Neural Networks

in many cases learning can not take place without an initial bias, where does this initial bias come from? \begin{equation} v = \Sigma^{n}_{i=1}w_{i}x_{i}+w_{0} \end{equation} bias can be considered (and implemented as) an axon with constant input or a separate parameter to Σ

Unlike a Σ neuron, a Π takes the product of it's inputs rather than the sum.

(see reading-activation-functions)

activation functions (φ) in increasing complexity – these will all be monotonically non-decreasing (biological plausibility)

• constant
• linear
• piecewise linear
• sigmoid \(\frac{1}{1+e^{-a(v+b)}}\)
  • equals 1 at +∞
  • equals 0 at -∞
  • where a controls slope and b controls intersection at origin
• hyperbolic tangent, a sigmoid shifted down so it equals -1 at -∞
• there is one which is the most biologically plausible \begin{equation} \phi(v) = \frac{v^{2}}{1+v^{2}} \end{equation}
• stochastic activation functions
  • in one the neuron fires with some probability
  • on the other type the output itself is a probability, however this is problematic because there is no way for a set of neurons to normalize their outputs, (probabilistic neural networks)

we like these to be

• monotonically non-decreasing
• bounded
• continuously differentiable

Rather than a sum or product of inputs, these take the difference between each input and its weight. So the larger the difference between the input vector \(\bar{x}\) and the weight vector \(\bar{w}\) the more active the neuron.

\begin{equation} v = |\bar{w}-\bar{x}|^{2} \end{equation}

and \begin{equation} y = e^{-v^{2}} \end{equation}

layers

• input layer l₀
• single layer has input and 1 processing layer (l₁)

recurrent system

• like a single-layer feed forward, but all output neurons are connected (lateral recurrent system)
• or with outputs from some l_i going back into some l_j s.t. j<i

recurrent systems can have very weird behavior, in terminology of control systems they are non-linear (see non-linear dynamical systems)

a variety of inputs can come from the same object (distance, orientation, etc…), needs to learn only this invariant information

How do you represent knowledge in a neural network? Some pattern of activation in the network.

• distance, ∀ \(\bar{x}\) ∃ some \(\bar{y}\) which is the activation of the network due to the input. Each \(\bar{y}\) can be thought of as a point in an n dimensional space (n neurons in the network). We can take the Euclidean distance between these points as measures of their similarities.
  • Manhattan distance is L₁ \begin{equation} L_1 = d_{kj} = |\bar{y}_{k} - \bar{y}_{j}| = \left(\sum^{n}_{l=1}{(y_{kl} - y_{lj})}^{2}\right)^{\frac{1}{2}} \end{equation}
  • Euclidean distance is L₂
  • another interesting one is L_∞
  • also dot product of the vectors is interesting
  • other metrics could be statistical, the mean of the activated values or something
  • edit or hamming distance

    A metric or distance must be

    positive

    d \geq 0

    triangular

    d₁₂+d₂₃ \geq d₁₃

    symmetric

    d₁₂ = d₂₁

• an important input should stimulate more neurons
• prior information can be built into the structure of the network or the pre-processor of the network
• three ways to handle invariants – it's very possible for your system to lock onto the wrong invariant if you're not careful in selection of test data
  1. structure
  2. training
  3. feature space transformation (preprocessing) e.g. we might calculate the moments of each image in a series of images

     moments

     the following are examples of moments, something \(\frac{y(x)}{m_o}\) for input images could be used to control for the overall brightness of the system, these can also be used for translation (e.g. x'=(x-m₁)), rotation, etc… assuming there's only one item of interest in the scene

     • m₀ = ∫_∞ y(x) dx (area under the curve)
     • m₁ = ∫_∞ y(x)x dx (expected value of x or mean)
     • m_n = ∫_∞ y(x)xⁿ dx

standard geometrical invariants

• translation – could add another layer that or's together a bunch of inputs from different locations
• rotation
• scale – ratios are invariant over different scales

Consider a neuron w/2 inputs x₁ and x₂ and for each combination we have a desired output value d.

table of desired behavior

x1	x2	d	y
…

In this example let error e equal \(\frac{1}{2}e^2=(d-y)^2\), the \(\frac{1}{2}\) is there for the kinetic energy analogy.

Kinetic Energy \begin{equation} e = \frac{1}{2}mv^2 \end{equation} So for the above, lets make our neuron linear, and have our desired value be either positive or negative.

We want to minimize the error using a gradient descent algorithm. \begin{equation} w_1(n+1) = w_1(n)-\eta\frac{\delta e^2(n)}{\delta w_1} \end{equation} where η provides a scaling factor to convert between units of error and units of weight. So what's our derivative? \begin{equation} \frac{\delta e^2}{\delta w_1} = 2e()x_1() \end{equation}

• learning algorithms
  
  1. supervised vs. unsupervised
  2. local vs. global
  3. statistical vs. deterministic
  4. memorization vs. generalization
  5. fast vs. slow, meaning the rate of weight change per experience, fast learning typically involves allot of forgetting (see stability plasticity dilemma)

