Supervised and Unsupervised learning and application to Neuroscience Cours CA6b-4

Machine Learning2 A Generic System System … … Input Variables: Hidden Variables: Output Variables: Training examples: Parameters:

Machine Learning3 A Generic System System … … Input Variables: Hidden Variables: Output Variables: Training examples: Parameters:

Machine Learning4 Different types of learning Supervised learning: 1.Classification (discrete y), 2.Regression (continuous y). Unsupervised learning (no target y). 1.Clustering (h = different groups of types of data). 2.Density estimation (h = parameters of probability dist.) 3.Reduction (h= a few latent variable describing high dimensional data). Reinforcement learning (y = actions).

Digit recognition (supervised) Handwritten Digit Recognition x: pixelized or pre-processed image. t: classs of pre-classified digits (training example.) y: digit class (computed by ML algorithm). h: contours, left/right handed…

Regression (supervised) Target output Parameters

Linear classifier ? Training examples

Linear classifier Decision boundary Heavyside function: 0 1

Linear classifier Decision boundary Heavyside function: 0 1

Assumptions Multivariate Gaussians Same covariance Two classes equiprobable

How do we compute the output? Positive: Class 1 Negative: Class 0 Orthogonal to decision boundary

How do we compute the output? Orthogonal to decision boundary

How do we learn the parameters? Orthogonal to decision boundary Linear discriminant analysis = Direct parameter estimation

How do we learn the parameters? Orthogonal to decision boundary Minimize mean-squared error:

How do we learn the parameters? Minimize mean-squared error: Gradient descent:

How do we learn the parameters? Minimize mean-squared error: Gradient descent: Stochastic gradient descent:

How do we learn the parameters? Stochastic gradient descent: Problem:is not differentiable

3. How do we learn the parameters? Solution: change y to expected class: The output is now the expected class Logistic function

3. How do we learn the parameters? Stochastic gradient descent:

3. How do we learn the parameters? Stochastic gradient descent: Always positive

3. How do we learn the parameters? Learning based on expected class: with Perceptron learning rule with

Application 1: Neural population decoding

w

How to find ? w w

Linear Discriminant Analysis (LDA) Covariance Matrix: Mean responses:

Inverse Covariance matrix Average neural responses when motion is right Average neural responses when motion is left Linear Discriminant Analysis (LDA) w

Neural network interpretation: Learning the connections with « Delta rule »: Each neuron is a classifier

Limitation of 1 layer perceptron: Linearly separable: ANDNon linearly separable: XOR

Extension: multilayer perceptron Towards a universal computer

Learning a multi-layer neural network with backprop Towards a universal computer

Extension: multilayer perceptron Towards a universal computer Initial error:

Extension: multilayer perceptron Towards a universal computer Backpropagate errors Initial error:

Extension: multilayer perceptron Towards a universal computer Backpropagate errors Apply delta rule: Initial error:

Big problem: overfitting... … Backprop was abandoned in the late eighties…

Compensate with very large datasets 9 th Order Polynomial … Resurgence of backprop with big data

Deep convulational networks Google: Image recognition, speech recognition. Trained on billions of examples…

Single neurons as 2 layer perceptron Poirazi and Mel, 2001, 2003

Regression (supervised) Target output Parameters

Regression in general Target output Basis functions

Gaussian noise assumption

How to learn the parameters? Gradient descent:

But: overfitting...

How to learn the parameters? Gradient descent:

Application 3: Neural coding: function approximation with tuning curves

“Classical view”: multiple spatial maps

Application 3: function approximation in sensorimotor area In Parietal cortex: Retinotopic cells gain modulated by eye position And also head position, arm position … Snyder and Pouget, 2000

Multisensory integration = multidirectional coordinate transform Experimental validation Model prediction: Pouget, Duhamel and Deneve, 2004 Avillac et al, 2005 Partially shifting tuning curves

Unsupervised learning …. First example of many

Principal component analysis Orthogonal basis

Principal component analysis (unsupervised learning) Orthogonal basis

Principal component analysis Orthogonal basis:Uncorrelated components: Note: not the same as independent

Principal component analysis and dimensionality reduction K<<N + “Noise”

Principal component analysis (unsupervised learning) Orthogonal basis N=2 K=1

One solution: eigenvalue decomposition of covariance matrix D D

How do we “learn” the parameters? K<<N Standard iterative method First component: other components:

PCA: gradient descent « Maximization » « Expectation » Generalized Oja rule

Natural images: Weights learnt by PCA

Application of PCA: analysis of large neural datasets Machens, Brody and Romo, 2010

Application of PCA: analysis of large neural datasets Time Frequency Machens, Brody and Romo, 2010