Langevin type dynamics for continuous and discrete systems Lukas - PowerPoint PPT Presentation

Langevin type dynamics for continuous and discrete systems Lukas Kades Cold Quantum Coffee - Heidelberg University - 17 April 2018

Structure ◮ Motivation ◮ Langevin dynamics for the 2D Ising model ◮ Demystification ◮ Results ◮ Applications http://www.kdnuggets.com Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 2

Motivation Langevin Dynamics Langevin equation: ∂ δ S ∂τ φ x ( τ ) = − δφ x ( τ ) + η x ( τ ) with Gaussian noise: http://mathsissmart.tumblr.com � η x ( τ ) , η x ′ ( τ ′ ) � η = 2 δ ( x − x ′ ) δ ( τ − τ ′ ) � η x ( τ ) � η = 0 Aim: Application of this ⇒ formalism on neuromorphic hardware http://www.fair-center.eu Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 3

Motivation Human Brain Project/BrainScaleS - Introduction ◮ Neuromorphic computing system ◮ 1 . 6 million neurons ◮ 0 . 4 billion dynamic synapses ◮ 10000 times faster than their biological Electronic Vision(s) Group, Heidelberg University archetypes Petrovici M.A., PhD Thesis, 2016 Electronic Vision(s) Group, Heidelberg University Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 4

Motivation Human Brain Project/BrainScaleS - Stochastic interference Ornstein-Uhlenbeck process: du eff ( t ) = Θ [ µ − u eff ( t )] + ση ( t ) dt with: 1 � � Θ = , µ = u leak + κ ( t , t s , i ) τ syn syn i spk s Free membrane potential u eff ( t ): http://www.kdnuggets.com and Petrovici M.A., PhD Thesis, 2016 Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 5

Motivation Comparison Langevin dynamics: BrainScaleS: ∂ δ S du eff ( t ) ∂τ φ x ( τ ) = − δφ x ( τ )+ η x ( τ ) = Θ [ µ − u eff ( t )]+ ση ( t ) dt ◮ Continuous system ◮ Effective two-state ⇔ ◮ Gaussian noise system ◮ Gaussian noise contribution contribution ◮ Coupled system ◮ Coupled system Langevin equation for a discrete two-state system? Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 6

How could a Langevin equation look like for the Ising model? Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 7

Langevin Dynamics for the 2D Ising Model 2D Ising Model ◮ N 2 states s i ∈ {− 1 , 1 } = {↓ , ↑} on a square lattice ◮ Hamiltonian: � � H = − J s i s j − h s i � i , j � i ◮ Second order phase transition ⇒ Can be mapped easily onto a Boltzmann machine with s i ∈ { 0 , 1 } : H = − 1 � � W ij s i s j − b i s i 2 ij i Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 8

Langevin Dynamics for the 2D Ising Model Heuristic Approach Langevin equation: ∂ δ S ∂τ φ x ( τ ) = − δφ x ( τ ) + η x ( τ ) ⇓ Discrete Langevin equation ( φ x := φ x ( τ ), φ ′ x := φ x ( τ + ǫ )): + √ ǫη x , x = φ x − ǫ δ S φ ′ δφ x ⇓ Identifications S = β H , φ x = s x : + √ ǫη i i = s i − ǫβ ∂ H s ′ ∂ s i Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 9

Langevin Dynamics for the 2D Ising Model Heuristic Approach Hamiltonian with s i ∈ {− 1 , 1 } = {↓ , ↑} : � � H = − J s i s j − h s i � i , j � i Langevin equation: � + √ ǫη i � s i − ǫβ ∂ H s ′ i = sign ∂ s i Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 10

Langevin Dynamics for the 2D Ising Model Numerical Results 1 Langevin equation: 0 . 8 |Magentization| � + √ ǫη i � s i − ǫβ ∂ H 0 . 6 s ′ i = sign ∂ s i 0 . 4 Standard MC ǫ = 0 . 2 0 . 2 ǫ = 0 . 4 ǫ = 0 . 6 ⇓ ǫ = 0 . 8 ǫ = 1 . 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 β Improved Langevin equation 1 with adapted noise j � = δ ( j − i ) δ ( t ′ − t ): 0 . 8 η ′ � ˜ η i , ˜ |Magentization| 0 . 6 � + √ ǫ ˜ � s i − ǫ β ∂ H s ′ 0 . 4 i = sign η i λ ( ǫ ) ∂ s i Standard MC ǫ = 0 . 2 0 . 2 ǫ = 0 . 4 ǫ = 0 . 6 ǫ = 0 . 8 ǫ = 1 . 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 β Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 11

I. Is this a Langevin equation for the Ising model? � + √ ǫ ˜ � s i − ǫ β ∂ H s ′ i = sign η i λ ( ǫ ) ∂ s i II. Why does this work? III. Is this a MCMC algorithm with the Boltzmann distribution P ( s ) ∝ exp( − β H ( s )) as equilibrium distribution? Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 12

To III.: Markov Chain Monte Carlo Algorithm? Markov Property Update rule: + √ ǫ ˜ � s i − ǫ β ∂ H � s ′ i = sign η i λ ( ǫ ) ∂ s i � x 1 2 π exp( − t 2 / 2)): Transition probabilities (Φ( x ) = −∞ dt √ √ ǫβ � � − 1 ∂ H W ( ↓→↑ ) = W ( ↓ | ↑ ) = Φ √ ǫ − λ ( ǫ ) ∂ s i √ ǫβ � � − 1 ∂ H √ ǫ + W ( ↑→↓ ) = W ( ↑ | ↓ ) = Φ λ ( ǫ ) ∂ s i ⇒ Markov property is fulfilled. Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 13

