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Game Theory Meets Statistical Mechanics in Deep Learning Design

Djamel Bouchaffra, Fayçal Ykhlef, Bilal Faye, Hanane Azzag, Mustapha Lebbah

TL;DR

A novel deep graphical representation that seamlessly merges principles of game theory with laws of statistical mechanics is presented and outperforms both multi-layer perceptron and convolutional neural network models in terms of efficiency and accuracy.

Abstract

We present a novel deep graphical representation that seamlessly merges principles of game theory with laws of statistical mechanics. It performs feature extraction, dimensionality reduction, and pattern classification within a single learning framework. Our approach draws an analogy between neurons in a network and players in a game theory model. Furthermore, each neuron viewed as a classical particle (subject to statistical physics' laws) is mapped to a set of actions representing specific activation value, and neural network layers are conceptualized as games in a sequential cooperative game theory setting. The feed-forward process in deep learning is interpreted as a sequential game, where each game comprises a set of players. During training, neurons are iteratively evaluated and filtered based on their contributions to a payoff function, which is quantified using the Shapley value driven by an energy function. Each set of neurons that significantly contributes to the payoff function forms a strong coalition. These neurons are the only ones permitted to propagate the information forward to the next layers. We applied this methodology to the task of facial age estimation and gender classification. Experimental results demonstrate that our approach outperforms both multi-layer perceptron and convolutional neural network models in terms of efficiency and accuracy.

Game Theory Meets Statistical Mechanics in Deep Learning Design

TL;DR

A novel deep graphical representation that seamlessly merges principles of game theory with laws of statistical mechanics is presented and outperforms both multi-layer perceptron and convolutional neural network models in terms of efficiency and accuracy.

Abstract

We present a novel deep graphical representation that seamlessly merges principles of game theory with laws of statistical mechanics. It performs feature extraction, dimensionality reduction, and pattern classification within a single learning framework. Our approach draws an analogy between neurons in a network and players in a game theory model. Furthermore, each neuron viewed as a classical particle (subject to statistical physics' laws) is mapped to a set of actions representing specific activation value, and neural network layers are conceptualized as games in a sequential cooperative game theory setting. The feed-forward process in deep learning is interpreted as a sequential game, where each game comprises a set of players. During training, neurons are iteratively evaluated and filtered based on their contributions to a payoff function, which is quantified using the Shapley value driven by an energy function. Each set of neurons that significantly contributes to the payoff function forms a strong coalition. These neurons are the only ones permitted to propagate the information forward to the next layers. We applied this methodology to the task of facial age estimation and gender classification. Experimental results demonstrate that our approach outperforms both multi-layer perceptron and convolutional neural network models in terms of efficiency and accuracy.

Paper Structure

This paper contains 23 sections, 1 theorem, 8 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Some Basics of Game Theory
  3. Simple Games
  4. Notion of Simple Coalition
  5. NEUROGAME: Game Theory Meets Statistical Mechanics
  6. Comparison between Conventional Deep Learning and NEUROGAME
  7. Computation of a Coalition Payoff
  8. Neurons Neighborhood System:
  9. Gibbs Distribution of a Configuration State:
  10. Generation of Configurations States:
  11. Layer Neighborhood System:
  12. The Payoff Function:
  13. The Concept of Strong Coalition:
  14. The Different Phases in NEUROGAME
  15. NEUROGAME Layers and Information Propagation
  16. ...and 8 more sections

Key Result

Theorem 1

For large values of $n$ (coalition size) and $i$ (iteration number during training), the function $\rho(S_{c_j}^i,i)$ will tend to increase, with the growth rate influenced by $ln(1+i)$.

Figures (4)

  • Figure 1: The training procedure of NEUROGAME showing the passage from M simple coalitions to p winning coalitions and then to p strong coalitions generated via the Shapley filtering process. The computation of the strong coalitions (integrated into a fully connected neural network) is repeated across all $k$ layers until NEUROGAME converges. The feature vector extracted at this convergence point is composed of activation values of the last optimal strong coalitions.
  • Figure 2: Comparison of training and validation losses and accuracies between MLP and NEUROGAME models. NEUROGAME shows better generalization performance, as evidenced by the lower validation loss and improved validation metrics.
  • Figure 3: Comparison of training and validation performance between CNN and NEUROGAME models for gender classification.
  • Figure 4: Comparison of training and validation performance between CNN and NEUROGAME for age classification.

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1: Shapley Threshold Behavior
  • proof