PDP Class January 16 2013 Goodness of Network States and their Probabilities Goodness of a network state How networks maximize goodness The Hopfield network and Rumelharts continuous version ID: 542531
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Slide1
The Essence of PDP: Local Processing, Global Outcomes
PDP Class
January
16, 2013Slide2
Goodness of Network States and their Probabilities
Goodness of a network state
How networks maximize goodness
The Hopfield network and
Rumelhart’s
continuous version
Stochastic networks: The Boltzmann Machine, and the relationship between goodness and probability
Equilibrium,
ergodicity
, and annealing
Exploring the relationship between Goodness and
Probability
in an ensemble of networksSlide3
Network Goodness and How to Increase itSlide4
The Hopfield Network
Assume symmetric weights.
Units have binary states [+1,-1]
Units are set into initial states
Choose a unit to update at random
If net > 0, then set state to 1.
Else set state to -1.Goodness always increases… or stays the same.Slide5
Rumelhart’s Continuous Version
Unit states have values between 0 and 1.
Units are updated asynchronously. Update is gradual,
according to the rule:
There are separate scaling parameters for external and
internal input:Slide6
The Cube Network
Positive weights have value +1
Negative weights have value -1.5
‘External input’ is implemented as a positive bias of .5 to all units.
These values are all scaled by the
istr
parameter in calculating
goodness
in the program (
istr
= 0.4).Slide7
Goodness Landscape of Cube NetworkSlide8
Rumelhart’s Room Schema Model
Units for attributes/objects found in rooms
Data: lists of attributes found in rooms
No room labels
Weights and biases:
Modes of use in simulation:Clamp one or more units, let the network settle
Clamp all units, let the network calculate the Goodness of a state (‘pattern’ mode)Slide9
Weights for all unitsSlide10
Goodness Landscape for Some RoomsSlide11
Slices thru landscape with three different starting pointsSlide12
The Boltzmann Machine:The Stochastic Hopfield Network
Units have binary states [0,1], Update is asynchronous. The activation
function is:
Assuming processing is ergodic: that is, it is possible to get from any state to any
other state, then when the state of the network reaches equilibrium,
the relative probability and relative goodness of two states are related as follows:
More generally, at equilibrium we have the Probability-Goodness Equation:
orSlide13
Simulated Annealing
Start with high temperature. This means it is easy to jump from state to state.
Gradually reduce temperature.
In the limit of infinitely slow annealing, we can guarantee that the network will be in the best possible state (or in one of them, if two or more are equally good).
Thus, the best possible interpretation can always be found (if you are patient)!Slide14
Exploring Probability Distributions over States
Imagine settling to a non-zero temperature, such as T = 0.5.
At this temperature, there’s still some probability of being in a state that is less than perfect.
Consider an ensemble of networks
At equilibrium (i.e. after enough cycles, possibly with annealing) the relative frequencies of being in the different states will approximate the relative probabilities given by the Probability-Goodness equation.
You will have an opportunity to explore this situation in the homework assignment