Sensitivity analysis of Wasserstein distributionally robust optimization problems

(a) Financial economics

We start with the simple example of risk-neutral pricing of a call option written on an underlying asset ${(S_t)}_{t\le T}$. Here, $T,K>0$ are the maturity and the strike, respectively, $f(x,a)={(S_{0}x-K)}^{+}$ and $\mu$ is the distribution of $S_T/S_0$. We set interest rates and dividends to zero for simplicity. In [25], the model $\mu$ is a lognormal distribution, i.e. $\log (S_T/S_0)\sim \mathcal{N}(-{\sigma }^{2}T/2,{\sigma }^{2}T)$ is Gaussian with mean $-{\sigma }^{2}T/2$ and variance ${\sigma }^{2}T$. In this case, $V(0)$ is given by the celebrated Black–Scholes formula. Note that this example is particularly simple since $f$ is independent of $a$. However, to ensure risk-neutral pricing, we have to impose a linear constraint on the measures in $B_{\delta }(\mu )$, giving

To compute its sensitivity we encode the constraint using a Lagrangian and apply theorem 2.2, see [9, remark 7.3, theorem 7.7]. For $p=2$, letting $k=K/S_0$ and ${\mu }_k=\mu ([k,\mathrm{\infty }))$, the resulting formula, see [9, example 7.10], is given by Let us specialize to the lognormal distribution of the Black–Scholes model above and denote the quantity in (3.1) as $\mathcal{R}BS(\delta )$. It may be computed exactly using methods from [26]. Figure 1 compares the exact value and the first-order approximation. We have ${\rm Y}=S_0\sqrt{\Phi (d_{-})(1-\Phi (d_{-}))}$, where $d_{-}=\log (S_0/K)-{\sigma }^{2}T/2/\sigma \sqrt{T}$ and $\Phi$ is the cdf of $\mathcal{N}(0,1)$ distribution. It is also insightful to compare ${\rm Y}$ with a parametric sensitivity. If instead of Wasserstein balls, we consider $\{\mathcal{N}(-{\tilde{\sigma }}^{2}T/2,{\tilde{\sigma }}^{2}T):|\sigma -\tilde{\sigma }|\le \delta \}$ the resulting sensitivity is known as the Black–Scholes Vega and given by $\mathcal{V}=S_0{\Phi }^{\prime }(d_{-}+\sigma \sqrt{T})$. We plot the two sensitivities in figure 2. It is remarkable how, for the range of strikes of interest, the non-parametric model sensitivity ${\rm Y}$ traces out the usual shape of $\mathcal{V}$ but shifted upwards to account for the idiosyncratic risk of departure from the lognormal family. More generally, given a book of options with payoff $f=f^{+}-f^{-}$ at time $T$, with $f^{+}$, $f^{-}\ge 0$, we could say that the book is ${\rm Y}$-neutral if the sensitivity ${\rm Y}$ was the same for $f^{+}$ and for $f^{-}$. In analogy to Delta-Vega hedging standard, one could develop a non-parametric model-agnostic Delta-Upsilon hedging. We believe these ideas offer potential for exciting industrial applications and we leave them to further research.

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We turn now to the classical notion of the optimized certainty equivalent (OCE) of [27]. It is a decision theoretic criterion designed to split a liability between today's and tomorrow’s payments. It is also a convex risk measure in the sense of [28] and covers many of the popular risk measures such as expected shortfall or entropic risk, see [29]. We fix a convex monotone function $l:\mathbb{R}\to \mathbb{R}$ which is bounded from below and $g:{\mathbb{R}}^d\to \mathbb{R}$. Here, $g$ represents the payoff of a financial position and $l$ is the negative of a utility function, or a loss function. We take $||\cdot ||=|\cdot |$ and refer to [9, lemma 7.14] for generic sufficient conditions for assumptions 2.1 and 2.3 to hold in this setup. The OCE corresponds to $V$ in (1.1) for $f(x,a)=l(g(x)-a)+a$ and $\mathcal{A}=\mathbb{R}$, $\mathcal{S}={\mathbb{R}}^d$. Theorems 2.2 and 2.4 yield the sensitivities

where, for simplicity, we took $p=q=2$ for the latter.

