# Stochastic Gradient Descent

This tutorial illustrates how to use the `stochastic_gradient_descent`

solver and different `DirectionUpdateRule`

s in order to introduce the average or momentum variant, see Stochastic Gradient Descent.

Computationally we look at a very simple but large scale problem, the Riemannian Center of Mass or Fréchet mean: For given points $p_i \in \mathcal M$, $i=1,\ldots,N$ this optimization problem reads

which of course can be (and is) solved by a gradient descent, see the introductionary tutorial. If $N$ is very large it might be quite expensive to evaluate the complete gradient. A remedy is, to evaluate only one of the terms at a time and choose a random order for these.

We first initialize the manifold (see [])

`using Manopt, Manifolds, Random, Colors`

and we define some colors from Paul Tol

```
black = RGBA{Float64}(colorant"#000000")
TolVibrantOrange = RGBA{Float64}(colorant"#EE7733") # Start
TolVibrantBlue = RGBA{Float64}(colorant"#0077BB") # a path
TolVibrantTeal = RGBA{Float64}(colorant"#009988") # points
```

And optain a large data set

```
n = 5000
σ = π / 12
M = Sphere(2)
x = 1 / sqrt(2) * [1.0, 0.0, 1.0]
Random.seed!(42)
data = [exp(M, x, random_tangent(M, x, Val(:Gaussian), σ)) for i in 1:n]
```

which looks like

```
asymptote_export_S2_signals("centerAndLargeData.asy";
points = [ [x], data],
colors=Dict(:points => [TolVibrantBlue, TolVibrantTeal]),
dot_sizes = [2.5, 1.0], camera_position = (1.,.5,.5)
)
render_asymptote("centerAndLargeData.asy"; render = 2)
```

Note that due to the construction of the points as zero mean tangent vectors, the mean should be very close to our initial point `x`

.

In order to use the stochastic gradient, we now need a function that returns the vector of gradients. There are two ways to define it in `Manopt.jl`

: as one function, that returns a vector or a vector of funtions.

The first variant is of course easier to define, but the second is more efficient when only evaluating one of the gradients. For the mean we have as a gradient

Which we define as

```
F(x) = 1 / (2 * n) * sum(map(p -> distance(M, x, p)^2, data))
∇F(x) = [∇distance(M, p, x) for p in data]
∇f = [x -> ∇distance(M, p, x) for p in data];
```

The calls are only slightly different, but notice that accessing the 2nd gradient element requires evaluating all logs in the first function. So while you can use both `∇F`

and `∇f`

in the following call, the second one is faster:

`@time x_opt1 = stochastic_gradient_descent(M, ∇F, x);`

` 18.525834 seconds (100.86 M allocations: 10.848 GiB, 15.17% gc time)`

versus

`@time x_opt2 = stochastic_gradient_descent(M, ∇f, x);`

` 0.130007 seconds (262.47 k allocations: 14.572 MiB)`

This result is reasonably close. But we can improve it by using a `DirectionUpdateRule`

, namely: On the one hand `MomentumGradient`

, which requires both the manifold and the initial value, in order to keep track of the iterate and parallel transport the last direction to the current iterate. you can also set a `vector_transport_method`

, if `ParallelTransport()`

is not available on your manifold. Here we simply do

```
@time x_opt3 = stochastic_gradient_descent(
M, ∇f, x; direction=MomentumGradient(M, x, StochasticGradient())
);
```

` 0.224690 seconds (455.08 k allocations: 30.037 MiB, 6.48% gc time)`

And on the other hand the `AverageGradient`

computes an average of the last `n`

gradients, i.e.

```
@time x_opt4 = stochastic_gradient_descent(
M, ∇f, x; direction=AverageGradient(M, x, 10, StochasticGradient())
);
```

` 0.400347 seconds (1.00 M allocations: 64.311 MiB, 4.06% gc time)`

note that the default `StoppingCriterion`

is a fixed number of iterations.

Note that since you can apply both also in the `Gradient`

case of `gradient_descent`

, both constructors have to know that internally the default avaluation of the Stochastic gradient (choosing one gradient $∇f_k$ at random) has to be specified.

For this small example you can of course also use a gradient descent with `ArmijoLinesearch`

, but it will be a little slower usually

`@time x_opt5 = gradient_descent(M, F, x -> sum(∇F(x)), x; stepsize=ArmijoLinesearch());`

` 3.226290 seconds (26.48 M allocations: 1.571 GiB, 13.54% gc time)`

but it is for sure faster than the variant above that evaluates the full gradient on every iteration, since stochastic gradient descent takes more iterations.

Note that all 5 of couse yield the same result

`[distance(M, x, y) for y in [x_opt1, x_opt2, x_opt3, x_opt4, x_opt5]]`

```
5-element Array{Float64,1}:
0.0
0.0
0.0
0.0
0.0
```