Hyperopt

A package to perform hyperparameter optimization. Currently supports random search, latin hypercube sampling and Bayesian optimization.

Usage

This package was designed to facilitate the addition of optimization logic to already existing code. I usually write some code and try a few hyper parameters by hand before I realize I have to take a more structured approach to finding good hyper parameters. I therefore designed this package such that the optimization logic is wrapped around existing code, and the user only has to specify which variables to optimize and candidate values (ranges) for these variables.

High-level example

In order to add hyper-parameter optimization to the existing pseudo code

a = manually_selected_value
b = other_value
cost = train_model(a,b)

we wrap it in @hyperopt like this

ho = @hyperopt for i = number_of_samples,
                   a = candidate_values,
                   b = other_candidate_values
cost = train_model(a,b)
end

Details

The macro @hyperopt takes a for-loop with an initial argument determining the number of samples to draw (i below).
The sample strategy can be specified by specifying the special keyword sampler = Sampler(opts...). Available options are RandomSampler(), LHSampler(), CLHSampler(dims=[Continuous(), Categorical(2), Continuous(), ...]), Hyperband(R=50, η=3, inner=RandomSampler()).
The subsequent arguments to the for-loop specifies names and candidate values for different hyper parameters (a = LinRange(1,2,1000), b = [true, false], c = exp10.(LinRange(-1,3,1000)) below).
A useful strategy to achieve log-uniform sampling is logarithmically spaced vector, e.g. c = exp10.(LinRange(-1,3,1000)).
In the example below, the parameters i,a,b,c can be used within the expression sent to the macro and they will hold a new value sampled from the corresponding candidate vector each iteration.

The resulting object ho::Hyperoptimizer holds all the sampled parameters and function values and has minimum/minimizer and maximum/maximizer properties (e.g., ho.minimizer). It can also be plotted using plot(ho) (uses Plots.jl). The exact syntax to use for various samplers is shown in the testfile, which should be fairly readable.

Full example

using Hyperopt

f(x,a,b=true;c=10) = sum(@. x + (a-3)^2 + (b ? 10 : 20) + (c-100)^2) # Function to minimize

# Main macro. The first argument to the for loop is always interpreted as the number of iterations (except for hyperband optimizer)
ho = @hyperopt for i=50,
            sampler = RandomSampler(), # This is default if none provided
            a = LinRange(1,5,1000),
            b = [true, false],
            c = exp10.(LinRange(-1,3,1000))
   print(i, "\t", a, "\t", b, "\t", c, "   \t")
   x = 100
   @show f(x,a,b,c=c)
end
1   3.910910910910911   false   0.15282140360258697     f(x, a, b, c=c) = 10090.288832348499
2   3.930930930930931   true    6.1629662551329405      f(x, a, b, c=c) = 8916.255534433481
3   2.7617617617617616  true    146.94918006248173      f(x, a, b, c=c) = 2314.282265997491
4   3.6666666666666665  false   0.3165924111983522      f(x, a, b, c=c) = 10057.226192959602
5   4.783783783783784   true    34.55719936762139       f(x, a, b, c=c) = 4395.942039196544
6   2.5895895895895897  true    4.985373463873895       f(x, a, b, c=c) = 9137.947692504491
7   1.6206206206206206  false   301.6334347259197       f(x, a, b, c=c) = 40777.94468684398
8   1.012012012012012   true    33.00034791125285       f(x, a, b, c=c) = 4602.905476253546
9   3.3583583583583585  true    193.7703337477989       f(x, a, b, c=c) = 8903.003911886599
10  4.903903903903904   true    144.26439512181574      f(x, a, b, c=c) = 2072.9615255755252
11  2.2332332332332334  false   119.97177354358843      f(x, a, b, c=c) = 519.4596697509966
12  2.369369369369369   false   117.77987011971193      f(x, a, b, c=c) = 436.52147646611473
13  3.2182182182182184  false   105.44427935261685      f(x, a, b, c=c) = 149.68779686009242
⋮

Hyperopt.Hyperoptimizer
  iterations: Int64 50
  params: Tuple{Symbol,Symbol,Symbol}
  candidates: Array{AbstractArray{T,1} where T}((3,))
  history: Array{Any}((50,))
  results: Array{Any}((50,))
  sampler: Hyperopt.RandomSampler


julia> best_params, min_f = ho.minimizer, ho.minimum
(Real[1.62062, true, 100.694], 112.38413353985818)

julia> printmin(ho)
a = 1.62062
b = true
c = 100.694

We can also visualize the result by plotting the hyperoptimizer

plot(ho)

This may allow us to determine which parameters are most important for the performance etc.

