Research Paper#Neural Networks, Optimization, Bayesian Inference🔬 ResearchAnalyzed: Jan 3, 2026 06:26
Gradient Descent as Implicit EM in Distance-Based Neural Models
Published:Dec 31, 2025 10:56
•1 min read
•ArXiv
Analysis
This paper provides a direct mathematical derivation showing that gradient descent on objectives with log-sum-exp structure over distances or energies implicitly performs Expectation-Maximization (EM). This unifies various learning regimes, including unsupervised mixture modeling, attention mechanisms, and cross-entropy classification, under a single mechanism. The key contribution is the algebraic identity that the gradient with respect to each distance is the negative posterior responsibility. This offers a new perspective on understanding the Bayesian behavior observed in neural networks, suggesting it's a consequence of the objective function's geometry rather than an emergent property.
Key Takeaways
- •Gradient descent on distance/energy-based objectives implicitly performs EM.
- •This unifies unsupervised learning, attention, and classification under a single mechanism.
- •Bayesian behavior in transformers is a consequence of objective geometry, not an emergent property.
- •Optimization and inference are the same process in these models.
Reference
“For any objective with log-sum-exp structure over distances or energies, the gradient with respect to each distance is exactly the negative posterior responsibility of the corresponding component: $\partial L / \partial d_j = -r_j$.”