Editing Hopfield network (section)

==Training==
Training a Hopfield net involves lowering the energy of states that the net should "remember". This allows the net to serve as a content addressable memory system, that is to say, the network will converge to a "remembered" state if it is given only part of the state. The net can be used to recover from a distorted input to the trained state that is most similar to that input. This is called associative memory because it recovers memories on the basis of similarity. For example, if we train a Hopfield net with five units so that the state (1, −1, 1, −1, 1) is an energy minimum, and we give the network the state (1, −1, −1, −1, 1) it will converge to (1, −1, 1, −1, 1). Thus, the network is properly trained when the energy of states which the network should remember are local minima. Note that, in contrast to [[Perceptron#Learning algorithm|Perceptron]] training, the thresholds of the neurons are never updated.

===Learning rules===
There are various different [[learning rule]]s that can be used to store information in the memory of the Hopfield network. It is desirable for a learning rule to have both of the following two properties:
* ''Local'': A learning rule is ''local'' if each weight is updated using information available to neurons on either side of the connection that is associated with that particular weight.
* ''Incremental'': New patterns can be learned without using information from the old patterns that have been also used for training. That is, when a new pattern is used for training, the new values for the weights only depend on the old values and on the new pattern.<ref name="storkey1991basins" />

These properties are desirable, since a learning rule satisfying them is more biologically plausible. For example, since the human brain is always learning new concepts, one can reason that human learning is incremental. A learning system that was not incremental would generally be trained only once, with a huge batch of training data.

===Hebbian learning rule for Hopfield networks===
[[Hebbian theory]] was introduced by Donald Hebb in 1949 in order to explain "associative learning", in which simultaneous activation of neuron cells leads to pronounced increases in synaptic strength between those cells.<ref>{{harvnb|Hebb|1949}}</ref> It is often summarized as "Neurons that fire together wire together. Neurons that fire out of sync fail to link".

The Hebbian rule is both local and incremental. For the Hopfield networks, it is implemented in the following manner when learning <math>n</math>
binary patterns:

<math> w_{ij}=\frac{1}{n}\sum_{\mu=1}^{n}\epsilon_{i}^\mu \epsilon_{j}^\mu </math>

where <math>\epsilon_i^\mu</math> represents bit i from pattern <math>\mu</math>.

If the bits corresponding to neurons i and j are equal in pattern <math>\mu</math>, then the product  <math> \epsilon_{i}^\mu \epsilon_{j}^\mu </math> will be positive. This would, in turn, have a positive effect on the weight <math>w_{ij} </math> and the values of i and j will tend to become equal. The opposite happens if the bits corresponding to neurons i and j are different.

===Storkey learning rule===
This rule was introduced by [[Amos Storkey]] in 1997 and is both local and incremental. Storkey also showed that a Hopfield network trained using this rule has a greater capacity than a corresponding network trained using the Hebbian rule.<ref name="storkey1997">{{cite book |last=Storkey |first=Amos |chapter=Increasing the capacity of a Hopfield network without sacrificing functionality |title=Artificial Neural Networks – ICANN'97 |year=1997 |citeseerx=10.1.1.33.103 |pages=451–6 |doi=10.1007/BFb0020196 |publisher=Springer |series= Lecture Notes in Computer Science |volume=1327 |isbn=978-3-540-69620-9}}</ref> The weight matrix of an attractor neural network{{clarify|reason=What is an attractor NN?|date=July 2019}} is said to follow the Storkey learning rule if it obeys:

<math> w_{ij}^{\nu} = w_{ij}^{\nu-1}
		    +\frac{1}{n}\epsilon_{i}^{\nu} \epsilon_{j}^{\nu} 
		    -\frac{1}{n}\epsilon_{i}^{\nu} h_{ji}^{\nu}
		    -\frac{1}{n}\epsilon_{j}^{\nu} h_{ij}^{\nu}
		    </math>

where <math> h_{ij}^{\nu} = \sum_{k=1~:~i\neq k\neq j}^{n} w_{ik}^{\nu-1}\epsilon_{k}^{\nu} </math> is a form of ''local field''<ref name="storkey1991basins">{{cite journal |last1=Storkey |first1=A.J. |first2=R. |last2=Valabregue |title=The basins of attraction of a new Hopfield learning rule |journal=Neural Networks |volume=12 |issue=6 |pages=869–876 |year=1999 |doi=10.1016/S0893-6080(99)00038-6 |pmid=12662662 |citeseerx=10.1.1.19.4681}}</ref> at neuron i.
		    
This learning rule is local, since the synapses take into account only neurons at their sides. The rule makes use of more information from the patterns and weights than the generalized Hebbian rule, due to the effect of the local field.