The Mathematical Odyssey to Artificial General Intelligence

Published in

Autonomous Agents

5 min readSep 6, 2023

The pursuit of Artificial General Intelligence (AGI) is not just a technological endeavor but a deeply mathematical one. While the current generation of Large Language Models (LLMs) has made commendable progress, they remain a mere shadow of the vast potential AGI promises. To truly understand the limitations of LLMs and the path forward, we must embark on a mathematical odyssey, drawing from a rich horizon of mathematical theories and concepts.

I present a few mathematical ideas in this blog with links to AI and Math research I have been working on my own.

The Mathematical Elegance of Graph Dynamical Systems

In my paper “Graph Dynamical Systems and Odd Analytic Coupling: Implications for Stability and Attention Mechanisms in Transformers” I offer a tantalizing glimpse into the world of energy-centric attention models.

Odd Analytic Functions: These functions, celebrated in complex analysis for their symmetry about the origin, can act as a powerful regularization mechanism in neural architectures. Their behavior is reminiscent of Euler’s formula, eix=cos(x)+isin(x), highlighting the harmonious relationship between exponential, trigonometric, and complex functions.
Gradient Dynamics: The exploration of the gradient structure of graph dynamical systems brings to mind the chaotic behavior observed in Lorenz systems. Just as the Lorenz system exhibits the butterfly effect, the flow of gradients during backpropagation in neural networks can significantly influence model stability and convergence.

Graph-based Attention: A Symphony of Tokens and Weights

Drawing parallels with graph theory, In my paper “Enhancing Attention with Graph Dynamical Systems and Odd Analytic Functions” I conceptualize tokens as vertices and attention weights as edges. This dynamical system on a graph is reminiscent of the Laplacian matrix in spectral graph theory.

Spectral Graph Theory: The eigenvalues of the Laplacian matrix, known as the spectrum of the graph, can reveal much about a graph’s structure. In the context of attention mechanisms, the spectrum could provide insights into the inherent relationships between tokens.

Energy Landscapes and the Quest for Optimal Attention

The concept of energy-based attention refinement is deeply rooted in statistical mechanics.

Boltzmann Distribution: In statistical mechanics, systems tend to evolve towards states of lower energy. Similarly, the attention mechanism, by minimizing its energy function, drives the system towards a state of optimal attention distribution.

Stability, Bifurcations, and the Dance of Equilibria

The paper’s discussion on equilibrium points and bifurcations is deeply evocative of catastrophe theory.

Catastrophe Theory: This mathematical theory studies the phenomena where continuous changes in parameters lead to sudden and drastic changes in system behavior. In the context of attention mechanisms, small shifts in external parameters might lead to paradigm shifts in attention distribution behaviors.

Topology and Neural Networks

Topology, the study of spaces and the properties preserved under continuous transformations, has profound implications for neural networks.

Persistent Homology: This concept from topological data analysis can be used to study the features of neural networks that persist across various scales. It could provide insights into the hierarchical structure of learned features.

Chaos Theory and Neural Dynamics

Chaos theory, which studies the behavior of dynamical systems highly sensitive to initial conditions, can offer insights into the training dynamics of neural networks.

Strange Attractors: These are sets to which a dynamical system evolves after a long time. Understanding the attractors of a neural network’s dynamics could provide insights into its long-term behavior and stability.

Information Theory and Model Generalization

Information theory, founded by Claude Shannon, can shed light on a model’s ability to generalize from training data.

Entropy and Mutual Information: By studying the entropy of model predictions and the mutual information between inputs and outputs, we can gain insights into the model’s uncertainty and its ability to generalize to new data.

The Fractional Laplacian: Bridging Scales and Domains

I wrote two papers on one of the research areas I am focused on in Fractional Laplacians:

The fractional Laplacian, a non-local differential operator, has recently emerged as a powerful mathematical tool with applications spanning various domains. Its ability to capture interactions over extended regions makes it particularly intriguing for modeling systems with long-range dependencies.

Non-local Interactions: Traditional differential operators, like the Laplacian, consider local interactions. In contrast, the fractional Laplacian captures the influence of distant points, introducing a memory effect. This non-local nature allows it to model phenomena that exhibit long-range dependencies, bridging the gap between local and global interactions.
Spectral Properties: In the Fourier space, the fractional Laplacian acts as a multiplier, emphasizing its influence over a spectrum of frequencies. This spectral representation provides a deeper understanding of the operator’s action and its ability to capture multi-scale interactions.
Boundary Behavior: The behavior of the fractional Laplacian near boundaries is of significant interest, especially when modeling phenomena in bounded domains. Understanding how the fractional Laplacian interacts with boundaries can provide insights into its applications in various physical and cognitive models.

Fractional Elliptic Problems and AGI

Fractional elliptic problems, which involve the fractional Laplacian, offer a fresh perspective on the challenges and potential breakthroughs in AGI modeling.

Memory Effects: Just as the fractional Laplacian introduces a memory effect, AGI systems can benefit from considering historical data and past interactions. This can lead to more robust decision-making and reasoning capabilities.
Modeling Long-term Dependencies: The ability of the fractional Laplacian to model long-term dependencies can be harnessed in neural networks and other AGI architectures. By incorporating non-local interactions, these models can capture richer patterns and relationships in data.
Enhancing Neural Dynamics: Introducing the fractional Laplacian into neural network dynamics can lead to more robust learning mechanisms. The non-local interactions can potentially mitigate challenges like the vanishing gradient problem, ensuring that activations from distant layers have a pronounced influence on the learning process.

Setting Boundaries with the Maximum Principle

The strong maximum principle, a foundational concept in the study of partial differential equations, provides insights into the behavior of solutions within a domain. For fractional elliptic problems, this principle offers a unique perspective on the behavior of solutions, especially in the context of AGI modeling.

Classical vs. Fractional: While the classical maximum principle provides information about the behavior of solutions without solving the differential equation explicitly, the fractional version introduces nuances due to the non-local nature of the fractional Laplacian.
Implications for Cognitive Modeling: Understanding the behavior of solutions to fractional elliptic problems can provide insights into cognitive processes, decision-making, and other aspects of AGI.

The Confluence of Mathematics and AGI

The journey to AGI is a confluence of diverse mathematical theories and concepts. From the elegant symmetries of complex analysis, intricate topologies of data spaces, to the non-local interactions of the fractional Laplacian, each mathematical insight brings us a step closer to understanding and achieving AGI. As we continue to weave these mathematical threads together, we pave a robust and enlightened path towards the realization of AGI. The odyssey is long, but with mathematics as our compass, the horizon is within reach.

Disclaimer

Freedom Preetham is an AI Researcher with background in math and quantum physics and working on genomics in particular. You are free to use and expand on this research idea as applicable to other domains. Attribution to Freedom Preetham is welcome if you find it useful.