https://github.com/google/jax
Tip revision: 81f9f8744ec5e3a7019be533d2c874721a87b265 authored by jax authors on 18 June 2023, 15:52:06 UTC
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Tip revision: 81f9f87
gaussian_process_regression.py
# Copyright 2018 The JAX Authors.
#
# 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.
"""A basic example demonstrating using JAX to do Gaussian process regression.
"""
from absl import app
from absl import flags
from functools import partial
from jax import grad
from jax import jit
from jax import vmap
from jax import config
import jax.numpy as jnp
import jax.random as random
import jax.scipy as scipy
import matplotlib.pyplot as plt
FLAGS = flags.FLAGS
def main(unused_argv):
numpts = 7
key = random.PRNGKey(0)
eye = jnp.eye(numpts)
def cov_map(cov_func, xs, xs2=None):
"""Compute a covariance matrix from a covariance function and data points.
Args:
cov_func: callable function, 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] = cov_func(xs[i], xs[j])`.
"""
if xs2 is None:
return vmap(lambda x: vmap(lambda y: cov_func(x, y))(xs))(xs)
else:
return vmap(lambda x: vmap(lambda y: cov_func(x, y))(xs))(xs2).T
def softplus(x):
return jnp.logaddexp(x, 0.)
# Note, writing out the vectorized form of the identity
# ||x-y||^2 = <x-y,x-y> = ||x||^2 + ||y||^2 - 2<x,y>
# for computing squared distances would be more efficient (but less succinct).
def exp_quadratic(x1, x2):
return jnp.exp(-jnp.sum((x1 - x2)**2))
def gp(params, x, y, xtest=None, compute_marginal_likelihood=False):
noise = softplus(params['noise'])
amp = softplus(params['amplitude'])
ls = softplus(params['lengthscale'])
ymean = jnp.mean(y)
y = y - ymean
x = x / ls
train_cov = amp*cov_map(exp_quadratic, x) + eye * (noise + 1e-6)
chol = scipy.linalg.cholesky(train_cov, lower=True)
kinvy = scipy.linalg.solve_triangular(
chol.T, scipy.linalg.solve_triangular(chol, y, lower=True))
if compute_marginal_likelihood:
log2pi = jnp.log(2. * 3.1415)
ml = jnp.sum(
-0.5 * jnp.dot(y.T, kinvy) -
jnp.sum(jnp.log(jnp.diag(chol))) -
(numpts / 2.) * log2pi)
ml -= jnp.sum(-0.5 * jnp.log(2 * 3.1415) - jnp.log(amp)**2) # lognormal prior
return -ml
if xtest is not None:
xtest = xtest / ls
cross_cov = amp*cov_map(exp_quadratic, x, xtest)
mu = jnp.dot(cross_cov.T, kinvy) + ymean
v = scipy.linalg.solve_triangular(chol, cross_cov, lower=True)
var = (amp * cov_map(exp_quadratic, xtest) - jnp.dot(v.T, v))
return mu, var
marginal_likelihood = partial(gp, compute_marginal_likelihood=True)
predict = partial(gp, compute_marginal_likelihood=False)
grad_fun = jit(grad(marginal_likelihood))
# Covariance hyperparameters to be learned
params = {"amplitude": jnp.zeros((1, 1)),
"noise": jnp.zeros((1, 1)) - 5.,
"lengthscale": jnp.zeros((1, 1))}
momentums = {k: p * 0. for k, p in params.items()}
scales = {k: p * 0. + 1. for k, p in params.items()}
lr = 0.01 # Learning rate
def train_step(params, momentums, scales, x, y):
grads = grad_fun(params, x, y)
for k in params:
momentums[k] = 0.9 * momentums[k] + 0.1 * grads[k][0]
scales[k] = 0.9 * scales[k] + 0.1 * grads[k][0]**2
params[k] -= lr * momentums[k]/jnp.sqrt(scales[k] + 1e-5)
return params, momentums, scales
# Create a really simple toy 1D function
y_fun = lambda x: jnp.sin(x) + 0.1 * random.normal(key, shape=(x.shape[0], 1))
x = (random.uniform(key, shape=(numpts, 1)) * 4.) + 1
y = y_fun(x)
xtest = jnp.linspace(0, 6., 200)[:, None]
for i in range(1000):
params, momentums, scales = train_step(params, momentums, scales, x, y)
if i % 50 == 0:
ml = marginal_likelihood(params, x, y)
print("Step: %d, neg marginal likelihood: %f" % (i, ml))
print(params)
mu, var = predict(params, x, y, xtest)
std = jnp.sqrt(jnp.diag(var))
plt.plot(x, y, "k.")
plt.plot(xtest, mu)
plt.fill_between(xtest.flatten(),
mu.flatten() - std * 2, mu.flatten() + std * 2)
if __name__ == "__main__":
config.config_with_absl()
app.run(main)