Optimizing restricted Boltzmann machine learning by injecting Gaussian noise to likelihood gradient approximation

Abstract

Restricted Boltzmann machines (RBMs) can be trained by applying stochastic gradient ascent to the objective function as the maximum likelihood learning. However, it is a difficult task due to the intractability of marginalization function gradient. Several methodologies have been proposed by adopting Gibbs Markov chain to approximate this intractability including Contrastive Divergence, Persistent Contrastive Divergence, and Fast Contrastive Divergence. In this paper, we propose an optimization which is injecting noise to underlying Monte Carlo estimation. We introduce two novel learning algorithms. They are Noisy Persistent Contrastive Divergence (NPCD), and further Fast Noisy Persistent Contrastive Divergence (FNPCD). We prove that the NPCD and FNPCD algorithms benefit on the average to equilibrium state with satisfactory condition. We have performed empirical investigation of diverse CD-based approaches and found that our proposed methods frequently obtain higher classification performance than traditional approaches on several benchmark tasks in standard image classification tasks such as MNIST, basic, and rotation datasets.

Appendix: Complete derivative of BG-RBM

Energy function:

$$E\left( \textbf{v},\textbf{h}\right)_{BG} = \sum\limits_{i = 1}^{n_{v}}\frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}}-\sum\limits_{j = 1}^{n_{h}}b_{j}h_{j} -\sum\limits_{i = 1}^{n_{v}}\sum\limits_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j} $$

1.1 Definition of conditional probability P(h|v)

$$\begin{array}{@{}rcl@{}} P(\textbf{h}|\textbf{v}) &=& \frac{P(\textbf{v},\textbf{h})}{P(\textbf{h})} = \frac{\frac{1}{Z}e^{-E(\textbf{v},\textbf{h})}}{\frac{1}{Z}{\sum}_{\textbf{h}} e^{-E(\textbf{v},\textbf{h})}} = \frac{e^{-E(\textbf{v},\textbf{h})}}{{\sum}_{\textbf{h}}e^{-E(\textbf{v},\textbf{h})}} \\ & =& \frac{e^{-\left( {\sum}_{i = 1}^{n_{v}}\frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}}-{\sum}_{j = 1}^{n_{h}}b_{j}h_{j} -{\sum}_{i = 1}^{n_{v}}{\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) }}{{\sum}_{h} e^{-\left( {\sum}_{i = 1}^{n_{v}}\frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}}-{\sum}_{j = 1}^{n_{h}}b_{j}h_{j} -{\sum}_{i = 1}^{n_{v}}{\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) }} \\ & =& \frac{e^{-{\sum}_{i = 1}^{n_{v}} \left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}}{{\sum}_{h} e^{-{\sum}_{i = 1}^{n_{v}} \left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}} \end{array} $$

Rewritting the above equation in term of product of expert model:

$$\begin{array}{@{}rcl@{}} P(\textbf{h}|\textbf{v}) & =& \frac{{\prod}_{j} e^{-{\sum}_{i = 1}^{n_{v}} \left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}}{{\prod}_{j} {\sum}_{h} e^{-{\sum}_{i = 1}^{n_{v}} \left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}} \\ & =& \prod\limits_{j} \frac{e^{\left( \frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}+b_{j}h_{j} \right) }}{{\sum}_{h} e^{\left( \frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}+b_{j}h_{j} \right) } } = \prod\limits_{j} P(h_{j}|\textbf{v}) \end{array} $$

For binary h_j ∈{0, 1}, P(h_j = 1) equals to:

$$P(h_{j} = 1) = \frac{e^{\left( \frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}+b_{j} \right) }}{e^{\left( \frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}+b_{j} \right) } + e^{0}} = sig\left( \frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}+b_{j}\right) $$

1.2 Definition of conditional probability P(v|h)

$$\begin{array}{@{}rcl@{}} P(\textbf{v}|\textbf{h}) & =& \frac{P(\textbf{v},\textbf{h})}{P(\textbf{v})} = \frac{\frac{1}{Z}e^{-E(\textbf{v},\textbf{h})}}{\frac{1}{Z}{\int}_{\textbf{v}} e^{-E(\textbf{v},\textbf{h})}dv} = \frac{e^{-E(\textbf{v},\textbf{h})}}{{\int}_{\textbf{v}}e^{-E(\textbf{v},\textbf{h})}dv} \\ & =& \frac{e^{-{\sum}_{i = 1}^{n_{v}} \left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}}{{\int}_{\textbf{v}} e^{-{\sum}_{i = 1}^{n_{v}} \left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}dv} \end{array} $$

