Curl Descent: Non-Gradient Learning Dynamics with Sign-Diverse Plasticity

2025-10-03

AI Summary

This work investigates whether biological neural networks employ intrinsically non-gradient learning mechanisms, specifically focusing on “curl” components—dynamical terms in learning that cannot be expressed as the gradient of any scalar objective function—induced by inhibitory–excitatory connectivity or Hebbian/anti-Hebbian synaptic plasticity. Method: Using analytical modeling of feedforward networks under the teacher–student framework, we systematically introduce sign-heterogeneous synaptic plasticity rules. Contribution/Results: We theoretically derive and empirically verify that anti-Hebbian plasticity explicitly introduces curl, breaking the gradient-descent paradigm. Weak curl preserves convergence stability, whereas strong curl accelerates escape from saddle points in specific architectures, enhancing optimization efficiency—outperforming standard gradient descent in certain regimes. These findings provide a key mechanistic foundation and empirical support for developing biologically plausible, non-gradient learning theories.

Technology Category

Machine Learning: Bio-inspired LearningNatural Language Processing: Learning & Optimization for NLPSearch and Optimization: Learning to Search

Application Category

Economics, Online Markets and Human Computation: Social networks and social learningGraph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Gradient-based algorithms are a cornerstone of artificial neural network training, yet it remains unclear whether biological neural networks use similar gradient-based strategies during learning. Experiments often discover a diversity of synaptic plasticity rules, but whether these amount to an approximation to gradient descent is unclear. Here we investigate a previously overlooked possibility: that learning dynamics may include fundamentally non-gradient "curl"-like components while still being able to effectively optimize a loss function. Curl terms naturally emerge in networks with inhibitory-excitatory connectivity or Hebbian/anti-Hebbian plasticity, resulting in learning dynamics that cannot be framed as gradient descent on any objective. To investigate the impact of these curl terms, we analyze feedforward networks within an analytically tractable student-teacher framework, systematically introducing non-gradient dynamics through neurons exhibiting rule-flipped plasticity. Small curl terms preserve the stability of the original solution manifold, resulting in learning dynamics similar to gradient descent. Beyond a critical value, strong curl terms destabilize the solution manifold. Depending on the network architecture, this loss of stability can lead to chaotic learning dynamics that destroy performance. In other cases, the curl terms can counterintuitively speed learning compared to gradient descent by allowing the weight dynamics to escape saddles by temporarily ascending the loss. Our results identify specific architectures capable of supporting robust learning via diverse learning rules, providing an important counterpoint to normative theories of gradient-based learning in neural networks.

Problem

• Investigating non-gradient curl components in neural learning dynamics
• Analyzing how curl terms affect stability and performance in networks
• Identifying architectures supporting robust learning via diverse plasticity rules

Innovation

• Non-gradient curl dynamics optimize loss functions
• Curl terms emerge from inhibitory-excitatory connectivity
• Curl components enable saddle escape and accelerated learning