...
    776.9 ms  ✓ MLDataDevices → MLDataDevicesChainRulesExt
  1 dependency successfully precompiled in 1 seconds. 41 already precompiled.
Precompiling MLDataDevicesFillArraysExt...
    410.7 ms  ✓ MLDataDevices → MLDataDevicesFillArraysExt
  1 dependency successfully precompiled in 0 seconds. 15 already precompiled.
Precompiling MLDataDevicesZygoteExt...
   1470.0 ms  ✓ MLDataDevices → MLDataDevicesZygoteExt
  1 dependency successfully precompiled in 2 seconds. 71 already precompiled.
Precompiling LuxZygoteExt...
   2639.9 ms  ✓ Lux → LuxZygoteExt
  1 dependency successfully precompiled in 3 seconds. 143 already precompiled.
Precompiling ComponentArraysZygoteExt...
   1551.2 ms  ✓ ComponentArrays → ComponentArraysZygoteExt
  1 dependency successfully precompiled in 2 seconds. 77 already precompiled.

Gradients

For our first example, consider a simple function computing $f (x) = \frac{1}{2} x^{T} x$ , where $\nabla f (x) = x$

julia

f(x) = x' * x / 2
∇f(x) = x  # `∇` can be typed as `\nabla<TAB>`
v = randn(rng, Float32, 4)

4-element Vector{Float32}:
 -0.4051151
 -0.4593922
  0.92155594
  1.1871622

Let's use ForwardDiff and Zygote to compute the gradients.

julia

println("Actual Gradient: ", ∇f(v))
println("Computed Gradient via Reverse Mode AD (Zygote): ", only(Zygote.gradient(f, v)))
println("Computed Gradient via Forward Mode AD (ForwardDiff): ", ForwardDiff.gradient(f, v))

Actual Gradient: Float32[-0.4051151, -0.4593922, 0.92155594, 1.1871622]
Computed Gradient via Reverse Mode AD (Zygote): Float32[-0.4051151, -0.4593922, 0.92155594, 1.1871622]
Computed Gradient via Forward Mode AD (ForwardDiff): Float32[-0.4051151, -0.4593922, 0.92155594, 1.1871622]

Note that AD.gradient will only work for scalar valued outputs.

Jacobian-Vector Product

I will defer the discussion on forward-mode AD to https://book.sciml.ai/notes/08-Forward-Mode_Automatic_Differentiation_(AD)_via_High_Dimensional_Algebras/. Here let us just look at a mini example on how to use it.

julia

f(x) = x .* x ./ 2
x = randn(rng, Float32, 5)
v = ones(Float32, 5)

5-element Vector{Float32}:
 1.0
 1.0
 1.0
 1.0
 1.0

Using DifferentiationInterface

While DifferentiationInterface provides these functions for a wider range of backends, we currently don't recommend using them with Lux models, since the functions presented here come with additional goodies like fast second-order derivatives.

Compute the jvp. AutoForwardDiff specifies that we want to use ForwardDiff.jl for the Jacobian-Vector Product

julia

jvp = jacobian_vector_product(f, AutoForwardDiff(), x, v)
println("JVP: ", jvp)

JVP: Float32[-0.877497, 1.1953009, -0.057005208, 0.25055695, 0.09351656]

Vector-Jacobian Product

Using the same function and inputs, let us compute the VJP.

julia

vjp = vector_jacobian_product(f, AutoZygote(), x, v)
println("VJP: ", vjp)

VJP: Float32[-0.877497, 1.1953009, -0.057005208, 0.25055695, 0.09351656]

Linear Regression

Finally, now let us consider a linear regression problem. From a set of data-points ${(x_{i}, y_{i}), i \in {1, \dots, k}, x_{i} \in R^{n}, y_{i} \in R^{m}}$ , we try to find a set of parameters $W$ and $b$ , s.t. $f_{W, b} (x) = W x + b$ , which minimizes the mean squared error:

L (W, b) ⟶ \sum_{i = 1}^{k} \frac{1}{2} ∥ y_{i} - f_{W, b} (x_{i}) ∥_{2}^{2}

We can write f from scratch, but to demonstrate Lux, let us use the Dense layer.

julia

model = Dense(10 => 5)

rng = Random.default_rng()
Random.seed!(rng, 0)

Random.TaskLocalRNG()

Let us initialize the parameters and states (in this case it is empty) for the model.

julia

ps, st = Lux.setup(rng, model)
ps = ComponentArray(ps)