• ECE506 is entirely dedicated to optimization

we have some function, and we want to find the minimum

• we can do gradient descent with \begin{equation*} \bar{\Delta}E = \frac{\delta E}{\delta w_1}\hat{w_1} + \frac{\delta E}{w_2}\hat{w_2} + \frac{\delta E}{w_3}\hat{w_3} \end{equation*} to update our weights with \begin{equation*} \bar{w}(n+1) = \bar{w}(n) - \eta\bar{\Delta}E \end{equation*} Using this we can prove that the error will not increase. Considering the single-dimension case with a Taylor expansion \begin{eqnarray*} E(w(n+1)) &=& E(w(n))+\frac{\delta E}{\delta w} (w(n+1)-w(n) + \ldots\\ &=& E(w(n)) - \eta \left(\frac{\delta E}{\delta w}\frac{\delta E}{\delta w}\right) \end{eqnarray*}
• using Newton's Method we can compute the \(\Delta w\) required to take us directly to the minimum \begin{eqnarray*} \Delta E &=& E(w(n+1)) - E(w(n))\\ &=& \frac{\delta E}{\delta w}\Delta w + \frac{1}{2}\frac{\delta^2 E}{\delta w^2}\Delta w^2\\ &=& 0 \end{eqnarray*} so solving for \(\Delta E\) we can get \begin{eqnarray*} \Delta E &=& \frac{-\frac{\delta E}{\delta w}}{\frac{1}{2}\frac{\delta^2 E}{\delta w^2}}\\ &=& -H^{-1}(n)\Delta E(n) \end{eqnarray*} where H is a Hessian (and NxN matrix of all possible partial derivatives of a N-length vector)

we have a set of training vectors

x

d

for each training vector we can do gradient descent of the weights towards that vector

incremental learning

• linear φ \begin{equation*} w(n+1) = w(n) + \eta e(n) x(n) \end{equation*}
• non-linear φ \begin{equation*} w(n+1) = w(n) + \eta \phi'(v(n)) e(n) x(n) \end{equation*}

an epic is a run through all of our training vectors

after an epic we can assess our progress as the overall error \begin{equation*} E(k) = \sum_1^N{e^2(n)} \end{equation*} to get our cumulative error in the same scale as our per-vector error we can take the root mean square (RMS) error \begin{equation*} E_{RMS}(k) = \sqrt{\frac{1}{N}\sum_1^N{e^2(n)}} \end{equation*}

HW 2.12

what is the question asking? these are two normalized Gaussians, we take the difference of these two Gaussians (Mexican hat). What happens if we translate this across along the x axis.

HW 2.10

two sums, write out the expression as the positive sum of the w_x's minus the sum of the c_y's or something… there are a number of ways this can be expressed

general

assume that the internal activation of a network under no input is set to 0

Eulerian integration

\(\frac{\delta y}{\delta t} = f(y)\), and we know the value at y=0. we can put this initial value in and use \(\frac{\delta y}{\delta t}\) to algebraically compute \(\delta y\) given some \(\delta t\). We can then just keep doing that.

• this weekend we'll get the API code
• the first step is to run a dumb agent that does nothing or provides a random sequence of actions, we then try to beat that
• so how could we use the LMS neuron for the project. we could use a competitive layer of winner take all neurons along our line of sight to select the brightest spot in our field of vision, then turn towards that spot.

  right before we eat an object we'll have a strong RGB input in our center neuron, we can treat this center neuron as an LMS neuron with a desired output of a positive \(\Delta energy\).

  this could be a simple starting architecture.

• our experimental setup should report both average length of lifetimes and standard deviations on this length over a number of trials – maybe even a t test?
• if we want to we can exceed the page limit with an appendix of additional figures
• would be good to try to compute the upper bound on the possible life-span given some assumptions
• we'll get a tentative outline

error minimization

• linear φ
• minimize \(\frac{1}{2}e^{2}(n)\) where \(e(n) = d(n) - y(n)\)

perceptron

• non-linear φ
• minimizing another criterion function aside from the squared error

with \begin{equation*} \phi = \left\{ \begin{array}{rl} 1 &: v > 0\\ -1 &: v \leq 0 \end{array} \right. \end{equation*} the only other neural network architecture that provably converges is adaptive resonance

• treat bias just like any other weight
• w₀ is the bias weights, which is updated like any other
• if error then update weights with \begin{equation*} \bar{w}(n+1) = \bar{w}(n) + \eta(\bar{w}(old)-\bar{x}) \end{equation*} or \begin{equation*} \bar{w}(n+1) = \bar{w}(n) +- \eta\bar{x} \end{equation*} where if error means \begin{equation*} if \left\{ \begin{array}{rcl} \bar{w}^{T}\bar{x} \geq 0 &and& \bar{x} \in c_{+1}\\ \bar{w}^{T}\bar{x} < 0 &and& \bar{x} \in c_{-1} \end{array} \right. \end{equation*}

for a multilayer feed-forward network we can use a matrix representation of the neurons and their weights, then the running of the neural network could be reduced to matrix multiplication.

the following computes the activation \begin{equation*} \bar{v}(n+1) = \bar{W}(n) \bar{y}(n) \end{equation*} and the output of the entire network \begin{equation*} \bar{y}(n+1) = \Phi(\bar{v}(n+1)) \end{equation*}

• learning internal weights, how to update weights which are further back in the network?

dependencies

E ← e ← y ← v ← w

partial error

\begin{equation*} e_i = d_i - y_i \end{equation*}

error

\begin{equation*} E(n) = \frac{1}{2}\Sigma_{i}{e^{2}_{i}(n)} \end{equation*}

output layer

\begin{eqnarray*} \Delta w_{ij} &=& - \eta \frac{\delta E}{\delta w_{ij}}\\ &=& - \eta e_{i}\frac{\delta e}{\delta w_{ij}}\\ &=& - \eta e_{i} \frac{\delta e_{i}}{\delta y_{i}}\frac{\delta y_i}{\delta w_{ij}}\\ &=& \eta e_i \phi_{i}' y_i \end{eqnarray*} where \(\phi'_i = \frac{\delta}{\delta v_i}\phi_i(v_i)\)

local gradient

how the overall error changes as the activation of a single neuron changes \(\delta_{i}=-\frac{\delta E}{\delta v_i}\) and hence the change in weights between any two neurons is as follows \begin{equation*} \Delta w_{ij} = \eta \delta_i y_i \end{equation*} this is like local Hebbian learning

