# Optimisers

Consider a simple linear regression. We create some dummy data, calculate a loss, and backpropagate to calculate gradients for the parameters `W`

and `b`

.

```
using Flux
W = rand(2, 5)
b = rand(2)
predict(x) = (W * x) .+ b
loss(x, y) = sum((predict(x) .- y).^2)
x, y = rand(5), rand(2) # Dummy data
l = loss(x, y) # ~ 3
θ = Flux.params(W, b)
grads = gradient(() -> loss(x, y), θ)
```

We want to update each parameter, using the gradient, in order to improve (reduce) the loss. Here's one way to do that:

```
η = 0.1 # Learning Rate
for p in (W, b)
p .-= η * grads[p]
end
```

Running this will alter the parameters `W`

and `b`

and our loss should go down. Flux provides a more general way to do optimiser updates like this.

```
using Flux: update!
opt = Descent(0.1) # Gradient descent with learning rate 0.1
for p in (W, b)
update!(opt, p, grads[p])
end
```

An optimiser `update!`

accepts a parameter and a gradient, and updates the parameter according to the chosen rule. We can also pass `opt`

to our training loop, which will update all parameters of the model in a loop. However, we can now easily replace `Descent`

with a more advanced optimiser such as `Adam`

.

## Optimiser Reference

All optimisers return an object that, when passed to `train!`

, will update the parameters passed to it.

`Flux.Optimise.update!`

— Function```
update!(opt, p, g)
update!(opt, ps::Params, gs)
```

Perform an update step of the parameters `ps`

(or the single parameter `p`

) according to optimizer `opt`

and the gradients `gs`

(the gradient `g`

).

As a result, the parameters are mutated and the optimizer's internal state may change. The gradient could be mutated as well.

`Flux.Optimise.Descent`

— Type`Descent(η = 0.1)`

Classic gradient descent optimiser with learning rate `η`

. For each parameter `p`

and its gradient `δp`

, this runs `p -= η*δp`

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights.

**Examples**

```
opt = Descent()
opt = Descent(0.3)
ps = Flux.params(model)
gs = gradient(ps) do
loss(x, y)
end
Flux.Optimise.update!(opt, ps, gs)
```

`Flux.Optimise.Momentum`

— Type`Momentum(η = 0.01, ρ = 0.9)`

Gradient descent optimizer with learning rate `η`

and momentum `ρ`

.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Momentum (
`ρ`

): Controls the acceleration of gradient descent in the prominent direction, in effect damping oscillations.

**Examples**

```
opt = Momentum()
opt = Momentum(0.01, 0.99)
```

`Flux.Optimise.Nesterov`

— Type`Nesterov(η = 0.001, ρ = 0.9)`

Gradient descent optimizer with learning rate `η`

and Nesterov momentum `ρ`

.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Nesterov momentum (
`ρ`

): Controls the acceleration of gradient descent in the prominent direction, in effect damping oscillations.

**Examples**

```
opt = Nesterov()
opt = Nesterov(0.003, 0.95)
```

`Flux.Optimise.RMSProp`

— Type`RMSProp(η = 0.001, ρ = 0.9, ϵ = 1.0e-8)`

Optimizer using the RMSProp algorithm. Often a good choice for recurrent networks. Parameters other than learning rate generally don't need tuning.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Momentum (
`ρ`

): Controls the acceleration of gradient descent in the prominent direction, in effect damping oscillations.

**Examples**

```
opt = RMSProp()
opt = RMSProp(0.002, 0.95)
```

`Flux.Optimise.Adam`

— Type`Adam(η = 0.001, β::Tuple = (0.9, 0.999), ϵ = 1.0e-8)`

Adam optimiser.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Decay of momentums (
`β::Tuple`

): Exponential decay for the first (β1) and the second (β2) momentum estimate.

**Examples**

```
opt = Adam()
opt = Adam(0.001, (0.9, 0.8))
```

`Flux.Optimise.RAdam`

— Type`RAdam(η = 0.001, β::Tuple = (0.9, 0.999), ϵ = 1.0e-8)`

Rectified Adam optimizer.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Decay of momentums (
`β::Tuple`

): Exponential decay for the first (β1) and the second (β2) momentum estimate.

**Examples**

