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Sampled-Data Wasserstein Distributionally Robust Control of Multiplicative Systems: A Convex Relaxation with Performance Guarantees

Chung-Han Hsieh

TL;DR

The paper tackles robust control of discrete-time, sampled-data systems with multiplicative noise under distributional uncertainty by jointly choosing a feedback policy and a sampling period. It introduces Wasserstein ambiguity sets to model distributional misspecification and develops a convex relaxation to overcome the nonconvex concave-max geometry that precludes exact minimax equality. The authors provide probabilistic performance guarantees, non-asymptotic duality-gap bounds, and long-run performance guarantees that connect finite-horizon optimization to asymptotic behavior, including a growth-rate floor for log utilities. The approach is validated on a log-optimal portfolio example, showing improved downside risk management and the value of adaptive sampling frequencies via a cutting-plane solution method.

Abstract

This paper investigates the robust optimal control of sampled-data stochastic systems with multiplicative noise and distributional ambiguity. We consider a class of discrete-time optimal control problems where the controller \emph{jointly} selects a feedback policy and a sampling period to maximize the worst-case expected concave utility of the inter-sample growth factor. Modeling uncertainty via a Wasserstein ambiguity set, we confront the structural obstacle of~``concave-max'' geometry arising from maximizing a concave utility against an adversarial distribution. Unlike standard convex loss minimization, the dual reformulation here requires a minimax interchange within the semi-infinite constraints, where the utility's concavity precludes exact strong duality. To address this, we utilize a general minimax inequality to derive a tractable convex relaxation. Our approach yields a rigorous lower bound that functions as a probabilistic performance guarantee. We establish an explicit, non-asymptotic bound on the resulting duality gap, proving that the approximation error is uniformly controlled by the Lipschitz-smoothness of the stage reward and the diameter of the disturbance support. Furthermore, we introduce necessary and sufficient conditions for \emph{robust viability}, ensuring state positivity invariance across the entire ambiguity set. Finally, we bridge the gap between static optimization and dynamic performance, proving that the optimal value of the relaxation serves as a rigorous deterministic floor for the asymptotic average utility rate almost surely. The framework is illustrated on a log-optimal portfolio control problem, which serves as a canonical instance of multiplicative stochastic control.

Sampled-Data Wasserstein Distributionally Robust Control of Multiplicative Systems: A Convex Relaxation with Performance Guarantees

TL;DR

The paper tackles robust control of discrete-time, sampled-data systems with multiplicative noise under distributional uncertainty by jointly choosing a feedback policy and a sampling period. It introduces Wasserstein ambiguity sets to model distributional misspecification and develops a convex relaxation to overcome the nonconvex concave-max geometry that precludes exact minimax equality. The authors provide probabilistic performance guarantees, non-asymptotic duality-gap bounds, and long-run performance guarantees that connect finite-horizon optimization to asymptotic behavior, including a growth-rate floor for log utilities. The approach is validated on a log-optimal portfolio example, showing improved downside risk management and the value of adaptive sampling frequencies via a cutting-plane solution method.

Abstract

This paper investigates the robust optimal control of sampled-data stochastic systems with multiplicative noise and distributional ambiguity. We consider a class of discrete-time optimal control problems where the controller \emph{jointly} selects a feedback policy and a sampling period to maximize the worst-case expected concave utility of the inter-sample growth factor. Modeling uncertainty via a Wasserstein ambiguity set, we confront the structural obstacle of~``concave-max'' geometry arising from maximizing a concave utility against an adversarial distribution. Unlike standard convex loss minimization, the dual reformulation here requires a minimax interchange within the semi-infinite constraints, where the utility's concavity precludes exact strong duality. To address this, we utilize a general minimax inequality to derive a tractable convex relaxation. Our approach yields a rigorous lower bound that functions as a probabilistic performance guarantee. We establish an explicit, non-asymptotic bound on the resulting duality gap, proving that the approximation error is uniformly controlled by the Lipschitz-smoothness of the stage reward and the diameter of the disturbance support. Furthermore, we introduce necessary and sufficient conditions for \emph{robust viability}, ensuring state positivity invariance across the entire ambiguity set. Finally, we bridge the gap between static optimization and dynamic performance, proving that the optimal value of the relaxation serves as a rigorous deterministic floor for the asymptotic average utility rate almost surely. The framework is illustrated on a log-optimal portfolio control problem, which serves as a canonical instance of multiplicative stochastic control.
Paper Structure (28 sections, 9 theorems, 51 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 28 sections, 9 theorems, 51 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation and Literature Review
  3. Sampled-Data Constraints and Friction
  4. Distributional Ambiguity
  5. The Concave-Max Geometry
  6. Contributions
  7. Preliminaries and Problem Formulation
  8. Notation
  9. Sampled-Data Multiplicative Dynamics
  10. Performance Criterion and State Positivity
  11. Standing Assumptions on Growth Factor
  12. Wasserstein Ambiguity Set
  13. Robust Viability and Admissible Controls
  14. Distributionally Robust Control Formulation
  15. Theoretical Results
  16. ...and 13 more sections

Key Result

Lemma 2.8

Fix a sampling period $n \ge 1$. Let $V_0 > 0$. If a control $u$ is admissible, i.e., $u \in \mathcal{U}_{\rm v}(n;\eta)$, then for any ambiguity radius $\varepsilon \ge 0$ and any distribution $\mathbb{F} \in \mathcal{B}_\varepsilon^{(p)}(\widehat{\mathbb{F}}_n)$, the state evolution satisfies

Figures (4)

  • Figure 1: Schematic of the sampled-data loop. The analysis uses the sampled state $V_k=V(t_k)$ and an aggregated disturbance $\mathcal{X}_{k, n}$ over $[t_k, t_{k+1})$ where $t_k = kn.$
  • Figure 1: Out-of-sample wealth trajectories comparing static sampling with fixed $n^*$ and adaptive sampling with dynamic $n^*_k$. The DRO strategies induce a conservative allocation during high-volatility regimes, effectively limits the downside risks.
  • Figure 2: Adaptive sampling scheme: realized sequence of selected sampling period $n_k^*$. The controller autonomously switches between high-frequency and low-frequency updates based on the trade-off between growth opportunities and friction (transaction costs).
  • Figure 3: Time-varying duality gap for both static- and adaptive-sampling controls. The gap remains consistently small across all sampling periods $n \in \{5,21,42,63\}$ and adaptive $n_k^*$.

Theorems & Definitions (33)

  • Remark 2.1: Dimensionality Mismatch
  • Remark 2.3
  • Example 2.4: Log-Optimal Portfolio Control
  • Definition 2.5: Wasserstein Metric
  • Definition 2.6: Calibrated Ambiguity Radii
  • Remark 2.7
  • Lemma 2.8: Robust Viability Condition
  • Proof 1
  • Definition 2.9: $(\varepsilon, \delta)$-Viability
  • Remark 2.10
  • ...and 23 more