# Stochastic Gradient Descent

Manopt.stochastic_gradient_descentFunction
stochastic_gradient_descent(M, gradF, x)

perform a stochastic gradient descent

Input

• M a manifold $\mathcal M$
• gradF – a gradient function, that either returns a vector of the subgradients or is a vector of gradients
• x – an initial value $x ∈ \mathcal M$

Optional

• cost – (missing) you can provide a cost function for example to track the function value
• evaluation – (AllocatingEvaluation) specify whether the gradient(s) works by allocation (default) form gradF(M, x) or MutatingEvaluation in place, i.e. is of the form gradF!(M, X, x) (elementwise).
• evaluation_order – (:Random) – whether to use a randomly permuted sequence (:FixedRandom), a per cycle permuted sequence (:Linear) or the default :Random one.
• stopping_criterion (StopAfterIteration(1000))– a StoppingCriterion
• stepsize (ConstantStepsize(1.0)) a Stepsize
• order_type (:RandomOder) a type of ordering of gradient evaluations. values are :RandomOrder, a :FixedPermutation, :LinearOrder
• order - ([1:n]) the initial permutation, where n is the number of gradients in gradF.
• retraction_method – (default_retraction_method(M)) a retraction(M,x,ξ) to use.

Output

• x_opt – the resulting (approximately critical) point of gradientDescent

OR

• options - the options returned by the solver (see return_options)
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Manopt.stochastic_gradient_descent!Function
stochastic_gradient_descent!(M, gradF, x)

perform a stochastic gradient descent in place of x.

Input

• M a manifold $\mathcal M$
• gradF – a gradient function, that either returns a vector of the subgradients or is a vector of gradients
• x – an initial value $x ∈ \mathcal M$

for all optional parameters, see stochastic_gradient_descent.

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## Options

Manopt.StochasticGradientDescentOptionsType
StochasticGradientDescentOptions <: AbstractStochasticGradientDescentOptions

Store the following fields for a default stochastic gradient descent algorithm, see also StochasticGradientProblem and stochastic_gradient_descent.

Fields

• x the current iterate
• stopping_criterion (StopAfterIteration(1000))– a StoppingCriterion
• stepsize (ConstantStepsize(1.0)) a Stepsize
• evaluation_order – (:Random) – whether to use a randomly permuted sequence (:FixedRandom), a per cycle permuted sequence (:Linear) or the default :Random one.
• order the current permutation
• retraction_method – (ExponentialRetraction()) a retraction(M,x,ξ) to use.

Constructor

StochasticGradientDescentOptions(x)

Create a StochasticGradientDescentOptions with start point x. all other fields are optional keyword arguments.

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Additionally, the options share a DirectionUpdateRule, so you can also apply MomentumGradient and AverageGradient here. The most inner one should always be.