Proposal about continuous learning

Description:

This issue proposes an enhancement to the Kolmogorov-Arnold Networks (KAN) architecture that involves the development of an automated method for converting trained models into symbolic activation functions. This approach aims to leverage the mathematical elegance and efficiency of symbolic computations to enhance the interpretability and efficiency of KAN models.

Motivation:

KANs introduce a novel method of utilizing learnable activation functions parametrized as splines, which replace traditional linear weight parameters. The adaptability of KAN to complex functions, as demonstrated in various mathematical and physical domains, showcases its potential. Extending this architecture to support symbolic activation functions could enable a new class of continuously learning models that evolve through various levels of complexity, starting from basic operations to more intricate computations like (mathematical) convolutions.

Pros:

Enhanced Efficiency: Symbolic representations can significantly reduce the computational overhead compared to traditional MLPs, particularly in environments where computational resources are constrained.
Continuous Learning Capability: By transforming learned behaviors into symbolic forms, models can seamlessly integrate new learning phases, potentially leading to a model that can evolve indefinitely without the need for retraining from scratch.
Improved Interpretability: Symbolic forms inherently allow for easier understanding and verification by humans, which is crucial for applications in fields requiring strict validation of model behavior, such as healthcare and physics.
Custom Library Development: This feature could pave the way for the creation of custom libraries of symbolic functions that can be reused across different models and applications, enhancing the modularity and reusability of neural network components.

Cons:

Complexity of Implementation: Developing an automated system to accurately convert learned numerical parameters into symbolic expressions involves significant challenges, particularly in ensuring the robustness and accuracy of the conversion process.
Potential Overfitting: While symbolic expressions can compactly represent learned behaviors, there's a risk of overfitting to specific patterns in the data, especially when the model complexity grows.
Scalability Issues: As the complexity of functions increases, the symbolic representation might become unwieldy or computationally intensive to evaluate, especially for real-time applications.
Limited Applicability: The effectiveness of symbolic activations may vary across different types of data and domains, potentially limiting the universality of this approach.

Suggested Implementation Strategy:

Phase 1: Develop a prototype that focuses on converting simple algebraic and trigonometric functions learned by KANs into symbolic form.
Phase 2: Test and refine the conversion algorithms using synthetic datasets designed to evaluate both the accuracy and computational efficiency of symbolic activations.
Phase 3: Expand the library of convertible functions based on feedback and testing, and integrate more complex functions such as those involved in convolution operations.
Phase 4: Release a beta version for community testing and feedback, targeting researchers and practitioners in domains that benefit from symbolic computation, such as symbolic regression, automated theorem proving, and complex system modeling.

Conclusion:

This proposal in my humble opinion sets the stage for a significant leap in the evolution of neural network architectures by integrating the robustness and interpretability of symbolic mathematics with the adaptive learning capabilities of KANs. I believe that this approach not only opens up new research avenues but also enhances the practical applicability of neural networks in scientifically rigorous domains.

I'd like to highlight an example that vividly illustrates the potential advantage of employing a continuous learning approach, like the one we propose with symbolic activation functions in KANs, over traditional MLP methods.

Consider a neural network tasked with learning to predict the trajectories of planets in a solar system—a classic problem that could benefit from understanding complex gravitational interactions. Typically, with an MLP, each planet's trajectory prediction would be approximated independently using large datasets containing positions over time, requiring the MLP to relearn from scratch if additional planets are introduced or if the dynamics slightly change, due to factors like added moons or asteroids.

In contrast, a KAN using symbolic activation functions could first learn simple gravitational relationships between two bodies (like the Earth and the Moon) in a symbolic form that explicitly captures Newton's law of universal gravitation. Once this relationship is learned symbolically, the model could be extended to incorporate more bodies (like adding Mars and its moons) without starting the learning process over. Instead, the model "continuously learns" by expanding its existing symbolic knowledge base. This approach not only saves computational resources by not requiring retraining from scratch but also enhances the model's adaptability to new data or scenarios.

Moreover, the symbolic form of the learned functions allows for direct interpretation and modification by scientists. For instance, if a new element like a satellite is introduced into the system, researchers can directly insert its influence into the symbolic equations without needing extensive new data for retraining. This adaptability and expandability exemplify the practical benefits of a continuous learning framework over a typical MLP setup, where adaptability often requires retraining or fine-tuning with a significant computational cost.

This example underscores the transformative potential of integrating symbolic activation functions into KANs, promising more adaptable, efficient, and interpretable models compared to the traditional MLP approaches in dynamically changing environments like space exploration or even financial markets where underlying relationships can be similarly complex and evolving.

Sorry to be pedantic, but more and more examples come to my mind, An illustrative example of the benefits offered by expanding the function space of activation functions in KANs, especially compared to traditional MLP approaches, can be seen in the task of image recognition—specifically in scenarios requiring the classification of objects across varying scales and orientations.

In a typical MLP setting, each layer’s fixed activation functions (like ReLU or sigmoid) limit the network's ability to adapt to new or varied data without comprehensive retraining. This is evident in tasks like recognizing objects in images where the same object may appear in different sizes or perspectives. MLPs often struggle with such variations unless explicitly trained on a wide range of transformations, which can be data and computation-intensive.

Conversely, by using KANs with a richer set of activation functions that are not just learnable but also symbolic, the network can potentially learn more generalized representations of objects. For instance, a KAN could learn a symbolic activation that encapsulates scaling and rotation invariances directly within its function. This means that once the KAN learns to recognize an object in one configuration, it can automatically recognize it in other configurations without needing additional data samples of each new orientation or scale.

This capability is fundamentally enabled by the engineering of a broader function space for the activation functions. Such activations can be designed to embody more complex mathematical transformations that are typical in advanced computer vision tasks. This not only reduces the need for extensive data augmentation during training but also enhances the network’s efficiency during inference, as the model can generalize well from limited data.

Moreover, these efficiency gains are not merely computational but also conceptual, as they allow the network to operate more intuitively in line with how human vision abstracts and recognizes patterns and shapes irrespective of orientation or scale. This approach significantly streamlines the deployment of neural networks in real-world applications, where computational resources are at a premium and adaptability is crucial.

This example should clearly demonstrates how KANs, through an enriched functional space of activation functions, offer substantial efficiency improvements over MLPs, particularly in tasks requiring high levels of generalization and adaptability. Such advancements underscore the potential of KANs to reshape the landscape of neural network design and application.

Thank you for the great insights! I really need to read this carefully this weekend, looking like treasure troves, but for now need to study for a deadline due Friday :(

This sounds a lot like the Closed Form Liquid Time Constant networks. https://github.com/raminmh/liquid_time_constant_networks https://github.com/raminmh/CfC

I wonder what @raminmh thinks about it?

This sounds a lot like the Closed Form Liquid Time Constant networks. https://github.com/raminmh/liquid_time_constant_networks https://github.com/raminmh/CfC

I wonder what @raminmh thinks about it?

Mmmh I only see similarity in having smaller "submodules" reused in a "big picture", which is a quite common practice often times proving its benefits, but that was not the overall picture.

KindXiaoming / pykan