from __future__ import print_function
import numpy as np
import timeit
import numdifftools as nd
import numdifftools.nd_algopy as nda
from algopy import dot
# from numpy import dot
from collections import OrderedDict
from numdifftools.core import MinStepGenerator, MaxStepGenerator
import matplotlib.pyplot as pyplot
[docs]class BenchmarkFunction(object):
def __init__(self, N):
A = np.arange(N * N, dtype=float).reshape((N, N))
self.A = np.dot(A.T, A)
def __call__(self, xi):
x = np.array(xi)
tmp = dot(self.A, x)
return 0.5 * dot(x * x, tmp)
[docs]def plot_errors(error_objects, problem_sizes, symbols):
ploterror = pyplot.semilogy
for title, funcs, results in error_objects:
pyplot.figure()
pyplot.title(title)
# ref_sol = results[0]
for i, method in enumerate(funcs):
ploterror(problem_sizes, results[i], symbols[i],
markerfacecolor='None', label=method)
pyplot.ylabel(r'Absolute error $\|g_{ref} - g\|$')
pyplot.xlabel('problem size $N$')
pyplot.ylim(loglimits(results))
pyplot.grid()
leg = pyplot.legend(loc=7)
frame = leg.get_frame()
frame.set_alpha(0.4)
pyplot.savefig(title.lower().replace(' ', '_') + '.png', format='png')
[docs]def plot_runtimes(run_time_objects, problem_sizes, symbols):
plottime = pyplot.loglog
for title, funcs, results in run_time_objects:
pyplot.figure()
pyplot.title(title)
for i, method in enumerate(funcs):
plottime(problem_sizes, results[i], symbols[i],
markerfacecolor='None', label=method)
pyplot.ylabel('time $t$')
pyplot.xlabel('problem size $N$')
pyplot.xlim(loglimits(problem_sizes))
pyplot.ylim(loglimits(results))
pyplot.grid()
leg = pyplot.legend(loc=2)
frame = leg.get_frame()
frame.set_alpha(0.4)
pyplot.savefig(title.lower().replace(' ', '_') + '.png', format='png')
[docs]def loglimits(data, border=0.05):
low, high = np.min(data), np.max(data)
scale = (high/low)**border
return low/scale, high*scale
fixed_step = MinStepGenerator(num_steps=1, use_exact_steps=True, offset=0)
epsilon = MaxStepGenerator(num_steps=14, use_exact_steps=True,
step_ratio=1.6, offset=0)
adaptiv_txt = '_adaptive_%d_%s_%d' % (epsilon.num_steps,
str(epsilon.step_ratio), epsilon.offset)
gradient_funs = OrderedDict()
nda_method = 'forward'
nda_txt = 'algopy_' + nda_method
gradient_funs[nda_txt] = nda.Jacobian(1, method=nda_method)
# gradient_funs['numdifftools'] = nd.Jacobian(1, **options)
for method in ['forward', 'central', 'complex']:
method2 = method + adaptiv_txt
gradient_funs[method] = nd.Jacobian(1, method=method, step=fixed_step)
gradient_funs[method2] = nd.Jacobian(1, method=method, step=epsilon)
HessianFun = 'Hessdiag'
ndcHessian = getattr(nd, HessianFun) # ndc.Hessian #
hessian_funs = OrderedDict()
hessian_funs[nda_txt] = getattr(nda, HessianFun)(1, method=nda_method)
for method in ['forward', 'central', 'complex']:
method2 = method + adaptiv_txt
hessian_funs[method] = ndcHessian(1, method=method, step=fixed_step)
hessian_funs[method2] = ndcHessian(1, method=method, step=epsilon)
[docs]def compute_gradients(gradient_funs, problem_sizes):
results_gradient_list = []
for N in problem_sizes:
print('N=', N)
num_methods = len(gradient_funs)
results_gradient = np.zeros((num_methods, 3))
ref_g = None
f = BenchmarkFunction(N)
for i, (_key, gradient_f) in enumerate(gradient_funs.items()):
t = timeit.default_timer()
gradient_f.f = f
preproc_time = timeit.default_timer() - t
t = timeit.default_timer()
x = 3 * np.ones(N)
val = gradient_f(x)
run_time = timeit.default_timer() - t
if ref_g is None:
ref_g = val
err = 0
norm_ref_g = np.linalg.norm(ref_g)
else:
err = np.linalg.norm(val - ref_g) / norm_ref_g
results_gradient[i] = run_time, err, preproc_time
results_gradient_list.append(results_gradient)
results_gradients = np.array(results_gradient_list) + 1e-16
print('results_gradients=\n', results_gradients)
return results_gradients
[docs]def compute_hessians(hessian_funs, problem_sizes):
print('starting hessian computation ')
results_hessian_list = []
hessian_N_list = problem_sizes
for N in hessian_N_list:
print('N=', N)
num_methods = len(hessian_funs)
results_hessian = np.zeros((num_methods, 3))
ref_h = None
f = BenchmarkFunction(N)
for i, (_key, hessian_f) in enumerate(hessian_funs.items()):
t = timeit.default_timer()
hessian_f.f = f
preproc_time = timeit.default_timer() - t
t = timeit.default_timer()
x = 3 * np.ones(N)
val = hessian_f(x)
run_time = timeit.default_timer() - t
if ref_h is None:
ref_h = val
err = 0.0
norm_ref_h = np.linalg.norm(ref_h.ravel())
else:
err = np.linalg.norm((val - ref_h).ravel()) / norm_ref_h
results_hessian[i] = run_time, err, preproc_time
results_hessian_list.append(results_hessian)
results_hessians = np.array(results_hessian_list) + 1e-16
print(hessian_N_list)
print('results_hessians=\n', results_hessians)
return results_hessians
if __name__ == '__main__':
problem_sizes = (4, 8, 16, 32, 64, 96)
symbols = ('-kx', ':k>', ':k<', '--k^', '--kv', '-kp', '-ks',
'b', '--b', '-k+')
results_gradients = compute_gradients(gradient_funs, problem_sizes)
results_hessians = compute_hessians(hessian_funs, problem_sizes)
print(results_gradients.shape)
run_time_objects = [('Jacobian run times',
gradient_funs, results_gradients[..., 0].T),
('Hessian run times',
hessian_funs, results_hessians[..., 0].T)]
error_objects = [('Jacobian errors',
gradient_funs, results_gradients[..., 1].T),
('Hessian errors',
hessian_funs, results_hessians[..., 1].T)]
plot_runtimes(run_time_objects, problem_sizes, symbols)
plot_errors(error_objects, problem_sizes, symbols)