Wasserstein Distributionally Robust Risk-Sensitive Estimation via Conditional Value-at-Risk
arXiv:2604.18546v1 [cs.LG] 20 Apr 2026
Feras Al Taha1 and Eilyan Bitar1 Abstract— We propose a distributionally robust approach to risk-sensitive estimation of an unknown signal x from an observed signal y. The unknown signal and observation are modeled as random vectors whose joint probability distribution is unknown, but assumed to belong to a given type-2 Wasserstein ball of distributions, termed the ambiguity set. The performance of an estimator is measured according to the conditional value-at-risk (CVaR) of the squared estimation error. Within this framework, we study the problem of computing affine estimators that minimize the worst-case CVaR over all distributions in the given ambiguity set. As our main result, we show that, when the nominal distribution at the center of the Wasserstein ball is finitely supported, such estimators can be exactly computed by solving a tractable semidefinite program. We evaluate the proposed estimators on a wholesale electricity price forecasting task using real market data and show that they deliver lower out-of-sample CVaR of squared error compared to existing methods.
I. I NTRODUCTION In many decision-making applications that rely on forecasts, guarding against rare but severe prediction errors is more important than minimizing average error alone. In power systems, for example, large errors in load or renewable generation forecasts can increase operating costs and, in extreme cases, threaten system reliability. In financial trading, failing to anticipate sharp price movements can lead to substantial losses or missed profit opportunities. Such considerations highlight the need for prediction models that explicitly guard against large errors, even at the expense of a modest increase in average prediction error. To better control the upper tail of the prediction error distribution, one can replace the commonly used mean squared error criterion with a risk-sensitive criterion, such as the conditional value-at-risk (CVaR) of the squared error. However, the effectiveness of such an approach depends on how well the assumed (nominal) distribution, typically estimated from data, captures the events that give rise to large prediction errors. If those events are misrepresented in frequency or severity, then minimizing CVaR under the nominal distribution may still leave the predictor exposed to substantial tail risk. To address this limitation, we incorporate distributional robustness into the risk-sensitive estimation framework by evaluating predictors according to their worstcase CVaR of squared error over a Wasserstein ball centered at the nominal distribution. This work was supported in part by the Cornell Atkinson Center for Sustainability and in part by the Bezos Earth Fund. 1 Feras Al Taha and Eilyan Bitar are with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853 USA
[email protected], [email protected]
Related Work: Robust and distributionally robust approaches to estimation have been studied extensively. For estimation problems in which the latent signal is observed via a linear measurement model with additive noise, robust affine estimators can be obtained by minimizing the worst-case mean squared error (MSE) over uncertainty sets describing the signal or noise, often resulting in tractable semidefinite programming (SDP) reformulations under suitable assumptions [1], [2], [3], [4], [5]. In the distributionally robust setting, one instead minimizes the worst-case MSE over ambiguity sets of probability distributions. Tractable formulations have been derived for ambiguity sets defined by spectral constraints on the signal and noise covariance matrices [6], [7] and divergence-based constraints on the signal and noise distributions [8], [9]. Most closely related to our work, [10] studies Wasserstein distributionally robust MSE minimization for linear measurement models and establishes optimality of affine estimators (among all possibly nonlinear estimators) when the nominal distribution is elliptical. Wasserstein-based distributionally robust optimization has also been applied in related settings such as Kalman filtering, linear regression, classification, and domain adaptation [11], [12], [13], [14], [15]. CVaR-based criteria have been used widely to incorporate risk sensitivity in machine learning