# Written by Francois Fleuret <francois@fleuret.org>
-import math
+import math, argparse
import matplotlib.pyplot as plt
+
import torch
######################################################################
-def compute_alpha(x, y, D, a = 0, b = 1, rho = 1e-11):
+parser = argparse.ArgumentParser(description='Example of double descent with polynomial regression.')
+
+parser.add_argument('--D-max',
+ type = int, default = 16)
+
+parser.add_argument('--nb-runs',
+ type = int, default = 250)
+
+parser.add_argument('--nb-train-samples',
+ type = int, default = 8)
+
+parser.add_argument('--train-noise-std',
+ type = float, default = 0.)
+
+parser.add_argument('--seed',
+ type = int, default = 0,
+ help = 'Random seed (default 0, < 0 is no seeding)')
+
+args = parser.parse_args()
+
+if args.seed >= 0:
+ torch.manual_seed(args.seed)
+
+######################################################################
+
+def pol_value(alpha, x):
+ x_pow = x.view(-1, 1) ** torch.arange(alpha.size(0)).view(1, -1)
+ return x_pow @ alpha
+
+def fit_alpha(x, y, D, a = 0, b = 1, rho = 1e-12):
M = x.view(-1, 1) ** torch.arange(D + 1).view(1, -1)
B = y
- if D+1 > 2:
- q = torch.arange(2, D + 1).view( 1, -1).to(x.dtype)
+ if D >= 2:
+ q = torch.arange(2, D + 1, dtype = x.dtype).view(1, -1)
r = q.view(-1, 1)
beta = x.new_zeros(D + 1, D + 1)
beta[2:, 2:] = (q-1) * q * (r-1) * r * (b**(q+r-3) - a**(q+r-3))/(q+r-3)
l, U = beta.eig(eigenvectors = True)
- Q = U @ torch.diag(l[:, 0].pow(0.5))
+ Q = U @ torch.diag(l[:, 0].clamp(min = 0) ** 0.5)
B = torch.cat((B, y.new_zeros(Q.size(0))), 0)
M = torch.cat((M, math.sqrt(rho) * Q.t()), 0)
- alpha = torch.lstsq(B, M).solution.view(-1)[:D+1]
-
- return alpha
+ return torch.lstsq(B, M).solution[:D+1, 0]
######################################################################
def phi(x):
- return 4 * (x - 0.5) ** 2 * (x >= 0.5)
+ return torch.abs(torch.abs(x - 0.4) - 0.2) + x/2 - 0.1
######################################################################
-torch.manual_seed(0)
+def compute_mse(nb_train_samples):
+ mse_train = torch.zeros(args.nb_runs, args.D_max + 1)
+ mse_test = torch.zeros(args.nb_runs, args.D_max + 1)
+
+ for k in range(args.nb_runs):
+ x_train = torch.rand(nb_train_samples, dtype = torch.float64)
+ y_train = phi(x_train)
+ if args.train_noise_std > 0:
+ y_train = y_train + torch.empty_like(y_train).normal_(0, args.train_noise_std)
+ x_test = torch.linspace(0, 1, 100, dtype = x_train.dtype)
+ y_test = phi(x_test)
-nb_train_samples = 7
-D_max = 16
-nb_runs = 250
+ for D in range(args.D_max + 1):
+ alpha = fit_alpha(x_train, y_train, D)
+ mse_train[k, D] = ((pol_value(alpha, x_train) - y_train)**2).mean()
+ mse_test[k, D] = ((pol_value(alpha, x_test) - y_test)**2).mean()
-mse_train = torch.zeros(nb_runs, D_max + 1)
-mse_test = torch.zeros(nb_runs, D_max + 1)
+ return mse_train.median(0).values, mse_test.median(0).values
+
+######################################################################
+# Plot the MSE vs. degree curves
-for k in range(nb_runs):
- x_train = torch.rand(nb_train_samples, dtype = torch.float64)
- y_train = phi(x_train)
- y_train = y_train + torch.empty(y_train.size(), dtype = y_train.dtype).normal_(0, 0.1)
- x_test = torch.linspace(0, 1, 100, dtype = x_train.dtype)
- y_test = phi(x_test)
+fig = plt.figure()