A related problem considers hedging strategies which minimize the expected loss of the hedged position, i.e. $f(x,a)=l(g(x)+⟨a,x-x_0⟩)$, where $\mathcal{A}={\mathbb{R}}^k$ and $(x_0,x)$ represent today's and tomorrow’s traded prices. We compute ${\rm Y}$ as

Furthermore we can combine loss minimization with OCE and consider $a=(H,m)\in {\mathbb{R}}^k\times \mathbb{R}$, $f(x,(h,m))=l(g(x)+⟨H,x-x_0⟩+m)-m$. This gives $V^{\prime }(0)$ as the infimum over $(H^{\star },m^{\star })\in {\mathcal{A}}_0^{\star }$ of The above formulae capture non-parametric sensitivity to model uncertainty for examples of key risk measurements in financial economics. To the best of our knowledge, this has not been achieved before.

Finally, we consider briefly the classical mean-variance optimization of [30]. Here $\mu$ represents the loss distribution across the assets and $a\in {\mathbb{R}}^d$, $\sum _{i=1}^d{a}_i=1$ are the relative investment weights. The original problem is to minimize the sum of the expectation and $\gamma$ standard deviations of returns $⟨a,X⟩$, with $X\sim \mu$. Using the ideas in [31, Example 2] and considering measures on ${\mathbb{R}}^d\times {\mathbb{R}}^d$, we can recast the problem as (1.1). While [31] focused on the asymptotic regime $\delta \to \mathrm{\infty }$, their non-asymptotic statements are related to our theorem 2.2 and either result could be used here to obtain that $V(\delta )\approx V(0)+\sqrt{1-{\gamma }^2}\delta$ for small $\delta$.

(b) Neural networks

We specialize now to quantifying robustness of neural networks (NN) to adversarial examples. This has been an important topic in machine learning since [32] observed that NN consistently misclassify inputs formed by applying small worst-case perturbations to a dataset. This produced a number of works offering either explanations for these effects or algorithms to create such adversarial examples, e.g. [33–39] to name just a few. The main focus of research works in this area, see [40], has been on faster algorithms for finding adversarial examples, typically leading to an overfit to these examples without any significant generalization properties. The viewpoint has been mainly pointwise, e.g. [32], with some generalizations to probabilistic robustness, e.g. [39].

In contrast, we propose a simple metric for measuring robustness of NN which is independent of the architecture employed and the algorithms for identifying adversarial examples. In fact, theorem 2.2 offers a simple and intuitive way to formalize robustness of NN: for simplicity consider a $1$-layer neural network trained on a given distribution $\mu$ of pairs $(x,y)$, i.e. $(A_1^{\star },A_2^{\star },b_1^{\star },b_2^{\star })$ solve

where the $inf$ is taken over $a=(A_1,A_2,b_1,b_2)\in \mathcal{A}={\mathbb{R}}^{k\times d}\times {\mathbb{R}}^{d\times k}\times {\mathbb{R}}^k\times {\mathbb{R}}^d$, for a given activation function $\sigma :\mathbb{R}\to \mathbb{R}$, where the composition above is understood componentwise. Set $f(x,y;A,b):=|y-(A_2(\cdot )+b_2)\circ \sigma \circ (A_1(\cdot )+b_1)(x){|}^p$. Data perturbations are captured by $\nu \in B_{\delta }^p(\mu )$ and (1.2) offers a robust training procedure. The first-order quantification of the NN sensitivity to adversarial data is then given by A similar viewpoint, capturing robustness to adversarial examples through the optimal transport lens, has been recently adopted by other authors. The dual formulation of (1.2) was used by [21] to reduce the training of neural networks to tractable linear programs. [41] modified (1.2) to consider a penalized problem $\underset{a\in \mathcal{A}}{inf}\underset{\nu \in \mathcal{P}(\mathcal{S})}{sup}{\int }_{\mathcal{S}}f(x,a)\hspace{0.17em}\nu (\text{d}x)-\gamma W_p(\mu ,\nu )$ to propose new stochastic gradient descent algorithms with inbuilt robustness to adversarial data.