and decision-making applications, including multi-armed bandits, reinforcement learning, robust control, fairness-aware learning, and supervised learning [16], [17], [18], [19], [20], [21], [22], to name a few applications. Distributionally robust CVaR minimization with quadratic objectives has also been studied in the context of stochastic control and portfolio optimization problems under moment-based or Gelbrich ambiguity sets [23], [24], [25]. However, to our knowledge, Wasserstein distributionally robust estimation with a CVaR of squared error criterion has not been studied previously. Summary of Contributions: We study the design of affine risk-sensitive estimators via distributionally robust optimization. Specifically, we formulate the estimation problem as the minimization of the worst-case CVaR of the squared estimation error over a type-2 Wasserstein ambiguity set centered at a given nominal distribution. Using duality for optimal transport problems, we derive an equivalent convex reformulation for the class of worst-case CVaR problems involving generic quadratic loss functions. This yields an exact reformulation of the estimation problem as a tractable semidefinite program when the nominal distribution is finitely supported. Numerical experiments on an electricity price forecasting task show that the proposed estimators
exhibit improved out-of-sample CVaR of estimation error relative to existing methods. Organization: The remainder of the paper is organized as follows. Section II formulates the distributionally robust risk-sensitive estimation problem. In Section III, we derive an equivalent reformulation for this problem as a semidefinite program. Numerical experiments are provided in Section IV. We conclude the paper in Section V. All proofs are provided in the appendix. Notation: Let R and R+ denote the set of real numbers and nonnegative real numbers, respectively. Let Sn denote the set of all symmetric matrices in Rn×n . Denote the cone of n × n real symmetric positive definite (resp. semidefinite) matrices by Sn++ (resp. Sn+ ). Given matrices A, B ∈ Sn , the relation A ≻ B (resp. A ⪰ B) means A − B ∈ Sn++ (resp. A − B ∈ Sn+ ). Given a scalar x ∈ R, we let (x)+ := max{0, x} represent the positive part of x. Let ∥ · ∥ denote the Euclidean norm. Let M(Rn ) be the collection of Borel probability measures on Rn with finite second moments. We use boldface symbols to denote random variables, and nonboldface symbols to denote particular values in the range of a random variable and other deterministic quantities. II. P ROBLEM F ORMULATION We now formulate the distributionally robust risk-sensitive estimation problem studied in this paper. To begin, suppose that x and y are random vectors taking values in Rn and Rm , respectively, and let d := n + m. Our objective is to estimate the latent vector x from the observation vector y. We assume that the joint distribution P ∈ M(Rd ) of z := (x, y) is unknown, but belongs to a prescribed family of distributions P ⊆ M(Rd ), which we refer to as the ambiguity set. In this paper, the ambiguity set is specified in terms of the type-2 Wasserstein distance, which is defined as follows. Definition 1: The type-2 Wasserstein distance between two distributions P1 , P2 ∈ M(Rd ) is defined as Z 2 W2 (P1 , P2 ) := inf ∥z1 − z2 ∥2 π(dz1 , dz2 ), π∈Π(P1 , P2 )
Rd ×Rd
where Π(P1 , P2 ) denotes the set of all joint distributions in M(Rd × Rd ) with marginal distributions P1 and P2 . Given a nominal distribution P0 ∈ M(Rd ), we define the ambiguity set P as the set of all distributions whose type-2 Wasserstein distance to P0 is at most r ≥ 0, i.e., P := P ∈ M(Rd ) W2 (P, P0 ) ≤ r . (1) The radius of the ambiguity set r represents the decision maker’s confidence in the accuracy of the nominal distribution. In particular, the ambiguity set becomes a singleton containing only the nominal distribution if the ambiguity set radius is decreased to zero. Having specified the distributional ambiguity model, we now introduce the class of estimators under consideration. We restrict our attention to affine estimators of the form ψ(y) = Ay + b,
(2)
where A ∈ Rn×m and b ∈ Rn . We let the set A denote the family of all affine mappings from Rm to Rn . To evaluate the quality of an estimator ψ ∈ A, we measure the discrepancy between the estimate ψ(y) and the target x via the squared error loss: ℓψ (z) := ∥x − ψ(y)∥2 .
