The training procedure of choice often is Stochastic Gradient Descent (SGD). It consists of a series of iterations where we update the parameters according to the equation $$ \begin{align} \boldsymbol{\theta}_{k+1} = \boldsymbol{\theta}_k - \eta \nabla \mathcal{C} (\boldsymbol{\theta}_k) \tag{15} \end{align} $$ at each \( k \)-th iteration. There are a range of variants of the algorithm which aim at making the learning rate \( \eta \) more adaptive so the method might be more efficient while remaining stable.
We now need the gradient of the cost function in order to minimize it. We find that $$ \begin{align} \frac{\partial \mathcal{C}(\{ \theta_i\})}{\partial \theta_i} &= \langle \frac{\partial E(\boldsymbol{x}; \theta_i)}{\partial \theta_i} \rangle_{data} + \frac{\partial \text{log} Z(\{ \theta_i\})}{\partial \theta_i} \tag{16}\\ &= \langle O_i(\boldsymbol{x}) \rangle_{data} - \langle O_i(\boldsymbol{x}) \rangle_{model}, \tag{17} \end{align} $$ where in order to simplify notation we defined the "operator" $$ \begin{align} O_i(\boldsymbol{x}) = \frac{\partial E(\boldsymbol{x}; \theta_i)}{\partial \theta_i}, \tag{18} \end{align} $$ and used the statistical mechanics relationship between expectation values and the log-partition function: $$ \begin{align} \langle O_i(\boldsymbol{x}) \rangle_{model} = \text{Tr} P_\theta(\boldsymbol{x})O_i(\boldsymbol{x}) = - \frac{\partial \text{log} Z(\{ \theta_i\})}{\partial \theta_i}. \tag{19} \end{align} $$