(c) Uncertainty quantification

In the context of UQ, the measure $\mu$ represents input parameters of a (possibly complicated) operation $G$ in a physical, engineering or economic system. We consider the so-called reliability or certification problem: for a given set $E$ of undesirable outcomes, one wants to control $\underset{\nu \in \mathcal{P}}{sup}\nu (G(x)\in E)$, for a set of probability measures $\mathcal{P}$. The distributionally robust adversarial classification problem considered recently by [42] is also of this form, with Wasserstein balls $\mathcal{P}$ around an empirical measure of $N$ samples. Using the dual formulation of [18], they linked the problem to minimization of the conditional value-at-risk and proposed a reformulation, and numerical methods, in the case of linear classification. We propose instead a regularized version of the problem and look for

for a given safety level $\alpha$. We thus consider the average distance to the undesirable set, $d(G(x),E):=\underset{e\in E}{inf}|G(x)-e|$, and not just its probability. The quantity $\delta (\alpha )$ could then be used to quantify the implicit uncertainty of the certification problem, where higher $\delta$ corresponds to less uncertainty. Taking statistical confidence bounds of the empirical measure in Wasserstein distance into account, see [14], $\delta$ would then determine the minimum number of samples needed to estimate the empirical measure.

Assume that $E$ is convex. Then $x\mapsto d(x,E)$ differentiable everywhere except at the boundary of $E$ with ${\mathrm{\nabla }}_{x}d(x,E)=0$ for $x\in E^o$ and $|{\mathrm{\nabla }}_{x}d(x,E)|=1$ for all $x\in {\overline{E}}^c$. Furthermore, assume $\mu$ is absolutely continuous w.r.t. Lebesgue measure on $\mathcal{S}$. Theorem 2.2, using [9, remark 7.3], gives a first-order expansion for the above problem:

In the special case ${\mathrm{\nabla }}_{x}G(x)=cI$ this simplifies to and the minimal measure $\nu$ pushes every point $G(x)$ not contained in $E$ in the direction of the orthogonal projection. This recovers the intuition of [43, theorem 1], which in turn relies on [17, corollary 2, example 7]. Note however that our result holds for general measures $\mu$. We also note that such an approximation could provide an ansatz for dimension reduction, by identifying the dimensions for which the partial derivatives are negligible and then projecting $G$ on to the corresponding lower-dimensional subspace (thus providing a simpler surrogate for $G$). This would be an alternative to a basis expansion (e.g. in orthogonal polynomials) used in UQ and would exploit the interplay between the properties of $G$ and $\mu$ simultaneously.

(d) Statistics

We discuss two applications of our results in the realm of statistics. We start by highlighting the link between our results and the so-called influence curves (IC) in robust statistics. For a functional $\mu \mapsto T(\mu )$ its IC is defined as

