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Provable local learning rule by expert aggregation for a Hawkes network

Sophie Jaffard, Samuel Vaiter, Alexandre Muzy, Patricia Reynaud-Bouret

TL;DR

A simple network of Hawkes processes is proposed as a cognitive model capable of learning to classify objects and it is proved mathematically that the network is able to learn on average and even asymptotically under more restrictive assumptions.

Abstract

We propose a simple network of Hawkes processes as a cognitive model capable of learning to classify objects. Our learning algorithm, named HAN for Hawkes Aggregation of Neurons, is based on a local synaptic learning rule based on spiking probabilities at each output node. We were able to use local regret bounds to prove mathematically that the network is able to learn on average and even asymptotically under more restrictive assumptions.

Provable local learning rule by expert aggregation for a Hawkes network

TL;DR

A simple network of Hawkes processes is proposed as a cognitive model capable of learning to classify objects and it is proved mathematically that the network is able to learn on average and even asymptotically under more restrictive assumptions.

Abstract

We propose a simple network of Hawkes processes as a cognitive model capable of learning to classify objects. Our learning algorithm, named HAN for Hawkes Aggregation of Neurons, is based on a local synaptic learning rule based on spiking probabilities at each output node. We were able to use local regret bounds to prove mathematically that the network is able to learn on average and even asymptotically under more restrictive assumptions.
Paper Structure (33 sections, 9 theorems, 106 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 33 sections, 9 theorems, 106 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. FRAMEWORK
  3. First notations and set-up
  4. The input layer
  5. The output layer
  6. Learning rule
  7. Gain formula
  8. Feasible weight family
  9. THEORETICAL RESULTS
  10. Average learning
  11. Limit behavior
  12. A CONCRETE EXAMPLE
  13. Framework
  14. Numerical results
  15. CONCLUSION
  16. ...and 18 more sections

Key Result

Theorem 3.3

Suppose Assumptions assumption reg and assump proportions hold. Let $\alpha \in (0,1]$. Suppose $\mathcal{Q}$ is non-empty. Then with probability greater than $1-\alpha$, where $E_{\text{tot}}(N,M,\alpha) = E_{\text{reg}}(M) + E(N,M,\alpha)$ with and

Figures (5)

  • Figure 1: Illustrative example of the network. The presented object excites the neurons encoding its features. Then it is classified in the class coded by the output neuron which spiked the most, here class $A$.
  • Figure 2: Numerical results with $M=2502$, $K=1$, $N=1000$, $p=0.2$, $q=0.3$, $\alpha^A=0.2$, $\alpha^B=0$, $\beta^j=2$ and $\eta^j=\frac{1}{\lvert \mathcal{O} \rvert}(2\frac{\ln(\lvert I^j \rvert)}{M})^{-1/2}$. Parameters of Component-Cue: $\lambda_w=0.005$ and $\phi=10$ (see details in Appendix \ref{['app cc']}). On the left, evolution of the proportion of correct classifications for HAN and HAN Solo with EWA and PWA and Component-Cue with time. A number $100$ of realizations were made; for each realization, a testing set of $500$ objects drawn randomly was generated. Then the network was trained for $278$ epochs, an epoch being a random sequence of the $9$ nature of objects. After each epoch the weights were frozen and the network performance was evaluated on the testing set. On the $x$-axis, number of epochs. On the $y$-axis, proportion of correctly classified objects of the testing set with confidence interval of level $0.9$. On the right, evolution of the weights of neurons $A$ and $B$ with time for one realization of HAN with EWA.
  • Figure 3: Comparison with the perceptron learning algorithm. Same parameters as in Figure \ref{['curves']}, learning rate $1$ for the perceptron. Zoom on the $10$ first epochs.
  • Figure 4: Evolution of the empirical spiking probabilities of neurons $A$ and $B$ with time by nature of object for the same realization of HAN with EWA as in Figure \ref{['curves']} (left). (Same parameters as in Figure \ref{['curves']}.)
  • Figure 5: Evolution of the proportion of correct classifications for HAN and HAN Solo with EWA and PWA and Component-Cue with time, with the same parameters as in Figure $2$. What changes here is that an epoch is a sequence of $9$ objects drawn randomly with replacement: all the natures of object are not necessarily presented during one epoch.

Theorems & Definitions (14)

  • Definition 2.1: Feasible weight family
  • Theorem 3.3: Oracle inequality
  • Theorem 3.4
  • Corollary 3.5
  • Proposition 8.1
  • Proposition 8.2
  • Proposition 13.1: Regret bound
  • proof
  • Proposition 13.2
  • proof
  • ...and 4 more