Now layer by layer

• Output Layer \begin{equation*} \Delta w_{ij} = - \eta \frac{\delta E}{\delta v_i} = \eta \delta_i y_i \end{equation*}
• Hidden Layer \begin{equation*} \Delta w_{jk} = - \eta \frac{\delta E}{\delta v_j}y_k \end{equation*} where \begin{equation*} \frac{\delta E}{\delta v_j} = \Sigma_i{e_i \frac{\delta e_i}{\delta v_j}} \end{equation*} and e changes with v, and y changes with v, and v_i changes with v_j&hellip; \begin{eqnarray*} \Delta w_{jk} &=& - \Sigma{e_i \phi'_i \frac{\delta v_i}{\delta y_i} \frac{\delta y_i}{\delta v_j}}\\ &=& - \Sigma{e_i \phi'_i w_{ij} \phi'_j}\\ &=& \eta y_k \phi'_j \Sigma{e_i \phi'_i w_{ij}}\\ &=& \eta \delta_j y_k \end{eqnarray*} finally we get \begin{equation*} \delta_j = \phi_{j}' \Sigma_i{\delta_i w_{ij}} \end{equation*}

Weight changes propagate back through the network, the δ_j is dependent on the sum of the δ_i's. The δ for each neuron need be computed only once.

This is a two-pass algorithm. for each pattern we pass through forward computing the v, y, and the φ' values, saving the y and φ' values. then on the way back we compute the e and φ' values to get the δ values of the output layer, and work backwards.

• forward pass
  • clamp on the inputs
  • compute activations for nodes and their outputs through the network, and store these
  • at the output we compute the errors e_i(n)
• backward pass
  • ∀ layers
    • compute the δ's, \(\delta_{j}(n) = \phi'_{j}\Sigma_{i}{\delta_{i}w_{ij}}\)
    • compute the \(\Delta w(n) = -\eta \delta_{j}y_{k}\)
    • calculate φ'
    • loop back to the next previous layer and repeat

• without momentum \begin{equation*} \Delta w_{ij}(n) = - \eta \delta_{j}(n) y_{k}(n) \end{equation*}
• with momentum \begin{equation*} \Delta w_{ij}(n) = \alpha \Delta w_{jk}(n-1) - \eta \delta_{j}(n) y_{k}(n) \end{equation*} if α and -η sum to one, then the above is a convex combination

1. hidden neurons: compute and store on the forward pass
   • \(v_j = \Sigma w_{jk}x_{k}\)
   • \(y_{i}=\phi(v_{i})\)
   • \(\phi'(v_{i})\)
2. output neurons: compute and store on the forward pass
   • \(v_{i}= \Sigma w_{ij}y_{j}\)
   • \(y_{i} = \phi(v_{i})\)
   • \(\phi'(v_{i})\)
3. then can compute the error as \(e_i = d_{i}-y_{i}\)
4. output neuron: backwards
   • \(\delta_{i} = e_{i}\phi'(v_{i})\)
   • \(\Delta w_{ij} = \eta \delta_{i} y_{j}\)
5. hidden neurons: backwards
   • \(\delta_{j} = \phi'(v_{j}) \Sigma_{i}{\delta_{i}w_{ij}}\)
   • \(\Delta w_{jk} = \eta \delta_{j} y_{k}\)
6. finally do one more forward pass through the network in which we add all of the \(\Delta w\) values to our weights

For back propagation with momentum we save the old \(\Delta w\) so that we can use it to calculate our new \(\Delta w\). \begin{equation*} \Delta w_{ij}(n) = \alpha \Delta w_{ij}(n-1) + \eta \delta_{i} y_{j}(n) \end{equation*}

• 2 inputs and 4 outputs
• for a full pass you can track a pattern error \(\frac{1}{2}\Sigma_{i}{e^{2}_{i}}\), however for a more intuitive metric it may be useful to look at the RMS error which is "of the same size" as the errors themselves \(\sqrt{\frac{1}{|i|}\Sigma_{i}{e^{2}_{i}}}\)
• the epic error could be taken as the RMS error over the entire set \(\sqrt{\frac{1}{|epic||i|}\Sigma_{n}\Sigma_{i}e_{i}^{2}(n)}\), we should plot these by epic
• after each epic we can turn off learning (backward pass) and compute the errors generated from the testing set of samples giving us another error (i.e. error_testing). We should plot both error_testing and error_training on the same scale (note we should re-run the training data w/o learning). This is called a generalization plot.
• the training error will monotonically decrease, however it is possible that the testing error could begin to rise if we're over-fitting the training data.
• would be good to look at both incremental and batch update of the weights (e.g. do or do not update mid-epic)
• we should also look at how the order of presentation affects the performance of the network (only has an effect when doing incremental weight updates)
• stopping criteria
  • some error threshold
  • testing error starts to increase
  • etc…
• 3 architectures, 3 numbers of nodes, possibly to vary η, α, and even breaking some connections or removing some neurons after training to see how the network holds up
• weight initialization is yet another thing we could vary across multiple runs. There could be many local minima which we could land in depending on our starting position (or initial weights). For a given output neuron (expanded using a Taylor's expansion) \begin{eqnarray*} y_{0} &=& \phi\left[\Sigma_{i}w_{oi}[\Sigma w_{ij}x_{j}]\right]\\ &=& \Sigma_{l}a_{l}(\Sigma_{i}w_{oi}\Sigma_{m}a_{n}(\Sigma w_{ij} x_{i})^{m})^{l} \end{eqnarray*} So you could have a very large number of local minima. Typically you want to pick your weights in a random distribution centered around 0 – small weights lead to large values of φ' and large changes in weight.

