CPSC540 - University of British Columbianando/540-2013/lectures/l6.pdf · CPSC540 Nando de Freitas...
Transcript of CPSC540 - University of British Columbianando/540-2013/lectures/l6.pdf · CPSC540 Nando de Freitas...
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CPSC540
Nando de FreitasJanuary 2013KPM Book Sections 4.3 and 15.2.
Gaussian ProcessesGaussian ProcessesGaussian ProcessesGaussian Processes
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Gaussian basics
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Gaussian basics
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Gaussian basics
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Multivariate Gaussian Theorem (see KPM)
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Gaussian basics
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Gaussian basics
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Gaussian basics
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GP: a distribution over functionsA GP is a Gaussian distribution over functions:
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Sampling from P(f)from __future__ import divisionimport numpy as npimport matplotlib.pyplot as pl
def kernel(a, b):""" GP squared exponential kernel """sqdist = np.sum(a**2,1).reshape(-1,1) + np.sum(b**2,1) - 2*np.dot(a, b.T)return np.exp(-.5 * sqdist)
n = 50 # number of test points.Xtest = np.linspace(-5, 5, n).reshape(-1,1) # Test points.K_ = kernel(Xtest, Xtest) # Kernel at test points.
# draw samples from the prior at our test points.L = np.linalg.cholesky(K_ + 1e-6*np.eye(n))f_prior = np.dot(L, np.random.normal(size=(n,10)))
pl.plot(Xtest, f_prior)
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GP posterior
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Active learning with GPs
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Noiseless GP regression
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Noiseless GP regression
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Effect of kernel width parameter
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Noisy GP regression
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Noisy GP regression
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Noisy GP regression and Ridge
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Noisy GP regression and Ridge
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Noisy GP regression and Ridge
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Learning the kernel parameters
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Numerical computation considerations
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Next lecture
In the next lecture, we capitalize on GPs to introduce active learning, Bayesian optimization and GP bandits.
For a recent article on bandits and Thompson sampling at work at Google, see:
http://analytics.blogspot.ca/2013/01/multi-armed-bandit-experiments.html
For an article on Bayesian optimization, see:
http://arxiv.org/abs/1012.2599