(3)
Rather than minimizing the mean squared error (MSE), however, we adopt a risk-sensitive criterion based on the conditional value-at-risk (CVaR), which measures the average loss in the upper tail of the loss distribution and therefore places greater emphasis on large estimation errors. The CVaR is formally defined as follows. Definition 2: Given a random vector z ∼ P ∈ M(Rd ) and a Borel-measurable loss function ℓ : Rd → R, the conditional value-at-risk (CVaR) of the loss ℓ(z) at a probability level α ∈ (0, 1] is defined as: 1 E ℓ(z) − τ . CVaRα ℓ(z) := inf τ + P P + τ ∈R α In this paper, we aim to find an affine estimator that minimizes the worst-case CVaR of the squared error loss over all probability distributions in the ambiguity set P, namely, (4) inf sup CVaRα P ℓψ (z) . ψ∈A P ∈P
This formulation subsumes several estimation frameworks as special cases. For example, when α = 1, the CVaR of the squared error coincides with its expectation, and problem (4) reduces to a distributionally robust mean squared error minimization problem of the kind studied in [10], [11]. At the other extreme, as α ↓ 0, the CVaR converges to the essential supremum of the loss, so that (4) approaches a robust estimation problem that minimizes the largest squared error that may be realized under any distribution in the ambiguity set. Finally, when the ambiguity set radius is zero, problem (4) reduces to an “ambiguity-free” CVaR minimization problem under the nominal distribution P0 . III. C ONVEX R EFORMULATION In this section, we derive an equivalent dual reformulation of the worst-case CVaR problem (the inner maximization) in (4). Using this dual reformulation, we go on to show that, when the nominal distribution is finitely supported, problem (4) can be equivalently reformulated as a tractable semidefinite program (SDP). A. Dual Reformulation of Worst-Case CVaR To solve problem (4), we first appeal to a strong duality result for generic worst-case risk problems over Wassersteinbased ambiguity sets. This result, adapted from [26, Theorem 5.22], gives conditions on the loss function under which strong duality is guaranteed to hold for the corresponding worst-case risk problem, and provides a characterization of the dual problem as a two-dimensional convex optimization problem. We then specialize this result to the class of quadratic loss functions considered in this paper. The required regularity condition is stated next.
Assumption 1: The loss function ℓ : Rd → R is upper semicontinuous and satisfies supP ∈P EP [(ℓ(z))+ ] < ∞ and EP [ℓ(z)] > −∞ for all P ∈ P and r > 0. Theorem 1 (Strong Duality): Let r > 0 and α ∈ (0, 1]. If the loss function ℓ : Rd → R satisfies Assumption 1, then 1 γr2 + EP0 [ϕ(τ, γ, z)] , sup CVaRα P ℓ(z) = inf τ + τ ∈R α P ∈P γ∈R+
(5) where ϕ : R × R × Rd → R ∪ {+∞} is defined as o n ϕ(τ, γ, z) := sup ℓ(v) − τ + − γ∥v − z∥2 .
B. Semidefinite Programming Reformulation Building on Theorem 2, we now show that problem (4) admits an equivalent SDP reformulation when the nominal distribution is finitely supported. Theorem 3 (SDP Reformulation): Let r > 0 and α ∈ (0, 1]. If the nominal distribution P0 ∈ M(Rd ) is uniformly distributed on the set {z1 , . . . , zN } ⊆ Rd , then problem (4) can be equivalently reformulated as the following SDP: inf τ +
(6)
v∈Rd
Building on Theorem 1, we now provide an equivalent reformulation of worst-case CVaR problems involving (possibly nonconvex) quadratic loss functions. Theorem 2: Let r > 0 and α ∈ (0, 1], and consider a family of quadratic worst-case CVaR problems given by ⊤ ⊤ sup CVaRα (7) P z Qz + 2q z , P ∈P
d
where Q ∈ S and q ∈ Rd . The optimal value of (7) is finite and equal to the optimal value of the following convex program: r2 2 ⊤ −1 − ∥z∥ , inf CVaRα (γz + q) Q (γz + q) + γ P0 γ γ∈Γ α (8) where Γ := {γ ∈ R+ | Qγ ≻ 0} and Qγ := γI − Q. When α = 1, the worst-case CVaR of the quadratic loss function in (7) reduces to its worst-case expectation. In this risk-neutral special case, the dual reformulation provided in Theorem 2 coincides with previously derived dual reformulations for quadratic worst-case expectation problems under type-2 Wasserstein ambiguity sets [27], [28]. A key difference, however, is that Theorem 2 establishes this dual equivalence under weaker assumptions on the nominal distribution. In particular, existing results require the nominal distribution to be either elliptical [27, Theorem 16] or absolutely continuous with respect to Lebesgue measure [28, Theorem 2], whereas Theorem 2 holds for any nominal distribution in M(Rd ). For α ∈ (0, 1), Nguyen et al. [25] study a related class of quadratic worst-case CVaR problems using ambiguity sets defined by the Gelbrich distance, which measures the discrepancy between distributions only through their means and covariance matrices.1 Consequently, the dual reformulation of the quadratic worst-case CVaR problem under Gelbrich ambiguity provided in [25, Theorems 9 and 10] depends on the nominal distribution only through its first two moments. In contrast, under type-2 Wasserstein ambiguity, the dual reformulation provided in Theorem 2 may depend on higherorder moments of the nominal distribution, beyond its mean and covariance. 1 As shown in [29], the Gelbrich distance between two distributions is a lower bound on their type-2 Wasserstein distance, with equality holding when the distributions being compared are multivariate Gaussian.
