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ha3-solution.py
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ha3-solution.py
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import numpy as np
import matplotlib.pyplot as plt
import torch.random
import torch.nn as nn
import torch.optim as optim
import tqdm
from scipy.stats import multivariate_normal
import scipy.linalg as linalg
EPS = 1e-9
class FeedforwardNetwork(nn.Module):
"""Neural network implemented using NumPy."""
def __init__(self, input_size, width=2000, beta=0.1, depth=2, only_optimize_last=False):
super(FeedforwardNetwork, self).__init__()
self.width = width
self.beta = beta
self.depth = depth
self.ws = [input_size] + [self.width for _l in range(self.depth)] + [1]
self.weights = []
self.biases = []
for i in range(depth + 1):
wi = nn.Parameter(torch.randn(self.ws[i], self.ws[i + 1]))
bi = nn.Parameter(torch.randn(self.ws[i + 1]))
if (not only_optimize_last) or (i == depth):
self.register_parameter('weight_{}'.format(i), wi)
self.register_parameter('bias_{}'.format(i), bi)
self.weights += [wi]
self.biases += [bi]
def forward(self, X):
"""Return output and gradient"""
Xf = self.get_features(X)
pred = 1/np.sqrt(self.ws[-2]) * Xf @ self.weights[-1] + self.beta * self.biases[-1]
return pred.flatten()
def get_features(self, X):
activ = X
for i in range(self.depth):
preactiv = 1/np.sqrt(self.ws[i]) * activ @ self.weights[i] + self.beta * self.biases[i]
activ = torch.relu(preactiv)
return activ
def train_and_predict(X_train, Y_train, X_test, net, epochs=1000, lr=1.0, sgd=False):
# Convert to pytorch
X_train_pth = torch.Tensor(X_train)
Y_train_pth = torch.Tensor(Y_train)
X_test_pth = torch.Tensor(X_test)
# Train
msg = 'Epoch = {} - loss = {:0.2f}'
if sgd:
pbar = tqdm.tqdm(initial=0, total=epochs, desc=msg.format(0, 0))
optimizer = optim.SGD(net.parameters(), lr=lr)
for i in range(epochs):
optimizer.zero_grad()
Y_pred = net(X_train_pth)
error = (Y_train_pth - Y_pred)
loss = 1/2 * (error * error).mean()
loss.backward()
optimizer.step()
# Update pbar
pbar.desc = msg.format(i, loss)
pbar.update(1)
pbar.close()
# Predict on test
with torch.no_grad():
Y_pred_test = net(X_test_pth).detach().numpy()
else:
with torch.no_grad():
# Train
Xf_train = net.get_features(X_train_pth).detach().numpy()
Xf_train = np.hstack([Xf_train, np.ones([Xf_train.shape[0], 1])])
estim_param, _resid, _rank, _s = linalg.lstsq(Xf_train, Y_train)
# Test
Xf_test = net.get_features(X_test_pth).detach().numpy()
Xf_test = np.hstack([Xf_test, np.ones([Xf_test.shape[0], 1])])
Y_pred_test = Xf_test @ estim_param
return Y_pred_test
def linkernel(x, beta=1):
n_pts, dim = x.shape
return 1 / dim * x @ x.T + beta**2 + EPS * np.eye(n_pts)
def approximate_kernel(C, fn, n_samples=1000):
y = multivariate_normal.rvs(size=n_samples, cov=C)
return 1 / n_samples * fn(y.T) @ fn(y)
def nngp_cov(x, n_layers, beta=1):
sigma = linkernel(x, beta=beta)
relu = lambda xx: np.maximum(xx, 0)
for _i in range(n_layers):
