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The Price of Distributional Robustness in Linear Quadratic Control
Andrea Martin, Giuseppe Belgioioso
TL;DR
The paper asks how much conservatism distributional robustness introduces when designing linear quadratic controllers from samples rather than knowing the true uncertainty distribution. It develops a Wasserstein DR analysis with sample-complexity guarantees and finds that suboptimality grows at most linearly with small Wasserstein radius and quadratically outside that regime.
Problem
The paper addresses how suboptimal a data-driven Wasserstein DR controller is relative to an oracle controller designed with knowledge of the true uncertainty distribution.
Method
The analysis uses an outer approximation of the Wasserstein ambiguity set, concentration bounds, and duality-based tractable optimization for zero-mean distributions.
Results
DR-controller suboptimality grows at most linearly with ρ for sufficiently small ambiguity and at most quadratically away from that regime, with a high-probability sample-complexity bound.
Takeaways & Limitations
The results quantify the performance price of Wasserstein robustness and identify how ambiguity radius and sample size govern the oracle-cost gap.
Abstract
from arXiv · showhide
Distributionally robust (DR) optimization seeks decisions that perform best under the most adverse law within a given ambiguity set, enabling the design of data-driven controllers with strong out-of-sample guarantees in the face of uncertainty. In this paper, we study the conservatism introduced by safeguarding against distributional ambiguity. Specifically, we consider the data-driven Wasserstein DR linear quadratic control problem, and we analyze the suboptimality of the corresponding solution relative to the oracle controller computed with foreknowledge of the underlying unknown uncertainty distribution. We present a sample complexity bound that characterizes the number of samples required to ensure that the true cost of the DR solution exceeds that of the oracle controller by at most a user-defined tolerance factor. Our analysis reveals that the suboptimality of the DR solution increases at most linearly with the Wasserstein radius for sufficiently small distributional ambiguity, and at most quadratically away from this local regime. Numerical simulations validate our bounds on the price of distributional robustness.
I. INTRODUCTION
The paper examines distributionally robust control when the uncertainty distribution is unknown and only samples are available. It focuses on quantifying the performance cost of Wasserstein robustness relative to an oracle controller.
- Unknown uncertainty distributions make classical stochastic optimal control difficult to apply and nominal estimates vulnerable to the optimizer’s curse.The optimizer’s curse arises because optimization can amplify estimation errors in the input model.
- Wasserstein ambiguity sets offer favorable computational and statistical properties, including tractable reformulations and out-of-sample guarantees for data-driven control.
- The central gap is how much performance the DR policy loses relative to an oracle controller that knows the true uncertainty distribution.
- The paper studies this price in data-driven Wasserstein DR linear quadratic control and provides quantitative answers about robustness, sampling, and computational tradeoffs.
II. PROBLEM STATEMENT
The problem considers finite-horizon linear feedback control with zero-mean stochastic disturbances whose true distribution is unknown but sampled from data. It then formulates Wasserstein DR control and asks how its cost compares with the oracle and how to choose the ambiguity radius and sample size.
- The true disturbance law P⋆ is zero-mean with nondegenerate covariance, but only N independent samples are available for controller design.The exogenous uncertainty vector collects initial-state, process, and measurement disturbances over the finite horizon.
- The controller is restricted to causal lower block-triangular linear feedback policies of the form u = Ky over a finite control horizon.
- Because P⋆ is not directly known, the paper replaces oracle design with a policy optimized against distributions near the centered empirical law in Wasserstein distance.
- The main questions are the DR policy’s suboptimality relative to the oracle and the radius and sample count needed to keep excess cost within tolerance ϵ.
III. MAIN RESULTS
The analysis bounds Wasserstein DR cost using an outer approximation of the ambiguity set and concentration results. It establishes global and local suboptimality behavior, derives sample-complexity scaling, and gives a semidefinite-programming solution route.
- A. Suboptimality and sample complexity analysis: The price of distributional robustness has a global quadratic upper bound in the Wasserstein radius ρ, while sufficiently small ρ yields a locally linear dependence.The bounds combine an outer approximation that ignores higher-order moments with concentration results for empirical measures and subgaussian variables.
- A. Suboptimality and sample complexity analysis: For sufficiently small ambiguity, the required sample complexity scales with 1/ϵ^d.
- B. Numerical implementation: The DR optimal control problem can be solved using semidefinite programming by restricting the adversary to zero-mean laws through duality.
A. Suboptimality and sample complexity analysis
The analysis converts Wasserstein distributional ambiguity into covariance perturbations, then combines deterministic cost bounds with concentration results to quantify DR suboptimality and sample requirements.
- Method: The closed-loop reformulation expresses system responses under causal linear feedback, enabling the quadratic objective to depend on the uncertainty distribution through its covariance matrix.The controller responses are parameterized by causal closed-loop maps, which yield an equivalent performance representation.
- Method: A Wasserstein ambiguity set is outer-approximated by distributions whose covariance differs from the empirical covariance by at most a radius-dependent operator-norm bound.This approximation neglects higher-order moment information and supports a deterministic upper bound on worst-case cost.
- Suboptimality bound: The resulting performance bound is tight as the Wasserstein radius vanishes, because the worst-case-cost upper bound converges to the nominal empirical cost.The bound is obtained through covariance perturbation and applies to any admissible causal feedback policy.
