WK7

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WK7 Hebbian Learning
CS 476 Networks of Neural Computation WK7
Hebbian Learning Dr. Stathis Kasderidis Dept. of
Computer Science University of Crete Spring
Semester, 2009
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Contents

• Introduction to Hebbian Learning
• Definitions on Pattern Association
• Pattern Association Network
• Formal Theory of Associations Building
  Correlations
• Examples
• Conclusions

Contents
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Hebbian Learning

• Hebbian Learning is a learning rule which is the
  oldest and most famous of all learning rules. It
  was postulated by Donald Hebb (1949) in his book
  (The Organisation of Behaviour)
• When an axon of cell A is near enough to excite
  a cell B and repeatedly or persistently takes
  part in firing it, some growth process or
  metabolic changes take place in one or both cells
  such as As efficiency as one of the cells firing
  B, is increased
• Hebb proposed this change as a basis of
  associative learning. We may expand this as a
  two-part rule

Hebb. Learn.
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Hebbian Learning-1

• If two neurons on either side of a synapse
  (connection) are activated simultaneously then
  the strength of that synapse is selectively
  increased.
• If two neurons on either side of a synapse are
  activated asynchronously, then that synapse is
  selectively weakened or eliminated.
• Such a synapse is called a Hebbian synapse. More
  precisely, we define a Hebbian synapse as a
  synapse that uses a time-dependent, highly local,
  and strongly interactive mechanism to increase
  synaptic efficiency as a function of the
  correlation between the pre-synaptic and
  post-synaptic activities.

Hebb. Learn.
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Hebbian Learning-2

• We analyse the four key mechanisms mentioned
  above
• Time dependent mechanism This mechanism refers
  to the fact that the modifications in the synapse
  depend on the exact time of occurrence of the
  pre-synaptic and post-synaptic signals
• Local mechanism By its very nature, a synapse is
  the transmission site where information-bearing
  signals are in spationtemporal contiguity. This
  locally available information is used by the
  synapse to produce a local modification that is
  input specific
• Interactive Mechanism The occurrence of a change
  in a synapse depends on signals on both

Hebb. Learn.
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Hebbian Learning-3

• sides of the synapse. That is, a Hebbian form of
  learning depends on a true interaction between
  the pre- and post-synaptic signals in the sense
  that we cannot make any prediction from either
  one of these two activities by itself. The
  interaction may be deterministic of stochastic
• Correlational mechanism The condition for a
  change in synaptic efficiency is the
  co-occurrence of pre- and post-synaptic signals.
  The correlation over time between the two signals
  is responsible for the synaptic change.

Hebb. Learn.
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Hebbian Learning-4

• We may classify synaptic modifications of a
  synapse as
• Hebbian which is a synapse that increases its
  strength when positively correlated pre- and
  post-synaptic signals are present and decreases
  its strength when these signals are either
  uncorrelated or negatively correlated
• Anti-Hebbian Such a synapse weakens positively
  correlated pre- and post-synaptic signals and
  strengthens negatively correlated signals
• Non-Hebbian It does not involve, in the
  modification of a synapse, any mechanism that is
  time dependent, highly local and strongly
  interactive in nature (as in the previous cases).

Hebb. Learn.
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Hebbian Learning-5

• To formulate the Hebbian rule mathematically we
  consider a weight wkj of a neuron k with pre-and
  post-synaptic signal denoted by xj and yk
  respectively. The adjustment to the weight wkj at
  time step n is given by
• ?wkj (n)F(yk(n), xj(n))
• where F(,) is a function of both signals. The
  above formula can take many specific forms.
  Typical examples are
• Hebbs hypothesis In the simplest case we have
  just the product of the two signals (it is also
  called the activity product rule)

Hebb. Learn.
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Hebbian Learning-6

• ?wkj (n)?yk(n) xj(n)
• where ? is a learning rate. This form emphasises
  the correlational nature of a Hebbian synapse.
  However this simple rule leads to an exponential
  growth of the weights (becomes unbounded). Thus
  we need to mechanism to stop the unbounded
  increase of the weights. One such is the
  following.
• Covariance hypothesis In this case we replace
  the product of pre- and post-synaptic signals
  with the departure of of the same signals from
  their respective average values over a certain
  time interval. If x and y is their

