## TAMU Neural Network 2 Learning Processes

2017-01-26

1. five basic learning rules
• error correction
• Hebbian
• memory-based
• competitive
• Boltzmann
• credit assignment problem
• supervised learning
• unsupervised learning
4. probabilistic and statistical aspects of learning

# learning rules

## error-correction learning

• input $x(n)$, output $y_k(n)$ and desired response or target output $d_k(n)$
• error signal $e_k(n) = d_k(n) - y_k(n)$
• $e_k(n)$ actuates a control mechanism that gradually adjust the synaptic weights, to minimize the cost function (or index of performance)
• cost function: $\epsilon(n) = 0.5 e_k^2(n)$
• when synaptic weights reach a steady state, learning is stopped.

• Widrow-Hoff rule, with learning rate $\eta$:
• With that, we can update the weights:

## memory-based learning

• All (or most) past experiences are explicitly stored, as input-target pairs $\{(x_i,d_i)\}_{i=1}^{N}$
• Two classes $C_1,C_2$
• Given a new input $x_{test}$, determine class based on local neighborhood of $x_{test}$.
• Criterion used for determining the neighborhood
• Learning rule applied to the neighborhood of the input, within the set of training examples.

### nearest neighbor

Nearest neighbor $x'_N \in X$ of $x_{test}$:
$min_i d(x_i,x_{test}) = d(x_N’,x_{test})$
where $d(a,b)$ is the Euclidean distance.

• $x_{test}$ is classified as the same class as $x'_N$

### k-nearest neighbor

• Identify k classified patterns that lie nearest to the test vector $x_{test}$, for some integer k.
• Assign $x_{test}$ to the class that is most frequently represented by the k neighbors (use majority vote)
• In effect, it is like averaging. It can deal with outliers

## Hebbian Learning

• If two neurons on either side of a synapse are activated simultaneously, the synapse is strengthened.
• If they are activate asynchronously, the synapse is weekened or eliminated.

Hebbian learning with learning rate $\eta$: $\Delta w_{kj}(n) = \eta y_k (n) x_j (n)$
Covariance rule: $\Delta w_{kj}(n) = \eta (y_k - \bar{y}) ( x_j - \bar{x})$

### covariance rule

• convergence to a nontrivial state
• prediction of both potentiation and depression
• observations
• weight enhanced when both pre- and post-synaptic activities are above average
• weight depressed when
• pre- more than average, post- less than average
• pre- less than average, post- more than average

## competitive learning

• Output neurons compete with each other for a chance to become active.
• Highly suited to discover statistically salient features (that may aid in classification)
• three basic elements:
• Same type of neurons with different weight sets, so that they respond differently to a given set of inputs
• A limit imposed on strength of each neuron
• Competition mechanism, to choose one winner: winner-takes-all neuron.

$\Delta w_{kj} = \eta (x_j-w_{kj}), \mbox{ if } k \mbox{ is the winner}$

Synaptic weight vector is moved toward the input vector.
Weight vectors converge toward local input clusters: clustering

## Boltzmann Learning

• Stochastic learning algorithm rooted in statistical mechanics
• Recurrent network, binary neurons (+1 or -1)
• Energy function $E = -0.5\sum_{j}\sum_{k,k \ne j} w_{kj} x_k x_j$
• Activation:
• Choose a random neuron k
• Flip state with a probability (given temperature T)
• $P(x_k \rightarrow -x_k) = (1+exp(-\Delta E_k/T))^{-1}$
• $\Delta E_k$ is the change in E due to the flip

### Boltzmann Machine

Types of neurons

• Visible: can be affected by the environment
• Hidden: isolated

Types of operations

• Clamped: visible neurons are fixed by environmental input and held constant
• Free-running: all neurons update their activity freely.

• Learning
• update weight during both clamped and free running condition
• Train weight $w_{kj}$ with various clamping input patterns
• After training is completed, present new clamping input pattern that is a partial input of one of the known vectors
• Let it run clamped on the new input (subset of visible neurons), and eventually it will complete the pattern

## credit assignment

Assign credit or blame for overall outcome to individual decisions.

• for outcomes of actions
• for actions to internal decisions

## learning without a teacher

two classes

• reinforcement learning/neurodynamic programming
• unsupervised learning/self-organization

### reinforcement

Goal is optimize the cumulative cost of actions.
In many cases, learning is under delayed reinforcement.

## pattern association

Associate key pattern with memorized pattern.

## pattern classification

Mapping between input pattern and a prescribled number of classes.

## function approximation

Nolinear input-output mapping.
System identification: learn function of an unknown system

## control

Control a plant, adjust plant input u so that the output of the plant y tracks the reference signal d.

## filtering/beamforming

Filtering: estimate quantity at time n, based on measurements up to time n
smoothing: estimate quantity at time n, based on measurements up to time n+a
prediction: estimate quantity at time n+a, based on measurements up to time n

# memory

q pattern pairs $(x_k,y_k)$:
Input (key) vector: $x_k$
Output (memorized) vector: $y_k$
By a weight matrix:
$y_k = W(k)x_k$

Let
$M = \sum_{k=1}^{q} W(k)=\sum_{k=1}^{q} y_k x_k^T$
If all $x_k$ are nomalized to length 1, then:
$M x_j = y_j$

• stationary environment: supervised learning can be used to obtain a relatively stable set of parameters
• nonstationary environment: parameters need to be adapted over time
• locally stationary

# statistical nature of learning

Target function: $f(x)$

Neural network realization of the function: $F(x,w)$ or $F(x,T)$

Random input vectors $X \in \{x_i\}_{i=1}^N$ and random output scalar values $D \in \{d_i\} _{i=1}^N$

Training set: $T=\{(x_i,d_i)\}_{i=1}^{N}$

regressive model: $D = f(X) + \epsilon$

Error term has zero mean: $E[\epsilon \vert x] = 0$

$E[D \vert x] = f(x)$

$E[\epsilon f(X)] = 0$

Cost function:
$\mathcal{E}(w)= \frac{1}{2}\sum_{i=1}^{N}(d_i - F(x_i,w))^2$
$\mathcal{E}(w)= \frac{1}{2}E_T[\epsilon^2] + E_T[\epsilon(f(x)-F(x,T))]+\frac{1}{2}E_T[(f(x)-F(x,T))^2]$

The first term is intrinsic error; second reduces to 0; we are interested in the third term.

## bias and variance

• bias: how much $F(x,T)$ differs from the true function $f(x)$, approximation error
• variance: the variance in $F(x,T)$ over entire training set $T$, estimation error

## VC dimension

Vapnik-Chervonenkis

Shattering: a dichotomy of a set S is a partition of S into two disjoint subsets.

A set of instances S is shattered by a function class $\mathcal{F}$ if and only if for every dichotomy of S there exists some function in $\mathcal{F}$ consistent with this dichotomy.

The VC dimension is the size of the largest finite subset shattered by that function.

At least one subset of a size can be shattered, then this size can be shattered by that $\mathcal{F}$.

If $\mathcal{F}$ is a set of lines, $VC(\mathcal{F}) = 3$

VC dimension increases:

• Training error decreases
• confidence interval increases
• sample complexity increases