# Finite Markov chains

In this post, we will look in more detail into an important class of Markov chains - Markov chains on finite state spaces. Many of the subtleties that are present when studying Markov chains in general state spaces do not appear in the finite case, while most of the key ideas and features of Markov … Continue reading Finite Markov chains

# Monte Carlo methods and Markov chains – an introduction

In our short series on machine learning, we have already applied sampling methods several times. We have used and implemented Gibbs sampling, and so far we have simply accepted that the approach works. Time to look at this in a bit more detail in order to understand why it works and what the limitations of … Continue reading Monte Carlo methods and Markov chains – an introduction

# Training a restricted Boltzmann machine on a GPU with TensorFlow

During the second half of the last decade, researchers have started to exploit the impressive capabilities of graphical processing units (GPUs) to speed up the execution of various machine learning algorithms (see for instance [1] and [2] and the references therein). Compared to a standard CPU, modern GPUs offer a breathtaking degree of parallelization - … Continue reading Training a restricted Boltzmann machine on a GPU with TensorFlow

# Training restricted Boltzmann machines with persistent contrastive divergence

In the last post, we have looked at the contrastive divergence algorithm to train a restricted Boltzmann machine. Even though this algorithm continues to be very popular, it is by far not the only available algorithm. In this post, we will look at a different algorithm known as persistent contrastive divergence and apply it to … Continue reading Training restricted Boltzmann machines with persistent contrastive divergence

# Learning algorithms for restricted Boltzmann machines – contrastive divergence

In the previous post on RBMs, we have derived the following gradient descent update rule for the weights. $latex \Delta W_{ij} = \beta \left[ \langle v_i \sigma(\beta a_j) \rangle_{\mathcal D} - \langle v_i \sigma(\beta a_j) \rangle_{P(v)} \right] &s=1$ In this post, we will see how this update rule can be efficiently implemented. The first thing … Continue reading Learning algorithms for restricted Boltzmann machines – contrastive divergence

# Restricted Boltzmann machines

In the previous post, we have seen that a Boltzmann machine as studied so far suffers from two deficiencies. First, training is very slow as we have to run a Gibbs sampler until convergence for every iteration of the gradient descent algorithm. Second, we can only see the second moments of the data distribution and … Continue reading Restricted Boltzmann machines

# Turn on the heating – from Hopfield networks to Boltzmann machines

In my recent post on Hopfield networks, we have seen that these networks suffer from the problem of spurious minima and that the deterministic nature of the dynamics of the network makes it difficult to escape from a local minimum. A possible approach to avoid this issue is to randomize the update rule. Intuitively, we want to … Continue reading Turn on the heating – from Hopfield networks to Boltzmann machines