fast classification using sparsely active spiking networksfpga prototype 97% test set classification...
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Fast classification using sparsely active spiking networks
Hesham MostafaInstitute of neural computation, UCSD
Artificial networks vs. spiking networks
input layer
hidden layer 1
hidden layer 2
output layer
......
backpropagation
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input layer
hidden layer 1
hidden layer 2
output layer
......
???
Multi-layer networks are extremelypowerful function approximators.
Backpropagation is the most effective method we know of to solve the credit assignment problem in deep artificial networks
How do we solve the credit assignmentproblem in multi-layer spiking networks?
Neural codes and gradient descent
Rate coding: ● Spike counts/rates are discrete quantities● Gradient is zero almost everywhere● Only indirect or approximate gradient
descent training possible
3
4
2
5
input output input output
t1
t2
t3
tOut
t1
t2
t3
tOut
Temporal coding● Spike times are analog quantities● Gradient of output spike time w.r.t input
spike times is well-defined and non-zero● Direct gradient descent training possible
The neuron model
dVmem(t)dt
=Isyn (t )
Isyn(t )=∑i
w i exp(−(t−t i))Θ(t−t i)
Θ(t−t i) : Step function
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Vm
em
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time0.0
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Isyn
(firing threshold is 1)
Non-leaky integrate and fire neuron
Exponentially decaying synaptic current
The neuron's transfer function
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em
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tout
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time0.0
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Isyn
w1w2
w3
w4
The neuron's transfer function
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Vm
em
t1 t2 t3 t4
tout
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
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Isyn
w1w2
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In general:
Where C is the causal set of inputspikes (input spikes that arrive before output spike)
The neuron's transfer function
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Vm
em
t1 t2 t3 t4
tout
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time0.0
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Isyn
w1w2
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In general:
Where C is the causal set of inputspikes (input spikes that arrive before output spike)
Time of the Lth output spike:
Change of variables
The neuron's transfer function then becomes piece-wise linear in the inputs(but not the weights):
Where is the non-linearity?
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vm
em
zout =w1z1 +w2z2 +w3z3
w1 +w2 +w3 −1
ln(zout)
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synapti
c cu
rrent ln(z1) ln(z2) ln(z3) ln(z4)
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em
zout =w1z1 +w2z2 +w3z3 +w4z4
w1 +w2 +w3 +w4 −1
ln(zout)
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time0.0
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synap
tic
curr
ent ln(z1) ln(z2) ln(z3)ln(z4)
Non-linearity arises due to the input dependence of the causal set of input spikes
The piecewise linear input-output relation is reminiscent of Rectified Linear Units(ReLU) networks
What is the form of computation implemented by the temporal dynamics?
....
● To compute zout:● Sort {z1,z2,..,zn} ● Find the causal set, C, by progressively considering more early spikes● Calculate
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vm
em
zout =w1z1 +w2z2 +w3z3
w1 +w2 +w3 −1
ln(zout)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
time0.0
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synapti
c cu
rrent ln(z1) ln(z2) ln(z3) ln(z4)
Can not be reduced to the conventional ANN neuron: zout=f (∑i
w i zi)
Backpropagation
To use backpropagation to train a multi-layer network, we need the derivatives of the neuron's output w.r.t:
Weights Inputs
Time of first spike encodes neuron’s value. Each neuron is allowed to spike only once in response to an input pattern:● Forces sparse activity. Training has to make maximum use of each spike● Allows quick classification response
Classification Tasks
● We can relate the time of any spike differentiably to the times of all spikes that caused it
● We can impose any differentiable cost function on the spike times of the output layer and use backpropagation to minimize cost across training set
● In a classification setting, use a loss function that encourages the output neuron representing the correct class to spike first
● Since we have an analytical input-output relation for each neuron, training can be done using conventional machine learning packages (Theano/Tensorflow)
MNIST task● Pixel values were binarized.● High intensity pixels spike early● Low intensity pixels spike late
Classification is extremely rapid
● A decision is made when the first output neuron spikes● A decision is made after only 25 spikes (on average) from the hidden layer
in the 768-800-10 network, i.e, only 3% of the hidden layer neurons contribute to each classification decision
FPGA prototype
● 97% test set classification accuracy on MNIST in a 784-600-10 network (8-bit weights)● Average number of spikes until classification: 139● Only 13% of input to hidden weights are looked up● Only 5% of hidden to output weights are looked up
0 100 200 300 400 500 600Timesteps to classification
0
200
400
600
800
1000
1200
1400
Count Mean : 167
Median : 162
0 20 40 60 80 100 120 140 160 180Number of hidden layer spikes before output spike
0
200
400
600
800
1000
1200
1400
Cou
nt Mean : 30
Median : 29
Acknowledgements
Institute ofneuroinformatics
Giacomo IndiveriTobi Delbruck
Gert CauwenberghsSadique SheikBruno Pedroi
Approximate learning
correct label
+
N7 N8 N9
- -
++
N5 N6
N2 N3
+ -- +
N4
N1
Output layer
Hidden layer
Input layer
● Update Hidden→Output weights to encourage the right neuron to spike first● Only update weights that actually contributed to output timings
Approximate learning
correct label
+
+1 -(-1)
-1 } -1 +1}
N7 N8 N9
- -
++
N5 N6
N2 N3
+ -- +
N4
N1
Output layer
Hidden layer
Input layer
● Backpropagate time deltas using only the sign of the weights● The final time delta at a hidden layer neuron can be obtained using 2
parallel popcount operations (count 1s in a bit vector) and a comparison.
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