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Statistical Properties of Robust Satisficing

Zhiyi Li, Yunbei Xu, Ruohan Zhan

TL;DR

The paper develops statistical theory for Robust Satisficing (RS), a robust optimization paradigm using a reference value $\tau$ and Wasserstein distance to the empirical distribution. It derives non-asymptotic, two-sided confidence intervals for the optimal loss $J^*$ and finite-sample generalization bounds for the RS optimizer, valid even under distribution shifts. A key result is the explicit relation between RS and DRO under Lipschitz losses, enabling a direct hyperparameter correspondence and showing RS is less sensitive to tuning than DRO. Numerical experiments reinforce that RS improves small-sample and shift-robust performance and provides practical advantages due to its simpler guarantees and global distribution consideration.

Abstract

The Robust Satisficing (RS) model is an emerging approach to robust optimization, offering streamlined procedures and robust generalization across various applications. However, the statistical theory of RS remains unexplored in the literature. This paper fills in the gap by comprehensively analyzing the theoretical properties of the RS model. Notably, the RS structure offers a more straightforward path to deriving statistical guarantees compared to the seminal Distributionally Robust Optimization (DRO), resulting in a richer set of results. In particular, we establish two-sided confidence intervals for the optimal loss without the need to solve a minimax optimization problem explicitly. We further provide finite-sample generalization error bounds for the RS optimizer. Importantly, our results extend to scenarios involving distribution shifts, where discrepancies exist between the sampling and target distributions. Our numerical experiments show that the RS model consistently outperforms the baseline empirical risk minimization in small-sample regimes and under distribution shifts. Furthermore, compared to the DRO model, the RS model exhibits lower sensitivity to hyperparameter tuning, highlighting its practicability for robustness considerations.

Statistical Properties of Robust Satisficing

TL;DR

The paper develops statistical theory for Robust Satisficing (RS), a robust optimization paradigm using a reference value and Wasserstein distance to the empirical distribution. It derives non-asymptotic, two-sided confidence intervals for the optimal loss and finite-sample generalization bounds for the RS optimizer, valid even under distribution shifts. A key result is the explicit relation between RS and DRO under Lipschitz losses, enabling a direct hyperparameter correspondence and showing RS is less sensitive to tuning than DRO. Numerical experiments reinforce that RS improves small-sample and shift-robust performance and provides practical advantages due to its simpler guarantees and global distribution consideration.

Abstract

The Robust Satisficing (RS) model is an emerging approach to robust optimization, offering streamlined procedures and robust generalization across various applications. However, the statistical theory of RS remains unexplored in the literature. This paper fills in the gap by comprehensively analyzing the theoretical properties of the RS model. Notably, the RS structure offers a more straightforward path to deriving statistical guarantees compared to the seminal Distributionally Robust Optimization (DRO), resulting in a richer set of results. In particular, we establish two-sided confidence intervals for the optimal loss without the need to solve a minimax optimization problem explicitly. We further provide finite-sample generalization error bounds for the RS optimizer. Importantly, our results extend to scenarios involving distribution shifts, where discrepancies exist between the sampling and target distributions. Our numerical experiments show that the RS model consistently outperforms the baseline empirical risk minimization in small-sample regimes and under distribution shifts. Furthermore, compared to the DRO model, the RS model exhibits lower sensitivity to hyperparameter tuning, highlighting its practicability for robustness considerations.
Paper Structure (28 sections, 8 theorems, 65 equations, 5 figures, 2 tables)

This paper contains 28 sections, 8 theorems, 65 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Set up
  4. Reference Value $\tau$
  5. Reformulation
  6. Statistical Properties
  7. Confidence Intervals of Optimal Loss
  8. Finite-Sample Generalization Error Bound
  9. Guarantees under Distribution Shift
  10. Numerical Experiments
  11. RS Performance in the Sampling Distribution
  12. RS Performance under Distribution Shift
  13. Connection to DRO under Lipschitz loss
  14. Hyperparameter correspondence
  15. Numerical sensitivity analysis
  16. ...and 13 more sections

Key Result

Theorem 1

Suppose Assumptions assump:light_tail & assump:lips hold. For any $N$, let $\beta_N$ be the confidence level. We have with probability at least $1-\beta_N$: where $r_N$, denoted as the "remainder", is solved from the below equation: with $c_1, c_2$ as positive constants that only depend on exponential decay rate $a$ and dimension $m$. Moreover, when choosing the confidence sequence $\{\beta_N\}$

Figures (5)

  • Figure 1: Performances across various sample sizes. RS outperforms the ERM baseline in small-sample regimes.
  • Figure 2: Performances across various degree of distribution shifts. RS outperforms the ERM baseline under distribution shifts.
  • Figure 3: Correspondence between RS torelance rate parameter $\epsilon$ and DRO radius parameter $r$.
  • Figure 4: function $r$-$\epsilon(m_u=10)$
  • Figure :

Theorems & Definitions (12)

  • Theorem 1: Confidence intervals of optimal loss
  • Lemma 1: Fragility Upper Bound
  • Remark 1
  • Remark 2
  • Corollary 2
  • Theorem 3
  • Remark 3
  • Corollary 4
  • Theorem 5: Distribution Shift
  • Remark 4
  • ...and 2 more