On the Oracle Complexity of Interpolation-Based Gradient Descent

2026-06-18 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

Piecewise polynomial interpolation-based gradient descent (PPI-GD) is a new inexact gradient method designed to improve the oracle complexity of gradient descent (GD) for empirical risk minimization (ERM). PPI-GD approximates the full gradient by querying a first-order oracle at equidistant points within the data domain, then constructing polynomial interpolants from these gradient samples over appropriately sized data patches. The analysis shows PPI-GD outperforms several GD variants for strongly convex and non-convex loss functions, particularly when the data space dimension is bounded by a polylogarithmic function of the number of training samples and the loss function is sufficiently smooth. This work also extends error analysis techniques for bicubic spline interpolants to d-variate tensor product polynomial interpolants.

Key takeaway

For AI Scientists optimizing empirical risk minimization with gradient descent, consider that data smoothness, not just parameter smoothness, can significantly impact optimization efficiency. Your choice of gradient method should account for the data space dimension and loss function smoothness. PPI-GD demonstrates that inexact gradient approximations, built on polynomial interpolation, can yield superior oracle complexity in specific regimes, suggesting a re-evaluation of traditional GD variants for certain problem types.

Key insights

PPI-GD improves gradient descent oracle complexity by approximating gradients via polynomial interpolation over data domain patches.

Principles

• Smoothness in training data can enhance GD oracle complexity.
• Inexact gradient methods can outperform full gradient variants.

Method

PPI-GD approximates full gradients by querying a first-order oracle at equidistant data points, then constructing polynomial interpolants from these samples over specific data domain patches.

Topics

• Gradient Descent
• Oracle Complexity
• Empirical Risk Minimization
• Polynomial Interpolation
• Optimization Algorithms
• Loss Functions

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.