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<!-- navigation toc: --> <li><a href="._BM-bs001.html#___sec0" style="font-size: 80%;">Types of Machine Learning, a repetition</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs002.html#___sec1" style="font-size: 80%;">Why Boltzmann machines?</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs003.html#___sec2" style="font-size: 80%;">An intermediate step, the Hopfield network and links to the Ising and Potts models</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs004.html#___sec3" style="font-size: 80%;">A brief review on Markov Chains, Metropolis and Gibbs sampling</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs005.html#___sec4" style="font-size: 80%;">Brownian motion and Markov processes</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs006.html#___sec5" style="font-size: 80%;">Brownian motion and Markov processes, Ergodicity and Detailed balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs007.html#___sec6" style="font-size: 80%;">Brownian motion and Markov processes, jargon</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs008.html#___sec7" style="font-size: 80%;">Brownian motion and Markov processes, sequence of ingredients</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs009.html#___sec8" style="font-size: 80%;">Applications: almost every field in science</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs010.html#___sec9" style="font-size: 80%;">Markov processes</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs011.html#___sec10" style="font-size: 80%;">Markov processes</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs012.html#___sec11" style="font-size: 80%;">Markov processes, the probabilities</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs013.html#___sec12" style="font-size: 80%;">Markov processes</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs014.html#___sec13" style="font-size: 80%;">An Illustrative Example</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs015.html#___sec14" style="font-size: 80%;">An Illustrative Example</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs016.html#___sec15" style="font-size: 80%;">An Illustrative Example, next step</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs017.html#___sec16" style="font-size: 80%;">An Illustrative Example, the steady state</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs018.html#___sec17" style="font-size: 80%;">An Illustrative Example, iterative steps</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs019.html#___sec18" style="font-size: 80%;">An Illustrative Example, what does it mean?</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs020.html#___sec19" style="font-size: 80%;">An Illustrative Example, understanding the basics</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs021.html#___sec20" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs022.html#___sec21" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs023.html#___sec22" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs024.html#___sec23" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs025.html#___sec24" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs026.html#___sec25" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs027.html#___sec26" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs028.html#___sec27" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs029.html#___sec28" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs030.html#___sec29" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs031.html#___sec30" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs032.html#___sec31" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs033.html#___sec32" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs034.html#___sec33" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs035.html#___sec34" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs036.html#___sec35" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs037.html#___sec36" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs038.html#___sec37" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs039.html#___sec38" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs040.html#___sec39" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs041.html#___sec40" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs042.html#___sec41" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs043.html#___sec42" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs044.html#___sec43" style="font-size: 80%;">Brief Summary</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs045.html#___sec44" style="font-size: 80%;">Gibbs sampling</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs046.html#___sec45" style="font-size: 80%;">Boltzmann Machines</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs047.html#___sec46" style="font-size: 80%;">Some similarities and differences from DNNs</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs048.html#___sec47" style="font-size: 80%;">Boltzmann machines (BM)</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs049.html#___sec48" style="font-size: 80%;">A standard BM setup</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs050.html#___sec49" style="font-size: 80%;">The structure of the RBM network</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs051.html#___sec50" style="font-size: 80%;">The network</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs052.html#___sec51" style="font-size: 80%;">Goals</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs053.html#___sec52" style="font-size: 80%;">Joint distribution</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs054.html#___sec53" style="font-size: 80%;">Network Elements, the energy function</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs055.html#___sec54" style="font-size: 80%;">Defining different types of RBMs</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs056.html#___sec55" style="font-size: 80%;">More about RBMs</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs057.html#___sec56" style="font-size: 80%;">Sampling: Metropolis sampling</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs058.html#___sec57" style="font-size: 80%;">Sampling: Gibbs sampling</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs059.html#___sec58" style="font-size: 80%;">Gaussian RBM</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs060.html#___sec59" style="font-size: 80%;">Cost function</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs061.html#___sec60" style="font-size: 80%;">Optimization / Training</a></li>
<!-- navigation toc: --> <li><a href="#___sec61" style="font-size: 80%;">More on RBMs</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs063.html#___sec62" style="font-size: 80%;">Which sampling to use</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs064.html#___sec63" style="font-size: 80%;">Kullback-Leibler relative entropy</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs065.html#___sec64" style="font-size: 80%;">Optimizing the cost function</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs066.html#___sec65" style="font-size: 80%;">Setting up for gradient descent calculations</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs067.html#___sec66" style="font-size: 80%;">More interpretations</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs068.html#___sec67" style="font-size: 80%;">Recent examples: RBMs for the quantum many body problem</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs069.html#___sec68" style="font-size: 80%;">Choose the right RBM</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs070.html#___sec69" style="font-size: 80%;">Representing the wave function</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs071.html#___sec70" style="font-size: 80%;">Choose the cost function</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs072.html#___sec71" style="font-size: 80%;">Running the codes</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs073.html#___sec72" style="font-size: 80%;">Energy as function of iterations, \( N=2 \) electrons</a></li>
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<h2 id="___sec61" class="anchor">More on RBMs </h2>
<p>
The data-dependent term in the gradient is known as the positive phase of the gradient, while the model-dependent term is known as the negative phase of the gradient. The aim of the training is to lower the energy of configurations that are near observed data points (increasing their probability), and raising the energy of configurations that are far from observed data points (decreasing their probability).
<p>
The gradient of the negative log-likelihood cost function of a Binary-Binary RBM is then
$$
\begin{align}
\frac{\partial \mathcal{C} (w_{ij}, a_i, b_j)}{\partial w_{ij}} =& \langle x_i h_j \rangle_{data} - \langle x_i h_j \rangle_{model}
\tag{23}\\
\frac{\partial \mathcal{C} (w_{ij}, a_i, b_j)}{\partial a_{ij}} =& \langle x_i \rangle_{data} - \langle x_i \rangle_{model}
\tag{24}\\
\frac{\partial \mathcal{C} (w_{ij}, a_i, b_j)}{\partial b_{ij}} =& \langle h_i \rangle_{data} - \langle h_i \rangle_{model}.
\tag{25}\\
\tag{26}
\end{align}
$$
To get the expecation values with respect to the <em>data</em>, we set the visible units to each of the observed samples in the training data, then update the hidden units according to the conditional probability found before. We then average over all samples in the training data to calculate expectation values with respect to the data.
<p>
<p>
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