Hebb. Learn.
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Hebbian Learning-7

• time-averaged value then the covariance form is
  defined by
• ?wkj (n)?(yk(n)-y) (xj(n)-x)
• The covariance hypothesis allows for the
  following
• Convergence to a non-trivial state, which is
  reached when xj(n)x or yk(n)y
• Prediction of both synaptic potentiation (i.e.
  increase in synaptic strength) and synaptic
  depression (i.e. decrease in synaptic strength).

Hebb. Learn.
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Pattern Association

• An associative memory is a brain-like distributed
  memory that learns associations. Association is a
  known and prominent feature of human memory.
• Association takes two forms
• Auto-association Here the task of a network is
  to store a set of patterns (vectors) by
  repeatedly presenting them to the network. The
  network subsequently is presented with a partial
  description or distorted (noisy) version of the
  original pattern stored in it, and the task is to
  retrieve (recall) that particular pattern.
• Hetero-association In this task we want to pair
  an arbitrary set of input patterns to an
  arbitrary set of output patterns.

Patt. Assoc.
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Pattern Association-1

• Auto-association involves the use of unsupervised
  learning (Hebbian, Hopfield) while
  hetero-association involves the use of
  unsupervised (Hebbian) or supervised learning
  (e.g. MLP/BP) approaches.
• Let xk denote a key pattern applied to an
  associative memory and yk denote a memorised
  pattern. The pattern association performed by the
  network is described by
• xk ? yk , k1,2,,q
• Where q is the number of patterns stored in the
  network. The key pattern xk acts as a stimulus
  that not only determines the storage location of
  memorised pattern yk but also holds the key for
  its retrieval.

Patt. Assoc.
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Pattern Association-2

• In an auto-associative memory, yk xk , so the
  input and output spaces have the same
  dimensionality. In a hetero-associative memory,
  yk ? xk , hence in this case the dimensionality
  of the output space may or may not equal the
  dimensionality of the input space.
• There are two phases involved in the operation of
  the associative memory
• Storage phase which refers to the training of
  the network in accordance with a suitable rule
• Recall phase which involves the retrieval of a
  memorised pattern in response to the presentation
  of a noisy version of a key pattern to the
  network.

Patt. Assoc.
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Pattern Association-3

• Let the stimulus x (input) represent a noisy
  version of a key pattern xj. This stimulus
  produces a response y (output). For perfect
  recall, we should find that y yj where yj is the
  memorised pattern associated with the key pattern
  xj. When y ? yj for x xj , the associative
  memory is said to have made an error in recall.
• The number q of patterns stored in an associative
  memory provides a direct measure of the storage
  capacity of the network. In designing an
  associative memory, we want to make the storage
  capacity q (expressed as a percentage of the
  total number N of neurons) as large as possible
  and yet insist that a large fraction of the
  patterns is recalled correctly.

Patt. Assoc.
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Pattern Association Network

• A pattern associator is a network which is able
  to learn hetero-associations of two patterns. A
  schematic representation is given below

Associator
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Pattern Association Network-1

• The net input that arrives to every unit is
  calculated as
• Where i is an output neuron and j an index of an
  input neuron. The dimensionality of the input
  space is N and of the output space is M. wij is
  the weight from neuron j to neuron i. aj is the
  activation of a neuron j.
• The activation of each neuron is produced by
  using a suitable threshold function and a
  threshold. For example we can assume that the
  activations are binary (i.e. either 0 or 1) and
  to achieve this we use the step

Associator
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Pattern Association Network-2

• function
• The training of the network takes place by using
  for example the Hebbian form. Thus what we have
  is a matrix of weights, with all of them zero
  initially, assuming an input pattern of (101010)
  and an output pattern (1100)

Associator
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Pattern Association Network-3

• If we assume a learning rate ?1 and after a
  single learning step we get
• To recall from the matrix we simply apply the
  input pattern and we perform matrix
  multiplication of the weight matrix with the
  input vector. We get in our example