Computing the $\mathrm{IC}$, if it exists, is in general hard and closed form solutions may be unachievable. However, for the so-called M-estimators, defined as optimizers for $V(0)$, for some $f$ (e.g. $f(x,a)=|x-a|$ for the median), we have under suitable assumptions on $f$, see [44, section 3.2.1]. In comparison, writing $T^{\delta }$ for the optimizer for $V(\delta )$, theorem 2.4 yields under assumption 2.3 and normalization $||{\mathrm{\nabla }}_{x}f(x,T(\mu ))|{|}_{L^p(\mu )}=1$. To investigate the connection let us Taylor-expand $\mathrm{IC}(y)$ around $x$ to obtain Choosing $y=x+\delta \mathrm{\nabla }{f}_x(x,T(\mu ))$ and integrating both sides over $\mu$ and dividing by $\delta$, we obtain the asymptotic equality for $\delta \to 0$ by (3.2). We conclude that considering the average directional derivative of IC in the direction of $\mathrm{\nabla }{f}_x(x,T(\mu ))$ gives our first-order sensitivity. For an interesting conjecture regarding the comparison of influence functions and sensitivities in KL-divergence, we refer to [45, Section 7.3] and [22, Section 3.4.2].

Our second application in statistics exploits the representation of the LASSO/Ridge regressions as robust versions of the standard linear regression. We consider $\mathcal{A}={\mathbb{R}}^k$ and $\mathcal{S}={\mathbb{R}}^{k+1}$. If instead of the Euclidean metric we take $||(x,y)|{|}_{\ast }=|x{|}_r{\mathbf{1}}_{\{y=0\}}+\mathrm{\infty }{\mathbf{1}}_{\{y\ne 0\}}$, for some $r>1$ and $(x,y)\in {\mathbb{R}}^k\times \mathbb{R}$, in the definition of the Wasserstein distance, then [19] showed that

holds, where $1/r+1/s=1$. The $\delta =0$ case is the ordinary least-squares regression. For $\delta >0$, the r.h.s. for $s=2$ is directly related to the Ridge regression, while the limiting case $s=1$ is called the square-root LASSO regression, a regularized variant of linear regression well known for its good empirical performance. Closed-form solutions to (3.3) do not exist in general and it is a common practice to use numerical routines to solve it approximately. Theorem 2.4 offers instead an explicit first-order approximation of $a_{\delta }^{\star }$ for small $\delta$. We denote by $a^{\star }$ the ordinary least-squares estimator and by $I$ the $k\times k$ identity matrix. Note that the first-order condition on $a^{\star }$ implies that $\int (y-⟨a^{\star },x⟩)x_i\mu (\text{d}x,\text{d}y)=0$ for all $1\le i\le k$. In particular, $V(0)=\int (y^2-⟨a^{\star },x⟩y)\mu (\text{d}x,\text{d}y)$ and $a^{\star }=D^{-1}\int yx\mu (\text{d}x,\text{d}y)$, where we assume the system is overdetermined so that $D=\int x{x}^T\hspace{0.17em}\mu (\text{d}x,\text{d}y)$ is invertible. A direct computation, see [9, example 8.2], yields For $s=2$, $h(a^{\star })=a^{\star }/|a^{\star }{|}_2$ and for $s=1$, $h(a^{\star })=\text{sign}(a^{\star })$ and hence¹ $a_{\delta }^{\star }$ is approximately respectively. This corresponds to parameter shrinkage: proportional for square-root Ridge and a shift towards zero for square-root LASSO. To the best of our knowledge, these are first such results and we stress that our formulae are valid in a general context and, in particular, parameter shrinkage depends on the direction through the $D^{-1}$ factor. Figure 3 compares the first-order approximation with the actual results and shows a remarkable fit. Furthermore, our results agree with what is known in the canonical test case for the (standard) Ridge and LASSO, see [46]. When $\mu ={\mu }_N$ is the empirical measure of $N$ i.i.d. observations, the data are centred and the covariates are orthogonal, i.e. $D=(1/N)I$. In that case, (3.5) simplifies to where $R^2$ is the usual coefficient of determination.

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The case of ${\mu }_N$ is naturally of particular importance in statistics and data science and we continue to consider it in the next subsection. In particular, we characterize the asymptotic distribution of $\sqrt{N}(a_{1/\sqrt{N}}^{\star }-a^{\star })$, where $a_{\delta }^{\star }\in {\mathcal{A}}_{\delta }^{\star }({\mu }_N)$ and $a^{\star }\in {\mathcal{A}}_0^{\star }({\mu }_{\mathrm{\infty }})$ is the optimizer of the non-robust problem for the data-generating measure. This recovers the central limit theorem of [47], a link we explain further in §4b.

