margin unlimited noisy threshold 0.10 noisy threshold 0.27 noisy threshold 0.35 noisy threshold 0.40
a b
Figure4.2: (a) Training error for different learn step sizes η.η = 0.002154, thin vertical line, was chosen for the following simulations as it minimizes the average training error reached within1000training cycles for the noisy threshold learning rule.
(b) Average number of training cycles required to reach zero training error.
both the margin and the stochastic learning rule do the same learn-ing steps to increase the margin Both increase the margins by movlearn-ing ϑ∗k+1andϑ∗k further away from the threshold. We expect that the noise on the threshold interferes with the learning of the task. The stronger the noise on the threshold the higher the probability that a learning step is moving a critical threshold in the wrong direction. This should lead to reduced convergence speed and issues when the noise is too strong. By setting κtrain to different values for both margin learning algorithms we study their dependence on this parameter.
Noise on the firing threshold before pattern presentation was im-plemented by drawing a new firing threshold from a uniform distri-bution in the interval [ϑ−κtrain,ϑ+κtrain]. With κtrain being the re-quested minimal margin. This simple stochastic learning rule is not limited to the multi-spike tempotron as it does not use the spike-threshold-surface to determine the current margin or the next learn-ing step. For instance R. Rubin, L. F. Abbott, and H. Sompolinsky, 2017 used a similar approach by drawing noise for the perceptron decision surface from a Gaussian distribution.
To compare the two methods for learning margins we use the syn-thetic embedded feature task and neuron parameters as described in section 2.5.1. The training pool consists of N = 1000 patterns gener-ated without noise. In each learning cycle every pattern is presented once. All simulations are run for 100different random seeds. To al-low for comparability between convergence times of the two learning methods we set the momentum parameter to zeroµ=0.0.
First we determined the optimal learning step sizeη for the noisy threshold algorithm with respect to the lowest training error reached within1000training cycles. The resulting parameterη=0.002154 was then used for tempotron, stochastic margin and margin learning rules to track how the training error, the amount of misclassified training patterns, and the critical threshold positionsϑ∗k+1 andϑk∗change over training time.ηmarginfor the margin learning rule was set to the same value asηfor the tempotron correction learn steps.
Figure 4.3 shows the results for both margin learning methods trained for different κtrain. Initially we set κtrain = 0.1 for both algo-rithms to compare them at a value ofκtrainthat both are able to reach.
For the margin learning rule we can set κtrain to infinity without it preventing the learning rule from solving the task.κincreases until it saturates at a maximum value for each training pattern. We measured the mean maximum margin for this task at around ¯κ =0.27. We used this value as κtrain for the stochastic margin algorithm as well as val-ues above it (κtrain = 0.35, κtrain = 0.40) to demonstrate the limits of the stochastic learning rule.
As expected the stochastic margin learning algorithm requires more training cycles to reach a zero tempotron error when κtrain is increased. Doing the same using the gradient based margin learning method does not negatively affect the mean amount of training cy-cles needed. Since every margin training step is designed to directly operate on the plateau in the spike-threshold-surface it requires less training cycles to reach the target margin. When setting κtrain to in-finity the learning rule increases the margin until it saturates with out negatively affecting the convergence time. But if the margin pa-rameter for the stochastic algorithm is set too high its performance starts to degrade, it becomes orders of magnitudes slower and not all simulation runs converge to a solution within the maximal number of training cycles. To achieve a maximal margin with the stochastic learning algorithm the margin parameter needs to be carefully tuned since a too small parameter wastes potential and a too high param-eter increases training time and the risk for unconverged simulation runs.
4.3 n o i s e r o b u s t n e s s
The mean κ value across all training patterns is not a good indica-tor for the robustness of the classifier. In maximal margin classifiers the optimal hyperplane is defined by the margins to the nearest data points in the training vectors. Similarly the smallestκacross all train-ing patterns is a better indicator for the robustness of the neural
classi-4.3 n o i s e r o b u s t n e s s 47
fier. In figure4.3panel (c) it can be seen that the meanκfor tempotron learning without margin is relatively high (>0.1) but individual pat-terns have a κclose to the firing threshold (<0.01) while the margin learning rules are able to increaseκfor all training patterns above 0.1.
To study how noise robustness and generalization performance of the tempotron with and without margin learning is impacted by the amount of available training data we measured the performance with different training pool sizes. Simulations were allowed to run until the trainings error reached zero and the average margin width, the distance between ϑ∗k+1 and ϑ∗k, saturated (< 1% increase in the last 250training cycles). Tempotron learn step size was set toη=1e−05, momentum is set to µ = 0.99, no noise was applied during training and test pool generation. We used three different learning step sizes for margin learning ηmargin = 1e−06, ηmargin = 5e−06 and ηmargin = 25e−06. Results are shown in figures4.4,4.5and4.6.
The minimum margin for both sides, ϑk∗+1 and ϑ∗k, of the multi-spike tempotron, are very close to the threshold (figure 4.4 (b)). Al-ready minimal threshold noise or additions and removal of input spikes can lead to misclassification of training patterns. This can be seen in figure 4.5(a), it shows the classification error on training pat-terns under 5 and 10% spike noise (random addition and removal of input spikes). Margin learning increases the observable margins around the threshold of training patterns which leads to drastically improved noise robustness of the resulting solutions. Generalization performance is only slightly improved, see figure4.5(b).
These results show that this margin learning rule stabilizes the found solution regardless of its ability to detect the target feature correctly.