N 1 X 1 2 γr + si α N i=1
(9a)
s.t. A ∈ Rn×m , b ∈ Rn , γ ∈ R+ , τ ∈ R, s ∈ RN +, ⊤ γI F ≻ 0, (9b) F I (τ + si + γ∥zi ∥2 ) γzi⊤ −b⊤ γzi γI F ⊤ ⪰ 0, ∀i = 1, . . . , N, −b F I (9c) where F := [−I, A], and the decision variables are A, b, γ, τ , and s. IV. E XAMPLE We consider an electricity price forecasting task where the goal is to predict the vector of hourly day-ahead (DA) energy prices x ∈ R24 ($/MWh) from the corresponding vector of hourly DA load forecasts y ∈ R24 (MW) provided by the independent system operator (ISO). This setting provides a natural testbed for evaluating risk-sensitive estimators, since electricity prices can exhibit sharp spikes that may not be well represented in historical data. Electricity Market Data: We use the same PJM electricity market dataset as in [30], spanning from May 1, 2013 to July 31, 2013. The first two months serve as the training set, and the final month is used for testing. We intentionally choose this test period to include price spikes that exceed those seen in the training data (as shown in Fig. 1), thereby providing a demanding out-of-sample assessment of tail-risk performance. As a preprocessing step, both prices and load forecasts are rescaled to the unit interval using min-max normalization. The nominal distribution P0 is then taken to be the empirical distribution of the normalized training data. Experiments and Discussion: We compute the proposed distributionally robust CVaR-based estimator (DR-CVA R) for a risk level α = 0.01 by solving problem (9), and compare its out-of-sample performance on the test set with that of the distributionally robust MSE estimator (DR-MSE), obtained by solving the same problem (9) with α = 1. To examine the effect of the ambiguity set radius r on out-of-sample performance, we vary r from 10−5 to 105 , recomputing both estimators for each radius value. Fig. 2 (top) depicts the out-of-sample performance of the two estimators, measured by the CVaR of the squared error at α = 0.01, as a function of the ambiguity set radius. As r ↓ 0, both estimators reduce to their nominal (non-robust) counterparts. As the radius r increases, both estimators
5
Test data
Out-of-sample CVaR
DA price ($/MWh)
300
Training data
250 200 150 100 50
#10 3
DR-CVaR DR-MSE
2.5 2
-5
0
10 May 01
May 15
May 29
Jun 12
Jun 26
Jul 10
5
10
Jul 24
10
Ambiguity set radius r
5
300
1.4
DA price ($/MWh)
DA load forecast (MW)
1.6 #10
1.2 1 0.8 0.6 0.4 May 01
May 15
May 29
Jun 12
Jun 26
Jul 10
Jul 24
Fig. 1. PJM’s hourly DA energy prices (top) and DA load forecasts (bottom) from May 1, 2013 to July 31, 2013. The dataset is split between training (before July 1, 2013) and testing (from July 1, 2013 onward) data.
benefit from distributional robustness and improve upon their nominal counterparts in terms out-of-sample CVaR. Across the entire range of radius values considered, the DR-CVA R estimator consistently achieves a lower out-of-sample CVaR than the DR-MSE estimator. This behavior is also visible in the forecasts themselves. As shown in Fig. 2 (bottom), the DR-CVaR estimator (for r = 0.01) produces slightly more accurate estimates of the DA prices on days exhibiting more pronounced price spikes. This reduction in tail risk comes with a modest loss in accuracy on days with more typical price patterns, reflecting the intended tradeoff of the risk-sensitive formulation. V. C ONCLUSION In this paper, we proposed a framework for distributionally robust risk-sensitive affine estimation in which the joint distribution of the unknown signal and observation is unknown, but assumed to belong to a type-2 Wasserstein ambiguity set. Within this framework, estimators are designed to minimize the worst-case CVaR of the squared estimation error over all distributions in the ambiguity set. By exploiting strong duality in Wasserstein distributionally robust optimization, we derived an equivalent convex reformulation of the resulting worst-case CVaR problem. In the important case where the nominal distribution is finitely supported, we obtain an equivalent reformulation of the proposed estimation problem as a tractable semidefinite program. A PPENDIX A. Proof of Theorem 2 The proof proceeds in three steps. First, we show that strong duality, as stated in Theorem 1, holds for the class of quadratic worst-case CVaR problems specified in (7).
