A Brief Survey of Modern Optimization for Statisticians

Area of Science:

• Computational Statistics
• Optimization
• Data Science

Background:

• Big data necessitates advanced computational statistical methods.
• Model parameter estimation frequently relies on optimization techniques.
• Incorporating nondifferentiable penalty terms (e.g., lasso) is crucial for model parsimony but complicates optimization.

Purpose of the Study:

• To survey key principles in algorithm design for high-dimensional optimization.
• To demonstrate the benefits of combining algorithmic principles.
• To provide an accessible overview for a broad audience, emphasizing derivation over abstract theory.

Main Methods:

• Focus on practical algorithm derivation for optimization problems.
• Illustrate principles with well-chosen examples.
• Downplay abstract convex calculus in favor of applied methods.

Main Results:

• Identified core principles for designing optimization algorithms.
• Demonstrated the effectiveness of a mixed-and-matched approach to these principles.
• Provided practical insights into handling nondifferentiable terms in statistical models.

Conclusions:

• A pragmatic, principle-driven approach to algorithm design is effective for high-dimensional optimization.
• Combining algorithmic strategies enhances problem-solving in computational statistics.
• The survey offers accessible guidance for statisticians and data scientists dealing with big data.