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  1. Home
  3. Gradient Descent As Loss Landscape Navigation: A Normative Framework For Deriving Learning Rules.
  1. Home
  3. Gradient Descent As Loss Landscape Navigation: A Normative Framework For Deriving Learning Rules.

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

John J Vastola1,2,3, Samuel J Gershman2,3, Kanaka Rajan1,3

  • 1Department of Neurobiology, Harvard Medical School.

Advances in Neural Information Processing Systems
|May 11, 2026

View abstract on PubMed

Summary
This summary is machine-generated.

This study frames learning rules as optimal control problems for navigating loss landscapes. It unifies gradient descent, momentum, and adaptive methods under a single theoretical framework for designing better machine learning algorithms.

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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.