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# Learning With Attractors

- Authors
- Name
- Jonas Kgomo
- @jonask

# Learning With Attractors

Initial conditions dictate how complex systems behave, in chaos theory this is the fundamental study, small differences in initial conditions can lead to drastically different outcomes over time. ML systems are sensitive to various initial conditions, ML models with different data sets bring forth different results.

Chaos theory is a branch of mathematics that studies the behavior of dynamic systems that are highly sensitive to initial conditions, often referred to as the butterfly effect. In chaos theory, it is often useful to study the long-term behavior of a system through the use of iterative maps and attractors. These tools can be used to visualize the evolution of the system and to identify patterns and structures within the system.

Machine learning involves the use of algorithms to analyze and learn from data, without being explicitly programmed to do so. Machine learning algorithms can be used to identify patterns and make predictions based on data.

Differential Equations Most physics systems can be modelled with differential equations , and specifically, ordinary differential equations (ODEs) and partial differential equations (PDEs). Oxford Mathematician Patrick Kidger in his thesis, shows that machine learning can be done with neural controlled differential equations.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin.

Initial Conditions In machine learning, the initial conditions of a model include the initial values of its parameters, the initial state of its internal states, and the initial data used for training.

For instance we may consider :

These initial conditions can have a significant impact on the model's performance and behavior, and should be carefully considered when designing and training a machine learning model.

Fluid turbulence develops through a strange attractor, a main concept of chaos theory. Iterative maps , attractors and cost functions A dynamic system is a system which evolution over time depends on its inputs (if any) and the value of its state. Perturbation theory studies this small changes , however there are other factors that affect a chaotic system, such as iterative maps and attractors.

There is a relationship between cost functions and iterative maps in that both can be used to study the behavior of a system over time. In machine learning, the optimization of a model's parameters often involves the use of iterative optimization algorithms, such as gradient descent, which involve the repeated application of a function (the cost function) to the model's parameters in order to improve the model's performance.

Overall, cost functions and iterative maps are both tools that can be used to study and optimize the behavior of a system over time. In the context of machine learning, they are often used together to optimize the parameters of a model and improve its performance.

Gradient Descent and Attractors Attractors in chaos theory are patterns or structures that emerge in the long-term behavior of a dynamic system. They can be used to visualize the evolution of a system and to identify patterns and structures within the system.

In gradient descent we want to move quickly in directions with small but consistent gradients(indetermintate initial conditions). Move slowly in directions with big but inconsistent gradients (predictibility). This is a guaranteed way to understand ML interpretability. A demonstration below considers the possibility of pairing two gradient descents with a Lorentz attractor.

A Lorentz attractor can be formulated as follows:

$\langle \dot{x},\dot{y}, \dot{z} \rangle = \sigma(y-x), x(\rho-z)-y , xy - \beta z$ ML is interested in what is the cost functions to end up at a rest state , a chaos question is what attractors or iterative maps do you use to end up at rest. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns,

interconnection (layers)

constant feedback loops, (cost functions)

repetition, (loss)

self-similarity, fractals, and (activation functions)

self-organization (agency)

Chaos ML Theory should therefore concern itself with reconciliation of ideas spanning both fields and exending already existing insights to understand global problems like truthfulness and interpretibility.

Observation and measurement The measurement problem in quantum mechanics refers to the difficulties in reconciling the probabilistic nature of quantum mechanics with our classical intuitive notions of measurement and observation. I wonder about how far this extends into measurements we make in RL environments: the measurement problem could potentially affect the learning process in several ways. For example, if the environment being modeled by the RL agent is itself a quantum system, the probabilistic nature of quantum mechanics could introduce uncertainty and unpredictability into the learning process. This could make it more difficult for the RL agent to learn the optimal policies for interacting with the environment.

Additionally, the measurement problem could potentially affect the accuracy and reliability of the data that is used to train the RL agent. If the data is collected through measurements of a quantum system, the probabilistic nature of quantum mechanics could introduce errors or uncertainties into the data. This could affect the RL agent's ability to learn from the data and to make accurate predictions about the environment.

Overall, the measurement problem in quantum mechanics could potentially have an impact on the learning process in RL environments, particularly if the environment being modeled is itself a quantum system.

Determinism and predictability Linear regression models are deterministic because they make predictions using a fixed set of coefficients (also called weights) that are learned from the training data.

There is some overlap between chaos theory and machine learning in that both can be used to study and analyze dynamic systems. Machine learning algorithms could be used to analyze data from a chaotic system and identify patterns or structures within the system that may not be easily visible to the naked eye. This article is a naive effort to inspire further archeological research into this studies