250
Realized price DR-CVaR forecast DR-MSE forecast
200 150 100 50 Jul 13
Jul 14
Jul 15
Jul 16
Jul 17
Fig. 2. Top: Out-of-sample CVaR of squared error versus the ambiguity set radius r for each estimator, evaluated on the test data. Bottom: Realized energy prices (solid gray); DR-CVA R price forecasts (solid black) for r = 0.01 and α = 0.01; and DR-MSE price forecasts (dotted red) for r = 0.08 from July 13 to July 22, 2013.
Second, we show that both the primal and dual optimal values are finite. Third, we prove the equivalence between problems (7) and (8). Throughout the proof, we denote the quadratic loss function by ℓ(z) := z ⊤ Qz + 2q ⊤ z. Step 1 (Establishing strong duality): By Theorem 1, strong duality holds if the given loss function satisfies Assumption 1. Since the loss function ℓ is quadratic, there exists a finite constant C > 0 such that |ℓ(z)| ≤ C 1 + ∥z∥2 for all z ∈ Rd . Moreover, for every P ∈ P, it holds that EP [∥z∥2 ] ≤ 2r2 + 2EP0 [∥z∥2 ]. Hence, for every P ∈ P, we have the uniform upper bound: EP [|ℓ(z)|] ≤ C 1+EP [∥z∥2 ] ≤ C 1+2r2 +2EP0 [∥z∥2 ] . Since EP0 [∥z∥2 ] < ∞ by assumption, taking the supremum over P ∈ P yields supP ∈P EP [|ℓ(z)|] < ∞. Thus, Assumption 1 is satisfied. Step 2 (Finiteness of primal and dual optimal values): We prove finiteness of the primal and dual optimal values by showing that they are bounded from above and below. First, the primal optimal value is bounded from below since sup CVaRα (10) P ℓ(z) ≥ EP0 [ℓ(z)] > −∞. P ∈P
The first inequality follows from the fact that P0 ∈ P and that the CVaR of a random variable is no smaller than its mean. The second inequality follows from Assumption 1, which the quadratic loss ℓ was shown to satisfy in Step 1. Next, we show that the dual optimal value is bounded from
above. By Theorem 1, we have the primal-dual equivalence:
the infimum in (11) as n o 1 τ + f (τ, γ) , τ ∈R α
1 sup CVaRα γr2 + EP0 [ϕ(τ, γ, z)] , P ℓ(z) = inf τ + τ ∈R α P ∈P
inf
γ∈R+
γ∈R+
(11) where ϕ(τ, γ, z) := sup (ℓ(v) − τ )+ − γ∥v − z∥2 .
(12)
v∈Rd
For any (τ, γ, z) ∈ R × R+ × Rd , it holds that ϕ(τ, γ, z) n o = sup max ℓ(v) − τ − γ∥v − z∥2 , −γ∥v − z∥2 v∈Rd (a)
(b)
= max
sup ℓ(v) − τ − γ∥v − z∥2 , sup −γ∥v − z∥2 v∈Rd
= max
v∈Rd
Equality (a) follows from interchanging the supremum over v with the max operator. Equality (b) follows from the fact that −γ∥v − z∥2 is maximized at v = z since γ ≥ 0. Now, note that for any γ ∈ Γ := {γ ∈ R+ | Qγ ≻ 0}, the quadratic function v 7→ −v ⊤ Qγ v + 2(γz + q)⊤ v is strictly concave in v. It, therefore, has a unique maximizer, which is given by v ⋆ = Q−1 γ (γz + q). Thus, for every γ ∈ Γ, the function ϕ(τ, γ, z) simplifies to 2 ϕ(τ, γ, z) = (γz + q)⊤ Q−1 γ (γz + q) − γ∥z∥ − τ + . (13) We use this fact to upper bound the dual optimal value as follows. Note that for any γ̄ ∈ Γ, it holds that inf τ +
τ ∈R γ∈R+
(a) 1
≤
α
(b) 1
1 γr2 + EP0 [ϕ(τ, γ, z)] α γ̄r2 + EP0 [ϕ(0, γ̄, z)]
γθ := θγ̄ + (1 − θ)λ̄(Q). Fix τ ∈ R. The function f (τ, γ) = γr2 + EP0 [ϕ(τ, γ, z)] is convex with respect to γ. Indeed, for each (τ, z) ∈ R × Rd , the map γ 7→ ϕ(τ, γ, z), given in (12), is the pointwise supremum of affine functions of γ, and is therefore convex; and taking expectation with respect to z preserves convexity. Using the convexity of f (τ, ·) and the fact that γθ ∈ Γ, we get inf f (τ, γ) ≤ f (τ, γθ ) ≤ θf (τ, γ̄) + (1 − θ)f (τ, λ̄(Q)).
