Gradient Descent as Loss Landscape Navigation: a Normative Framework for Deriving Learning Rules

Area of Science:

• Machine Learning
• Optimization Theory
• Computational Neuroscience

Background:

• Learning rules are fundamental to improving model performance but are often empirically chosen rather than theoretically derived.
• Understanding the optimality conditions for various learning rules remains a significant challenge in artificial intelligence.

Purpose of the Study:

• To develop a unified theoretical framework for understanding and deriving optimal learning rules.
• To investigate the assumptions under which different learning rules are optimal.
• To provide a principled foundation for designing novel adaptive algorithms.

Main Methods:

• Casting learning rules as policies for navigating loss landscapes within an optimal control problem framework.
• Analyzing the emergence of known learning rules (e.g., gradient descent, momentum, Adam) under specific assumptions.
• Connecting continual learning strategies to task uncertainty within the proposed framework.

Main Results:

• Demonstrated that gradient descent arises from short-horizon optimization.
• Showed momentum emerges from longer-horizon planning.
• Explained natural gradients via parameter space geometry and adaptive optimizers like Adam via Bayesian inference.
• Characterized continual learning as optimal responses to task uncertainty.

Conclusions:

• The proposed optimal control framework unifies diverse learning rules under a single objective.
• This provides a principled approach for designing more effective and adaptive machine learning algorithms.
• The framework clarifies the computational structure underlying machine learning optimization.