```
opt = RAdam()
opt = RAdam(0.001, (0.9, 0.8))
```

`Flux.Optimise.AdaMax`

— Type`AdaMax(η = 0.001, β::Tuple = (0.9, 0.999), ϵ = 1.0e-8)`

AdaMax is a variant of Adam based on the ∞-norm.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Decay of momentums (
`β::Tuple`

): Exponential decay for the first (β1) and the second (β2) momentum estimate.

**Examples**

```
opt = AdaMax()
opt = AdaMax(0.001, (0.9, 0.995))
```

`Flux.Optimise.AdaGrad`

— Type`AdaGrad(η = 0.1, ϵ = 1.0e-8)`

AdaGrad optimizer. It has parameter specific learning rates based on how frequently it is updated. Parameters don't need tuning.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights.

**Examples**

```
opt = AdaGrad()
opt = AdaGrad(0.001)
```

`Flux.Optimise.AdaDelta`

— Type`AdaDelta(ρ = 0.9, ϵ = 1.0e-8)`

AdaDelta is a version of AdaGrad adapting its learning rate based on a window of past gradient updates. Parameters don't need tuning.

**Parameters**

- Rho (
`ρ`

): Factor by which the gradient is decayed at each time step.

**Examples**

```
opt = AdaDelta()
opt = AdaDelta(0.89)
```

`Flux.Optimise.AMSGrad`

— Type`AMSGrad(η = 0.001, β::Tuple = (0.9, 0.999), ϵ = 1.0e-8)`

The AMSGrad version of the Adam optimiser. Parameters don't need tuning.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Decay of momentums (
`β::Tuple`

): Exponential decay for the first (β1) and the second (β2) momentum estimate.

**Examples**

```
opt = AMSGrad()
opt = AMSGrad(0.001, (0.89, 0.995))
```

`Flux.Optimise.NAdam`

— Type`NAdam(η = 0.001, β::Tuple = (0.9, 0.999), ϵ = 1.0e-8)`

NAdam is a Nesterov variant of Adam. Parameters don't need tuning.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Decay of momentums (
`β::Tuple`

): Exponential decay for the first (β1) and the second (β2) momentum estimate.

**Examples**

```
opt = NAdam()
opt = NAdam(0.002, (0.89, 0.995))
```

`Flux.Optimise.AdamW`

— Function`AdamW(η = 0.001, β::Tuple = (0.9, 0.999), decay = 0)`

AdamW is a variant of Adam fixing (as in repairing) its weight decay regularization.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Decay of momentums (
`β::Tuple`

): Exponential decay for the first (β1) and the second (β2) momentum estimate. `decay`

: Decay applied to weights during optimisation.

**Examples**

```
opt = AdamW()
opt = AdamW(0.001, (0.89, 0.995), 0.1)
```

`Flux.Optimise.OAdam`

— Type`OAdam(η = 0.0001, β::Tuple = (0.5, 0.9), ϵ = 1.0e-8)`

OAdam (Optimistic Adam) is a variant of Adam adding an "optimistic" term suitable for adversarial training.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Decay of momentums (
`β::Tuple`

): Exponential decay for the first (β1) and the second (β2) momentum estimate.

**Examples**

```
opt = OAdam()
opt = OAdam(0.001, (0.9, 0.995))
```

`Flux.Optimise.AdaBelief`

— Type`AdaBelief(η = 0.001, β::Tuple = (0.9, 0.999), ϵ = 1.0e-8)`

The AdaBelief optimiser is a variant of the well-known Adam optimiser.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. - Decay of momentums (
`β::Tuple`

): Exponential decay for the first (β1) and the second (β2) momentum estimate.

**Examples**

```
opt = AdaBelief()
opt = AdaBelief(0.001, (0.9, 0.8))
```

## Optimiser Interface

Flux's optimisers are built around a `struct`

that holds all the optimiser parameters along with a definition of how to apply the update rule associated with it. We do this via the `apply!`

function which takes the optimiser as the first argument followed by the parameter and its corresponding gradient.

In this manner Flux also allows one to create custom optimisers to be used seamlessly. Let's work on this with a simple example.

