[Paper Review] Regret Analysis of Global Optimization in Univariate Functions with Lipschitz Derivatives

Kaan Gökcesu, Hakan Gökcesu|arXiv (Cornell University)|Aug 24, 2021

Advanced Bandit Algorithms Research48 references4 citations

TL;DR

This paper analyzes cumulative regret in global optimization of univariate functions using lower bounding algorithms like Piyavskii-Shubert. For Lipschitz continuous functions with parameter $L$, it establishes $O(L\log T)$ cumulative regret; for Lipschitz smooth functions with parameter $H$, it achieves $O(H)$ regret, extending to broader function classes.

ABSTRACT

In this work, we study the problem of global optimization in univariate loss functions, where we analyze the regret of the popular lower bounding algorithms (e.g., Piyavskii-Shubert algorithm). For any given time $T$, instead of the widely available simple regret (which is the difference of the losses between the best estimation up to $T$ and the global optimizer), we study the cumulative regret up to that time. With a suitable lower bounding algorithm, we show that it is possible to achieve satisfactory cumulative regret bounds for different classes of functions. For Lipschitz continuous functions with the parameter $L$, we show that the cumulative regret is $O(L\log T)$. For Lipschitz smooth functions with the parameter $H$, we show that the cumulative regret is $O(H)$. We also analytically extend our results for a broader class of functions that covers both the Lipschitz continuous and smooth functions individually.

Motivation & Objective

• To analyze cumulative regret in global optimization of univariate functions, moving beyond simple regret.
• To evaluate the performance of lower bounding algorithms such as Piyavskii-Shubert in minimizing cumulative regret.
• To derive tight regret bounds for functions with Lipschitz continuous and Lipschitz smooth derivatives.
• To extend the analysis to a broader class of functions that generalizes both Lipschitz continuous and smooth functions.

Proposed method

• Adopt a cumulative regret framework instead of simple regret, measuring total loss over time $T$.
• Apply lower bounding algorithms that construct confidence intervals based on derivative Lipschitz constraints.
• Use the Piyavskii-Shubert algorithm as a core method for function approximation under Lipschitz derivative assumptions.
• Derive regret bounds by analyzing the convergence rate of the lower bound to the global minimum.
• Generalize the results to a broader function class by combining Lipschitz continuity and smoothness properties.

Experimental results

Research questions

• RQ1 What is the cumulative regret behavior of lower bounding algorithms in univariate global optimization?
• RQ2 How does the cumulative regret scale with time $T$ for Lipschitz continuous functions with parameter $L$?
• RQ3 What is the cumulative regret bound for Lipschitz smooth functions with parameter $H$?
• RQ4 Can the regret analysis be extended to a broader class of functions that includes both Lipschitz continuous and smooth functions?

Key findings

• For univariate functions with Lipschitz continuous derivatives of parameter $L$, the cumulative regret is bounded by $O(L\log T)$.
• For functions with Lipschitz smooth derivatives of parameter $H$, the cumulative regret is bounded by $O(H)$, independent of $T$.
• The regret bounds are derived using a lower bounding algorithm that exploits derivative Lipschitz constraints to refine search intervals.
• The analysis extends to a broader class of functions that generalizes both Lipschitz continuous and smooth functions, maintaining the same regret order.