Distributionally Robust Optimisation with Bayesian Ambiguity Sets

Venue: NeurIPS_cc_2024_Workshop_BDU
Published: 2024-09-03 22:16:57

Abstract: Decision making under uncertainty is challenging since the data-generating process (DGP) is often unknown. Bayesian inference proceeds by estimating the DGP through posterior beliefs about the model's parameters. However, minimising the expected risk under these posterior beliefs can lead to sub-optimal decisions due to model uncertainty or limited, noisy observations. To address this, we introduce Distributionally Robust Optimisation with Bayesian Ambiguity Sets (DRO-BAS) which hedges against uncertainty in the model by optimising the worst-case risk over a posterior-informed ambiguity set. We show that our method admits a closed-form dual representation for many exponential family members and showcase its improved out-of-sample robustness against existing Bayesian DRO methodology in the Newsvendor problem.

Main paper: 
 introduction: 
 Decision-makers are regularly confronted with the problem of optimising an objective under uncertainty. Let x ∈ R d be a decision-making variable that minimises a stochastic objective function f : R d × Ξ → R, where Ξ is the data space and let P ⋆ ∈ P(Ξ) be the data-generating process (DGP) where P(Ξ) is the space of Borel distributions over Ξ. In practice, we do not have access to P ⋆ but to n independently and identically distributed (i.i.d.) observations D := ξ 1:n ∼ P ⋆ . Without knowledge of the DGP, model-based inference considers a family of models P Θ := {P θ : θ ∈ Θ} ⊂ P(Ξ)where each P θ has probability density function p(ξ|θ) for parameter space Θ ⊆ R k . In a Bayesian framework, data D is combined with a prior π(θ) to obtain posterior beliefs about θ through Π(θ|D).Bayesian Risk Optimisation (Wu et al., 2018) then solves a stochastic optimisation problem:min x∈R d E θ∼Π(θ|D) [E ξ∼P θ [f (x, ξ)]].(1) However, our Bayesian estimator is likely different from the true DGP due to model and data uncertainty: the number of observations may be small; the data noisy; or the prior or model may be misspecified. The optimisation problem (1) inherits any estimation error, and leads to overly optimistic decisions on out-of-sample scenarios even if the estimator is unbiased: this phenomenon is called the optimiser's curse (Kuhn et al., 2019) . For example, if the number of observations is small and the prior is overly concentrated, then the decision is likely to be overly optimistic.To hedge against the uncertainty of the estimated distribution, the field of Distributionally Robust Optimisation (DRO) minimises the expected objective function under the worst-case distribution that lies in an ambiguity set U ⊂ P(Ξ). Discrepancy-based ambiguity sets contain distributions that are close to a nominal distribution in the sense of some discrepancy measure such as the Kullback-Leibler (KL) divergence (Hu and Hong, 2013) , Wasserstein distance (Kuhn et al., 2019) or Maximum Mean Discrepancy (Staib and Jegelka, 2019) . For example, some model-based methods ( Iyengar et al., 2023; Michel et al., 2021 Michel et al., , 2022) ) consider a family of parametric models and create discrepancy-based ambiguity sets centered on the fitted model. However, uncertainty about the parameters is not captured in these works, which can lead to a nominal distribution far away from the DGP when the data is limited. The established framework for capturing such uncertainty is Bayesian inference.The closest work to ours, using parametric Bayesian inference to inform the optimisation problem, is Bayesian DRO (BDRO) by Shapiro et al. (2023) . BDRO constructs discrepancy-based ambiguity sets with the KL divergence and takes an expected worst-case approach, under the posterior distribution. More specifically, let U ϵ (P θ ) := {Q ∈ P(Ξ) : d KL (Q∥P θ ) ≤ ϵ} be the ambiguity set centered on distribution P θ with parameter ϵ ∈ [0, ∞) controlling the size of the ambiguity set. Under the expected value of the posterior, Bayesian DRO solves:min x∈R d E θ∼Π(θ|D)supEQUATIONwhereE θ∼Π(θ|D) [Y ] := Θ Y (θ)Π(θ | D) dθ denotes the expectation of random variable Y : Θ → R with respect to Π(θ | D).A decision maker is often interested in protecting against and quantifying the worst-case risk, but BDRO does not correspond to a worst-case risk analysis. Moreover, the BDRO dual problem is a two-stage stochastic problem that involves a double expectation over the posterior and likelihood. To get a good approximation of the dual problem, a large number of samples are required, which increases the solve time of the dual problem.We introduce DRO with Bayesian Ambiguity Sets (DRO-BAS), an alternative optimisation objective for Bayesian decision-making under uncertainty, based on a posterior-informed ambiguity set.The resulting problem corresponds to a worst-case risk minimisation over distributions with small expected deviation from the candidate model. We go beyond ball-based ambiguity sets, which are dependent on a single nominal distribution, by allowing the shape of the ambiguity set to be informed by the posterior. For many exponential family models, we show that the dual formulation of DRO-BAS is an efficient single-stage stochastic program. 
