4  Sequential Parameter Optimization: Using scipy Optimizers

As a default optimizer, spotpython uses differential_evolution from the scipy.optimize package. Alternatively, any other optimizer from the scipy.optimize package can be used. This chapter describes how different optimizers from the scipy optimize package can be used on the surrogate. The optimization algorithms are available from https://docs.scipy.org/doc/scipy/reference/optimize.html

import numpy as np
from math import inf
from spotpython.fun.objectivefunctions import Analytical
from spotpython.spot import spot
from scipy.optimize import shgo
from scipy.optimize import direct
from scipy.optimize import differential_evolution
from scipy.optimize import dual_annealing
from scipy.optimize import basinhopping
from spotpython.utils.init import fun_control_init, design_control_init, optimizer_control_init, surrogate_control_init

4.1 The Objective Function Branin

The spotpython package provides several classes of objective functions. We will use an analytical objective function, i.e., a function that can be described by a (closed) formula. Here we will use the Branin function. The 2-dim Branin function is \[ y = a (x_2 - b x_1^2 + c x_1 - r) ^2 + s (1 - t) \cos(x_1) + s, \] where values of \(a\), \(b\), \(c\), \(r\), \(s\) and \(t\) are: \(a = 1\), \(b = 5.1 / (4\pi^2)\), \(c = 5 / \pi\), \(r = 6\), \(s = 10\) and \(t = 1 / (8\pi)\).

It has three global minima: \(f(x) = 0.397887\) at \((-\pi, 12.275)\), \((\pi, 2.275)\), and \((9.42478, 2.475)\).

Input Domain: This function is usually evaluated on the square \(x_1 \in [-5, 10] \times x_2 \in [0, 15]\).

from spotpython.fun.objectivefunctions import Analytical
lower = np.array([-5,-0])
upper = np.array([10,15])
fun = Analytical(seed=123).fun_branin

4.2 The Optimizer

Differential Evolution (DE) from the scikit.optimize package, see https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.differential_evolution.html#scipy.optimize.differential_evolution is the default optimizer for the search on the surrogate. Other optimiers that are available in spotpython, see https://docs.scipy.org/doc/scipy/reference/optimize.html#global-optimization.

  • dual_annealing
  • direct
  • shgo
  • basinhopping

These optimizers can be selected as follows:

from scipy.optimize import differential_evolution
optimizer = differential_evolution

As noted above, we will use differential_evolution. The optimizer can use 1000 evaluations. This value will be passed to the differential_evolution method, which has the argument maxiter (int). It defines the maximum number of generations over which the entire differential evolution population is evolved, see https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.differential_evolution.html#scipy.optimize.differential_evolution

TensorBoard

Similar to the one-dimensional case, which is discussed in Section 7.5, we can use TensorBoard to monitor the progress of the optimization. We will use a similar code, only the prefix is different:

fun_control=fun_control_init(
                    lower = lower,
                    upper = upper,
                    fun_evals = 20,
                    PREFIX = "04_DE_"
                    )
surrogate_control=surrogate_control_init(
                    n_theta=len(lower))
spot_de = spot.Spot(fun=fun,
                    fun_control=fun_control,
                    surrogate_control=surrogate_control)
spot_de.run()
spotpython tuning: 3.8004662117718677 [######----] 55.00% 
spotpython tuning: 3.8004662117718677 [######----] 60.00% 
spotpython tuning: 3.159024883515257 [######----] 65.00% 
spotpython tuning: 3.133916697143885 [#######---] 70.00% 
spotpython tuning: 2.8926749183116236 [########--] 75.00% 
spotpython tuning: 0.4190219407803557 [########--] 80.00% 
spotpython tuning: 0.401871440801683 [########--] 85.00% 
spotpython tuning: 0.39926034519166187 [#########-] 90.00% 
spotpython tuning: 0.39926034519166187 [##########] 95.00% 
spotpython tuning: 0.39926034519166187 [##########] 100.00% Done...

<spotpython.spot.spot.Spot at 0x15538e240>

4.2.1 TensorBoard

If the prefix argument in fun_control_init()is not None (as above, where the prefix was set to 04_DE_) , we can start TensorBoard in the background with the following command:

tensorboard --logdir="./runs"

We can access the TensorBoard web server with the following URL:

http://localhost:6006/

The TensorBoard plot illustrates how spotpython can be used as a microscope for the internal mechanisms of the surrogate-based optimization process. Here, one important parameter, the learning rate \(\theta\) of the Kriging surrogate is plotted against the number of optimization steps.

TensorBoard visualization of the spotpython optimization process and the surrogate model.