(see convergence heuristics in the text book)

• you can look at the variance of your training data, and get a feel for what the σ_w of your weights should be
• for our homework assignment, the most effective solution will be to center our input data on the origin, and force a unit standard deviation on the input – this will keep us from saturating our neurons thereby reducing their information capacity to binary on/off. Note: this is a moment transformation, subtracting first moment and dividing by second moment.
• also, maybe set target values to something achievable (e.g. 0.1 and 0.9 instead of the asymptotic 0 and 1)

(p.208 in the text)

For certain types of bounded functions over a finite domain ∃ a single-hidden-layer feed forward neural network which can arbitrarily approximate that function (i.e. ∀ ε ∃ a number n s.t. a network with n neurons in the hidden layer can approximate the function).

\begin{equation*} F(\bar{x}) = \Sigma_{i=1}^{n}\alpha_{j}\phi_{j}(\Sigma w_{ji}x_{i}+b_{i}) \end{equation*}

This is like a Fourier Transform, a sum of orthonormal parts to approximate an arbitrary function.

We could also for example take the partial of a (the slope of the sigma function) of a neuron, and use pack-propagation to adjust these values.

We can take partials of our inputs \(\Delta x_{i} = \eta \frac{\delta E}{\delta x_{i}}\) to guess what x would likely give us any particular output y. You would need an initial guess of inputs, but for any initial guess back-prop could be used to move form the guess input to another input which is more appropriate for a particular desired output.

• hidden layers can be used as feature detectors. if you force a large amount of data through a small hidden layer then the data will be compressed through that layer which will require discovery of structures in the data to achieve the compression. such hidden layers are sometimes called "feature detectors".
• Auto-associative network that maps inputs to identical outputs. This could also be used for compression or encryption. the bottleneck hidden layer could be considered a compressed (and probably unintuitive encryption) of the inputs.
• Introduce weight sharing where each neuron in the hidden layer shares the same weight structure (e.g. mexican hat). This could be used to for example build an apple detector over images, no matter where the apple is present in the original image the same weight pattern will be present near that part of the image.
• prediction: say we have a time series, we can take a series of values as input, and then take a single later value as the desired output. In this way we can train a predictor.

                +--------+
   +----------->|   NN   |
   |            |        | output
   |  input     |        |--------+
   |       +--->|        |        |
   |       |    +--------+        |
   |       |                      |
   |       |                      v
  -----------------------------------time-series-->
  

  The "prediction company" formed out of the Santa Fe institute doing things like this for financial prediction.

• In practice we won't know how to build a network, i.e. how many layers and how many neurons in the layers. We want to limit the complexity of the network. You can add a penalty term s.t. when a weight has too much penalty it is removed.
  • you can start big and cut things out, intermittently remove all small weights from the network, this won't remove neurons or layers but it will simplify the network
  • you can add weights. start with a single neuron, doing learning with the standard Δ-rule. whenever an input results in a large error a new neuron is introduced which reduces that error and is connected to every existing neuron. The network is then trained through normal backwards propagation. These can work very well.
  • GA, the chromosome is generally the adjacency matrix of the neural network (with a set number of neurons constant across the entire species). This matrix could be linearized out into one long vector. Then simple mutation and any length-preserving method of crossover can be used. More generally any method of graph crossover could be used.

φ-separability

a data set is φ-separable if ∃ a function φ which separates the classes of the set

Covers Theorem: Any set of data with two classes (dichotomy) is more likely to be linearly separable the higher the dimension of the space in which the data is embedded.

as we non-linearly map our data into a higher dimensional space the linear separability of the data will increase. eventually we can just use a single perceptron to learn the data.

we can use radial basis neurons to perform this non-linear mapping

if we take a set of i functions φ_i, then we can use these i functions to map a point in 2 dimensions to a point in 2i dimensions by passing each coordinate through all i functions.

• H = {x_1&hellip;x_n}
• Dichotomy = (H₁, H₂)
• a set of functions \(\bar{\phi}(x)\)

a dichotomy is φ-separable if ∃ \(\bar{w}\) s.t.

• \(\bar{w}^{T}\bar{\phi}(x) > 0\) if \(\bar{x} \in H_1\)
• \(\bar{w}^{T}\bar{\phi}(x) \leq 0\) if \(\bar{x} \in H_2\)

• \(\phi_1 = e^{-|x-t_{1}|^{2}}\) where \(t_1=(1,1)\)
• \(\phi_2 = e^{-|x-t_{2}|^{2}}\) where \(t_2=(0,0)\)

x	φ1	φ2
1,1	1.0	0.13
0,1	0.36	0.36
1,0	0.36	0.36
0,0	0.13	1.0

the non-linearly separable classes of Xor are mapped by φ₁ and φ₂ to a new plane in which they are separable

An interpolative function will pass through all data points (even if it doesn't accurately reflect the behavior of the original function between the given data points).

