## How to improve Julia's performance using just in time compilation (JIT)

Problem Description:

I have been playing with JAX (automatic differentiation library in Python) and Zygote (the automatic differentiation library in Julia) to implement Gauss-Newton minimisation method.

I came upon the `@jit`

macro in Jax that runs my Python code in around 0.6 seconds compared to ~60 seconds for the version that does not use `@jit`

.

Julia ran the code in around 40 seconds. Is there an equivalent of `@jit`

in Julia or Zygote that results is a better performance?

Here are the codes I used:

**Python**

```
from jax import grad, jit, jacfwd
import jax.numpy as jnp
import numpy as np
import time
def gaussian(x, params):
amp = params[0]
mu = params[1]
sigma = params[2]
amplitude = amp/(jnp.abs(sigma)*jnp.sqrt(2*np.pi))
arg = ((x-mu)/sigma)
return amplitude*jnp.exp(-0.5*(arg**2))
def myjacobian(x, params):
return jacfwd(gaussian, argnums = 1)(x, params)
def op(jac):
return jnp.matmul(
jnp.linalg.inv(jnp.matmul(jnp.transpose(jac),jac)),
jnp.transpose(jac))
def res(x, data, params):
return data - gaussian(x, params)
@jit
def step(x, data, params):
residuals = res(x, data, params)
jacobian_operation = op(myjacobian(x, params))
temp = jnp.matmul(jacobian_operation, residuals)
return params + temp
N = 2000
x = np.linspace(start = -100, stop = 100, num= N)
data = gaussian(x, [5.65, 25.5, 37.23])
ini = jnp.array([0.9, 5., 5.0])
t1 = time.time()
for i in range(5000):
ini = step(x, data, ini)
t2 = time.time()
print('t2-t1: ', t2-t1)
ini
```

**Julia**

```
using Zygote
function gaussian(x::Union{Vector{Float64}, Float64}, params::Vector{Float64})
amp = params[1]
mu = params[2]
sigma = params[3]
amplitude = amp/(abs(sigma)*sqrt(2*pi))
arg = ((x.-mu)./sigma)
return amplitude.*exp.(-0.5.*(arg.^2))
end
function myjacobian(x::Vector{Float64}, params::Vector{Float64})
output = zeros(length(x), length(params))
for (index, ele) in enumerate(x)
output[index,:] = collect(gradient((params)->gaussian(ele, params), params))[1]
end
return output
end
function op(jac::Matrix{Float64})
return inv(jac'*jac)*jac'
end
function res(x::Vector{Float64}, data::Vector{Float64}, params::Vector{Float64})
return data - gaussian(x, params)
end
function step(x::Vector{Float64}, data::Vector{Float64}, params::Vector{Float64})
residuals = res(x, data, params)
jacobian_operation = op(myjacobian(x, params))
temp = jacobian_operation*residuals
return params + temp
end
N = 2000
x = collect(range(start = -100, stop = 100, length= N))
params = vec([5.65, 25.5, 37.23])
data = gaussian(x, params)
ini = vec([0.9, 5., 5.0])
@time for i in range(start = 1, step = 1, length = 5000)
ini = step(x, data, ini)
end
ini
```

## Solution – 1

Your Julia code doing a number of things that aren’t idiomatic and are worsening your performance. This won’t be a full overview, but it should give you a good idea to start.

The first thing is passing `params`

as a `Vector`

is a bad idea. This means it will have to be heap allocated, and the compiler doesn’t know how long it is. Instead, use a `Tuple`

which will allow for a lot more optimization. Secondly, don’t make `gaussian`

act on a `Vector`

of `x`

s. Instead, write the scalar version and broadcast it. Specifically, with these changes, you will have

```
function gaussian(x::Number, params::NTuple{3, Float64})
amp, mu, sigma = params
# The next 2 lines should probably be done outside this function, but I'll leave them here for now.
amplitude = amp/(abs(sigma)*sqrt(2*pi))
arg = ((x-mu)/sigma)
return amplitude*exp(-0.5*(arg^2))
end
```

## Solution – 2

One straightforward way to speed this up is to use ForwardDiff not Zygote, since you are taking a gradient of a vector of length 3, many times. Here this gets me from 16 to 3.5 seconds, with the last factor of 2 involving `Chunk(3)`

to improve type-stability. Perhaps this can be improved further.

```
function myjacobian(x::Vector, params)
# return rand(eltype(x), length(x), length(params)) # with no gradient, takes 0.5s
output = zeros(eltype(x), length(x), length(params))
config = ForwardDiff.GradientConfig(nothing, params, ForwardDiff.Chunk(3))
for (i, xi) in enumerate(x)
# grad = gradient(p->gaussian(xi, p), params)[1] # original, takes 16s
# grad = ForwardDiff.gradient(p-> gaussian(xi, p)) # ForwardDiff, takes 7s
grad = ForwardDiff.gradient(p-> gaussian(xi, p), params, config) # takes 3.5s
copyto!(view(output,i,:), grad) # this allows params::Tuple, OK for Zygote, no help
end
return output
end
# This needs gaussian.(x, Ref(params)) elsewhere to use on many x, same params
function gaussian(x::Real, params)
# amp, mu, sigma = params # with params::Vector this is slower, 19 sec
amp = params[1]
mu = params[2]
sigma = params[3] # like this, 16 sec
T = typeof(x) # avoids having (2*pi)::Float64 promote everything
amplitude = amp/(abs(sigma)*sqrt(2*T(pi)))
arg = (x-mu)/sigma
return amplitude * exp(-(arg^2)/2)
end
```

However, this is still computing many small gradient arrays in a loop. It could easily compute one big gradient array instead.

While in general Julia is happy to compile loops to something fast, loops that make individual arrays tend to be a bad idea. And this is especially true for Zygote, which is fastest on matlab-ish whole-array code.

Here’s how this looks, it gets me under 1s for the whole program:

```
function gaussian(x::Real, amp::Real, mu::Real, sigma::Real)
T = typeof(x)
amplitude = amp/(abs(sigma)*sqrt(2*T(pi)))
arg = (x-mu)/sigma
return amplitude * exp(-(arg^2)/2)
end
function myjacobian2(x::Vector, params) # with this, 0.9s
amp = fill(params[1], length(x))
mu = fill(params[2], length(x))
sigma = fill(params[3], length(x)) # use same sigma & different x value at each row:
grads = gradient((amp, mu, sigma) -> sum(gaussian.(x, amp, mu, sigma)), amp, mu, sigma)
hcat(grads...)
end
# Check that it agrees:
myjacobian2(x, params) ≈ myjacobian(x, params)
```

While this has little effect on the speed, I think you probably also want `op(jac::Matrix) = Hermitian(jac'*jac) jac'`

rather than `inv`

.