Escaping Saddle Points in Constant Dimensional Spaces: an - - PowerPoint PPT Presentation

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Dec 07, 2022 232 likes β€’293 views

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 A discrete time stochastic process { :

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Escaping Saddle Points in Constant Dimensional Spaces: an Agent-based Modeling Perspective

Grant Schoenebeck, University of Michigan Fang-Yi Yu, Harvard University

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Reinforced random walk with 𝐺

A discrete time stochastic process {π‘Œπ‘™: 𝑙 = 0, 1, … } in ℝ𝑒 that admits the following representation, π‘Œπ‘™+1 βˆ’ π‘Œπ‘™ = 1 π‘œ 𝐺 π‘Œπ‘™ + 𝑉𝑙

  • Agent based models with π‘œ agents

– Evolutionary games – Dynamics on social networks

  • Heuristic local search algorithms with uniform step size 1/π‘œ

π‘Œπ‘™ 1 π‘œ 𝐺(π‘Œπ‘™) π‘Œπ‘™+1 1 π‘œ 𝑉𝑙

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Node Dynamic πŽπ„(𝐻, 𝑔

𝑢𝑬, π‘ŒπŸ)[SY18]

  • Fixed a (weighted) graph 𝐻 = (π‘Š, 𝐹)
  • pinion set {0,1}, an update function

π’ˆπ‘Άπ‘¬

  • Given an initial configuration

π‘Œ0:V ↦ {0,1}

  • At round t,
  • A node v is picked uniformly at random
  • 𝒀𝒖 π’˜ = 1 w.p. π’ˆπ‘Άπ‘¬ π’”π’€π’–βˆ’πŸ π’˜

; = 0 otherwise

π‘ π‘Œπ‘’βˆ’1 𝑀 = 1 7

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Gradient-like dynamics

Converges to an attracting fixed-point region in O(π‘œ log π‘œ) steps.

If

  • Noise, 𝑉𝑙

– Martingale difference – bounded – Noisy

  • Expected difference, 𝐺 ∈ π’Ÿ2

– Fixed points are hyperbolic – Potential function