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)
  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!(x, x̄)

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

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.

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 = 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 dampening 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 dampening oscillations.

Examples

opt = Nesterov()

opt = Nesterov(0.003, 0.95)

Flux.Optimise.RMSProp — Type

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)

Flux.Optimise.ADAM — Type

ADAM(η = 0.001, β::Tuple = (0.9, 0.999))

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))

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))

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)

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)

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))

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))

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))

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))

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 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(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.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

In this manner 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 initalize a cosine annealing schedule to varying 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)

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.

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())

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(WeigthDecay(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.