# 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 = 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:

using Flux.Optimise: update!

η = 0.1 # Learning Rate
for p in (W, b)
update!(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.

opt = Descent(0.1) # Gradient descent with learning rate 0.1

for p in (W, b)
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!(x, x̄)

Update the array x according to x .-= x̄.

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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.

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Flux.Optimise.DescentType
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 = params(model)

gs = gradient(ps) do
loss(x, y)
end

Flux.Optimise.update!(opt, ps, gs)
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Flux.Optimise.MomentumType
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 dampening oscillations.

Examples

opt = Momentum()

opt = Momentum(0.01, 0.99)
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Flux.Optimise.NesterovType
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 dampening oscillations.

Examples

opt = Nesterov()

opt = Nesterov(0.003, 0.95)
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Flux.Optimise.RMSPropType
RMSProp(η = 0.001, ρ = 0.9)

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 dampening oscillations.

Examples

opt = RMSProp()

opt = RMSProp(0.002, 0.95)
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Flux.Optimise.ADAMType
ADAM(η = 0.001, β::Tuple = (0.9, 0.999))

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))
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Flux.Optimise.RADAMType
RADAM(η = 0.001, β::Tuple = (0.9, 0.999))

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))
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Flux.Optimise.AdaMaxType
AdaMax(η = 0.001, β::Tuple = (0.9, 0.999))

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))
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Flux.Optimise.ADAGradType
ADAGrad(η = 0.1)

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)
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Flux.Optimise.ADADeltaType
ADADelta(ρ = 0.9)

Parameters

• Rho (ρ): Factor by which the gradient is decayed at each time step.

Examples

opt = ADADelta()

opt = ADADelta(0.89)
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Flux.Optimise.AMSGradType
AMSGrad(η = 0.001, β::Tuple = (0.9, 0.999))

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))
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Flux.Optimise.NADAMType
NADAM(η = 0.001, β::Tuple = (0.9, 0.999))

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))
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Flux.Optimise.ADAMWFunction
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)
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## 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 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 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(0.001, 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.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

In this manner it is possible to compose optimisers for some added flexibility.

## Decays

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

Flux.Optimise.ExpDecayType
ExpDecay(η = 0.001, decay = 0.1, decay_step = 1000, clip = 1e-4)

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.

Examples

To apply exponential decay to an optimiser:

Optimiser(ExpDecay(..), Opt(..))

opt = Optimiser(ExpDecay(), ADAM())
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Flux.Optimise.InvDecayType
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.

Examples

Optimiser(InvDecay(..), Opt(..))
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