sigma = approximate_kernel(sigma, relu) + beta ** 2
return sigma
def ntk_cov(x, n_layers, beta=1):
sigma = linkernel(x, beta)
theta = sigma
relu = lambda xx: np.maximum(xx, 0)
relu_deriv = lambda xx: np.array(xx > 0, dtype=float)
for _i in range(n_layers):
sigma = approximate_kernel(sigma, relu) + beta ** 2
sigma_dot = approximate_kernel(sigma, relu_deriv)
theta = theta * sigma_dot + sigma
return sigma, theta
def conditioning(c, y_train):
n_train = len(y_train)
if n_train == 0:
return np.zeros(c.shape[0]), c
cov_train = c[:n_train, :n_train]
cov_test_train = c[n_train:, :n_train]
cov_test = c[n_train:, n_train:]
inv_cov = np.linalg.inv(cov_train)
m = cov_test_train @ inv_cov @ y_train
cc = cov_test - cov_test_train @ inv_cov @ cov_test_train.T
return m, cc
def ntk_limit(c, ntkc, y_train):
n_train = len(y_train)
if n_train == 0:
return np.zeros(c.shape[0]), c
# Get submatrices
cov_train = c[:n_train, :n_train]
cov_test_train = c[n_train:, :n_train]
cov_test = c[n_train:, n_train:]
# Get ntk submatrices
ntk_cov_train = ntkc[:n_train, :n_train]
ntk_cov_test_train = ntkc[n_train:, :n_train]
inv_ntkc = np.linalg.inv(ntk_cov_train)
m = ntk_cov_test_train @ inv_ntkc @ y_train
cc = cov_test + ntk_cov_test_train @ inv_ntkc @ cov_train @ inv_ntkc @ ntk_cov_test_train.T + \
- ntk_cov_test_train @ inv_ntkc @ cov_test_train.T - cov_test_train @ inv_ntkc @ ntk_cov_test_train.T
return m, cc
if __name__ == "__main__":
N_test = 50
n_layers = 4
beta = 0.1
epochs = 1000
width = 1000
lr = 1
tp = 'nngp' # CHANGE HERE: 'nngp' gives part 1; 'ntk' gives part 2.
n_runs = 5
# Train data
gamma_train = np.array([-2, -1.2, -0.4, 0.5, 1.8])
X_train = np.stack([np.cos(gamma_train), np.sin(gamma_train)]).T
Y_train = X_train.prod(axis=1)
# Test data
gamma_test = np.linspace(-np.pi, np.pi, N_test)
X_test = np.stack([np.cos(gamma_test), np.sin(gamma_test)]).T
Y_test = X_test.prod(axis=1)
# Define neural network
Y_test_pred = []
for i in range(n_runs):
tqdm.tqdm.write('run={}'.format(i))
torch.manual_seed(i)
net = FeedforwardNetwork(2, beta=beta, depth=n_layers, width=width,
only_optimize_last=(tp=='nngp'))
Y_test_pred += [train_and_predict(X_train, Y_train, X_test, net, lr=lr, epochs=epochs,
sgd=(tp!='nngp'))]
# Plot on test data
for i, y in enumerate(Y_test_pred):
plt.plot(gamma_test, y, color='black')
plt.plot(gamma_test, Y_test)
plt.plot(gamma_train, Y_train, '*', color='black', ms=10)
# Compute covariance matrix
if tp == 'nngp':
c = nngp_cov(np.vstack([X_train, X_test]), n_layers=n_layers, beta=beta)
m, cc = conditioning(c, Y_train)
else:
c, ntkc = ntk_cov(np.vstack([X_train, X_test]), n_layers=n_layers, beta=beta)
m, cc = ntk_limit(c, ntkc, Y_train)
y_gp = multivariate_normal.rvs(size=5, mean=m.flatten(), cov=cc)
plt.plot(gamma_test, y_gp.T, color=str(0.4), alpha=0.8)
for nn in [3, 2, 1]:
plt.fill_between(gamma_test, m - nn * np.sqrt(np.diag(cc)),
m + nn * np.sqrt(np.diag(cc)), color=str(0.4 + 0.15 * nn), alpha=0.5)
plt.plot(gamma_train, Y_train, '*', color='black', ms=10)
plt.show()