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kernel_lsq.py
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/
kernel_lsq.py
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# Copyright 2018 Google LLC
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from functools import partial
from six.moves import xrange
import numpy.random as npr
import jax.numpy as np
from jax.config import config
from jax.experimental import optimizers
from jax import grad, jit, make_jaxpr, vmap
def gram(kernel, xs):
'''Compute a Gram matrix from a kernel and an array of data points.
Args:
kernel: callable, maps pairs of data points to scalars.
xs: array of data points, stacked along the leading dimension.
Returns:
A 2d array `a` such that `a[i, j] = kernel(xs[i], xs[j])`.
'''
return vmap(lambda x: vmap(lambda y: kernel(x, y))(xs))(xs)
def minimize(f, x, num_steps=10000, step_size=0.000001, mass=0.9):
opt_init, opt_update, get_params = optimizers.momentum(step_size, mass)
@jit
def update(i, opt_state):
x = get_params(opt_state)
return opt_update(i, grad(f)(x), opt_state)
opt_state = opt_init(x)
for i in xrange(num_steps):
opt_state = update(i, opt_state)
return get_params(opt_state)
def train(kernel, xs, ys, regularization=0.01):
gram_ = jit(partial(gram, kernel))
gram_mat = gram_(xs)
n = xs.shape[0]
def objective(v):
risk = .5 * np.sum((np.dot(gram_mat, v) - ys) ** 2.0)
reg = regularization * np.sum(v ** 2.0)
return risk + reg
v = minimize(objective, np.zeros(n))
def predict(x):
prods = vmap(lambda x_: kernel(x, x_))(xs)
return np.sum(v * prods)
return jit(vmap(predict))
if __name__ == "__main__":
n = 100
d = 20
# linear kernel
linear_kernel = lambda x, y: np.dot(x, y)
truth = npr.randn(d)
xs = npr.randn(n, d)
ys = np.dot(xs, truth)
predict = train(linear_kernel, xs, ys)
print('MSE:', np.sum((predict(xs) - ys) ** 2.))
def gram_jaxpr(kernel):
return make_jaxpr(partial(gram, kernel))(xs)
rbf_kernel = lambda x, y: np.exp(-np.sum((x - y) ** 2))
print()
print('jaxpr of gram(linear_kernel):')
print(gram_jaxpr(linear_kernel))
print()
print('jaxpr of gram(rbf_kernel):')
print(gram_jaxpr(rbf_kernel))