The type Hyperoptimizer is iterable, it iterates for the specified number of iterations, each iteration providing a sample of the parameter vector, e.g.

ho = Hyperoptimizer(10, a = LinRange(1,2,50), b = [true, false], c = randn(100))
for (i,a,b,c) in ho
    println(i, "\t", a, "\t", b, "\t", c)
end

1   1.2244897959183674  false   0.8179751164732062
2   1.7142857142857142  true    0.6536272580487854
3   1.4285714285714286  true    -0.2737451706680355
4   1.6734693877551021  false   -0.12313108128547606
5   1.9795918367346939  false   -0.4350837079334295
6   1.0612244897959184  true    -0.2025613848798039
7   1.469387755102041   false   0.7464858339748051
8   1.8571428571428572  true    -0.9269021128132274
9   1.163265306122449   true    2.6554272337516966
10  1.4081632653061225  true    1.112896676939024

If used in this way, the hyperoptimizer can not keep track of the function values like it did when @hyperopt was used. To manually store the same data, consider a pattern like

ho = Hyperoptimizer(10, a = LinRange(1,2), b = [true, false], c = randn(100))
for (i,a,b,c) in ho
    res = computations(a,b,c)
    push!(ho.history, [a,b,c])
end

Categorical variables

RandomSampler and CLHSampler support categorical variables which do not have a natural floating point representation, such as functions:

@hyperopt for i=20, fun = [tanh, σ, relu]
    train_network(fun)
end
# or
@hyperopt for i=20, sampler=CLHSampler(dims=[Categorical(3), Continuous()]),
                    fun   = [tanh, σ, relu],
                    param = LinRange(0,1,20)
    train_network(fun, param)
end

Which sampler to use?

RandomSampler is a good baseline and the default if none is chosen. Hyperband(R=50, η=3, inner=RandomSampler()) runs the expression with varying amount of resources, allocating more resources to promising hyperparameters. See below for more info on Hyperband.

If number of iterations is small, LHSampler work better than random search. Caveat: LHSampler needs all candidate vectors to be of equal length, i.e.,

hob = @hyperopt for i=100, sampler = LHSampler(),
                            a = LinRange(1,5,100),
                            b = repeat([true, false],50),
                            c = exp10.(LinRange(-1,3,100))
    f(a,b,c=c)
end

where all candidate vectors are of length 100. The candidates for b thus had to be repeated 50 times.

The categorical CLHSampler circumvents this

hob = @hyperopt for i=100,
                    sampler=CLHSampler(dims=[Continuous(), Categorical(2), Continuous()]),
                    a = LinRange(1,5,100),
                    b = [true, false],
                    c = exp10.(LinRange(-1,3,100))
    f(a,b,c=c)
end

Hyperband

Hyperband(R=50, η=3, inner=RandomSampler()) Implements Hyperband: A Novel Bandit-Based Approach to Hyperparameter Optimization. The maximum amount of resources is given by R and the parameter η roughly determines the proportion of trials discarded between each round of successive halving. When using Hyperband the expression inside the @hyperopt macro takes the form of the following pseudocode

ho = @hyperopt for resources=50, sampler=Hyperband(R=50, η=3, inner=RandomSampler()), a = LinRange(1,5,1800), c = exp10.(LinRange(-1,3,1800))
    if state === nothing # Query if state is initialized
        res = optimize(resources, a, b) # if state is uninitialized, start a new optimization using the selected hyper parameters
    else
        res = optimize(resources, state=state) # If state has a value, continue the optimization from the state
    end
    minimum(res), get_state(res) # return the minimum value and a state from which to continue the optimization
end

the resources are increased by defining a variable resources inside each loop, which grows according to the hyperband algorithm. How to interpret resources is entirely up to the user - it can be a time limit, the maximum number of iterations, or anything else.