- Sample complexity: Under a light-tailed true distribution, concentration results provide a high-probability Wasserstein radius ensuring that the true law belongs to the ambiguity set.The required concentration constants depend on the tail parameters and dimension, with the analysis simplifying the displayed formula under d > 4.
- Suboptimality bound: The DR policy’s excess true cost grows at most linearly in the Wasserstein radius for sufficiently small ambiguity and at most quadratically outside that regime.The local expansion uses the covariance perturbation term 2β̂ρ + O(ρ^2).
- Sample complexity: The sample-complexity corollary gives a radius and sample threshold that guarantee, with probability at least 1 − ζ, a DR price of robustness no greater than ε.The required sample count is decreasing in the radius-dependent guarantee and has small-radius scaling described up to logarithmic factors in ζ^-1.
B. Numerical implementation
The paper addresses the difficulty of directly solving the nonconvex, infinite-dimensional Wasserstein DR control problem by optimizing over closed-loop maps and reformulating the adversary’s subproblem through duality.
- B. Numerical implementation: The method optimizes over causal closed-loop maps instead of controller gains to address the nonconvex control formulation.The closed-loop maps are constrained to be causal and lie in an affine subspace.
- B. Numerical implementation: Duality theory yields a finite-dimensional reformulation of the adversary’s problem over zero-mean distributions in the Wasserstein ambiguity set.The resulting formulation supports tractable computation of the DR policy.
- B. Numerical implementation: Theorem 2 characterizes the DR optimal policy through a convex optimization problem.The formulation includes nonnegative dual variables and constraints indexed by the available samples.
IV. EXPERIMENTS
Experiments evaluate Wasserstein DR controllers on an open-loop unstable system using Gaussian uncertainty and compare empirical suboptimality, theoretical bounds, distributional distance, and computation time. The results qualitatively support the theory while revealing diminishing returns from additional samples and dimensionality-related looseness in the sample bound.
- IV. EXPERIMENTS: The theoretical bound captures the growth rate of empirical suboptimality despite the experiment violating some assumptions and requiring prohibitively many samples for the stated guarantee.The experiment uses a 14-dimensional uncertainty vector, while the tail and sample requirements of the theory are not fully met.
- IV. EXPERIMENTS: For N = 50 and N = 100, empirical suboptimality initially decreases as ρ increases, consistent with DR mitigating the optimizer’s curse under limited data.This behavior is reported for the experiment’s Gaussian uncertainty setting.
- IV. EXPERIMENTS: As N grows, the theoretical curve becomes nearly parallel to empirical suboptimality for ρ approximately greater than 10^-2.This comparison is made in Figure 1 across increasing training-set sizes.
- IV. EXPERIMENTS: For larger N, empirical suboptimality increases monotonically with ρ, as predicted by the theoretical bound.Larger datasets make the empirical distribution better approximate the true law.
- IV. EXPERIMENTS: Increasing the number of training samples yields diminishing performance returns while substantially increasing the computational cost of solving the semidefinite program.Figure 2 reports average evaluation time as a function of N, with error bars computed over 10 independent draws.
- IV. EXPERIMENTS: For sufficiently large ρ, empirical suboptimality converges to approximately the same value across training-set sizes.The relative improvement from increasing N decreases over the tested range ρ ∈ [10^-4, 2].
V. CONCLUSION
The paper quantifies the price of distributional robustness in data-driven Wasserstein DR linear quadratic control and connects its theory to computation and experiments.
- V. CONCLUSION: Suboptimality relative to the oracle controller grows at most linearly with ρ for sufficiently small ambiguity and at most quadratically outside that regime.The result follows from an outer approximation of Wasserstein ambiguity sets that ignores higher-order moment information.
- V. CONCLUSION: A sample-complexity bound ensures, with high probability, that the DR solution’s suboptimality remains below a prescribed level.
- V. CONCLUSION: The Wasserstein DR control problem admits a tractable semidefinite-program reformulation, and numerical experiments illustrate the qualitative behavior predicted by the bounds.
APPENDIX
The appendix reformulates the Wasserstein distributionally robust control problem through coupling measures, duality, and semidefinite optimization. It establishes strong duality under a generalized Slater condition and derives the finite-dimensional reformulation using the quadratic loss structure and system-level parametrization.
- Duality: The proof reduces the Wasserstein adversary’s optimization to a Lagrangian dual whose supremum over coupling measures yields the distributionally robust subproblem.The derivation uses the coupling representation, linear constraints on the transport plan, and standard conic duality results.
- Semidefinite reformulation: Because the closed-loop state and input trajectories depend linearly on ξ, the tilted loss remains quadratic and admits an explicit finite-dimensional reformulation.System-level parametrization and relaxation of the quadratic equality constraint lead to a linear matrix inequality via the Schur complement.
- Coupling reformulation: The adversary’s subproblem is rewritten as an optimization over nonnegative coupling measures supported on the empirical support and the uncertainty space.The coupling constraints encode probability normalization, the empirical first marginal, the Wasserstein distance bound, and the zero-mean condition.
- Duality: Strong duality follows because the centered empirical distribution and strictly positive constraint components satisfy the generalized Slater condition for every ρ > 0 and N ∈ N.The resulting conic linear program therefore admits a strong dual formulation.
- Semidefinite reformulation: The resulting semidefinite optimization problem preserves the optimal closed-loop map while replacing the quadratic equality constraint with an equivalent relaxation.The constructed solution is shown to retain feasibility and the same objective value, after which the Schur complement produces the LMI formulation.