Escaping Saddle Points in Constant Dimensional Spaces: an Agent-based Modeling Perspective Grant Schoenebeck, University of Michigan Fang-Yi Yu , Harvard University
Reinforced random walk with πΊ A discrete time stochastic process {π π : π = 0, 1, β¦ } in β π that admits the following representation, π π π π+1 β π π = 1 1 π πΊ π π + π π π πΊ(π π ) β’ Agent based models with π agents 1 π π π β Evolutionary games π π+1 β Dynamics on social networks β’ Heuristic local search algorithms with uniform step size 1/π
Node Dynamic ππ(π», π πΆπ¬ , π π ) [SY18] β’ Fixed a (weighted) graph π» = (π, πΉ) π π π’β1 π€ = 1 7 opinion set {0,1} , an update function π πΆπ¬ β’ Given an initial configuration π 0 :V β¦ {0,1} β’ At round t, β’ A node v is picked uniformly at random β’ π π π = 1 w.p. π πΆπ¬ π π πβπ π ; = 0 otherwise
Gradient-like dynamics Converges to an attracting fixed-point region in O(π log π) steps. If β’ Noise, π π β Martingale difference β bounded β Noisy β’ Expected difference, πΊ β π 2 β Fixed points are hyperbolic β Potential function
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