F is interpolative if F(x_i)=d_i ∀ i∈[1..N]

\begin{equation*} F(x) = \Sigma_1^{n}w_{i}\phi_{i}(x-x_{i}) \end{equation*}

In matrix notation \(\Phi \bar{w} = \bar{d}\) or \begin{equation*} \left[ \begin{array}{ccc} \phi_{11} & \ldots & \phi_{1N}\\ \vdots & \ddots & \vdots\\ \phi_{N1} & \ldots & \phi_{NN} \end{array} \right] \left[ \begin{array}{c} w_{1}\\ \vdots\\ w_{N} \end{array} \right] = \left[ \begin{array}{c} d_{1}\\ \vdots\\ d_{N} \end{array} \right] \end{equation*} Then you can find the weight matrix in a single matrix inversion

A perceptron allows you to split a space with a hyperplane, however a radial basis neuron allows you to split a space with a hypersphere.

• Application
  Let's say we want to fit a bunch of data with radial basis neurons using Gaussian functions for our φ's. If we have 10,000 data points and we only want 10 hidden neurons, then we could just pick 10 points at random to be the centers of our neurons. Random works very well because we tend to sample from dense sample areas.

  Since 10 does not equal 10,000 we can't use simple matrix inversion to learn our weights because a 10x10,000 matrix is not square.

  we could also use

  • clustering algorithms, or
  • gradient descent – the free parameters in this approach above are the weights and the centers of the spheres, we could then computer the partial errors over δ-t and δ-w and descend on both the centers and weights

(RBF learning strategies in the book p.320)

types

1. random centers
2. self organizing centers
3. supervised – gradient descent centers

lets assume they're Gaussians for this example \begin{equation*} \phi(|\bar{x}-\bar{t}|) = exp(-\frac{|\bar{x}-\bar{t}|}{2\sigma^{-2}}) \end{equation*} So for a network with a hidden layer of m radial basis neurons all connected to all inputs and all feeding into a single perceptron the output is \begin{equation*} y=\Sigma_{i=0}^{m}w_{i}\phi(|\bar{x}-\bar{t_{i}}|) \end{equation*} initial values for our Gaussians

1. Under type (1) we set the centers t_i to random input points, to pick σ we want some overlap between the Gaussians positioned around different centers, but we also want different values for points closer to a particular center. So to approximate a good value for σ we can find the maximum distance between any two pairs of centers, and then we can set \(\sigma^{2} \simeq \frac{d_{max}^{2}}{\sqrt{m}}\).

   Why do we divide this by \(\sqrt{m}\) (which is related to the dimensionality of the space)? Not sure, but this is an accepted heuristic.

   Then to select the values for w_i,

   Once these initial values are selected we can use gradient descent to improve the values of w_i ∀ i. The centers and the σ's are fixed, so we only need update the weights.

2. we select the centers using something like k-means. For k-means we
   1. pick a value of k (this is somewhat unsatisfying, we could start with only 1 center, and then add new centers whenever the max center to x_i distance is over some preset threshold)
   2. randomly pick k centers (possibly from the data)
   3. loop over the data (x_i's) and associate each x with the closest center
   4. for each center we move it towards the center of mass of the associated x_i's by some learning parameter η
   5. keep doing it until the centers stop moving
   6. the standard deviations of these centers are easy to compute, just compute the standard deviation of the x_i's assigned to each center
3. we can bring in a desired value, compute an error, and take the partial errors along the weights, and then we can do gradient descent on each of the hidden radial basis neurons.

   If we think of every radial basis neuron as a function like \(\phi(\bar{x},\bar{t_{i}},\sigma_{i})\) then we can take partials of all three of the arguments to φ, e.g. \begin{equation*} t_{i}(new) = t_{i}(old) - \eta \frac{\delta e^{2}}{\delta \bar{t_{i}}} \end{equation*} We're still guessing at the number of hidden nodes, but having made that choice this works very well at moving around the centers of these nodes.

   We can use a covariance matrix to allow σ to vary across dimensions, we could then learn these m_i² elements of the covariance matrix using gradient descent.

Hebbian learning in a winner take all system with a laterally connected output layer learning can be localized to the winner for any particular input.

highly recommended book ← should purchase

Title	Self Organization and Associative Memory
Author	T. Kohonen
ISBN	3-540-18314-0

also this is Chapter 9 in the text.

The locations of neurons relative to each other in some space will begin to become important.

Imagine that all output neurons are embedded in a two dimensional planar grid. Imagine local inhibition of these output neurons on this 2D grid.

A self organizing map is a neural network which is continuous s.t. ∀ η ∃ δ s.t. if the output of two inputs x₁ and x₂ is within δ in the output space (our 2D plane) then x₁ and x₂ are within η in the input space.

typical representation of these 2D output neural networks

These neurons often have alternating competitive and cooperative phases. In normal competitive Hebbian learning only the winner has his weights changed, however in the Kohonen self-organizing map there is also a cooperative phase in which the winner shares learning with his neighbors, they share slightly less with their neighbors, etc…

competitive

first we find the winner by computing the cross product of the input \(\bar{x}\) with each weight vector \(\bar{w_{i}}\) and the winner is the neuron with the smallest inner product. \begin{equation*} J(\bar{x})=min_{j \in i,n}(||\bar{x} - \bar{w_{i}}||) \end{equation*} so the neuron with the weight vector closest to the x vector is the winner. We then move this weight vector slightly (by η) closer to the x vector.

cooperative

For a generic neuron in the network \begin{equation*} \bar{w_{j}}(n+1) = \bar{w_{j}}(n) + \eta h(J(x))(\bar{x}(n) - \bar{w_{j}}(n)) \end{equation*} where the \(h\) function need to return the highest values for those neurons nearest to the winner with unity value right on the winner and fairly quickly dropping to zero as we move away from the winner. We can specify \(h\) as \begin{equation*} h(J(\bar{x}),j) = \left\{ \begin{array}{rl} 1 & j=J(\bar{x})\\ f(||\bar{r}_{J(\bar{x})} - \bar{r}_{j}||) & j \neq J(\bar{x}) \end{array} \right. \end{equation*} or actually more like \begin{equation*} h(J,j) = exp\left(\frac{-d(J,j)^{2}}{2\sigma^{2}}\right) \end{equation*} where the distance metric \begin{equation*} d(J,j) = ||\bar{r}_{J} - \bar{r}_{j}|| \end{equation*} and where σ is used to control the spread of the weight propagation.