(e) Out-of-sample error

A benchmark of paramount importance in optimization is the so-called out-of-sample error, also known as the prediction error in statistical learning. Consider the setup above when ${\mu }_N$ is the empirical measure of $N$ i.i.d. observations sampled from the ‘true’ distribution $\mu ={\mu }_{\mathrm{\infty }}$ and take, for simplicity, $||\cdot ||=|\cdot {|}_s$, with $s>1$. Our aim is to compute the optimal $a^{\star }$ which solves the original problem (1.1). However, we only have access to the training set, encoded via ${\mu }_N$. Suppose we solve the distributionally robust optimization problem (1.2) for ${\mu }_N$ and denote the robust optimizer $a_{\delta }^{\star ,N}$. Then the out-of-sample error

quantifies the error from using $a_{\delta }^{\star ,N}$ as opposed to the true optimizer $a^{\star }$.

While this expression seems to be hard to compute explicitly for finite samples, theorem 2.4 offers a way to find the asymptotic distribution of a (suitably rescaled version of) the out-of-sample error. We suppose the assumptions in theorem 2.4 are satisfied and note that the first order condition for $a^{\star }$ gives ${\mathrm{\nabla }}_{a}V(0,a^{\star })=0$. Then, a second-order Taylor expansion gives

for some $\tilde{a}$ (coordinate-wise) between $a^{\star }$ and $a_{\delta }^{\star ,N}$. Now we write where we define $a^{\star ,N}$ as the optimizer of the non-robust problem (1.1) with $\mu$ replaced by ${\mu }_N$. In particular, the $\delta$-method for M-estimators implies that where $H\sim \mathcal{N}(0,\int {({\mathrm{\nabla }}_{a}f(x,a^{\star }))}^T{\mathrm{\nabla }}_{a}f(x,a^{\star })\hspace{0.17em}\mu (\text{d}x))$ and $\Rightarrow$ denotes the convergence in distribution. On the other hand, for a fixed $N\in \mathbb{N}$, theorem 2.4 applied to ${\mu }_N$ yields where Almost surely (w.r.t. sampling of ${\mu }_N$), we know that ${\mu }_N\to \mu$ in $W_p$ as $N\to \mathrm{\infty }$, and under the regularity and growth assumptions on $f$ in [9, equation (8.2)] we check that ${\mathrm{\Delta }}_N\to 0$ a.s., see [9, example 8.4] for details. In particular, taking $\delta =1/\sqrt{N}$ and combining the above with (3.7) we obtain This recovers the central limit theorem of [47], as discussed in more detail in §4b below. Together, (3.6) and (3.9) give us the a.s. asymptotic behaviour of the out-of-sample error These results also extend and complement [48, Prop. 17]. [48] investigate when the distributionally robust optimizers $a_{\delta }^{\star ,N}$ yield, on average, better performance than the simple in-sample optimizer $a^{\star ,N}$. To this end, they consider the expectation, over the realizations of the empirical measure ${\mu }_N$ of This is closely related to the out-of-sample error and our derivations above can be easily modified. The first-order term in the Taylor expansion no longer vanishes and, instead of (3.6), we now have which holds, e.g. if for any $r>0$, there exists $c>0$ such that $\sum _{i=1}^k|{\mathrm{\nabla }}_a{\mathrm{\nabla }}_{a_i}f(x,a)|\le c(1+|x{|}^p)$ for all $x\in \mathcal{S}$, $|a|\le r$. Combined with (3.8), this gives asymptotics in small $\delta$ for a fixed $N$. For quadratic $f$ and taking $q\uparrow \mathrm{\infty }$, we recover the result in [48, Prop. 17], see [9, example 8.4] for details.

