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Closing the Gaps: Optimality of Sample Average Approximation for Data-Driven Newsvendor Problems

Jiameng Lyu, Shilin Yuan, Bingkun Zhou, Yuan Zhou

TL;DR

This work analyzes the regret of Sample Average Approximation for data-driven newsvendor problems with general convex costs, establishing optimality under both $\alpha$-global and $(\alpha,\beta)$-local strong convexity. It introduces a gradient-approximation technique and a novel inverted-hat hard-instance class to prove a tight $\Theta(\log T/\alpha + 1/(\alpha\beta))$ upper bound under $(\alpha,\beta)$-local strong convexity, and a matching $\Omega(\log T/\alpha)$ lower bound under $\alpha$-global strong convexity. These results close longstanding gaps between upper and lower bounds and provide deeper insight into how local versus global curvature governs long-run regret in data-driven inventory decisions. The methodological contributions, including the gradient-approximation framework and the inverted-hat construction, may extend to broader data-driven stochastic optimization problems and influence design of robust data-driven policies.

Abstract

We study the regret performance of Sample Average Approximation (SAA) for data-driven newsvendor problems with general convex inventory costs. In literature, the optimality of SAA has not been fully established under both α-global strong convexity and (α,β)-local strong convexity (α-strongly convex within the β-neighborhood of the optimal quantity) conditions. This paper closes the gaps between regret upper and lower bounds for both conditions. Under the (α,β)-local strong convexity condition, we prove the optimal regret bound of Θ(\log T/α+ 1/ (αβ)) for SAA. This upper bound result demonstrates that the regret performance of SAA is only influenced by αand not by βin the long run, enhancing our understanding about how local properties affect the long-term regret performance of decision-making strategies. Under the α-global strong convexity condition, we demonstrate that the worst-case regret of any data-driven method is lower bounded by Ω(\log T/α), which is the first lower bound result that matches the existing upper bound with respect to both parameter αand time horizon T. Along the way, we propose to analyze the SAA regret via a new gradient approximation technique, as well as a new class of smooth inverted-hat-shaped hard problem instances that might be of independent interest for the lower bounds of broader data-driven problems.

Closing the Gaps: Optimality of Sample Average Approximation for Data-Driven Newsvendor Problems

TL;DR

This work analyzes the regret of Sample Average Approximation for data-driven newsvendor problems with general convex costs, establishing optimality under both -global and -local strong convexity. It introduces a gradient-approximation technique and a novel inverted-hat hard-instance class to prove a tight upper bound under -local strong convexity, and a matching lower bound under -global strong convexity. These results close longstanding gaps between upper and lower bounds and provide deeper insight into how local versus global curvature governs long-run regret in data-driven inventory decisions. The methodological contributions, including the gradient-approximation framework and the inverted-hat construction, may extend to broader data-driven stochastic optimization problems and influence design of robust data-driven policies.

Abstract

We study the regret performance of Sample Average Approximation (SAA) for data-driven newsvendor problems with general convex inventory costs. In literature, the optimality of SAA has not been fully established under both α-global strong convexity and (α,β)-local strong convexity (α-strongly convex within the β-neighborhood of the optimal quantity) conditions. This paper closes the gaps between regret upper and lower bounds for both conditions. Under the (α,β)-local strong convexity condition, we prove the optimal regret bound of Θ(\log T/α+ 1/ (αβ)) for SAA. This upper bound result demonstrates that the regret performance of SAA is only influenced by αand not by βin the long run, enhancing our understanding about how local properties affect the long-term regret performance of decision-making strategies. Under the α-global strong convexity condition, we demonstrate that the worst-case regret of any data-driven method is lower bounded by Ω(\log T/α), which is the first lower bound result that matches the existing upper bound with respect to both parameter αand time horizon T. Along the way, we propose to analyze the SAA regret via a new gradient approximation technique, as well as a new class of smooth inverted-hat-shaped hard problem instances that might be of independent interest for the lower bounds of broader data-driven problems.
Paper Structure (13 sections, 11 theorems, 55 equations, 2 figures, 1 table)

This paper contains 13 sections, 11 theorems, 55 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Technical Contributions
  3. Brief Literature Review
  4. Problem Formulations
  5. Upper Bound of SAA
  6. Proof of Theorem \ref{['tm:tm_main_regret_ub']}
  7. Lower Bound
  8. Proof of Theorem \ref{['thm:lbgloabal']}
  9. Conclusion
  10. Some Useful Technical Lemmas
  11. Omitted Proofs in Section \ref{['sec:Upper bound of SAA']}
  12. Omitted Proofs in Section \ref{['sec:lower bound']}
  13. Constant Regret under Uniform Distributions

Key Result

Theorem 1

Suppose Assumptions ass:demand bounded and ass:saa-solution hold, and $C(x)$ satisfies the $(\alpha,\beta)$-local strong convexity condition. We have where $K_1= 2\widebar{D}\sqrt{K_4}$, $K_2 = 8\sqrt{2}\widebar{D}\sqrt{K_4}(4C_0 B +C_1)+ 8\widebar{D}\sqrt{K_4}/(4C_0B+C_1)$, $K_3 = 2K_4$, and $K_4 = (4C_0B+C_1)^2+2B^2+\sqrt{2\pi}B(4C_0B+C_1)$.

Figures (2)

  • Figure 1: Our inverted-hat-shape hard instance using cosin functions to connect segments.
  • Figure 2: Another choice of hard instance using straight lines to connect segments.

Theorems & Definitions (16)

  • Remark 1
  • Definition 1: Global strong convexity
  • Definition 2: Local strong convexity
  • Theorem 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Theorem 2
  • Corollary 1
  • Lemma 2
  • ...and 6 more