For the third step, the problem is that it is not trivial finding which combination of hi and Jij generates a specific combination of mj and cij. This is known as the ‘inverse Ising problem’, which can be solved by using a simple gradient descent rule18:3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{c}{h}_{i}\leftarrow {h}_{i}+\mu ({m}_{i}^{\ast }-{m}_{i}^{m})\\ {J}_{ji}\leftarrow {J}_{ji}+\mu ({c}_{ij}^{\ast }-{c}_{ij}^{m})\end{array}$$\end{document}hi←hi+μ(mi⁎−mim)Jji←Jji+μ(cij⁎−cijm)where μ is a constant learning rate, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m}_{i}^{\ast }$$\end{document}mi⁎ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ij}^{\ast }$$\end{document}cij⁎ are the reference mean and correlations of the learning algorithm, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m}_{i}^{m}$$\end{document}mim and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{ij}^{m}$$\end{document}cijm are the mean and correlations of the model for the current values of hi and Jij. Generally, performing each learning step is computationally expensive, since it requires summing over all possible states of s, although approximate methods such as Monte Carlo sampling are generally used to speed up learning. Similarly, we compute the approximate correlations by simulating our networks using the Glauber dynamics in Equation 1 for a number of steps.