(a) Discussion of related literature

Let us first remark, that while theorem 2.2 offers some superficial similarities to a classical maximum theorem, which is usually concerned with continuity properties of $\delta \mapsto V(\delta )$, in this work, we are instead interested in the exact first derivative of the function $\delta \mapsto V(\delta )$. Indeed, the convergence $\underset{\delta \to 0}{lim}\underset{\nu \in B_{\delta }(\mu )}{sup}\int f(x)\hspace{0.17em}\nu (\text{d}x)=\int f(x)\hspace{0.17em}\mu (\text{d}x)$ follows for all $f$ satisfying $f(x)\le c(1+|x{|}^p)$ directly from the definition of convergence in Wasserstein metric (e.g. [49, Def. 6.8]). In conclusion, the main issue is to quantify the rate of this convergence by calculating the first derivative $V^{\prime }(\delta )$.

Our work investigates model uncertainty broadly conceived: it includes errors related to the choice of models from a particular (parametric or not) class of models as well as the mis-specification of such a class altogether (or indeed, its absence). In the decision theoretic literature, these aspects are sometimes referred to as model ambiguity and model mis-specification, respectively, see [50]. However, seeing our main problem (1.2) in decision theoretic terms is not necessarily helpful as we think of $f$ as given and not coming from some latent expected utility type of problem. In particular, our actions $a\in \mathcal{A}$ are just constants.

In our work, we decided to capture the uncertainty in the specification of $\mu$ using neighbourhoods in the Wasserstein distance. As already mentioned, other choices are possible and have been used in past. Possibly, the most often used alternative is the relative entropy, or the Kullblack–Leibler divergence. In particular, it has been used in this context in economics, see [51]. To the best of our knowledge, the only comparable study of sensitivities with respect to relative entropy balls is [22], see also [45] allowing for additional marginal constraints. However, this only considered the specific case $f(x,a)=f(x)$ where the reward function is independent of the action. Its main result is

where $B_{\delta }^{KL}(\mu )$ is a ball of radius ${\delta }^2$ centred around $\mu$ in KL-divergence, ${\mathrm{Var}}_{\mu }(f(X))$ and ${\kappa }_3(f(X))$ denote the variance and kurtosis of $f$ under the measure $\mu$ respectively. In particular, the first-order sensitivity involves the function $f$ itself. By contrast, our theorem 2.2 states $V^{\prime }(\delta )={(\int {(\hspace{0.17em}{f}^{\mathrm{\prime }}(x))}^2\hspace{0.17em}\mu (\text{d}x))}^{1/2}$ and involves the first derivative $f^{\prime }$. In the trivial case of a point mass $\mu ={\delta }_x$, we recover the intuitive sensitivity $V^{\prime }(\delta )=|f^{\mathrm{\prime }}(x)|$, while the results of [22] do not apply for this case. We also note that [22] requires exponential moments of the function $f$ under the baseline measure $\mu$, while we only require polynomial moments. In particular, in applications in econometrics (or any field in which $\mu$ typically has fat tails), the scope of application of the corresponding results might then be decisively different. We remark however, that this requirement can be substantially weakened (to the existence of polynomial moments) when replacing KL-divergences by $\alpha$-divergences, e.g. [52,53]. We expect a sensitivity analysis similar to [22] to hold in this setting. However, to the best of our knowledge no explicit results seem to be available in the literature.