γ∈Γ
Letting θ ↓ 0 gives inf γ∈Γ f (τ, γ) ≤ f (τ, λ̄(Q)) since f (τ, γ̄) is finite. Thus, the point γ = λ̄(Q) may also be excluded from the feasible set in (15) without changing the value of the infimum. We have shown that the feasible set for γ in (15) may be restricted from R+ to Γ without changing the value of the infimum. It follows that 1 1 γr2 + EP0 [ϕ(τ, γ, z)] inf τ + f (τ, γ) = inf τ + τ ∈R τ ∈R α α γ∈Γ
γ∈R+
α
2 γ̄r2 + EP0 [((γz + q)⊤ Q−1 γ (γz + q) − γ∥z∥ )+ ]
< ∞.
(14)
=
where f (τ, γ) := γr2 + EP0 [ϕ(τ, γ, z)]. First, it is straightforward to see that if γ < λ̄(Q) (equivalently Qγ ̸⪰ 0), then ϕ(τ, γ, z) = ∞ for all (τ, z) ∈ R×Rd . Hence, f (τ, γ) = ∞ for any τ ∈ R and γ < λ̄(Q), so these values of γ may be excluded from the feasible set in problem (15) without changing the value of the infimum. Next, we show that the point γ = λ̄(Q) may also be excluded from the feasible set in problem (15) without changing the value of the infimum. We only consider the case where λ̄(Q) ≥ 0; otherwise, the point γ = λ̄(Q) < 0 already lies outside the feasible set of problem (15). Let γ̄ ∈ Γ and for each θ ∈ (0, 1), define
sup −v ⊤ Qγ v + 2(γz + q)⊤ v − γ∥z∥2 − τ, 0 . v∈Rd
(c)
Inequality (a) follows from taking τ = 0 and γ = γ̄. Equality (b) follows from the identity in (13), and (c) follows from the assumption that the nominal distribution P0 has a finite second moment. Combining (10) and (14), it follows that that the primal and dual optimal values are finite.
(15)
1 = inf τ + γr2 τ ∈R α γ∈Γ 2 + EP0 (γz + q)⊤ Q−1 γ (γz + q) − γ∥z∥ − τ + . Here, the second equality follows from (13). Finally, the infimum in the last line above can be shown to be equal to problem (8) by applying the change of variables τ = τ̄ − γr2 /α, and using the definition of CVaR. B. Proof of Theorem 3
Step 3 (Reformulating the dual problem): To complete the proof, it remains to show that the right-hand side of (11) reduces to (8). To this end, we show that the infimum in (11) is unchanged if the feasible set for the decision variable γ is restricted from R+ to the set Γ := {γ ∈ R+ | Qγ ≻ 0} = {γ ∈ R+ | γ > λ̄(Q)}, where Qγ := γI −Q and λ̄(Q) denotes the largest eigenvalue of the matrix Q. To show this, it will be convenient to rewrite
We rewrite the squared error loss for an estimator ψ ∈ A under a realization z ∈ Rd as ℓψ (z) = z ⊤ F ⊤ F z + 2b⊤ F z + b⊤ b, where F := [−I, A]. Taking Q = F ⊤ F and q = F ⊤ b, and applying Theorem 2, we can reformulate the worst-case CVaR of the squared error loss as follows ⊤ ⊤ ⊤ ⊤ sup CVaRα P (z F F z + 2b z) + b b
P ∈P
n ⊤ ⊤ ⊤ −1 CVaRα (γz + F ⊤ b) P0 (γz + F b) (γI − F F ) γ∈R+ o + γ(α−1 r2 − ∥z∥2 ) + b⊤ b γI ≻ F ⊤ F .