dro with bayesian ambiguity sets: 
 We propose the following DRO-BAS objective:EQUATIONwhere Q ∈ P(Ξ) is a distribution in the ambiguity set, f x (ξ) := f (x, ξ) is the objective function,D : P(Ξ) × P(Ξ) → R is a divergence, and ϵ ∈ [0, ∞) is a tolerance level.The ambiguity set is informed by the posterior distribution Π by considering all probability measures Q ∈ P(Ξ) which are ϵ-away from P θ in expectation, with ϵ dictating the desired amount of risk in the decision.The shape of our ambiguity set is flexible and driven by the posterior distribution. This is contrary to standard ambiguity sets which correspond to a ball around a fixed nominal distribution. The DRO-BAS problem (3) is still a worst-case approach, keeping with DRO tradition, instead of BDRO's expected worst-case formulation (2), see Figure 1 . p(ξ | θ) Π(θ | τ n ) θn G(τ n ) N (ξ | µ, σ 2 ) N (µ | µ n , σ 2 n ) µ n , σ 2 σ 2 n 2σ 2 N (ξ | µ, λ -1 ) NG(µ, λ | µ n , κ n , α n , β n ) µ n , βn αn 1 2 1 κn + ln α n -ψ(α n ) Exp(ξ | θ) Ga(θ | α n , β n ) αn βn ln α n -ψ(α n )The EQUATION)Exact closed-form solutions of DRO-BAS can be obtained for a wide range of exponential family models with conjugate priors. When the likelihood distribution is a member of the exponential family, a conjugate prior also belongs to the exponential family (Gelman et al., 1995, Ch. 4.2) . In this setting, before we prove the main result, we start with an important Lemma.Lemma 1. Let p(ξ | θ) be an exponential family likelihood and π(θ), Π(θ | D) a conjugate priorposterior pair, also members of the exponential family. Let τ 0 , τ n ∈ T be hyperparameters of the prior and posterior respectively, where T is the hyperparameter space. Let θn ∈ Θ depend upon τ n and let G : T → R be a function of the hyperparameters. If the following identity holds:E θ∼Π [ln p(ξ | θ)] = ln p(ξ | θn ) -G(τ n ),(5) then the expected KL-divergence can be written as:EQUATIONThe condition in ( 5) is a natural property of many exponential family models, some of which are showcased in Table 1 . Future work aims to prove this for all exponential family models. It is straightforward to establish the minimum tolerance level ϵ min required to obtain a non-empty ambiguity set.Since the KL divergence is non-negative, under the condition of Lemma 1, for any Q ∈ P(Ξ):EQUATIONWe are now ready to prove our main result.Theorem 1. Suppose the conditions of Lemma 1 hold and ϵ ≥ ϵ min (n) as in ( 7). LetEQUATIONTo guarantee that the DRO-BAS objective upper bounds the expected risk under the DGP, the decision-maker aims to choose ϵ large enough so that P ⋆ is contained in the ambiguity set. The condition in (5) yields a closed-form expression for the optimal radius ϵ ⋆ by noting that:ϵ ⋆ = E θ∼Π [d KL (P ⋆ ∥P θ )] = d KL (P ⋆ , Pθ n ) + G(τ n ).(9) If the model is well-specified, and hence P ⋆ and Pθ n belong to the same exponential family, it is straightforward to obtain ϵ ⋆ based on the prior, posterior and true parameter values. We give examples in Appendix A. In practice, since the true parameter values are unknown, we can approximate ϵ ⋆ using the observed samples. It follows that for any ϵ ≥ ϵ ⋆ ≥ ϵ min (n): 3 The Newsvendor Problem Experiment setup. We evaluate DRO-BAS against the BDRO framework on a univariate Newsvendor problem with a well-specified univariate Gaussian likelihood with unknown mean and variance (Appendix D showcases a misspecified setting). The goal is to choose an inventory level 0 ≤ x ≤ 