A (simple) working example using Hyperband and Optim is given below, where the resources are used to control the maximum calls to the objective function:

using Optim
f(a;c=10) = sum(@. 100 + (a-3)^2 + (c-100)^2)
hohb = @hyperopt for resources=50, sampler=Hyperband(R=50, η=3, inner=RandomSampler()), a = LinRange(1,5,1800), c = exp10.(LinRange(-1,3,1800))
    if !(state === nothing)
        a,c = state
    end
    res = Optim.optimize(x->f(x[1],c=x[2]), [a,c], SimulatedAnnealing(), Optim.Options(f_calls_limit=round(Int, resources)))
    Optim.minimum(res), Optim.minimizer(res)
end
plot(hohb)

and a more complicated example that also explores different Optim optimizers as the inner optimizer is

hohb = @hyperopt for resources=50, sampler=Hyperband(R=50, η=3, inner=RandomSampler()),
    algorithm = [SimulatedAnnealing(), ParticleSwarm(), NelderMead(), BFGS(), NewtonTrustRegion()],
    a = LinRange(1,5,1800),
    c = exp10.(LinRange(-1,3,1800))
    if state !== nothing
        algorithm, x0 = state
    else
        x0 = [a,c]
    end
    println(resources, " algorithm: ", typeof(algorithm).name.name)
    res = Optim.optimize(x->f(x[1],c=x[2]), x0, algorithm, Optim.Options(time_limit=resources+1, show_trace=false))
    Optim.minimum(res), (algorithm, Optim.minimizer(res))
end

Function / vector interface

Hyperband can also be called by itself with a more standard optimizer interface. In this case, the objective function takes a scalar resources and a vector of parameters, and returns the objective value and a vector of parameters.

Example:

using Hyperopt
using Optim: optimize, Options, minimum, minimizer
f(a;c=10) = sum(@. 100 + (a-3)^2 + (c-100)^2)

objective = function (resources::Real, pars::AbstractVector)
    res = optimize(x->f(x[1],c=x[2]), pars, SimulatedAnnealing(), Options(time_limit=resources/100))
    minimum(res), minimizer(res)
end

candidates = (a=LinRange(1,5,300), c=exp10.(LinRange(-1,3,300))) # A vector of vectors also works, but parameters will not get nice names in plots
hohb = hyperband(objective, candidates; R=50, η=3, threads=true)

BOHB

BOHB: Robust and Efficient Hyperparameter Optimization at Scale refines Hyperband by replacing the random sampler by a bayesian-optimization-based sampler. Now you can use it by simply replace the sampler in Hyperband as BOHB(dims=[<dims>...])

Examples

Below in an example without BOHB, which should be familiar from previous examples:

using Optim
hb = @hyperopt for i=18, sampler=Hyperband(R=50, η=3, inner=RandomSampler()), a = LinRange(1,5,800), c = exp10.(LinRange(-1,3,1800))
    if state !== nothing
        a,c = state
    end
    res = Optim.optimize(x->f(x[1],c=x[2]), [a,c], NelderMead(), Optim.Options(f_calls_limit=round(Int, i)))
    Optim.minimum(res), Optim.minimizer(res)
end

To use BOHB, simply replace the inner sampler. Here we change from RandomSampler to BOHB.
Remember to specify dimension types for BOHB!

bohb = @hyperopt for i=18, sampler=Hyperband(R=50, η=3, inner=BOHB(dims=[Hyperopt.Continuous(), Hyperopt.Continuous()])), a = LinRange(1,5,800), c = exp10.(LinRange(-1,3,1800))
    if state !== nothing
        a,c = state
    end
    res = Optim.optimize(x->f(x[1],c=x[2]), [a,c], NelderMead(), Optim.Options(f_calls_limit=round(Int, i)))
    Optim.minimum(res), Optim.minimizer(res)
end

When using BOHB, a Kernel Density Estimator will estimate hyperparameters that balance exploration and exploitation based on previous observations. It does this by setting the state variable to a tuple of the estimated set of hyperparameters, at certain points within the loop. As a consequence, the state returned (Optim.minimizer(res) in the previous example) needs to be a tuple holding the values for each hyperparameter in order ((a, c) in the previous example).

Note - BOHB currently only handles Continuous variables, see issue #80 for a discussion on adding support for categorical variables.

Parallel execution

The macro @phyperopt works in the same way as @hyperopt but distributes all computation on available workers. The usual caveats apply, code must be loaded on all workers etc.
- @phyperopt accepts an optional second argument which is a pmap-like function. E.g. (args...,) -> pmap(args...; on_error=...).
The macro @thyperopt uses ThreadPools.tmap to evaluate the objective on all available threads. Beware of high memory consumption if your objective allocates a lot of memory.

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