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Best of Both Worlds Guarantees for Smoothed Online Quadratic Optimization

Neelkamal Bhuyan, Debankur Mukherjee, Adam Wierman

TL;DR

This work advances smoothed online quadratic optimization by delivering a complete stochastic analysis and a robust best-of-both-worlds solution. It derives LAI as the online-optimal policy for martingale minimizers via dynamic programming, achieving an $\mathcal{O}(T)$ total cost and distribution-agnostic performance. It also characterizes the stochastic limitations of the adversarial-optimized ROBD and fixed interpolation strategies, and introduces lai($\gamma$), a tunable algorithm that attains near-optimal performance in both stochastic and adversarial environments. The results illuminate a practical path to switching-efficient online optimization across uncertain dynamics, with theoretical guarantees and empirical validation. The findings have implications for smart grids, adaptive control, and data-center management where both robustness and low switching costs are crucial.

Abstract

We study the smoothed online quadratic optimization (SOQO) problem where, at each round $t$, a player plays an action $x_t$ in response to a quadratic hitting cost and an additional squared $\ell_2$-norm cost for switching actions. This problem class has strong connections to a wide range of application domains including smart grid management, adaptive control, and data center management, where switching-efficient algorithms are highly sought after. We study the SOQO problem in both adversarial and stochastic settings, and in this process, perform the first stochastic analysis of this class of problems. We provide the online optimal algorithm when the minimizers of the hitting cost function evolve as a general stochastic process, which, for the case of martingale process, takes the form of a distribution-agnostic dynamic interpolation algorithm (LAI). Next, we present the stochastic-adversarial trade-off by proving an $Ω(T)$ expected regret for the adversarial optimal algorithm in the literature (ROBD) with respect to LAI and, a sub-optimal competitive ratio for LAI in the adversarial setting. Finally, we present a best-of-both-worlds algorithm that obtains a robust adversarial performance while simultaneously achieving a near-optimal stochastic performance.

Best of Both Worlds Guarantees for Smoothed Online Quadratic Optimization

TL;DR

This work advances smoothed online quadratic optimization by delivering a complete stochastic analysis and a robust best-of-both-worlds solution. It derives LAI as the online-optimal policy for martingale minimizers via dynamic programming, achieving an total cost and distribution-agnostic performance. It also characterizes the stochastic limitations of the adversarial-optimized ROBD and fixed interpolation strategies, and introduces lai(), a tunable algorithm that attains near-optimal performance in both stochastic and adversarial environments. The results illuminate a practical path to switching-efficient online optimization across uncertain dynamics, with theoretical guarantees and empirical validation. The findings have implications for smart grids, adaptive control, and data-center management where both robustness and low switching costs are crucial.

Abstract

We study the smoothed online quadratic optimization (SOQO) problem where, at each round , a player plays an action in response to a quadratic hitting cost and an additional squared -norm cost for switching actions. This problem class has strong connections to a wide range of application domains including smart grid management, adaptive control, and data center management, where switching-efficient algorithms are highly sought after. We study the SOQO problem in both adversarial and stochastic settings, and in this process, perform the first stochastic analysis of this class of problems. We provide the online optimal algorithm when the minimizers of the hitting cost function evolve as a general stochastic process, which, for the case of martingale process, takes the form of a distribution-agnostic dynamic interpolation algorithm (LAI). Next, we present the stochastic-adversarial trade-off by proving an expected regret for the adversarial optimal algorithm in the literature (ROBD) with respect to LAI and, a sub-optimal competitive ratio for LAI in the adversarial setting. Finally, we present a best-of-both-worlds algorithm that obtains a robust adversarial performance while simultaneously achieving a near-optimal stochastic performance.
Paper Structure (39 sections, 27 theorems, 175 equations, 2 figures, 2 algorithms)

This paper contains 39 sections, 27 theorems, 175 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model and Preliminaries
  3. Adversarial SOQO
  4. Stochastic SOQO
  5. Algorithms and Main Results
  6. Optimal Stochastic SOQO
  7. Stochastic Analysis of ROBD and Fixed Interpolation Algorithms
  8. Adversarial Analysis of Lazy Adaptive Interpolation Algorithms
  9. A Near-Optimal Algorithm for Stochastic and Adversarial SOQO
  10. Numerical Experiments
  11. Experiments in purely stochastic environments
  12. Experiments in stochastic and adversarial environments
  13. Experiments in stochastic and adversarial environments
  14. Concluding Remarks
  15. Detailed Literature Review
  16. ...and 24 more sections

Key Result

Theorem 3.1

Lazy Adaptive Interpolation lai is online optimal in the stochastic setting eqn:minimizer_martingale with total cost where $C_L = \frac{A + 2I - \sqrt{A^2 + 4A}}{2}$ and $v_0 = x_0$.

Figures (2)

  • Figure 1: Regret of lai(1) and robd for martingale minimizers with light and heavy tails
  • Figure 2: Behavior of lai(1) and robd for a mixed sequence of minimizers. In each figure, the topmost plot corresponds to $\left\{0.3^i \right\}_{i=0}^9$, the middle one represents $\left\{0.45^i \right\}_{i=0}^9$, and the bottom-most plot pertains to $\left\{0.5^i \right\}_{i=0}^9$.

Theorems & Definitions (49)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Definition 3.6
  • Theorem 3.7
  • ...and 39 more