4.4 Show the Progress

spot_de.plot_progress(log_y=True)

spot_de.surrogate.plot()

4.5 Exercises

4.5.1 dual_annealing

Tip: Selecting the Optimizer for the Surrogate

We can run spotpython with the dual_annealing optimizer as follows:

spot_da = spot.Spot(fun=fun,
                    fun_control=fun_control,
                    optimizer=dual_annealing,
                    surrogate_control=surrogate_control)
spot_da.run()
spot_da.print_results()
spot_da.plot_progress(log_y=True)
spot_da.surrogate.plot()
spotpython tuning: 3.800450053998908 [######----] 55.00% 
spotpython tuning: 3.800450053998908 [######----] 60.00% 
spotpython tuning: 3.158868878483762 [######----] 65.00% 
spotpython tuning: 3.1342300163360486 [#######---] 70.00% 
spotpython tuning: 2.893119335801825 [########--] 75.00% 
spotpython tuning: 0.41578200014582833 [########--] 80.00% 
spotpython tuning: 0.40201943256176875 [########--] 85.00% 
spotpython tuning: 0.3991976498681318 [#########-] 90.00% 
spotpython tuning: 0.3991976498681318 [##########] 95.00% 
spotpython tuning: 0.3991976498681318 [##########] 100.00% Done...

min y: 0.3991976498681318
x0: 3.1505143339793134
x1: 2.2985181488234647

4.5.2 direct

  • Describe the optimization algorithm
  • Use the algorithm as an optimizer on the surrogate
Tip: Selecting the Optimizer for the Surrogate

We can run spotpython with the direct optimizer as follows:

spot_di = spot.Spot(fun=fun,
                    fun_control=fun_control,
                    optimizer=direct,
                    surrogate_control=surrogate_control)
spot_di.run()
spot_di.print_results()
spot_di.plot_progress(log_y=True)
spot_di.surrogate.plot()
spotpython tuning: 3.812970247994418 [######----] 55.00% 
spotpython tuning: 3.812970247994418 [######----] 60.00% 
spotpython tuning: 3.162514679816068 [######----] 65.00% 
spotpython tuning: 3.1189615135325983 [#######---] 70.00% 
spotpython tuning: 2.6597698275013038 [########--] 75.00% 
spotpython tuning: 0.3984917773445744 [########--] 80.00% 
spotpython tuning: 0.3984917773445744 [########--] 85.00% 
spotpython tuning: 0.3984917773445744 [#########-] 90.00% 
spotpython tuning: 0.3984917773445744 [##########] 95.00% 
spotpython tuning: 0.3984917773445744 [##########] 100.00% Done...

min y: 0.3984917773445744
x0: 3.137860082304525
x1: 2.3010973936899863

4.5.3 shgo

  • Describe the optimization algorithm
  • Use the algorithm as an optimizer on the surrogate
Tip: Selecting the Optimizer for the Surrogate

We can run spotpython with the direct optimizer as follows:

spot_sh = spot.Spot(fun=fun,
                    fun_control=fun_control,
                    optimizer=shgo,
                    surrogate_control=surrogate_control)
spot_sh.run()
spot_sh.print_results()
spot_sh.plot_progress(log_y=True)
spot_sh.surrogate.plot()
spotpython tuning: 3.800455654373023 [######----] 55.00% 
spotpython tuning: 3.800455654373023 [######----] 60.00% 
spotpython tuning: 3.1589998607289376 [######----] 65.00% 
spotpython tuning: 3.134192072951082 [#######---] 70.00% 
spotpython tuning: 2.896589509267746 [########--] 75.00% 
spotpython tuning: 0.433945961604028 [########--] 80.00% 
spotpython tuning: 0.40260825513499476 [########--] 85.00% 
spotpython tuning: 0.39928786197917887 [#########-] 90.00% 
spotpython tuning: 0.39928786197917887 [##########] 95.00% 
spotpython tuning: 0.39928786197917887 [##########] 100.00% Done...

min y: 0.39928786197917887
x0: 3.1517408652058885
x1: 2.2972000572204814

4.5.4 basinhopping

  • Describe the optimization algorithm
  • Use the algorithm as an optimizer on the surrogate
Tip: Selecting the Optimizer for the Surrogate

We can run spotpython with the direct optimizer as follows:

spot_bh = spot.Spot(fun=fun,
                    fun_control=fun_control,
                    optimizer=basinhopping,
                    surrogate_control=surrogate_control)
spot_bh.run()
spot_bh.print_results()
spot_bh.plot_progress(log_y=True)
spot_bh.surrogate.plot()
spotpython tuning: 3.8004536838842267 [######----] 55.00% 
spotpython tuning: 3.8004536838842267 [######----] 60.00% 
spotpython tuning: 3.1590078408373294 [######----] 65.00% 
spotpython tuning: 3.1342008291295436 [#######---] 70.00% 
spotpython tuning: 2.8939639248159903 [########--] 75.00% 
spotpython tuning: 0.4191913455845153 [########--] 80.00% 
spotpython tuning: 0.4019076839214648 [########--] 85.00% 
spotpython tuning: 0.3992176203377902 [#########-] 90.00% 
spotpython tuning: 0.3992176203377902 [##########] 95.00% 
spotpython tuning: 0.3992176203377902 [##########] 100.00% Done...

min y: 0.3992176203377902
x0: 3.150847266242321
x1: 2.2981098602302974

4.5.5 Performance Comparison

Compare the performance and run time of the 5 different optimizers:

  • differential_evolution
  • dual_annealing
  • direct
  • shgo
  • basinhopping.

The Branin function has three global minima:

  • \(f(x) = 0.397887\) at
    • \((-\pi, 12.275)\),
    • \((\pi, 2.275)\), and
    • \((9.42478, 2.475)\).
  • Which optima are found by the optimizers?
  • Does the seed argument in fun = Analytical(seed=123).fun_branin change this behavior?

4.6 Jupyter Notebook

Note