Rewritting the above equation in term of product of expert model:

$$P(\textbf{v}|\textbf{h}) = \frac{{\prod}_{i} e^{-\left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}}{{\prod}_{i} {\int}_{\textbf{v}} e^{- \left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}dv} $$

Simplifying the denominator:

$$\begin{array}{@{}rcl@{}} P(\textbf{v}|\textbf{h}) & = & \frac{{\prod}_{i} e^{\!-\left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}}{{\prod}_{i} {\int}_{\textbf{v}} e^{-\left( \frac{1}{2{\sigma_{i}^{2}}}({v_{i}^{2}} - 2v_{i}a_{i} + {a_{i}^{2}})-{\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}} W_{ij}h_{j} \right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j} } dv } \\ & = & \frac{{\prod}_{i} e^{-\left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}}{{\prod}_{i} e^{\left( \frac{-{a_{i}^{2}}}{2{\sigma_{i}^{2}}}\right)+{\sum}_{j = 1}^{n_{h}}b_{j}h_{j}} {\int}_{\textbf{v}}e^{\frac{1}{2{\sigma_{i}^{2}}}\left( -{v_{i}^{2}} + 2v_{i}a_{i} \right)+{\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}} dv} \\ & = & \frac{{\prod}_{i} e^{-\left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}}{{\prod}_{i} e^{\left( \frac{-{a_{i}^{2}}}{2{\sigma_{i}^{2}}}\right)+{\sum}_{j = 1}^{n_{h}}b_{j}h_{j}} {\int}_{\textbf{v}}e^{\frac{1}{2{\sigma_{i}^{2}}}\left( -{v_{i}^{2}}\right)} e^{v_{i} \left( \frac{a_{i}}{{\sigma_{i}^{2}}} + {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}} W_{ij}h_{j} \right)^{2} } dv} \end{array} $$

Integrating the denominator w.r.t v:

$$\begin{array}{@{}rcl@{}} & =& \frac{{\prod}_{i} e^{-\left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}}{{\prod}_{i} e^{\left( \frac{-{a_{i}^{2}}}{2{\sigma_{i}^{2}}}\right)+{\sum}_{j = 1}^{n_{h}}b_{j}h_{j}} e^{\frac{{\sigma_{i}^{2}} \left( \frac{a_{i}}{{\sigma_{i}^{2}}} + {\sum}_{j = 1}^{n_{h}}\frac{W_{ij}}{{\sigma_{i}^{2}}}h_{j}\right)^{2} }{2}} \left( \sqrt{2{\sigma_{i}^{2}}\pi}\right)} \\ & =& \frac{{\prod}_{i} e^{-\left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}}}{\left( \sigma_{i}\sqrt{2\pi}\right) {\prod}_{i} e^{\frac{1}{2} \left( {\sum}_{j = 1}^{n_{h}}W_{ij}h_{j} \right)^{2}+{\sum}_{j = 1}^{n_{h}}b_{j}h_{j}+\frac{a_{i}W_{ij}h_{j}}{{\sigma_{i}^{2}}}} } \end{array} $$

Simplifying the above equation:

$$\begin{array}{@{}rcl@{}} & =& \prod\limits_{i} \frac{1}{\sigma_{i}\sqrt{2\pi}} e^{-\left( \frac{(v_{i}-a_{i})^{2}}{2{\sigma_{i}^{2}}} - {\sum}_{j = 1}^{n_{h}}\frac{v_{i}}{{\sigma_{i}^{2}}}W_{ij}h_{j}\right) + {\sum}_{j = 1}^{n_{h}}b_{j}h_{j}-\left( \frac{1}{2} \left( {\sum}_{j = 1}^{n_{h}}W_{ij}h_{j} \right)^{2}+{\sum}_{j = 1}^{n_{h}}b_{j}h_{j}+\frac{a_{i}W_{ij}h_{j}}{{\sigma_{i}^{2}}}\right) } \\ & =& \prod\limits_{i} \frac{1}{\sigma_{i}\sqrt{2\pi}}e^{-\frac{1}{2{\sigma_{i}^{2}}} \left( v_{i} - \left( a_{i}+{\sum}_{j = 1}^{n_{h}}W_{ij}h_{j}\right) \right)^{2}} \end{array} $$

The above equation is a probability density function of Gaussian distribution with mean $v_{i} - \left (a_{i}+{\sum }_{j = 1}^{n_{h}}W_{ij}h_{j}\right )$ and variance ${\sigma _{i}^{2}}$.