So how do you pick η and σ? At least initially you will want σ to allow the weight updates to touch most weights. Generally you will want to diminish both η and σ as time progresses.

pictured in the input space the following pretty much amounts to, whenever an input lands in the input space, all weights shift slightly towards that input with the degree of the weights movement based upon its closeness to the input (or rather its closeness to the closest weight).

There are some very cool demonstrations of and visualizations of these sorts of systems available on-line.

we can turn our Canonical model into a continuous time differential equation, in which case we get \begin{equation*} \frac{\delta v_{i}(t)}{\delta t} = \Sigma_{i} w_{ij}(t) y_{i}(t) \end{equation*} This is similar to an RC circuit in which the input is the battery and the neuron is the capacitor.

Given this equilibrium equation… \begin{equation*} \frac{\delta v_{i}(t)}{\delta t} = \Sigma_{i} w_{ij}(t) y_{i}(t) - \frac{v_{i}(t)}{\tau} \end{equation*} When we first add input if the sum term is a positive constant, then the δ will initially be positive (because v_i is initially 0), however eventually δ=0 when the sum equals the v_i. This would be one time step in our model.

we also have… \begin{equation*} y_{i}(t) = \phi(v_{i},t) \end{equation*} these two equations together form a coupled system of non-linear equations.

some points

1. if φ is linear then it is a coupled system of linear equations and would be more amenable to analysis
2. there is no learning in the above, it gets considerably more complicated in this case \begin{equation*} \frac{\delta w_{ij}(t)}{\delta t} = F(?) - \frac{w_{ij}(t)}{\tau} \end{equation*} ultimately this is system of three highly related equations
3. this system of equations can generally be bounded in space as the weights and activations won't normally go off to ∞

\begin{equation*} \frac{\delta x_{j}(t)}{\delta t} = F_{i}(\bar{x}(t)) \end{equation*}

autonomous

there is no explicit time term inside of the main function

non-autonomous

the function "F" does depend on time in some way e.g. \(F_{i}(\bar{x}(t),t)\)

because non-autonomous systems are very difficult we will focus on systems without an explicit time term.

so fixed points in the state space of this system of equations will correspond to states in which the neural network has ceased to learn and has learned some input, this can be thought of as a memory.

it is possible to linearize these differential equations in some bounded areas of space.

to study the stability of our "memory" attractors we'll be using Lyapunov's Theory (direct method) to determine if these attractors exist.

Lyapunov's Theory, assume ∃ V(x) which is

1. continuous
2. V(x^star)=0 where x^^star is an attractor
3. V(x)>0 when x ≠ x^star

In the range of the attractor (in its basin) \(\frac{\delta v(x)}{\delta t}<0\), this defines the region of the attractor (i.e. where \(\frac{\delta v(x)}{\delta t}<0\) is true).

the application of dynamic systems analysis to neural networks

• additive model \begin{eqnarray*} \frac{\delta v_{j}(t)}{\delta t} = -v_{j}(t)+\Sigma_{i=1}^{n}w_{ji}(t)\phi(v_{j}(t))+I_{j}\\ y_{j}(t) = \phi(v_{j}(t)) \end{eqnarray*}
• alternate additive model \begin{equation*} \frac{\delta y_{j}(t)}{\delta t} = -y_{j}(t)+\phi(\Sigma_{i=1}^{n} w_{ij}y_{i}) \end{equation*}

\begin{equation*} \frac{\delta v_{j}}{\delta t}=-\frac{v_{j}}{\tau}+\Sigma_{i=1}^{n} w_{ji} \phi(v_{i}(t)) + I_{j} \end{equation*}

with

• w_ji = - w_ij
• the φ function is invertible
• and ∃ a Lyapunov's function \begin{equation*} - v = - \frac{1}{2} \Sigma_{i=1}^{n} \Sigma_{j=1}^{n} w_{ji}y_{j}y_{i}+ ORG-LIST-END \Sigma_{j=1}^{n}\frac{1}{\tau}\int_{0}^{y}\phi^{-1}_{j}(y)\delta y - \Sigma_{j=1}^{n}I_{j}y_{j} \end{equation*}
• \(\frac{\delta v}{\delta t} < 0\)

               +--------------------+      dipole layer, each neuron is a pair
F2             |o o o o o o o o o o |   <- connected as a flip-flop, and all
               +--------------------+      pairs connected as winner-take-all

                                               +----+   gain control
             Full Bidirectional connectivity   | GC |<- fully connected
              (top down weights are binary)    +----+   to F1 and F2
         
    +---------------------------------------------+
F1  |o o o o o o o o o o o o o o o o o o o o o o o|
    +---------------------------------------------+    +---+    vigilance,
      ^        ^    ^           ^      ^     ^         | e | <- fully connected
      |        |    |           |      |     |         +---+    to F1 and F0
    +---------------------------------------------+
F0  |o o o o o o o o o o o o o o o o o o o o o o o|
    +---------------------------------------------+
       ^         ^       ^       ^           ^
       |         |       |       |           |