Associator
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Pattern Association Network-4

• If we assume a threshold of 2 we can get the
  correct answer (1100) using a step function as
  activation function.
• We can learn multiple associations using the same
  weight matrix. For example assume that a new
  input vector (110001) is given with corresponding
  output

Associator
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Pattern Association Network-5

• vector as (0101). In this case after a single
  presentation (with ?1) we will get an updated
  weight matrix
• Again we can get the correct output vectors when
  we introduce the corresponding input vector

Associator
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Pattern Association Network-6

• Again by using the threshold of 2 and a step
  function we can get the correct answers of (1100)
  and (0101).
• However, keep in mind that there is only a
  limited number of patterns which can be stored
  before perfect recall fails. Typical capacity of
  an associator network is 20 of the total number
  of neurons.

Associator
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Pattern Association Network-7

• Recall accuracy reflects the similarity of a key
  pattern with the stored patterns. The network can
  generalise in the sense when an input pattern is
  not exactly the same with any of the stored
  patterns, then it returns the (stored) patterns
  which more closely resembles the input.
• Properties of pattern associators
• Generalisation
• Fault Tolerance
• Distributed representations are necessary for
  generalisation and fault tolerance

Associator
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Pattern Association Network-8

• Prototype extraction and noise removal
• Speed
• Interference is not necessarily a bad thing (it
  is the basis of generalisation).

Associator
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Correlations

• We have stated that the simple Hebb form creates
  unbounded weights. One way to overcome this
  problem was the covariance rule. A second one is
  the Ojas rule. The latter rule has the benefit
  that is closely related to the principal
  components analysis method.
• Let us restate the Hebb form for a single linear
  unit in the output layer and for an input vector
  with dimension larger than 1
• ?wi ?V?i
• Where V is the activation of the output unit, and
  ?i is the activation of input neuron i. ? is the
  learning rate.

Correlations
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Correlations-1

• This rule as it stands does not have any
  (non-trivial) stable fixed point. To see this,
  let us assume for the moment that it there are
  (hypothetically) some fixed points. (A fixed
  point is a pair of (V, ?) such that lt ?wgt0). In
  this case we will have
• 0lt ?wigtltV?igtlt?jwj?j?igt?jCijwjCw
• Where the angle brackets indicate an average over
  the input distribution P(?) and we have defined
  the correlation matrix C by
• Cij?lt?i ?jgt
• Or Cij?lt? ?Tgt

Correlations
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Correlations-2

• Several things should be noted for C
• C is not the covariance matrix of the input,
  which would be defined in terms of the means ?ilt
  ?igt as lt(?i - ?i)(?j - ?j)gt
• C is symmetric, i.e. Cij Cji which implies that
  the eigenvalues are real and the eigenvectors can
  be taken as orthogonal
• Because of the outer product form, C is positive
  semi-definite, thus all its eigenvalues are
  positive or zero.
• Now let us return to the equation
• Cw 0

Correlations
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Correlations-3

• This equation says that w is an eigenvector of C
  with eigenvalue 0. But this will never be stable
  because C has some positive eigenvalues. Thus we
  conclude that there are only unstable fixed
  points for the plain Hebb learning procedure.
• One can prevent the divergence of the Hebbian
  learning by constraining the growth of the weight
  vector w. There are several methods how this can
  be achieved
• One way is to renormalise the new vector, wiawi
  , of all the weights after each update, choosing
  as such that w1

Correlations
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Correlations-4

• Another way is to clip the value of the weight at
  a lower and higher bound, in other words to
  constrain the value of the weight to higher or
  lower value when tries to cross over these
  values, i.e.
• w- ? wi ? w
• Another way is to use the Ojas rule. This will
  examine next.
• Oja has modified the plain Hebb rule in such a
  way so as to make possible the weight vector to
  approach a constant length w1, without having
  to do any renormalisation by hand.
• Moveover, w approaches an eigenvector of C with
  largest eigenvalue ?max. We call this maximal

Correlations
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Correlations-5

• eigenvector.
• Ojas modification corresponds to adding a weight
  decay proportional to V2 to the plain Hebb rule
• ?wi ?V(?i-Vwi)
• Note that this form looks like a delta rule where
  the correction ?wi depends on the difference of
  the actual input and the backpropagated output.
• We state some properties of Ojas rule without
  any proof
• Unit length w1
• Eigenvector direction w lies in the maximal

Correlations
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Correlations-6

• eigenvector direction of C
• Variance maximisation w lies in a direction that
  maximises ltV2gt
• Other rules exist in the literature about the
  modification of the plain Hebb rule. In most
  cases these are more complex forms.