50 of a perishable product with unknown customer demand ξ ∈ R that minimises the Analysis. Figure 2 shows that, for small sample size N = 25, 100, our framework dominates BDRO in the sense that DRO-BAS forms a Pareto front for the out-of-sample mean-variance tradeoff of the objective function f . That is, for any ϵ 1 , let m BDRO (ϵ 1 ) and v BDRO (ϵ 1 ) be the out-of-sample mean and variance respectively of BDRO: then there exists ϵ 2 with out-of-sample mean m BAS (ϵ 2 )E ξ∼P ⋆ [f (x, ξ)] ≤ sup Q:E θ∼Π [dKL(Q∥P θ )]≤ϵ E ξ∼Q [f x (ξ)].cost function f (x, ξ) = h max(0, x -ξ) + b max(0, ξ -x),and variance v BAS (ϵ 2 ) of BAS-DRO such that m BAS (ϵ 2 ) < m BDRO (ϵ 1 ) and v BAS (ϵ 2 ) < v BDRO (ϵ 1 ).When N = 900, Figure 2 shows DRO-BAS and BDRO lie roughly on the same Pareto front. To summarise, BDRO requires more samples N than DRO-BAS for good out-of-sample performance, likely because BDRO must evaluate a double expectation over the posterior and likelihood, whilst DRO-BAS only samples from p(ξ | θn ). For fixed N , the solve times for DRO-BAS and BDRO are broadly comparable (see Appendix C). 
discussion: 
 We proposed a novel approach to Bayesian decision-making under uncertainty through a DRO ob- The Supplementary Material is organised as follows: Appendix A provides details for all the exponential family models discussed in Table 1 , while Appendix B contains the proofs of all mathematical results appearing in the main text. Appendix C provides additional experimental details for the Newsvendor problem in Section 3. Finally, in Appendix D we present experimental results for the newsvendor problem example with a misspecified model.We derive the values of G(τ n ) and θn for different likelihoods and conjugate prior/posterior in Table 1 . Each subsection contains a corollary with the result in Table 1 .A.1 Gaussian model with unknown mean and known varianceLet the random variable ξ be univariate and have continuous support. We assume the variance σ 2 of ξ is known. We estimate the mean of a univariate Gaussian distribution with known variance σ 2 .The example can be found in (Bishop, 2006, Section 2.3.6) . We define our parameter θ to be the unknown mean µ and we place a Gaussian prior π(µ) over it.Definition 1 (Gaussian with unknown mean and known variance). The likelihood isp(ξ | µ) = N (µ, σ 2 ), the prior over µ is π(µ) = N (µ 0 , σ 2 0 ), and the conjugate posterior is Π(µ | D) = N (µ | µ n , σ 2 n ),whereµ n := σ 2 nσ 2 0 + σ 2 µ 0 + nσ 2 0 nσ 2 0 + σ 2 μ 1 σ 2 n := 1 σ 2 0 + n σ 2 μ := 1 n n i=1 ξ i . Lemma 2. Let µ n ∈ R and σ, σ n ∈ R + . Then E µ∼N (µn,σ 2 n ) log N (µ, σ 2 ) = log N (µ n , σ 2 ) - σ 2 n 2σ 2 .Proof. The result is a special case of (Stratos, 2023, Lemma H.10) .Corollary 1. When the likelihood is a Gaussian distribution with unknown mean and known variance σ 2 and the prior and posterior are Gaussian distributions (see Definition 1), then Theorem 1holds with θn = µ n and G(τ n ) = σ 2 n 2σ 2 .Proof. Lemma 2 shows that the condition (5) in Lemma 1 holds, thus Theorem 1 follows.Tolerance level ϵ In the well-specified case, where we assume that P ⋆ := P ⋆ θ for some θ ⋆ ∈ Θ, it is easy to obtain the required size of the ambiguity set exactly. Let θ ⋆ := µ ⋆ and P ⋆ := N (µ ⋆ , σ 2 ).By Corollary 1 it follows that:EQUATION)So for a fixed finite sample ξ1:n ∼ P ⋆ , if ϵ ≥ ϵ ⋆ := (µ ⋆ -µn) 2 +σ 2 n 2σ 2, it follows that DRO-BAS upper-bounds the target optimisation objective:E ξ∼P ⋆ [f x (ξ)] ≤ sup Q:E θ∼Π(•|ξ 1:n ) [dKL(Q ∥ P θ )]≤ϵ E ξ∼Q [f x (ξ)].In practice, since µ ⋆ is unknown, ϵ ⋆ can be approximated by using the sample mean.In this section, we consider a Bayesian model that estimates the unknown mean and variance of a uni-variate Gaussian distribution. We define our model in Definition 2, then prove some preliminary results for the normal-gamma distribution, before proving our main result in Lemma 4.Definition 2 (Unknown mean and variance of a Gaussian). Following (Murphy, 2023, Chapter 3.4 .3), we place a normal-gamma prior over the mean µ and precision λ = σ -2 . The normalgamma prior is the conjugate to a Gaussian likelihood and results in a normal-gamma posterior distribution.Likelihood:p(ξ | µ, λ) = N (ξ | µ, λ -1 ) Prior: π(µ, λ) = N G(µ, λ | µ 0 , κ 0 , α 0 , β 0 ) = N (µ | µ 0 , (κ 0 λ) -1 ) • Ga(λ | α 0 , β 0 ) Posterior: π(µ, λ | D) = N G(µ, λ | µ n , κ n , α n , β n ) = N (µ | µ n , (λκ n ) -1 ) • Ga(λ | α n , β n )whereµ n := κ 0 µ 0 + n ξn n + κ 0 , κ n := κ 0 + n, α n := α 0 + n 2 , ξn = 1 n n i-1 ξ i , β n := β 0 + 1 2 n i=1 (ξ i -ξn ) 2 + κ 0 n( ξn -µ 0 ) 2 2(κ 0 + n) .In Lemma 4, we derive condition Equation ( 5) for a Gaussian model with unknown mean and unknown variance. Before proceeding, we need to define the gamma and digamma functions and recall the moments of the normal-gamma distribution. E N G [ln λ] = ψ(α n ) -ln β n , E N G [λ] = α n β n , E N G [λµ] = µ n α n β n , E N G [λµ 2 ] = 1 κ n + µ 2 n α n β n .where ψ : N → R is the digamma function from Definition 3.Lemma 4. Let µ n ∈ R and κ n , α n , β n ∈ R + . Then E N G(µ,λ|µn,κn,αn,βn) ln N (ξ | µ, λ -1 ) = ln N ξ | µ n , β n α n - 1 2 1 κ n + ln α n -ψ(α n )where ψ : N → R is the digamma function from Definition 3.Proof. First, observe that the natural logarithm of the Gaussian distribution may be re-written asln N (ξ | µ, λ -1 ) = ln λ 1 2 (2π) 1 2 exp - λ 2 (ξ -µ) 2 = 1 2 ln λ - 1 2 ln 2π - λ 2 (ξ -µ) 2 = 1 2 ln λ - 1 2 ln 2π - 1 2 λξ 2 + λµξ - 1 2 λµ 2 .In what follows, for shorthand, we denote the expectation E N G(µ,λ|µn,κn,αn,βn) as E µ,λ∼N G :E µ,λ∼N G ln N (ξ | µ, λ -1 ) (i) = E µ,λ∼N G 1 2 ln λ - 1 2 ln 2π - 1 2 λξ 2 + λµξ - 1 2 λµ 2 (ii) = - 1 2 ln 2π + 1 2 E µ,λ∼N G [ln λ] - 1 2 ξ 2 • E µ,λ∼N G [λ] + ξ • E µ,λ∼N G [λµ] - 1 2 E µ,λ∼N G λµ 2 (iii) = - 1 2 ln 2π + 1 2 (ψ(α n ) -ln β n ) - 1 2 ξ 2 α n β n + ξµ n α n β n - 1 2 1 κ n + µ 2 n α n β n (iv) = - 1 2 ln 2π + 1 2 ψ(α n ) -ln β n - 1 κ n - α n 2β n (ξ -µ n ) 2 (v) = - 1 2 ln 2π - 1 2 ln β n -ln α n + ln α n -ψ(α n ) + 1 κ n + ln exp - 1 2 βn αn (ξ -µ n ) 2 (vi) = - 1 2 ln α n -ψ(α n ) + 1 κ n - 1 2 ln 2π β n α n + ln exp - 1 2 βn αn (ξ -µ n ) 2 (vii) = - 1 2 1 2α n + I αn + 1 κ n + ln   1 2π βn αn exp - 1 2 βn αn (ξ -µ n ) 2   (viii) = - 1 2 ln α n -ψ(α n ) + 1 κ n + ln N ξ | µ n , β n α nwhere in equation (i) we take the expectation over the normal-gamma distribution; (ii) we apply linearity of expectation; (iii) we use the moment-generating functions from Lemma 3; (iv) we complete the square; (v) we add and subtract ln α n ; (vi) and (vii) we re-arrange and apply log identities;and finally in (viii) we use the definition of a Gaussian probability density function.Corollary 2. When the likelihood is a Gaussian distribution with unknown mean and variance, and the conjugate prior and posterior are normal-gamma distributions (see Definition 2), then Theorem 1 holds with θn = (µ n , βn αn ) and G(τ n ) = 1 2 ln α n -ψ(α n ) + 1 κn .Proof. Lemma 4 