On Contrastive Divergence Learning
On Contrastive Divergence Learning
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发表时间：
2005
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通讯作者：
M. A. Carreira-Perpiñán;Geoffrey E. Hinton
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作者：
M. A. Carreira-Perpiñán;Geoffrey E. Hinton
马尔可夫随机场的最大似然(ML)学习具有挑战性，因为它需要对具有指数数量的项的平均值进行估计。马尔可夫链蒙特卡罗方法通常需要很长时间才能收敛到无偏估计，但Hinton(2002)表明，如果马尔可夫链只运行几个步骤，学习仍然可以很好地工作，并且它近似最小化一个不同的函数，称为对比发散(CD)。CD学习已经成功地应用于各种类型的随机场。在这里，我们研究CD学习的属性，并表明它提供了总体上的有偏估计，但偏差通常非常小。因此，可以使用快速CD学习来接近ML解决方案，然后可以使用缓慢的ML学习来微调CD解决方案。考虑向量x上的概率分布(假设离散的w.l.o.g。)参数Wp(x；W)=1Z(W)e(1)，其中Z(W)=∑x e−E(x；W)是归一化常数，E(x；W)是能量函数。这类随机场分布已经发现了许多实际应用(Li，2001；Winkler，2002；Teh等人，2003；他等人，2004)。给定iid样本X={xn}n=1，参数W的最大似然(ML)学习可以通过梯度上升来完成：W=W+η∂L(W；X)∂W|
Maximum-likelihood (ML) learning of Markov random fields is challenging because it requires estimates of averages that have an exponential number of terms. Markov chain Monte Carlo methods typically take a long time to converge on unbiased estimates, but Hinton (2002) showed that if the Markov chain is only run for a few steps, the learning can still work well and it approximately minimizes a different function called “contrastive divergence” (CD). CD learning has been successfully applied to various types of random fields. Here, we study the properties of CD learning and show that it provides biased estimates in general, but that the bias is typically very small. Fast CD learning can therefore be used to get close to an ML solution and slow ML learning can then be used to fine-tune the CD solution. Consider a probability distribution over a vector x (assumed discrete w.l.o.g.) and with parameters W p(x;W) = 1 Z(W) e (1) where Z(W) = ∑ x e −E(x;W) is a normalisation constant and E(x;W) is an energy function. This class of random-field distributions has found many practical applications (Li, 2001; Winkler, 2002; Teh et al., 2003; He et al., 2004). Maximum-likelihood (ML) learning of the parameters W given an iid sample X = {xn}n=1 can be done by gradient ascent: W = W + η ∂L(W;X ) ∂W ∣