# Optimisation Rules

Flux builds in many optimisation rules for use with train! and other training functions.

The mechanism by which these work is gradually being replaced as part of the change from "implicit" dictionary-based to "explicit" tree-like structures. At present, the same struct (such as Adam) can be used with either form, and will be automatically translated.

For full details of how the new interface works, see the Optimisers.jl documentation.

For full details on how the old "implicit" interface worked, see the Flux 0.13.6 manual.

## Optimiser Reference

All optimisers return an object that, when passed to train!, will update the parameters passed to it.

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 = Flux.params(model)

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 damping 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 damping oscillations.

Examples

opt = Nesterov()

opt = Nesterov(0.003, 0.95)
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Flux.Optimise.RMSPropType
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)
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Flux.Optimise.AdamType
Adam(η = 0.001, β::Tuple = (0.9, 0.999), ϵ = 1.0e-8)

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), ϵ = 1.0e-8)

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), ϵ = 1.0e-8)

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, 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)
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Flux.Optimise.AdaDeltaType
AdaDelta(ρ = 0.9, ϵ = 1.0e-8)

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), ϵ = 1.0e-8)

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), ϵ = 1.0e-8)

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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Flux.Optimise.OAdamType
OAdam(η = 0.0001, β::Tuple = (0.5, 0.9), ϵ = 1.0e-8)

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))
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Flux.Optimise.AdaBeliefType
AdaBelief(η = 0.001, β::Tuple = (0.9, 0.999), ϵ = 1.0e-8)

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))
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## 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.OptimiserType
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.

Note

This will be replaced by Optimisers.OptimiserChain in Flux 0.14.

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## 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
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)
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.ExpDecayType
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.

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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.

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))
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Flux.Optimise.WeightDecayType
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())
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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.ClipValueType
ClipValue(thresh)

Clip gradients when their absolute value exceeds thresh.

Note

This will be replaced by Optimisers.ClipGrad in Flux 0.14.

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