a fast learning algorithm for deep belief nets

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Complementary priors

One obstacle that has hindered the ability to make inference in directed belief nets is the "explaining away" phenomenon: it is extremely difficult, in general, to compute the posterior distribution over hidden variables in a dense directed belief network.

The authors describe a way to cancel out the explaining away phenomenon in a hidden layer by using additional hidden layers to create what they refer to as "complementary priors". The idea behind this is that in a logistic belief net (a network where the probability of turning on a unit is a logistic function of the weighted states of its immediate ancestors), if there is only one hidden layer, then the posterior distribution is independent because it is created by the likelihood term coming from the data. Thus the complementary priors can be set so that they precisely make the posterior distribution factorial, and simplify the computation of posterior distributions.

A fast, greedy learning algorithm

The main contribution of this paper is a fast greedy algorithm that can learn weights for a deep belief network. The idea of the algorithm is to construct multi-layer directed networks, one layer at a time. As each new layer is added, the overall generative model improves. The essence of the algorithm is similar to the concept of boosting, where the same weak learner is repeatedly used, but with different weighting on the data vector each time; however, in this case, it is the representation of the data vector that changes each time the weak learner is used. Here, the weak learner is an undirected graphical model.

The figure below shows a hybrid network where the top two layers have undirected connections and the layers below have directed connections in both directions.

[Technical issues -- figure should appear here]

In the above diagram, the weight matrix [math]W_0\,[/math] can be learned, to some level of accuracy, by assuming that all weight matrices are equal and treating the entire network as a Restricted Boltzmann Machine (RBM). Once [math]W_0\,[/math] is learned, [math]W_0^T\,[/math] can be used to map the data to a higher level in the first hidden layer, and a similar process can be repeated.

In each stage, the higher level weight matrices would have to be modified. The following greedy algorithm is proposed:

  1. Learn [math]W_0\,[/math] assuming all the weight matrices are tied.
  2. Freeze [math]W_0\,[/math] and use[math]W_0^T\,[/math] to infer factorial approximate posterior distributions over the states of the variables in the first hidden layer. Do this even though subsequent changes in the higher level weights mean that the inference is no longer always correct.
  3. Keep all higher weight matrices tied to each other, but untie them from [math]W_0\,[/math]. In this setting, learn an RBM for the higher level states, using results of the data having [math]W_0\,^T[/math] applied as a transformation.

The authors of the paper are able to show that if this greedy algorithm is used to change higher-level weight matrices, then the generative model is guaranteed to improve.

The up-down algorithm

The greedy learning algorithm is an effective and (relatively) rapid way to learn the weights in the deep belief network, but will not necessarily guarantee high quality weights. In order to obtain better weights, the "up-down" method has been suggested; this is a contrastive version the "wake-sleep" method proposed in a previous paper of 1995, although without some of the drawbacks.

The idea is that, after weights have been learned in such a way that the posterior in each layer must be approximated with a factorial distribution given the values of the preceding layer, the upward "recognition" weights are untied from the downward "generative" weights. Then, higher-level weights can be used to influence lower-level ones.

Each "up-pass" consists of using the recognition weights to stochastically pick states for each hidden variable, and then adjusting the generative weights using the following maximum likelihood learning rule:

[math]\frac{\partial \log p(v^0)}{\partial w_{ij}^{00}} = \langle h_j^0(v_i^0 - \hat{v_i^0})\rangle [/math]

The "down-pass" is similar, in that it iterates through layers and adjusts weight matrices, although the iteration begins at the top layers and propagates along the top-down generative connections, and it is the bottom-up recognition weights that are modified. There is some discussion on how the performance may be superior to the similar "wake-sleep" process, if the implementation carries a particular "contrastive" quality.

Performance on MNIST

The training method was applied on a deep belief net consisting of three hidden layers and approximately 1.7 million weights in an experiment ot classify MNIST data.