To understand the relative technical difficulties and merits, it is insightful to go into the details of the statements. In fact, in the case of relative entropy and the one-period set-up we are considering, the exact form of the optimizing density can be determined exactly (see [22, Proposition 3.1]) up to a one-dimensional Langrange parameter. This is well known and is the reason behind the usual elegant formulae obtained in this context. But this then reduces the problem in [22] to a one-dimensional problem, which can be well-approximated via a Taylor approximation. By contrast, when we consider balls in the Wasserstein distance, the form of the optimizing measure is not known (apart from some degenerate cases). In fact, a key insight of our results is that the optimizing measure can be approximated by a deterministic shift in the direction ${(x+f^{\mathrm{\prime }}(x)\delta )}_{\ast }\mu$ (this is, in general, not exact but only true as a first-order approximation). The reason for these contrastive starting points of the analyses is the fact that Wasserstein balls contain a more heterogeneous set of measures, while in the case of relative entropy, exponentiating $f$ will always do the trick. We remark however that this is not true for the finite-horizon problems considered in [22, Section 3.2] any more, where the worst-case measure is found using an elaborate fixed-point equation.

A point which further emphasizes the fact that the topology introduced by the Wasserstein metric is less tractable is the fact that

where $H(\pi \mid R^{\epsilon })=\int \log (\frac{d\pi }{d{R}^{\epsilon }})\hspace{0.17em}d\pi$ is the relative entropy and for some normalizing constant $c_0>0$ (e.g. [54]). This is known as the entropic optimal transport formulation and has received considerable interest in the ML community in the past years (e.g. [55]). In particular, the Wasserstein distance can be approximated by relative entropy, but only with respect to reference measures on the product space. As we consider optimization over $\nu$ above it amounts to changing the reference measure. In consequence, the topological structure imposed by Wasserstein distances is more intricate compared to relative entropy, but also more flexible.

The other well-studied distance is the Hellinger distance. [24] calculates influence curves for the minimum Hellinger distance estimator $a^{\mathrm{Hell},\star }$ on a countable sample space. Their main result is that for the choice $f(x,a)=\log (\ell (x,a))$ (where ${(\ell (x,a))}_{a\in \mathcal{A}}$ is a collection of parametric densities)

the product of the inverse Fisher information matrix and the score function, which is the same as for the classical maximum-likelihood estimator. Denote by ${\mu }_N$ the empirical measure of $N$ data samples and by $a^{\mathrm{Hell},\star }(N)$ the corresponding minimum Hellinger distance estimator for ${\mu }_N$. In particular, the above result then implies the same CLT as for M-estimators given by where $H\sim \mathcal{N}(0,\int {\mathrm{\nabla }}_{a}f{(x,a^{\mathrm{Hell},\star })}^T{\mathrm{\nabla }}_{a}f(x,a^{\mathrm{Hell},\star })\hspace{0.17em}\mu (\text{d}x)).$ As we discuss in the next section, our theorem 2.4 yields a similar CLT, namely Thus the Wasserstein worst-case approach leads to a shift of the mean of the normal distribution in the direction compared to the non-robust case. In the simple case $\mu =\mathcal{N}(0,{\sigma }^2)$ with standard deviation $\sigma >0,$ we obtain the MLE ${\sigma }^{\star ,N}=\frac{1}{N}\sum _{k=1}^N{X}_i^2$. We can directly compute (for $a=\sigma$) that Thus the robust approach accounts for a shift of $1/{({\sigma }^{\star })}^2$ (of order 1 if mulitplied with inverse Fisher information) to account for a possibly higher variance in the underlying data. In particular, in our approach, the so-called neutral spaces considered (e.g. [56], eqn (21)]) as should also take this shift into account, i.e. their definition should be adjusted to Lastly, let us mention another situation when our approach provides directly interpretable insights in the context of a parametric family of models. Namely, if one considers a family of models $\mathcal{P}$ such that the worst-case model in the Wasserstein ball remains in $\mathcal{P}$, i.e. ${(x+f^{\mathrm{\prime }}(x)\delta )}_{\ast }\mu \in \mathcal{P}$, then considering (the first-order approximation to) model uncertainty over Wasserstein balls actually reduces to considerations within the parametric family. While uncommon, such a situation would arise, for example, for a scale-location family $\mathcal{P}$, with $\mu \in \mathcal{P}$ and a linear/quadratic $f$.