Expectation Propagation with Stochastic Kinetic Model in Complex Interaction Systems
Technological breakthroughs allow us to collect data with increasing spatiotemporal resolution from complex interaction systems. The combination of highresolution observations, expressive dynamic models, and efficient machine learning algorithms can lead to crucial insights into complex interaction dynamics and the functions of these systems. In this paper, we formulate the dynamics of a complex interacting network as a stochastic process driven by a sequence of events, and develop expectation propagation algorithms to make inferences from noisy observations. To avoid getting stuck at a local optimum, we formulate the problem of minimizing Bethe free energy as a constrained primal problem and take advantage of the concavity of dual problem in the feasible domain of dual variables guaranteed by duality theorem. Our expectation propagation algorithms demonstrate better performance in inferring the interaction dynamics in complex transportation networks than competing models such as particle filter, extended Kalman filter, and deep neural networks.
This paper considers the variational approach to the inference problem on certain type of temporal graphical models. It defines a convex formulation of the Bethe free energy, resulting in a method with optimal convergence guarantees. The authors derive Expectation-Propagation fixed point equations and gradient-based updates for solving the optimization problem. A toy example is used to illustrate the approach in the context of transportation dynamics and it is shown that the proposed method outperforms sampling, extended Kalman Filtering and a neural network method.
In the variational formulation, the optimization problem is generally non-convex, due to the entropy terms. Typical variational methods bound the concave part linearly and employ an EM-like algorithm. The key contribution of this paper is to work out a dual formulation on the natural parameters so that the dual problem becomes convex. The derivations seem correct and this looks like an important contribution, if I have not missed some important detail. Authors should elaborate on the generality and applicability of their approach to other models.
I think it would be useful to compare their method with the standard double-loop approach [11] both in terms of describing the differences and in numerical comparison (in terms of error and cpu-time).
The authors state that they are minimizing the Bethe free energy, in found a bit confusing that they call this a mean field forward and backward in the algorithm.
Regarding the experiments, the state representation goes from a factorized state with the state of an agent being one component to location states in which the identities of the agents are lost and the components become strongly coupled. This is OK, but to me it contradicts a bit the motivation in the introduction, in which the authors state "we are not satisfied with only a macroscopic description of the system (...) the goal is (..) to track down the actual underlying dynamics".
Some details about the other methods would be good to know. For example, how many particles N where used for PF? How sensitive are the results (accuracy and time) w.r.t N? Also it is not clear how the NNs are trained. Is the output provided the actual hidden state?
The paper looks too dense. I think substantial part in the Introduction is not necessary and more space could be gain so that formulas do not need to be in reduced font and figure 1 can be made larger.
Other minor comments:
It should be stated in the beginning that x is continuous variable unless otherwise stated
Eq (3) no need to define g_v, since it is not used in the rest of the paper
line 130: better $h(x_{t-1},c_v)$ than $h(x_{t},c_v)$
Eq (4) I think $1-\tau$ should be $(1-\tau)$
Equations between 139-140. \psi(|x_{t-1,1}) -> \psi(x_{t-1,1})
Equations between 139-140. "F_dual" -> "F_Dual"
line 104: "dynamics" -> "dynamic"
line 126: remove "we have that"
line 193: -> "by strong duality both solutions exist and are unique"
line 194: "KKT conditions gives" -> "KKT conditions give"
Use \eqref{} instead of \ref{} for equations
Algorithm 1: the name can be improved
Algorithm 1: Gradient ascent: missing eqrefs in Eqs. ...
Algorithm 1: Gradient ascent: missing parenthesis in all $\langle f(x_t \rangle$
Figure 1
line 301: "outperforms against" -> "outperform"
line 302: "scales" -> "scale"