           Input I is binary vector

Neurons

GC

is on unless it is inhibited, any inputs turn it off

F0

these neurons are binary on or off depending on their input

F1

these are threshold neurons, which have three inputs (one from F0, one from GC and one from each neuron in F2 which is initially only 1), it requires at least 2 inputs to turn on

F2

each neuron is a flip-flop pair of neurons with the whole layer connected into a winner-take all network, let T_i be the weights up from F1 to neuron i in F2, then the downward weights of i B_i are always a binary scaling of T_i

e

the vigilance neuron checks the Hamming distance between the activation of F0 and the activation of F1, to see if the top-down weights (defining the center of the active neuron in F2) is close enough to the input (the activation in F1), we can calculate this with \begin{equation*} \frac{||T_{1}-I||}{||I||} \geq \rho \end{equation*} where T₁ is the activation pattern of F1 and I is the input vector. e turns on when the activation is less than ρ. When e turns on it activates the flip-flop neuron of the active neuron in F2 which latches off F2, and when the effects of F2 turning off propagates through the network and the original input activation propagates back up to F2, a new neuron is recruited to activate in response to the input.

Lets look at the behavior of this system

1. initially only GC is turned on
2. lets apply a pattern
3. the F0 neurons of the 1 inputs will activate
4. the F1 neurons related to the active F0 neurons will activate (because they now have two inputs)
5. the active F1 neurons then push activation to the recruited neurons in F2, and the winner turns on
6. when the winner turns on it begins activating GC and activating back from F2 to F1 (according to its top-down weight vector)
7. without GC on all neurons in F1 which aren't activated by the active neuron in F2 turn off, those neurons in F1 which are turned on are now the intersection of those in the input pattern and those with top-down weights from the active neuron in F2

This is like leader clustering where we test for closest and close-enough. The closest part of this is controlled by the winner-take-all formation in F2, the Vigilance neuron e is responsible for the close-enough test.

We then learn the weights between F1 and F2 with Hebbian learning. All weights are held between 1 and 0, as the activation resonates then those neurons in F1 which resonate with the active neuron in F2 will have their weights to and from F2 tend to 1 and all others have their weights tend to 0. This is similar to template matching in the pattern matching literature.

           k=1             nf2
     +---------------------------+
   F2|     ------------------    | \   
     |                           |  |
     |        T                  |  |
     |         k                 |  |
     |                           |  |
   F1|  -----------------------  |  |
     |                           |\ /
     |                           | e
   F0|  -----------------------  |/
     |                           |
     +---------------------------+

closest

How do we compute the closest neuron in F2?

The activation normalized by the number of up weights (size of the template) \begin{equation*} k = argmax_{k=1,nf2}\left\{\frac{||I \wedge T_{k}||}{\beta + ||T_{k}||}\right\} \end{equation*} β has the effect of breaking ties in favor of the larger template, β is typically set to \(\frac{1}{nf0}\).

close enough

How do we decide if this is close enough

\begin{equation*} \frac{||I \wedge T_{k}||}{||I||} \ge \rho \end{equation*} If the above is not true, then we reset and add a new neuron to F2

If it is true then we learn and update T_k using the gated Hebbian we discussed last time

↑ the above is the entirety of the ART algorithm ↑

some properties:

• this can all be described by a set of coupled differential equations
• the order of presentation of training points affects the learned structure

we can prove this using compliment coding – meaning ∀ inputs I we concatenate I and its compliment I_C (I,I_C). The size of this concatenation will always be equal to the number of bits in I.

This will converge (number of nodes and values of weights) in 1 epic.

thermometer code

• normalize your inputs to between zero and 1
• then multiply each normalize value by nf0 and represent it by that many consecutive 1's and pad the rest with zeros

this real value case with unary encoding can be very easily visualized geometrically where every neuron in F2 becomes a box in the number of dimensions as there are real values in the input

• we replace ∧ with a fuzzy-∧ which is just min, it returns the min of its inputs (which don't have to be 0 or 1 but can be any real value between 0 and 1).
• similarly compliment becomes 1- so the compliment of (0.1,0.3) becomes (0.9,0.7)

Not very popular, is the case where the F2 row is not winner-take-all but rather can have multiple neurons turn on.

for supervised learning

                   x
                  / \
                 /   \
                x     x Association Matrix
                 \   /
                  \ /
                   X


    +----------+      +----------+
    |          |      |          |
    |    A     |      |    B     |
    |          |      |          |
    |          |      |          |
    +----------+      +----------+
        x                  d

x and d are the inputs and the desired values, these train A and B which are both ART-1 architectures, and their outputs go into an association matrix in which the related outputs are associated.

nonlinearity

in the weighting between neurons, each neuron is non-linear as are their sum

input-output mapping

ideal for supervised learning

adaptable

easily re-weighted to adapt to a changing environment, however shouldn't change to fast to fleeting disturbance this is called the stability-plasticity dilemma

evidential response

can return confidence along with it's clarifications

contextual information

each neuron can potentially affect each other neuron allowing for natural spread of contextual information

fault tolerant

naturally distributed so if any single neuron is damaged the global behavior is not impaired but not drastically affected robust, graceful degradation

VLSI

massively parallel, take advantage of parallel hardware

uniform

regardless of problem domain the same structure (interconnected neurons) is used, allows for modularity and composability

biological analogy

as an analog of a biological system, many good ideas can be borrowed from nature

(see notes-activation-functions)

Threshold

\begin{equation} \phi(v) = \left\{ \begin{array}{lcl} 1 &if& v \geq 0\\ 0 &if& v < 0 \end{array} \right. \end{equation}