shows that the condition (5) in Lemma 1 holds, thus Theorem 1 follows.Tolerance level ϵ In the well-specified case, where we assume that P ⋆ := P ⋆ θ for some θ ⋆ ∈ Θ, it is easy to obtain the required size of the ambiguity set exactly. Let θ ⋆ := (µ ⋆ , λ ⋆-1 ) and P ⋆ := N (µ ⋆ , λ ⋆-1 ). Using Corollary 2 we obtain:E µ,λ∼N G(µ,λ|µn,κn,αn,βn) d KL (P ⋆ , N (ξ | µ, λ -1 )) = d KL P ⋆ ∥ N µ n , β n α n + 1 2 1 κ n + ln α n -ψ(α n ) = ln λ ⋆ β n α n + λ ⋆-1 + (µ ⋆ -µ n ) 2 2 βn αn - 1 2 + 1 2 1 κ n + ln α n -ψ(α n ) = 1 2 ln (λ ⋆ β n ) + λ ⋆-1 + (µ ⋆ -µ n ) 2 ) 2 βn αn -1 + 1 κ n -ψ(α n ) .A α n = α 0 + n and β n = β 0 + ξi∈D ξ i .Lemma 5. When the likelihood is an exponential distribution with gamma prior and posterior (see Definition 4), thenE Ga(θ|αn,βn) [ln Exp(ξ | θ)] = ln Exp ξ | α n β n + ψ(α n ) -ln α n .Proof. Starting from the left-hand side, we take the log of the PDF of the exponential distribution, then use the logarithm expectation of the gamma distribution, and finally re-arrange using log identities:E Ga(θ|αn,βn) [ln Exp(ξ | θ)] = E Ga(θ|αn,βn) [ln θ -θξ] = ψ(α n ) -ln β n - α n β n ξ = ψ(α n ) -ln α n + ln α n β n - α n β n ξ = ψ(α n ) -ln α n + ln α n β n exp - α n β n ξ = ψ(α n ) -ln α n + ln Exp ξ | α n β n .The last line follows by the definition of the PDF of the exponential distribution.Corollary 3. When the likelihood is an exponential distribution with gamma prior and posterior, then Theorem 1 holds with θn = αn βn andG(τ n ) = ψ(α n ) -ln α n .Proof. Lemma 5 shows that the condition (5) in Lemma 1 holds, thus Theorem 1 follows.Tolerance level ϵ In the well-specified case, where we assume that P ⋆ := P ⋆ θ for some θ ⋆ ∈ Θ, it is easy to obtain the required size of the ambiguity set exactly. Let θ ⋆ be the true rate parameter, i.e.P ⋆ := Exp(θ ⋆ ). Using Corollary 3 we obtain:E Ga(θ|αn,βn) [d KL (P ⋆ , Exp(θ)] = d KL P ⋆ ∥ Exp α n β n + ψ(α n ) -ln(α n ) = ln(θ ⋆ ) -ln α n β n + α n β n θ ⋆ -1 + ψ(α n ) -ln(α n ).B.1 Proofs of DRO-BAS upper bound in Equation ( 4)Before proving the required upper bound, we recall the definition of the KL divergence and its convex conjugate.Definition 5 (KL-divergence). Let µ, ν ∈ P(Ξ) and assume µ is absolutely continuous with respect to ν (µ ≪ ν). The KL-divergence of µ with respect to ν is defined as:d KL (µ∥ν) := Ξ ln µ(dξ) ν(dξ) µ(dξ).Lemma 6 (Conjugate of the KL-divergence). Let ν ∈ P(Ξ) be non-negative and finite. The convexconjugate d ⋆ KL (•∥ν) of d KL (•∥ν) is d ⋆ KL (•∥ν)(h) = ln Ξ exp(h)dν .Proof. See Proposition 28 and Example 7 in Agrawal and Horel (2021) .Proof of Equation 4. The result follows from a standard Lagrangian duality argument and an application of Jensen's inequality. More specifically, we introduce a Lagrangian variable γ ≥ 0 for the expected-ball constraint on the left-hand side of (4) as follows:supQ:E θ∼Π [dKL(Q∥P θ )]≤ϵ E Q [f x ] (i) ≤ inf γ≥0 sup Q∈P(Ξ) E Q [f x ] + γϵ -γE Π [d KL (Q∥P θ )] (ii) = inf γ≥0 γϵ + sup Q∈P(Ξ) E Q [f x ] -E Π [γd KL (Q∥P θ )] (iii) = inf γ≥0 γϵ + (E Π [γd KL (•∥P θ )]) ⋆ (f x ) (iv) ≤ inf γ≥0 γϵ + E Π (γd KL (•∥P θ )) ⋆ (f x ) (v) = inf γ≥0 γϵ + E Π γ ln E P θ exp f x γ .Inequality (i) holds by weak duality. Equality (ii) holds by linearity of expectation and a simple rearrangement. Equality (iii) holds by the definition of the conjugate function. Inequality (iv) holds by Jensen's inequality(E[•]) ⋆ ≤ E[(•) ⋆ ] because the conjugate is a convex function. Equality (v)holds by Lemma 6 and the fact