ComponentVector{Float32}(weight = Float32[-0.48351598 0.29944375 0.44048917 0.5221656 0.20001543 0.1437841 4.8317274f-6 0.5310851 -0.30674052 0.034259234; -0.04903387 -0.4242767 0.27051234 0.40789893 -0.43846482 -0.17706361 -0.03258145 0.46514034 0.1958431 0.23992883; 0.45016125 0.48263642 -0.2990853 -0.18695377 -0.11023762 -0.4418456 0.40354207 0.25278285 0.18056087 -0.3523193; 0.05218964 -0.09701932 0.27035674 0.12589 -0.29561827 0.34717593 -0.42189494 -0.13073668 0.36829436 -0.3097294; 0.20277858 -0.51524514 -0.22635892 0.18841726 0.29828635 0.21690917 -0.04265762 -0.41919118 0.071482725 -0.45247704], bias = Float32[-0.04199602, -0.093925126, -0.0007736237, -0.19397983, 0.0066712513])

Set problem dimensions.

julia

n_samples = 20
x_dim = 10
y_dim = 5

Generate random ground truth W and b.

julia

W = randn(rng, Float32, y_dim, x_dim)
b = randn(rng, Float32, y_dim)

5-element Vector{Float32}:
 -0.9436797
  1.5164032
  0.011937321
  1.4339262
 -0.2771789

Generate samples with additional noise.

julia

x_samples = randn(rng, Float32, x_dim, n_samples)
y_samples = W * x_samples .+ b .+ 0.01f0 .* randn(rng, Float32, y_dim, n_samples)
println("x shape: ", size(x_samples), "; y shape: ", size(y_samples))

x shape: (10, 20); y shape: (5, 20)

For updating our parameters let's use Optimisers.jl. We will use Stochastic Gradient Descent (SGD) with a learning rate of 0.01.

julia

using Optimisers, Printf

Define the loss function

julia

lossfn = MSELoss()

println("Loss Value with ground true parameters: ", lossfn(W * x_samples .+ b, y_samples))

Loss Value with ground true parameters: 9.3742405e-5

We will train the model using our training API.

julia

function train_model!(model, ps, st, opt, nepochs::Int)
    tstate = Training.TrainState(model, ps, st, opt)
    for i in 1:nepochs
        grads, loss, _, tstate = Training.single_train_step!(
            AutoZygote(), lossfn, (x_samples, y_samples), tstate
        )
        if i % 1000 == 1 || i == nepochs
            @printf "Loss Value after %6d iterations: %.8f\n" i loss
        end
    end
    return tstate.model, tstate.parameters, tstate.states
end

model, ps, st = train_model!(model, ps, st, Descent(0.01f0), 10000)

println("Loss Value after training: ", lossfn(first(model(x_samples, ps, st)), y_samples))

Loss Value after      1 iterations: 7.80465555
Loss Value after   1001 iterations: 0.12477568
Loss Value after   2001 iterations: 0.02535537
Loss Value after   3001 iterations: 0.00914141
Loss Value after   4001 iterations: 0.00407581
Loss Value after   5001 iterations: 0.00198415
Loss Value after   6001 iterations: 0.00101147
Loss Value after   7001 iterations: 0.00053332
Loss Value after   8001 iterations: 0.00029203
Loss Value after   9001 iterations: 0.00016878
Loss Value after  10000 iterations: 0.00010551
Loss Value after training: 0.00010546855

Appendix

julia

using InteractiveUtils
InteractiveUtils.versioninfo()

if @isdefined(MLDataDevices)
    if @isdefined(CUDA) && MLDataDevices.functional(CUDADevice)
        println()
        CUDA.versioninfo()
    end

    if @isdefined(AMDGPU) && MLDataDevices.functional(AMDGPUDevice)
        println()
        AMDGPU.versioninfo()
    end
end

Julia Version 1.11.5
Commit 760b2e5b739 (2025-04-14 06:53 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 16 default, 0 interactive, 8 GC (on 16 virtual cores)
Environment:
  JULIA_CPU_THREADS = 16
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/01872db4-8c79-43af-ab7d-12abac4f24f6
  JULIA_PKG_SERVER = 
  JULIA_NUM_THREADS = 16
  JULIA_CUDA_HARD_MEMORY_LIMIT = 100%
  JULIA_PKG_PRECOMPILE_AUTO = 0
  JULIA_DEBUG = Literate

This page was generated using Literate.jl.

Trusted by

Julia & Lux for the Uninitiated ​

Arrays ​

CUDA Arrays ​

(Im)mutability ​

Managing Randomness ​

Automatic Differentiation ​

Gradients ​

Jacobian-Vector Product ​

Vector-Jacobian Product ​

Linear Regression ​

Appendix ​

Julia & Lux for the Uninitiated

Arrays

CUDA Arrays

(Im)mutability

Managing Randomness

Automatic Differentiation

Gradients

Jacobian-Vector Product

Vector-Jacobian Product

Linear Regression

Appendix