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Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Ulisse Ferrari Institut de la Vision, Sorbonne Universités, UPMC New Frontiers in Non-equilibrium Physics 2015

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Outlook of the seminar 1 Introduction with an application of pairwise Ising Model to Neuroscience 2 Maximal Entropy model and the Vanilla (Standard) Learning Algorithm 3 Approximate Newton Method 4 The Long-Time Limit: Stochastic Dynamics 5 Properties of the Stationary Distribution 6 Conclusions and Perspectives

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Model Inference : Finding the probability distribution reproducing the data system statistics.

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Model Inference : Finding the probability distribution reproducing the data system statistics. Useful for characterizing the behavior of systems of many, strongly correlated, units: neurons, proteins, virus, species distribution, bird flocks but...

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Model Inference : Finding the probability distribution reproducing the data system statistics. Useful for characterizing the behavior of systems of many, strongly correlated, units: neurons, proteins, virus, species distribution, bird flocks but... which distribution?

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Model Inference : Finding the probability distribution reproducing the data system statistics. Useful for characterizing the behavior of systems of many, strongly correlated, units: neurons, proteins, virus, species distribution, bird flocks but... which distribution? Maximum Entropy (MaxEnt) Inference: Search for the largest entropy distribution satisfying a set of constraints.

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Example: pairwise Ising Model Given binary units data-set of B configurations of N units: � B � { σ i ( b ) } N i = 1 b = 1 Find the MaxEnt model reproducing single and pairwise correlations: � σ i � MODEL = � σ i � DATA ≡ 1 � b σ i ( b ) B � σ i σ j � MODEL = � σ i σ j � DATA ≡ 1 � b σ i ( b ) σ j ( b ) B

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Example: pairwise Ising Model Given binary units data-set of B configurations of N units: � B � { σ i ( b ) } N i = 1 b = 1 Find the MaxEnt model reproducing single and pairwise correlations: � σ i � MODEL = � σ i � DATA ≡ 1 � b σ i ( b ) B � σ i σ j � MODEL = � σ i σ j � DATA ≡ 1 � b σ i ( b ) σ j ( b ) B Finely tune the parameters { h , J } of the pairwise Ising model: � � � P h , j ( σ ) = exp i h i σ i + � ij J ij σ i σ j / Z [ h , J ]

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction In vivo Pre-Frontal Cortex Recording:

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction In vivo Pre-Frontal Cortex Recording: 97 experimental sessions of: Peyrache et al. Nat. Neurosci. (2009)

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Ising Model Inference σ i ( b ) = 1 if neuron i spiked during time-bin b Ask to reproduce neurons firing rates and correlations. Schneidman et al. Nature 2006; Cocco, Monasson ,PRL (2011)

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Ising Model Inference ⇒ ⇒ ⇒ σ i ( b ) = 1 if neuron i spiked during time-bin b Ask to reproduce neurons firing rates and correlations. Schneidman et al. Nature 2006; Cocco, Monasson ,PRL (2011)

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Ising Model Inference ⇒ ⇒ ⇒ σ i ( b ) = 1 if neuron i spiked during time-bin b Ask to reproduce neurons firing rates and correlations. 97 × 3 couplings network sets ( 97 × { PRE, TASK , POST } ) Schneidman et al. Nature 2006; Cocco, Monasson ,PRL (2011)

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Learning related coupling Adjustement � � � � � J TASK − J PRE J POST − J PRE A = sign · ij ij ij ij i , j : J TASK , J POST � 0

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Learning related coupling Adjustement � � � � � J TASK − J PRE J POST − J PRE A = sign · ij ij ij ij i , j : J TASK , J POST � 0

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Introduction Learning related coupling Adjustement � � � � � J TASK − J PRE J POST − J PRE A = sign · ij ij ij ij i , j : J TASK , J POST � 0

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Maximal Entropy Models and the Vanilla (standard) Learning Algorithm Maximal Entropy Models and the Vanilla (standard) 1 Learning Algorithm Approximated Newton Method 2 The Long-Time Limit: Stochastic Dynamics 3 4 Properties of the Stationary Distribution

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Maximal Entropy Models and the Vanilla (standard) Learning Algorithm General MaxEnt Given a list of D observables to reproduce { Σ a ( σ ) } D a = 1 (generic functions of the system units) Find the MaxEnt model parameters { X a } D a = 1 � � � P X ( σ ) = exp a X a Σ a ( σ ) / Z [ X ] reproducing the observables averages: � Σ a � DATA ≡ P a = Q a [ X ] ≡ � Σ a � X

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Maximal Entropy Models and the Vanilla (standard) Learning Algorithm Equivalent to log-likelihood maximization: X ∗ = arg max X � � � � logL[ X ] ≡ arg max X X · P − log Z [ X ]

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Maximal Entropy Models and the Vanilla (standard) Learning Algorithm Equivalent to log-likelihood maximization: X ∗ = arg max X � � � � logL[ X ] ≡ arg max X X · P − log Z [ X ] in fact: � � d ∇ a logL[ X ] = X · P − log Z [ X ] = P a − Q a [ X ] dX a

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Maximal Entropy Models and the Vanilla (standard) Learning Algorithm Equivalent to log-likelihood maximization: X ∗ = arg max X � � � � logL[ X ] ≡ arg max X X · P − log Z [ X ] in fact: � � d ∇ a logL[ X ] = X · P − log Z [ X ] = P a − Q a [ X ] dX a Cannot be solved analytically. Ackley, Hinton and Sejnowski (Vanilla Gradient): X t + 1 = X t + δ X VG δ X VG = α ( P − Q [ X t ]) ; t t

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Maximal Entropy Models and the Vanilla (standard) Learning Algorithm Equivalent to log-likelihood maximization: X ∗ = arg max X � � � � logL[ X ] ≡ arg max X X · P − log Z [ X ] in fact: � � d ∇ a logL[ X ] = X · P − log Z [ X ] = P a − Q a [ X ] dX a Cannot be solved analytically. Ackley, Hinton and Sejnowski (Vanilla Gradient): X t + 1 = X t + δ X VG δ X VG = α ( P − Q [ X t ]) ; t t If 0 < P a < 1 for all a = 1 , . . . D , the problem is well posed: X ∗ exists and is unique and the dynamics converges (for infinitesimally small α )

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Maximal Entropy Models and the Vanilla (standard) Learning Algorithm A 2-dimensional example: 2 ( u − u ∞ ) 2 − b logL [ u , v ] = − a 2 ( v − v ∞ ) 2

Approximated Newton Algorithm for the Ising Model Inference Speeds Up Convergence, Performs Optimally and Avoids Over-fitting Maximal Entropy Models and the Vanilla (standard) Learning Algorithm A 2-dimensional example: 2 ( u − u ∞ ) 2 − b logL [ u , v ] = − a 2 ( v − v ∞ ) 2