Correlations
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Examples

• Ex1- Hippocampal Model There has been strong
  support up today to suggest that the brain area
  known as hipocampus uses a Hebb style learning
  for forming episodic memories.
• A model which captures the interactions of the
  hippocampus (DG / CA3 /CA1) with the immediate
  surrounding regions (Entorhinal cortex,
  Subiculum) and the neocortex areas is given below

Examples
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Examples-1
Examples
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Examples-2

• The module details are as follows
• Entorhinal cortex 600 neurons, each with 200
  synapses and sparseness0.05
• DG 1000 with 60 synapses each and
  sparseness0.05
• CA3 1000 neurons each with
• 200 recurrent synapses (from other CA3 neurons)
• 120 synapses from Entorhinal cortex
• 4 synapses from DG
• With a sparseness0.05

Examples
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Examples-3

• CA1 1000 neurons 200 synapses each and
  sparseness0.01
• Sparseness is the number of activated neurons
  when a new stimulus arrives. This is determined
  by true data from the rat hippocampal area.
• Input is coming to Entorhinal cortex
• The connections from Ent. Cortex ? DG are trained
  using Hebbian learning
• DG is a competitive network
• CA3 is an auto-association network
• CA3 recurrent connections use Hebbian learning

Examples
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Examples-4

• The connections CA3 ? CA1 are trained with a
  Hebbian rule
• CA1 is a competitive network
• The connections from CA1 ? Ent. Coertx use
  Hebbian learning
• Simulations of the model showed that one-shot
  learning is possible and it matched well a number
  of experimental data.

Examples
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Examples-5

• Ex2 VisNet This network is a model of
  biological vision and tries to solve the problem
  of position and view invariant representations
  built from multiple views of the same object,
  e.g. a human face.
• It uses a hierarchical layered structure where
  the neurons of a top layer are connected to
  neurons of a previous layer by using receptive
  fields of appropriate size. The fields are
  progressively becoming wider as we move along the
  hierarchy.
• In each layer we have an array of 32x32 cells,
  which use lateral inhibition in a competitive
  network arrangement.

Examples
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Examples-6

• Forward connections from one layer to another are
  trained by Hebbian-style learning.
• Each cell receives 100 conenctions from the
  previous layer with 67 probability that a
  connections is coming from within 4 cells of the
  distribution centre.
• The architecture is shown below

Examples
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Examples-7
Examples
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Examples-8

• The input to the model is an image of a face
  which is then is convoluted with appropriate
  filters so as to recognise different orientations
  and edges in the input image. This corresponds
  roughly to V1 brain area.
• The learning law that is used, is a Hebbian rule
  with a memory trace
• ?wkj (n)?ak(n) mj(n)
• mi(n)(1-?)ai(n) ?mi(n-1)
• Where ? is a constant which determines the
  contribution of memory and of current activation.
  ai(n)is the activation of the neuron at time n
  and is calculated in the usual way.

Examples
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Examples-9

• The model successfully provides recognition of
  faces in different angles and positions in the
  input image. For more details one has to see the
  literature (Rolls Treves, 1998)

Examples
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Conclusions

• Hebbian learning is the oldest learning law
  discovered in neural networks
• It is used mainly in order to build associators
  of patterns.
• The original Hebb rule creates unbounded weights.
  For this reason there are other forms which try
  to correct this problem. There are also temporal
  forms of the Hebbian rule. A hybrid case is the
  memory case presented before in the VisNet case.
• It has wide applications in pattern association
  problems and models of computational neuroscience
  cognitive science.

Conclusions