1.3 Derivative of log-likelihood function

$$\begin{array}{@{}rcl@{}} \frac{\partial\ln P(\textbf{v})}{\partial{W_{ij}}} & =& \frac{\partial\ln{\sum}_{\textbf{h}}e^{-E(\textbf{v},\textbf{h})} }{\partial{W_{ij}}}-\frac{\partial\ln{\sum}_{\textbf{h}}{\sum}_{\textbf{v}}e^{-E(\textbf{v},\textbf{h})} }{\partial{W_{ij}}} \\ & =& \frac{1}{{\sum}_{\textbf{h}}e^{-E(\textbf{v},\textbf{h})}} \left( {\sum}_{\textbf{h}}e^{-E(\textbf{v},\textbf{h})}.-\frac{\partial E(\textbf{v},\textbf{h})}{\partial{W_{ij}}} \right) \\&&- \frac{1}{{\sum}_{\textbf{v},\textbf{h}}e^{-E(\textbf{v},\textbf{h})}} \left( {\sum}_{\textbf{v},\textbf{h}}e^{-E(\textbf{v},\textbf{h})}.-\frac{\partial E(\textbf{v},\textbf{h})}{\partial{W_{ij}}} \right) \\ & =& - {\sum}_{\textbf{h}}\frac{e^{-E(\textbf{v},\textbf{h})}}{{\sum}_{\textbf{h}} e^{-E(\textbf{v},\textbf{h})}}.\left( \frac{\partial E(\textbf{v},\textbf{h})}{\partial{W_{ij}}} \right) \\&&+ {\sum}_{\textbf{v},\textbf{h}}\frac{e^{-E(\textbf{v},\textbf{h})}}{{\sum}_{\textbf{v},\textbf{h}} e^{-E(\textbf{v},\textbf{h})}}.\left( \frac{\partial E(\textbf{v},\textbf{h})}{\partial{W_{ij}}} \right) \\ & =& -{\sum}_{\textbf{h}} P(\textbf{h}|\textbf{v}).\frac{\partial E(\textbf{v},\textbf{h})}{\partial{W_{ij}}} + {\sum}_{\textbf{v},\textbf{h}} P(\textbf{h},\textbf{v}).\frac{\partial E(\textbf{v},\textbf{h})}{\partial{W_{ij}}} \end{array} $$

1.4 Derivative of log-likelihood approximation

$$\begin{array}{@{}rcl@{}} \frac{\partial\ln P(\textbf{v})}{\partial{W_{ij}}} & \simeq& -\sum\limits_{\textbf{h}} P(\textbf{h}|\textbf{v}) \frac{\partial E(\textbf{v},\textbf{h})}{\partial{W_{ij}}} + {\sum}_{\tilde{\textbf{v}}} P(\tilde{\textbf{v}}) {\sum}_{\tilde{\textbf{h}}} P(\tilde{\textbf{h}}|\tilde{\textbf{v}}) \frac{\partial E(\tilde{\textbf{v}},\tilde{\textbf{h}})}{\partial{W_{ij}}} \\ & \simeq& \sum\limits_{\textbf{h}} P(\textbf{h}|\textbf{v})\frac{v_{i}h_{j}}{\sigma^{2}} - {\sum}_{\tilde{\textbf{v}}}P(\tilde{\textbf{v}}) {\sum}_{\tilde{\textbf{h}}} P(\tilde{\textbf{h}}|\tilde{\textbf{v}})\frac{\tilde{v_{i}}\tilde{h_{j}}}{\sigma^{2}} \\ & \simeq& \sum\limits_{\textbf{h}} P(h=+ 1|\textbf{v})\frac{v_{i}}{\sigma^{2}}-{\sum}_{\tilde{\textbf{v}}}P(\tilde{\textbf{v}}){\sum}_{\tilde{\textbf{h}}}P(\tilde{h} = + 1|\tilde{\textbf{v}})\frac{\tilde{v_{i}}}{\sigma^{2}} \end{array} $$

By applying this procedure to all biases, we obtain the gradients as follows:

$$\begin{array}{@{}rcl@{}} \frac{\partial\ln P(\textbf{v})}{\partial{W_{ij}}} & \simeq& \frac{v_{i} h_{j} - \tilde{v}_{i} \tilde{h}_{j}}{\sigma^{2}} \end{array} $$

$$\begin{array}{@{}rcl@{}} \frac{\partial\ln P(\textbf{v})}{\partial{a_{i}}} & \simeq& \frac{v_{i} - \tilde{v}_{i} }{\sigma^{2}} \\ \frac{\partial\ln P(\textbf{v})}{\partial{b_{j}}} & \simeq& h_{j} - \tilde{h}_{j} \end{array} $$