that for γ ≥ 0 and function ϕ, (γϕ) ⋆ (y) = γϕ ⋆ (y/γ).Starting from the left-hand side, we haveE Π [d KL (Q∥P θ )] (i) = E θ∼π(θ|D) Ξ q(ξ) ln q(ξ) p(ξ | θ) dξ (ii) = E θ∼π(θ|D) Ξ q(ξ) ln (q(ξ)) -q(ξ) ln (p(ξ | θ)) dξ (iii) = Ξ q(ξ) ln (q(ξ)) -q(ξ) • E θ∼π(θ|D) [ln (p(ξ | θ))] dξ (iv) = Ξ q(ξ) ln (q(ξ)) -q(ξ) • ln p(ξ | θn ) -G(τ n ) dξ (v) = Ξ q(ξ) ln q(ξ) p(ξ | θn ) dξ + Ξ q(ξ) • G(τ n ) dξ (vi) = d KL (Q∥Pθ n ) + E Q [G(τ n )] (vii) = d KL (q(ξ)∥p(ξ | θn )) + G(τ n ).where (vii) holds by the expected value of a constant.We begin by restating the Lagrangian dual from the proof of Equation ( 4), but with the added claim that strong duality holds between the primal and dual problems:EQUATIONProof. The conditions under which our claim of strong duality holds will be proved later. Next, we substitute the right-hand side of equation (vi) above into the dual problem in (11):sup Q:E θ∼Π [dKL(Q ∥ P θ )]≤ϵ E ξ∼Q [f x (ξ)] = inf γ≥0 γϵ + sup Q∈P(Ξ) Ξ f x (ξ)q(ξ) dξ -γ d KL Q ∥ p(ξ | θn ) + G(τ n ) = inf γ≥0 γϵ + γG(τ n ) + γ d KL • ∥ p(ξ | θn ) ⋆ (f x (ξ)) = inf γ≥0 γ(ϵ -G(τ n )) + γ ln E ξ∼p(ξ| θn) exp f x (ξ) γ ,where the second and third equality holds by the definition of the conjugate of the KL-divergence and by Lemma 6.Finally, it remains to argue that strong duality holds. First, note that the primal problem is a concave optimisation problem with respect to distribution Q. Second, when ϵ > G(τ n ), then distribution Q = p(ξ | θn ) is a strictly feasible point to the primal constraint becauseE θ∼Π [d KL ( Q ∥ P θ )] = 0 < ϵ -G(τ n ).We provide additional details about our Newsvendor experiment in Section 3 when P ⋆ is a Gaussian distribution with µ ⋆ = 25 and σ 2 ⋆ = 100.Hyperparameters. The prior and posterior are normal-gamma distributions. We set the prior hyperparameters to be µ 0 = 0 and κ 0 , α 0 , β 0 = 1. The derivation of the hyperparameters can be found in Definition 2.Values of ϵ min and ϵ ⋆ . From Table 1 and equation ( 7), the value of ϵ min is 0.047. From equation (9), the average value of ϵ ⋆ over all J seeds is 0.089 with standard deviation 0.048.Implementation. We implemented the dual problems for DRO-BAS (Theorem 1) and BDRO (Shapiro et al., 2023) in Python using CVXPY version 1.5.2 and the MOSEK solver version 10.1.28.Our implementation uses disciplined parametrized programming (Agrawal et al., 2019) which -after an initial warm start for seed j = 1 -allows us to solve subsequent seeds j = 2, . . . , J rapidly (see Table 2 ). We used a 12-core Dual Intel Xeon E5-2643 v3 @ 3.4 Ghz with 128GB RAM.Out-of-sample mean and variance. For a given ϵ and seed j, let the optimal solution be x (j) (ϵ).We calculate the out-of-sample mean m(j) (ϵ) = E ξ∼ P(j) m [f (x (j) (ϵ), ξ)] and variance v (j) (ϵ) = Var ξ∼ P(j) m [f (x (j) (ϵ), ξ)]of the objective under the empirical test distribution P(j) m . For a given ϵ, the out-of-sample mean m(ϵ) and variance v(ϵ) across all seeds ism(ϵ) = 1 J J j=1 m (j) (ϵ) v(ϵ) = 1 J J j=1 v (j) (ϵ) + 1 J -1 J j=1 m (j) (ϵ) -m(ϵ) 2 .The out-of-sample variance v(ϵ) is equal to the mean of the variances v (j) (ϵ) plus the variance of the means m (j) (ϵ) (Gotoh et al., 2021) .Solve time. On the initial warm-start seed j = 1, for each N , DRO-BAS solves the dual problem from Theorem 1 faster than the BDRO dual problem. For example, when N = 900, DRO-BAS solves problems in 0.27 seconds on average, whilst BDRO solves problem in 5.56 seconds. These results suggest that, for fixed N , if the solve is started from scratch with no warm start, then DRO-BAS will solve instances faster than BDRO. For seeds j = 2, . . . , J, disciplined Table 2: Average (AVG) and standard deviation (STD) of the solve time for initial warm start on seed j = 1 and for subsequent seeds j = 2 . . . , J. Distribution P ⋆ is a Gaussian N (25, 100). parametrized programming (DPP) significantly speeds up the solve for BDRO: the average solve time for seeds j = 2, . . . , J is 0.40 seconds when N = 900. In contrast, DPP does not speed up the solve for DRO-BAS: the average solve time for seeds j = 2, . . . , J is 0.40 when N = 900.We conjecture that the speed up for BDRO using DPP is because BDRO has N θ Lagrangian dual variables compared to DRO-BAS having exactly one Lagrangian dual variable. BDRO then benefits from the warm start because it can reuse the presolve effort spent on the N θ dual variables spent during the warm start.In this section, we present additional experiments when the data-generating process P ⋆ is a truncated normal distribution. The truncated normal has mean µ ⋆ = 10 and variance σ 2 ⋆ = 100. The likelihood is a Gaussian distribution, so our model is misspecified. The conjugate prior and posterior are still normal-gamma distributions with the same hyperparameters as Section 3. The experimental setup is also the same as Section 3: the values of ϵ, N , n, m, and J are all specified the same.Analysis. When the likelihood is misspecified, Figure 3 shows the out-of-sample mean-variance tradeoff is again a Pareto front. This is the same conclusion as the well-specified case in Section 3.Furthermore, when N = 900, DRO-BAS has a small advantage on the mean-variance tradeoff.Solve time. Table 3 shows the same conclusions about the solve time from Appendix C can be made about the solve time for the truncated normal data-generating process.Table 3 : Average (AVG) and standard deviation (STD) of the solve time for initial warm start on seed j = 1 and for subsequent seeds j = 2 . . . , J. Distribution P ⋆ is a truncated normal.Solve time in seconds AVG (STD) j = 1 j = 2, . . . , J N DRO-BAS BDRO DRO-BAS BDRO 25 0.03 (0.00) 0.07 (0.02) 0.01 (0.00) 0.01 (0.00) 100 0.03 (0.00) 0.14 (0.02) 0.02 (0.01) 0.03 (0.00) 900 0.27 (0.02) 5.54 (0.04) 0.40 (0.11) 0.23 (0.01) 
 
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Reviews: review: title: title: Very interesting theoretically sound approach for robust optimization
main_review: main_review: # Summary

This is an excellent work providing a new formulation for distributionally robust optimization in a principled manner via Bayesian ambiguity sets. The work improves over the previous related literature by providing a more suitable/principled objective (Eq. 3), which truly expresses the ambiguity set based on the current Bayesian posterior (effectively, using the correct ordering for where the expectation is taken). Then it provides solutions for some cases for the KL divergence and members of the exponential family that satisfy condition in Eq. 5. Finally, it empirically shows the method on one benchmark problem, the Newsvendor Problem.

# Comments

This paper is top-notch. Despite my lack of familiarity with the specific area, it was easy to follow along and I find the rationale and execution very compelling. Robust optimization and decision making (which naturally includes robust estimation) under posterior indeterminacy (which could be due to model misspecification) is a research area of major interest and this paper provides a significant and original contribution, to my knowledge. I think it's an excellent contribution and of broad interest especially for this venue.