Piecewise Linear

\begin{equation} \phi(v) = \left\{ \begin{array}{lcl} 1 &if& v \geq +\frac{1}{2}\\ v &if& +\frac{1}{2} > v > -\frac{1}{2}\\ 0 &if& v \leq -\frac{1}{2} \end{array} \right. \end{equation}

Sigmoid

\begin{equation} \phi(v) = \frac{1}{1+exp(-av)} \end{equation}

Stochastic

\begin{equation} \phi(v) = \left\{ \begin{array}{lcl} +1 &\text{with probability}& P(v)\\ -1 &\text{with probability}& 1-P(v) \end{array} \right. \end{equation} where \begin{equation} \phi(v) = \frac{1}{1+exp(-v/\tau)} \end{equation} where τ is a pseudo temperature used to control the amount of noise in the system

single-layer feed-forward

single layer of input nodes which connect to a single layer of output nodes, acyclic

multi-layer feed-forward

like the above but with hidden layers between the input and output layer, these can be particularly useful when the size of the input layer is large. These are fully connected when every node on a layer is connected to every node on the adjacent layers

recurrent networks

neural networks which has as least one feedback loop, which will typically involve unit-delay elements. feedback loops have significant effect on the behavior of the network

It is difficult to talk about knowledge being represented inside of a neural network, as any such knowledge will be stored implicitly in the structure of the network. As such any knowledge which a neural network may contain about it's world (inputs and outputs) is directed toward acting in the world.

Knowledge Rules

1. similar inputs from similar outputs should result in similar (e.g. by Euclidean distance of a vector of neuron activation) internal states and should thus be classified similarly
2. opposite of (1) items in different categories should be given different internal representations
3. important features should be allotted a large number of neurons in the network
4. prior information and invariance should be built into the design of the network
   • biologically plausible
   • smaller size
     • limits the search space of the network
     • faster information transition through the network
     • cheaper to build becu

no general rules.

two ad-hoc rules for prior information

1. Restricting network architecture through use of local connections known as receptive fields
2. Constraining the choice of weights through weight-sharing

invariance, e.g. same object but viewed form different angles, same voice but spoken loudly or softly

methods of entraining invariance into a network

1. invariance by structure
2. invariance by training
3. invariant feature space, assuming ∃ features which are constant across all variants of the same input

• Error-correction Learning
  
  • directed to a particular neuron
  • the output of that neuron is compared to the desired output
  • the difference error is used to re-weight the inputs to the neuron
  • this process continues until the neural net hits some steady state

  (this is implemented with back-propagation)

• Memory-based Learning
  
  • each input is associated with an output
  • when a new unknown input is seen it is classified as the same value as the nearest (Euclidean) known input

• Hebbian Learning
  associative learning
  1. if the neurons on either side of an axon are activated simultaneously then the strength of the axon is increased
  2. if the neurons are activated asynchronously then the synapse is weakened or eliminated

  this sort of learning is

  • time-dependent
  • local
  • interactive (depends on both side of the synapse)
  • conjunctional or correlational

  strong physiological evidence that associative Hebbian learning takes place in the brain

• Competitive Learning
  
  • each input neuron is attached to every output neuron
  • random weights on all of these attachments
  • for each set of inputs the output neuron with the highest activation is the winner, and all of it's active input connections are strengthened
  • this process continues until the weights stabilize

  with k output neurons this performs similar to k-means clustering, in which each output neuron is finally associated with a cluster of similar inputs

• Boltzman Learning
  the neural network or Boltzman machine is characterized by an energy function. \begin{equation} E = -\frac{1}{2}\Sigma_{j}\Sigma_{k \neq j}w_{kj}x_{k}x_{j} \end{equation} the machine operates by selecting a neuron at random during the learning process and flipping it's output with probability \begin{equation} P(x_{k} \rightarrow -x_{k}) = \frac{1}{1 + exp(-\Delta E_{k}/\tau)} \end{equation} two running conditions
  • clamped condition in which the visible neurons (attached to the environment) can't be changed
  • free-running condition in which all neurons can be flipped

  let

  • \(p_{kj}^{+}\) denote the correlation between neurons j and k when in clamped condition
  • \(p_{kj}^{-}\) denote the correlation j and k when free-running condition

  then the change in weight is \begin{equation} \Delta w_{jk} = \eta(p_{kj}^{+} - p_{kj}^{-}), j \neq k \end{equation}

how to assign credit and blame to inner portions of a neural network, this is sometimes complicated through temporal delay, requiring assignment of credit to past actions/events

these first two require a critic

• in reinforcement learning the system matches input to output to maximize some scalar performance measure.
• in delayed reinforcement learning the system attempts to minimize a cost-to-go function, which is the cost of a sequence of actions taken over some sequence of inputs. learn from the results of actions. related to dynamic programming

an example of unsupervised learning: a two-layer neural network in which the first layer is the input layer and the second the competitive layer, s.t. the neurons in the competitive layer compete with each other to respond to an input.

• pattern association
  

  associative memory

  distributed memory which learns by association

  auto-association

  (unsupervised) a set of patterns are repeatedly presented to the network, and it tries to store them s.t. when presented with a noisy version of a pattern it returns the original pattern

  hetero-association

  (supervised) arbitrary set of input patterns are paired with another arbitrary set of output patterns

• pattern recognition
  assignments of inputs to classes

  

  classical design for pattern association network

• function approximation
  Attempt to match some function \(d = f(x)\) where x is the input vector and d is the desired output. Supervised learning can be used to train the network to match \(f\).