Risk Comparisons in Linear Regression: Implicit Regularization Dominates Explicit Regularization
arXiv:2509.17251v2 Announce Type: replace-cross Abstract: Existing theory suggests that for linear regression problems categorized by capacity and source conditions, gradient descent (GD) is always minimax optimal, while both ridge regression and online stochastic gradient descent (SGD) are polynomially suboptimal for certain categories of such problems. Moving beyond minimax theory, this work provides instance-wise comparisons of the finite-sample risks for these algorithms on any well-specified linear regression problem. Our analysis yields three key findings. First, GD dominates ridge regre
This research, published in 2026, represents ongoing advancements in theoretical machine learning, refining the understanding of fundamental algorithms.
Improved theoretical understanding of linear regression algorithms like gradient descent and ridge regression can lead to more efficient and reliable AI model development.
The focus shifts from broad minimax optimality to instance-wise finite-sample risk comparisons, revealing hidden strengths and weaknesses of common optimization methods.
- · AI researchers and practitioners
- · Machine learning framework developers
- · Industries relying on statistical modeling
- · Developers solely relying on minimax theory
- · Inefficient AI model implementations
Refined understanding of implicit regularization in core machine learning algorithms.
Development of new algorithms or modifications that leverage these insights for improved performance.
Enhanced efficiency and robustness of AI systems across various applications due to optimized underlying statistical methods.
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Read at arXiv cs.LG