To III.: Markov Chain Monte Carlo Algorithm? Ergodicity and Detailed Balance ◮ Ergodicity ⇒ Yes! ◮ Detailed balance equation: W ( ↑→↓ ) P ( ↑ ) = W ( ↓→↑ ) P ( ↓ ) W ( ↓→↑ ) = P ( ↑ ) ! ⇒ P ( ↓ ) = exp [ − β ( E ( ↑ ) − E ( ↓ ))] W ( ↑→↓ ) ⇒ Detailed balance equation seems to be satisfied. Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 14

To II.: Why does this work? ◮ Limit between the cumulative Gaussian distribution and the exponential function: √ ǫ � � − 1 √ ǫ − Φ λ ( ǫ ) x ∝ exp( − x ) ◮ Symmetry properties of the Ising model ( H = − J � � i , j � s i s j − h � i s i ): E ( ↑ ) = − E ( ↓ ) = ∂ H ∂ s i Detailed balance equation: √ ǫβ � � − 1 ∂ H Φ √ ǫ − W ( ↓→↑ ) λ ( ǫ ) ∂ s i � = exp [ − β ( E ( ↑ ) − E ( ↓ ))] ⇒ W ( ↑→↓ ) = √ ǫβ � − 1 ∂ H Φ √ ǫ + λ ( ǫ ) ∂ s i Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 15

To I.: Is this is a Langevin equation for the Ising model? � + √ ǫ ˜ � s i − ǫ β ∂ H s ′ i = sign η i λ ( ǫ ) ∂ s i ⇒ No! It is a Langevin like MCMC algorithm ⇒ with Gaussian noise input. Did we learn something? Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 16

Result I: Generalised Relations Cumulative Gaussian Distribution 10 ǫ = 0 . 3 ǫ = 0 . 7 ǫ = 1 . 0 ǫ = 2 . 0 exp( x ) 1 n ǫ ( x ) 12 8 n ǫ ( x ) 0 . 1 4 0 − 6 − 4 − 2 0 2 4 6 x 0 . 01 − 6 − 4 − 2 0 2 4 6 x √ ǫ + √ ǫ x √ ǫϕ � � � � − 1 − 1 Φ √ ǫ λ ǫ = exp( x )+ O ( ǫ x 2 ) , lim with: λ ǫ = Φ( − 1 � � √ ǫ ) − 1 ǫ → 0 Φ √ ǫ Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 17

Result I: Generalised Relations Derivatives of the Cumulative Gaussian Distribution 10 m = 1, ǫ = 0 . 2 m = 3, ǫ = 0 . 2 m = 10, ǫ = 0 . 2 m = 30, ǫ = 0 . 2 exp( x ) 1 n ǫ ( x ) 12 8 n ǫ ( x ) 0 . 1 4 0 − 6 − 4 − 2 0 2 4 6 x 0 . 01 − 6 − 4 − 2 0 2 4 6 x √ ǫ + √ ǫ t � � � ∂ m − 1 � ∂ t m Φ � He m ( − 1 / √ ǫ ) � t = − √ ǫ He m − 1( − 1 / √ ǫ ) x = exp( x ) + O ( ǫ x 2 ) lim √ ǫ + √ ǫ t ) ∂ m ∂ t m Φ( − 1 � ǫ → 0 � t =0 Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 18

Result II: MCMC Algorithm based on Gaussian Noise Generalised update rule: 2 λ ( ǫ )∆ E ( ν, s i ) + √ ǫη T � � ǫβ s ′ i = s i + ( ν − s i )Θ − 1 − i with a proposal state ν . For the Ising model this is equivalent to: + √ ǫ ˜ � � s i − ǫ β ∂ H s ′ ⇔ i = sign η i λ ( ǫ ) ∂ s i Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 19

Application I: Numerical Results for Other Models 9 Standard MC ǫ = 0 . 2 8 ǫ = 0 . 5 ǫ = 1 . 0 q-Potts model: ǫ = 1 . 5 7 ǫ = 2 . 0 6 Specific Heat � 5 H p = − J p δ s i , s j , 4 � i , j � 3 2 with s i ∈ { 1 , 2 , . . . , q } . 1 0 0 . 8 0 . 9 1 1 . 1 1 . 2 1 . 3 1 . 4 β Clock model: 3 . 5 Standard MC ǫ = 0 . 2 ǫ = 0 . 5 3 ǫ = 1 . 0 ǫ = 1 . 5 � H c = − J c cos ( θ i − θ j ) , ǫ = 2 . 0 2 . 5 Specific Heat � i , j � 2 1 . 5 with θ i = 2 π n q ) and n ∈ 1 { 1 , 2 , . . . , q } . 0 . 5 0 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 β Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 20

Application II: Langevin Machine Scheme: Identifications: ◮ W ′ 1 ii = √ ǫ √ ǫ ◮ b ′ i = λ ( ǫ ) b i √ ǫ ◮ W ′ ij = λ ( ǫ ) W ij Implicit update rule: s ′ i = sign( u i + ˜ η i ) 1 Activation function: p i ( ↑ ) = 1 + exp(2 H i ( ↑ )) ⇒ Alternative implementation of the Boltzmann machine with a different update dynamic and self interaction Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 21

Does this help for a computation on the neuromorphic hardware of the BrainScaleS project? ⇒ No! Still different dynamics. Lukas Kades Langevin type dynamics for continuous and discrete systems 17 April 2018 22