= inf
Using the assumption that P0 is uniformly distributed on the set {z1 , . . . , zN }, the inner maximization in problem (4) can be rewritten as the following convex program: N
1 X inf τ̄ + b b + max γ(α−1 r2 − ∥zi ∥2 ) − τ̄ αN i=1 ⊤
+ (γzi + F ⊤ b)⊤ (γI − F ⊤ F )−1 (γzi + F ⊤ b), 0
s.t. τ̄ ∈ R, γ ∈ R+ , γI − F ⊤ F ≻ 0. By using a change of variable τ̄ = τ + γr2 /α − b⊤ b and putting each term in the summation over i in epigraph form, this convex program can be equivalently reformulated as: inf τ +
N 1 2 1 X γr + si α N i=1
s.t. τ ∈ R, γ ∈ R+ , γI − F ⊤ F ≻ 0, si ≥ max (γzi + F ⊤ b)⊤ (γI − F ⊤ F )−1 (γzi + F ⊤ b) + b⊤ b − γ∥zi ∥2 − τ, 0 ∀i = 1, . . . , N. It follows from the Schur complement condition for positive semidefiniteness that the constraint γI − F ⊤ F ≻ 0 is equivalent to the linear matrix inequality (9b). Moreover, each constraint involving the variable si can be expressed as two constraints: si ≥ 0 and si ≥ b⊤ b − γ∥zi ∥2 − τ + (γzi + F ⊤ b)⊤ (γI − F ⊤ F )−1 (γzi + F ⊤ b) si + γ∥zi ∥2 + τ − b⊤ b (γzi + F ⊤ b)⊤ ⇐⇒ ⪰ 0, (16) γzi + F ⊤ b γI − F ⊤ F where the last equivalence follows from using the Schur complement condition for positive semidefiniteness since γI − F ⊤ F ≻ 0. To prove equivalence between the matrix inequality (16) and (9c), it is helpful to rewrite (16) as ⊤ ⊤ ⊤ −b −b si + γ∥zi ∥2 + τ γzi⊤ − ⪰ 0. γzi γI F⊤ F⊤ Using the Schur complement condition once again, it follows that the above constraint is equivalent to (9c). This proves that problem (4) is equivalent to the SDP (9). R EFERENCES [1] Y. C. Eldar, A. Beck, and M. Teboulle, “A minimax Chebyshev estimator for bounded error estimation,” IEEE Transactions on Signal Processing, vol. 56, no. 4, pp. 1388–1397, 2008. [2] A. Juditsky and A. Nemirovski, “Near-optimality of linear recovery in Gaussian observation scheme under ∥ · ∥22 -loss,” The Annals of Statistics, vol. 46, no. 4, pp. 1603–1629, 2018. [3] A. Beck and Y. C. Eldar, “Regularization in regression with bounded noise: A Chebyshev center approach,” SIAM Journal on Matrix Analysis and Applications, vol. 29, no. 2, pp. 606–625, 2007. [4] Y. C. Eldar, A. Ben-Tal, and A. Nemirovski, “Robust mean-squared error estimation in the presence of model uncertainties,” IEEE Transactions on Signal Processing, vol. 53, no. 1, pp. 168–181, 2004. [5] A. Beck, Y. C. Eldar, and A. Ben-Tal, “Mean-squared error estimation for linear systems with block circulant uncertainty,” SIAM Journal on Matrix Analysis and Applications, vol. 29, no. 3, pp. 712–730, 2007.
