Every Local Minimum Value Is the Global Minimum Value of Induced Model in Nonconvex Machine Learning.

Area of Science:

• Machine Learning
• Optimization Theory
• Deep Learning Theory

Background:

• Nonconvex optimization is prevalent in machine learning, posing theoretical challenges.
• Understanding the properties of local minima in nonconvex models is crucial for reliable training.
• Existing theories often rely on convex assumptions or handcrafted bases, limiting applicability.

Purpose of the Study:

• To theoretically analyze the global optimality of local minima in nonconvex machine learning.
• To establish a theoretical equivalence between nonconvex and convex optimization under specific conditions.
• To provide a unified framework for analyzing various deep learning architectures.

Main Methods:

• Mathematical proofs under mild assumptions.
• Analysis of the perturbable gradient basis model.
• Geometric insights into optimization landscapes.

Main Results:

• Every local minimum in nonconvex models achieves the globally optimal value for the perturbable gradient basis model.
• Nonconvex machine learning is theoretically as supported as convex machine learning at differentiable local minima.
• Results are directly applicable to deep neural networks without method modification.

Conclusions:

• The findings bridge the gap between theory and practice in nonconvex machine learning.
• This work offers a unified theoretical foundation for analyzing deep neural networks, deep residual networks, and overparameterized models.
• The results contribute to the theoretical understanding of representation learning.