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# Analysis of kinetic models for label switching and stochastic gradient descent

• *Corresponding author: Alex Rossi

The authors gratefully acknowledge the support by the RTG 2339 "Interfaces, Complex Structures and Singular Limits" of the German Research Foundation (DFG)

• In this paper we provide a novel approach to the analysis of kinetic models for label switching, which are used for particle systems that can randomly switch between gradient flows in different energy landscapes. Besides problems in biology and physics, we also demonstrate that stochastic gradient descent, the most popular technique in machine learning, can be understood in this setting, when considering a time-continuous variant.

Our analysis is focusing on the case of evolution in a collection of external potentials, for which we provide analytical and numerical results about the evolution as well as the stationary problem.

Mathematics Subject Classification: Primary: 35A01, 35A02, 35R09, 45K05; Secondary: 68T07.

 Citation:

• Figure 1.  Solutions with fixed label and convergence in semilogarithmic scale towards stationary states given by $d = 1$ with quadratic potentials $f(x, s) = \frac s2|x - s|^2$ and $f(x, s) = sx^2$, $S = \left\{ {1, 2} \right\}$ and $\mu = Bern(p)$ compared with the estimate provided by theorem 3.2

Figure 2.  Stationary state of example 4 with $v = (0, 1)$ and $p = 0.7$. The red dots represent the support of this state

Figure 3.  Stationary state of example 4 with v = (1; 1) and p = 0:7. The red dots represent the support of this state

Figure 4.  Solutions for a fixed label to equation (2) with $f(x, s) = \frac s2|x - s|^2$: in the first two figures the solutions with $K = 10$ and $K = 10^9$, in the figure below the corresponding solution of the gradient flow as discussed in subsection 5.1

Figure 5.  Convergence in semilogarithmic scale given by $d = 1$ with quadratic potentials $f(x, s) = \frac s2|x - s|^2$ and $f(x, s) = sx^2$, $K(t) = 1 + t$, $S = \left\{ {1, 2} \right\}$ and $\mu = Bern(0.5)$ towards $\delta_{5/3}$ and $\delta_0$ compared with the theoretical estimate of subsection 5.3

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