In terms of limitations, the obvious ones would be addressing a broader class of distributions or providing tools or suggestions for solving Eq. 3 in more general cases. The authors might want to discuss the possible limitations of using the KL as a divergence, and possibilities of using other divergences (e.g., it's not clear if that would be possible; the current results seem to rely heavily on various KL properties).
title: title: Review of DRO-BAS
main_review: main_review: The paper introduces a novel objective function for decision-making under uncertainty, which is a worst-case expected loss. The worst case is determined from a set of distributions that, a posteriori, are expected to belong to an ambiguity set of the data-generating process. The authors present an upper bound for this objective function and, for some exponential family models, derive a closed-form expression. They also empirically compare the out-of-sample mean-variance trade-off of their approach with the Bayesian DRO method described in Shapiro (2023).

I found the paper both interesting and well-written, with a potentially promising idea. However, based on the presented results, I'm unsure of the concrete advantages it offers over Bayesian DRO. The advantage seems limited to cases where a small sample is drawn from the posterior and likelihood in Bayesian DRO, and the proposed objective function has a closed-form expression. That said, when this closed-form expression holds, it doesn't seem particularly computationally demanding to draw a larger sample from the posterior and likelihood, allowing Bayesian DRO optimization to achieve the same Pareto front.
title: title: Review
main_review: main_review: This paper introduces a new method, DRO-BAS, designed to improve decision-making under uncertainty by mitigating the risks associated with noisy data and model misspecification. By extending traditional Bayesian DRO approaches, the authors propose optimising the worst-case risk using ambiguity sets informed by posterior distributions. They demonstrate the effectiveness of DRO-BAS through theoretical proofs and a numerical example using the Newsvendor problem.

Pros:
- DRO-BAS showcases better out-of-sample performance than existing Bayesian DRO (BDRO) techniques, particularly in scenarios with small sample sizes. Figure 2 clearly illustrates that DRO-BAS dominates BDRO in mean-variance tradeoff when data is limited.
- The paper provides a clear theoretical foundation, including a closed-form dual representation for a wide range of exponential family models.
- The DRO-BAS method simplifies the dual formulation into a single-stage stochastic program, reducing computational complexity. 

Cons:
- While the DRO-BAS approach works well for exponential family models, its extension to more general, high-dimensional problems is only briefly discussed as future work. This limits its immediate applicability to broader real-world decision-making problems.
- While DRO-BAS outperforms BDRO for small sample sizes, the performance gap diminishes as the number of samples increases. For large datasets, both methods show similar out-of-sample performance. 
- The method introduces additional complexity in selecting the ambiguity set's tolerance level.