```
mutable struct Momentum
eta
rho
velocity
end
Momentum(eta::Real, rho::Real) = Momentum(eta, rho, IdDict())
```

The `Momentum`

type will act as our optimiser in this case. Notice that we have added all the parameters as fields, along with the velocity which we will use as our state dictionary. Each parameter in our models will get an entry in there. We can now define the rule applied when this optimiser is invoked.

```
function Flux.Optimise.apply!(o::Momentum, x, Δ)
η, ρ = o.eta, o.rho
v = get!(o.velocity, x, zero(x))::typeof(x)
@. v = ρ * v - η * Δ
@. Δ = -v
end
```

This is the basic definition of a Momentum update rule given by:

\[v = ρ * v - η * Δ w = w - v\]

The `apply!`

defines the update rules for an optimiser `opt`

, given the parameters and gradients. It returns the updated gradients. Here, every parameter `x`

is retrieved from the running state `v`

and subsequently updates the state of the optimiser.

Flux internally calls on this function via the `update!`

function. It shares the API with `apply!`

but ensures that multiple parameters are handled gracefully.

## Composing Optimisers

Flux defines a special kind of optimiser simply called `Optimiser`

which takes in arbitrary optimisers as input. Its behaviour is similar to the usual optimisers, but differs in that it acts by calling the optimisers listed in it sequentially. Each optimiser produces a modified gradient that will be fed into the next, and the resultant update will be applied to the parameter as usual. A classic use case is where adding decays is desirable. Flux defines some basic decays including `ExpDecay`

, `InvDecay`

etc.

`opt = Optimiser(ExpDecay(1, 0.1, 1000, 1e-4), Descent())`

Here we apply exponential decay to the `Descent`

optimiser. The defaults of `ExpDecay`

say that its learning rate will be decayed every 1000 steps. It is then applied like any optimiser.

```
w = randn(10, 10)
w1 = randn(10,10)
ps = Params([w, w1])
loss(x) = Flux.Losses.mse(w * x, w1 * x)
loss(rand(10)) # around 9
for t = 1:10^5
θ = Params([w, w1])
θ̄ = gradient(() -> loss(rand(10)), θ)
Flux.Optimise.update!(opt, θ, θ̄)
end
loss(rand(10)) # around 0.9
```

It is possible to compose optimisers for some added flexibility.

`Flux.Optimise.Optimiser`

— Type`Optimiser(a, b, c...)`

Combine several optimisers into one; each optimiser produces a modified gradient that will be fed into the next, and this is finally applied to the parameter as usual.

## Scheduling Optimisers

In practice, it is fairly common to schedule the learning rate of an optimiser to obtain faster convergence. There are a variety of popular scheduling policies, and you can find implementations of them in ParameterSchedulers.jl. The documentation for ParameterSchedulers.jl provides a more detailed overview of the different scheduling policies, and how to use them with Flux optimizers. Below, we provide a brief snippet illustrating a cosine annealing schedule with a momentum optimiser.

First, we import ParameterSchedulers.jl and initialize a cosine annealing schedule to vary the learning rate between `1e-4`

and `1e-2`

every 10 steps. We also create a new `Momentum`

optimiser.

```
using ParameterSchedulers
opt = Momentum()
schedule = Cos(λ0 = 1e-4, λ1 = 1e-2, period = 10)
for (eta, epoch) in zip(schedule, 1:100)
opt.eta = eta
# your training code here
end
```

`schedule`

can also be indexed (e.g. `schedule(100)`

) or iterated like any iterator in Julia.

ParameterSchedulers.jl schedules are stateless (they don't store their iteration state). If you want a *stateful* schedule, you can use `ParameterSchedulers.Stateful`

:

```
using ParameterSchedulers: Stateful, next!
schedule = Stateful(Cos(λ0 = 1e-4, λ1 = 1e-2, period = 10))
for epoch in 1:100
opt.eta = next!(schedule)
# your training code here
end
```

ParameterSchedulers.jl allows for many more scheduling policies including arbitrary functions, looping any function with a given period, or sequences of many schedules. See the ParameterSchedulers.jl documentation for more info.

## Decays

Similar to optimisers, Flux also defines some simple decays that can be used in conjunction with other optimisers, or standalone.

`Flux.Optimise.ExpDecay`

— Type`ExpDecay(η = 0.001, decay = 0.1, decay_step = 1000, clip = 1e-4, start = 1)`

Discount the learning rate `η`

by the factor `decay`

every `decay_step`

steps till a minimum of `clip`

.

**Parameters**

- Learning rate (
`η`

): Amount by which gradients are discounted before updating the weights. `decay`

: Factor by which the learning rate is discounted.`decay_step`

: Schedule decay operations by setting the number of steps between two decay operations.`clip`

: Minimum value of learning rate.- 'start': Step at which the decay starts.

See also the Scheduling Optimisers section of the docs for more general scheduling techniques.

**Examples**

`ExpDecay`

is typically composed with other optimizers as the last transformation of the gradient:

`opt = Optimiser(Adam(), ExpDecay(1.0))`

Note: you may want to start with `η=1`

in `ExpDecay`

when combined with other optimizers (`Adam`

in this case) that have their own learning rate.

`Flux.Optimise.InvDecay`

— Type`InvDecay(γ = 0.001)`

Apply inverse time decay to an optimiser, so that the effective step size at iteration `n`

is `eta / (1 + γ * n)`

where `eta`

is the initial step size. The wrapped optimiser's step size is not modified.

See also the Scheduling Optimisers section of the docs for more general scheduling techniques.

**Examples**

`InvDecay`

is typically composed with other optimizers as the last transformation of the gradient:

```
# Inverse decay of the learning rate
# with starting value 0.001 and decay coefficient 0.01.
opt = Optimiser(Adam(1f-3), InvDecay(1f-2))
```

`Flux.Optimise.WeightDecay`

— Type`WeightDecay(λ = 0)`

Decay weights by $λ$. Typically composed with other optimizers as the first transformation to the gradient, making it equivalent to adding $L_2$ regularization with coefficient $λ$ to the loss.

**Examples**

`opt = Optimiser(WeightDecay(1f-4), Adam())`

## Gradient Clipping

Gradient clipping is useful for training recurrent neural networks, which have a tendency to suffer from the exploding gradient problem. An example usage is

`opt = Optimiser(ClipValue(1e-3), Adam(1e-3))`

`Flux.Optimise.ClipValue`

— Type`ClipValue(thresh)`

Clip gradients when their absolute value exceeds `thresh`

.

`Flux.Optimise.ClipNorm`

— Type`ClipNorm(thresh)`

Clip gradients when their L2 norm exceeds `thresh`

.

# Optimisers.jl

Flux re-exports some utility functions from `Optimisers.jl`

and the complete `Optimisers`

package under the `Flux.Optimisers`

namespace.

`Optimisers.destructure`

— Function`destructure(model) -> vector, reconstructor`

Copies all `trainable`

, `isnumeric`

parameters in the model to a vector, and returns also a function which reverses this transformation. Differentiable.

**Example**

```
julia> v, re = destructure((x=[1.0, 2.0], y=(sin, [3.0 + 4.0im])))
(ComplexF64[1.0 + 0.0im, 2.0 + 0.0im, 3.0 + 4.0im], Restructure(NamedTuple, ..., 3))
julia> re([3, 5, 7+11im])
(x = [3.0, 5.0], y = (sin, ComplexF64[7.0 + 11.0im]))
```

If `model`

contains various number types, they are promoted to make `vector`

, and are usually restored by `Restructure`

. Such restoration follows the rules of `ChainRulesCore.ProjectTo`

, and thus will restore floating point precision, but will permit more exotic numbers like `ForwardDiff.Dual`

.

If `model`

contains only GPU arrays, then `vector`

will also live on the GPU. At present, a mixture of GPU and ordinary CPU arrays is undefined behaviour.

`Optimisers.trainable`

— Function`trainable(x::Layer) -> NamedTuple`

This should be overloaded to make optimisers ignore some fields of every `Layer`

, which would otherwise contain trainable parameters. (Elements such as functions and sizes are always ignored.)

The default is `Functors.children(x)`

, usually a NamedTuple of all fields, and `trainable(x)`

must contain a subset of these.

`Optimisers.isnumeric`

— Function`isnumeric(x) -> Bool`

Returns `true`

on any parameter to be adjusted by Optimisers.jl, namely arrays of non-integer numbers. Returns `false`

on all other types.

Requires also that `Functors.isleaf(x) == true`

, to focus on e.g. the parent of a transposed matrix, not the wrapper.