[6] Y. C. Eldar, “Robust competitive estimation with signal and noise covariance uncertainties,” IEEE Transactions on Information Theory, vol. 52, no. 10, pp. 4532–4547, 2006. [7] A. Beck, A. Ben-Tal, and Y. C. Eldar, “Robust mean-squared error estimation of multiple signals in linear systems affected by model and noise uncertainties,” Mathematical Programming, vol. 107, no. 1, pp. 155–187, 2006. [8] B. C. Levy and R. Nikoukhah, “Robust least-squares estimation with a relative entropy constraint,” IEEE Transactions on Information Theory, vol. 50, no. 1, pp. 89–104, 2004. [9] M. Zorzi, “On the robustness of the Bayes and Wiener estimators under model uncertainty,” Automatica, vol. 83, pp. 133–140, 2017. [10] V. A. Nguyen, S. Shafieezadeh-Abadeh, D. Kuhn, and P. Mohajerin Esfahani, “Bridging Bayesian and minimax mean square error estimation via Wasserstein distributionally robust optimization,” Mathematics of Operations Research, vol. 48, no. 1, pp. 1–37, 2023. [11] S. Shafieezadeh Abadeh, V. A. Nguyen, D. Kuhn, and P. M. Mohajerin Esfahani, “Wasserstein distributionally robust Kalman filtering,” Advances in Neural Information Processing Systems, vol. 31, 2018. [12] S. Shafieezadeh Abadeh, P. M. Mohajerin Esfahani, and D. Kuhn, “Distributionally robust logistic regression,” Advances in Neural Information Processing Systems, vol. 28, 2015. [13] S. Shafieezadeh-Abadeh, D. Kuhn, and P. M. Esfahani, “Regularization via mass transportation,” Journal of Machine Learning Research, vol. 20, no. 103, pp. 1–68, 2019. [14] L. Aolaritei, S. Shafiee, and F. Dörfler, “Wasserstein distributionally robust estimation in high dimensions: performance analysis and optimal hyperparameter tuning,” Math. Programming, pp. 1–85, 2026. [15] B. Taskesen, M.-C. Yue, J. Blanchet, D. Kuhn, and V. A. Nguyen, “Sequential domain adaptation by synthesizing distributionally robust experts,” in International Conference on Machine Learning. PMLR, 2021, pp. 10 162–10 172. [16] A. Sani, A. Lazaric, and R. Munos, “Risk-aversion in multi-armed bandits,” Advances in Neural Info. Processing Systems, vol. 25, 2012. [17] Y. Chow, A. Tamar, S. Mannor, and M. Pavone, “Risk-sensitive and robust decision-making: a CVaR optimization approach,” Advances in Neural Information Processing Systems, vol. 28, 2015. [18] Y. Chow, M. Ghavamzadeh, L. Janson, and M. Pavone, “Riskconstrained reinforcement learning with percentile risk criteria,” Journal of Machine Learning Research, vol. 18, no. 167, pp. 1–51, 2018. [19] R. Williamson and A. Menon, “Fairness risk measures,” in International Conf. on Machine Learning. PMLR, 2019, pp. 6786–6797. [20] T. Soma and Y. Yoshida, “Statistical learning with conditional value at risk,” arXiv preprint arXiv:2002.05826, 2020. [21] Y. Laguel, K. Pillutla, J. Malick, and Z. Harchaoui, “Superquantiles at work: Machine learning applications and efficient subgradient computation,” Set-Valued and Variational Analysis, vol. 29, no. 4, pp. 967–996, 2021. [22] B. P. Van Parys, D. Kuhn, P. J. Goulart, and M. Morari, “Distributionally robust control of constrained stochastic systems,” IEEE Transactions on Automatic Control, vol. 61, no. 2, pp. 430–442, 2015. [23] M. P. Chapman and L. Lessard, “Toward a scalable upper bound for a CVaR-LQ problem,” IEEE Control Systems Letters, vol. 6, pp. 920– 925, 2021. [24] M. Kishida and A. Cetinkaya, “Risk-aware linear quadratic control using conditional value-at-risk,” IEEE Transactions on Automatic Control, vol. 68, no. 1, pp. 416–423, 2022. [25] V. A. Nguyen, S. Shafiee, D. Filipović, and D. Kuhn, “Mean-covariance robust risk measurement,” arXiv preprint arXiv:2112.09959, 2021. [26] D. Kuhn, S. Shafiee, and W. Wiesemann, “Distributionally robust optimization,” Acta Numerica, vol. 34, p. 579–804, 2025. [27] D. Kuhn, P. M. Esfahani, V. A. Nguyen, and S. Shafieezadeh-Abadeh, “Wasserstein distributionally robust optimization: Theory and applications in machine learning,” in Operations research & management science in the age of analytics. Informs, 2019, pp. 130–166. [28] F. Al Taha, S. Yan, and E. Bitar, “A distributionally robust approach to regret optimal control using the Wasserstein distance,” in 62nd IEEE Conference on Decision and Control. IEEE, 2023, pp. 2768–2775. [29] M. Gelbrich, “On a formula for the L2 Wasserstein metric between measures on Euclidean and Hilbert spaces,” Mathematische Nachrichten, vol. 147, no. 1, pp. 185–203, 1990. [30] P. Donti, B. Amos, and J. Z. Kolter, “Task-based end-to-end model learning in stochastic optimization,” Advances in Neural Information Processing Systems, vol. 30, 2017.