Overview of flow based MCM
Problem
The MCMC method suffers from Critical Slowing Down, that is near the critical point physical simulations become really time-expensive. This is due to the local nature of the method and the high auto-correlations between sites near the critical point; the flow based MCMC (Albergo, Kanwar, and Shanahan 2019), on the other hand, samples a global configuration at every markovian time step, so it can generate improved auto-correlations and does not suffer from critical slowing down.
Method and self-training
The paper proposes to use Normalizing Flows, which learn a map \(f^{-1}\) (\(f\) is called flow) to evaluate the target distribution \(\tilde{p}_{f}\) and use it in the [[Metropolis algorithm]] to sample the true distribution of our system. The main advantage of this approach is the self-training, that is we do not need real data to train the model, but only the action of the system which enters in the (shifted) Kullback-Leibler (KL) divergence, that is the loss that we want to minimize.
Real NVP flow
The main idea of this paper is enclosed in this diagram:
where \(z\) are samples drawn from a simple (prior) distribution \(r\): then \(f^{-1}\) is a change of variables allowing us to sample \(\phi\) according to the new distribution \(\tilde{p}_{f}\), which is close to the effective, physical, distribution \[ p(\phi) = \frac{e^{ -S(\phi) }}{Z}, \] with \(S\) a quartic action.
Having said that, we construct the map \(f\) using the real non-volume-preserving (NVP) flow approach, that is we costruct \(f\) by composition of affine coupling layers \(\{g_{i}\}\) \[ f(\phi) = g_{1} \circ g_{2} \circ \dots g_{n}(\phi). \]
The proposal distribution \(\tilde{p}_{f}\) is then given by \[ \tilde{p}_{f}(\phi) = r(f(\phi)) \left|\det \frac{ \partial f(\phi) }{ \partial \phi } \right|. \] Affine couplings also solve the problem of computing the jacobian.