- for D in range(D_max + 1):
- alpha = compute_alpha(x_train, y_train, D)
- X_train = x_train.view(-1, 1) ** torch.arange(D + 1).view(1, -1)
- X_test = x_test.view(-1, 1) ** torch.arange(D + 1).view(1, -1)
- mse_train[k, D] = ((X_train @ alpha - y_train)**2).mean()
- mse_test[k, D] = ((X_test @ alpha - y_test)**2).mean()
+ax = fig.add_subplot(1, 1, 1)
+ax.set_yscale('log')
+ax.set_ylim(1e-5, 1)
+ax.set_xlabel('Polynomial degree', labelpad = 10)
+ax.set_ylabel('MSE', labelpad = 10)
-mse_train = mse_train.median(0).values
-mse_test = mse_test.median(0).values
+ax.axvline(x = args.nb_train_samples - 1,
+ color = 'gray', linewidth = 0.5, linestyle = '--')
+ax.text(args.nb_train_samples - 1.2, 1e-4, 'Nb. params = nb. samples',
+ fontsize = 10, color = 'gray',
+ rotation = 90, rotation_mode='anchor')
+
+mse_train, mse_test = compute_mse(args.nb_train_samples)
+
+ax.plot(torch.arange(args.D_max + 1), mse_train, color = 'blue', label = 'Train error')
+ax.plot(torch.arange(args.D_max + 1), mse_test, color = 'red', label = 'Test error')
+
+ax.legend(frameon = False)
+
+fig.savefig('dd-mse.pdf', bbox_inches='tight')
+
+plt.close(fig)
######################################################################
+# Plot some examples of train / test
-torch.manual_seed(4) # I picked that for pretty
+torch.manual_seed(9) # I picked that for pretty
-x_train = torch.rand(nb_train_samples, dtype = torch.float64)
+x_train = torch.rand(args.nb_train_samples, dtype = torch.float64)
y_train = phi(x_train)
-y_train = y_train + torch.empty(y_train.size(), dtype = y_train.dtype).normal_(0, 0.1)
+if args.train_noise_std > 0:
+ y_train = y_train + torch.empty_like(y_train).normal_(0, args.train_noise_std)
x_test = torch.linspace(0, 1, 100, dtype = x_train.dtype)
y_test = phi(x_test)
-for D in range(D_max + 1):
+for D in range(args.D_max + 1):
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.set_title(f'Degree {D}')
ax.set_ylim(-0.1, 1.1)
- ax.plot(x_test, y_test, color = 'blue', label = 'Test values')
- ax.scatter(x_train, y_train, color = 'blue', label = 'Training examples')
+ ax.plot(x_test, y_test, color = 'black', label = 'Test values')
+ ax.scatter(x_train, y_train, color = 'blue', label = 'Train samples')
- alpha = compute_alpha(x_train, y_train, D)
- X_test = x_test.view(-1, 1) ** torch.arange(D + 1).view(1, -1)
- ax.plot(x_test, X_test @ alpha, color = 'red', label = 'Fitted polynomial')
+ alpha = fit_alpha(x_train, y_train, D)
+ ax.plot(x_test, pol_value(alpha, x_test), color = 'red', label = 'Fitted polynomial')
ax.legend(frameon = False)
fig.savefig(f'dd-example-{D:02d}.pdf', bbox_inches='tight')
-######################################################################
-
-fig = plt.figure()
-
-ax = fig.add_subplot(1, 1, 1)
-ax.set_yscale('log')
-ax.set_xlabel('Polynomial degree', labelpad = 10)
-ax.set_ylabel('MSE', labelpad = 10)
-
-ax.axvline(x = nb_train_samples - 1, color = 'gray', linewidth = 0.5)
-ax.plot(torch.arange(D_max + 1), mse_train, color = 'blue', label = 'Train error')
-ax.plot(torch.arange(D_max + 1), mse_test, color = 'red', label = 'Test error')
-
-ax.legend(frameon = False)
-
-fig.savefig('dd-mse.pdf', bbox_inches='tight')
+ plt.close(fig)
######################################################################
projects / pytorch.git / blobdiff