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Neurons and neural networks II. Hopfield network

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Perceptron recap• key ingredient: adaptivity of the system• unsupervised vs supervised learning• architecture for discrimination: single neuron — perceptron• error function & learning rule• gradient descent learning & divergence• regularisation• learning as inference

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Interpreting learning as inferenceSo far: optimization wrt. an objective function

where

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Interpreting learning as inferenceSo far: optimization wrt. an objective function

where

What’s this quirky regularizer, anyway?

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Interpreting learning as inferenceLet’s interpret y(x,w) as a probability:

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Interpreting learning as inferenceLet’s interpret y(x,w) as a probability:

in a compact form:

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Interpreting learning as inferenceLet’s interpret y(x,w) as a probability:

in a compact form:

the likelihood of the input data can be expressed with the original error function function

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Interpreting learning as inferenceLet’s interpret y(x,w) as a probability:

in a compact form:

the likelihood of the input data can be expressed with the original error function function

the regularizer has the form of a prior!

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Interpreting learning as inferenceLet’s interpret y(x,w) as a probability:

in a compact form:

the likelihood of the input data can be expressed with the original error function function

the regularizer has the form of a prior!

what we get in the objective function M(w): the posterior distribution of w:

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Interpreting learning as inference Relationship between M(w) and the posterior

interpretation: minimizing M(w) leads to finding the maximum a posteriori estimate wMP

The log probability interpretation of the objective function retains:additivity of errors, whilekeeping the multiplicativity of probabilities

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Interpreting learning as inference Properties of the Bayesian estimate

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Interpreting learning as inference Properties of the Bayesian estimate

• The probabilistic interpretation makes our assumptions explicit:by the regularizer we imposed a soft constraint on the learned parameters, which expresses our prior expecations.

• An additional plus:beyond getting wMP we get a measure for learned parameter uncertainty

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Interpreting learning as inference Demo

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Interpreting learning as inference Demo

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Interpreting learning as inference Demo

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Interpreting learning as inference Demo

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Interpreting learning as inference Making predictions

Up to this point the goal was optimization:

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Interpreting learning as inference Making predictions

Up to this point the goal was optimization:

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Interpreting learning as inference Making predictions

Up to this point the goal was optimization:

Are we equally confident in the two predictions?

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Interpreting learning as inference Making predictions

Up to this point the goal was optimization:

Are we equally confident in the two predictions?

The Bayesian answer exploits the probabilistic interpretation:

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Interpreting learning as inference Calculating Bayesian predictions

Predictive probability:

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Interpreting learning as inference Calculating Bayesian predictions

Predictive probability:

Likelihood:

Weight posterior

Partition function:

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Interpreting learning as inference Calculating Bayesian predictions

Predictive probability:

Likelihood:

Weight posterior

Partition function:

Finally:

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Interpreting learning as inference Calculating Bayesian predictions

How to solve the integral?

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Interpreting learning as inference Calculating Bayesian predictions

How to solve the integral?Bad news: Monte Carlo integration is needed

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Interpreting learning as inference Calculating Bayesian predictions

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Interpreting learning as inference Calculating Bayesian predictions

Original estimate

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Interpreting learning as inference Calculating Bayesian predictions

Original estimate Bayesian estimate

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Interpreting learning as inference Gaussian approximation

Taylor expansion around the MAP estimate

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Interpreting learning as inference Gaussian approximation

The Gaussian approximation:

Taylor expansion around the MAP estimate

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Interpreting learning as inference Gaussian approximation

The Gaussian approximation:

Taylor expansion around the MAP estimate

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Neural networks Unsupervised learning

Unsupervised learning: what is it about?

Capacity of a single neuron is limited: certain data can only be learnedSo far, we used a supervised learning paradigm: a teacher was necessaryto teach an input-output relation

Hopfield networks try to cure both

Hebb rule: an enlightening example

assuming 2 neurons and a weight modification process:

This simple rule realizes an associative memory!

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Neural networks The Hopfield network

Architecture: a set of I neuronsconnected by symmetric synapses of weight wij no self connections: wii=0output of neuron i: xi

Activity rule:

Synchronous/ asynchronous update

Learning rule:

;

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Neural networks The Hopfield network

Architecture: a set of I neuronsconnected by symmetric synapses of weight wij no self connections: wii=0output of neuron i: xi

Activity rule:

Synchronous/ asynchronous update

Learning rule:

alternatively, a continuous network can be defined as:

;

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Neural networks Stability of Hopfield network

Are the memories stable?

Necessary conditions: symmetric weights; asynchronous update

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Neural networks Stability of Hopfield network

Are the memories stable?

Necessary conditions: symmetric weights; asynchronous update

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Neural networks Stability of Hopfield network

Are the memories stable?

Necessary conditions: symmetric weights; asynchronous update

Robust against perturbation ofa subset of weights

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Neural networks Capacity of Hopfield network

How many traces can be memorized by a network of I neurons?

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Neural networks Capacity of Hopfield network