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Robust Risk Under Evolving Uncertainty: A Wasserstein Counterpart of the Entropic Value-at-Risk

Deep Kumar Ganguly, Jan Křetínský

arXiv:2608.19073v1cs.AIcs.LGstat.ML

TL;DR

Relative-entropy risk cannot represent catastrophes outside the nominal support, limiting safety under evolving uncertainty. This paper replaces that geometry with Wasserstein ambiguity and shows that belief entropy yields a contracting risk attitude while preserving sensitivity to reachable catastrophes.

  • Problem

    Relative-entropy ambiguity sets cannot represent catastrophes assigned zero nominal probability, creating a safety gap as learned models become overconfident.

  • Method

    The paper defines WEVaR, proves its transport-based dual and risk-hierarchy placement, and drives its Wasserstein radius by belief entropy in robust dynamic programming.

  • Results

    At fixed confidence, the entropic value-at-risk changes by exactly 0, while WEVaR at a fixed radius changes by 807.5 as catastrophe loss grows from 50 to 1000.

  • Takeaways & Limitations

    Belief entropy provides a state-dependent caution dial that contracts toward risk-neutral behavior as observations sharpen while retaining sensitivity to reachable catastrophes.

  • Takeaways & Limitations

    Extending the closed-form operator to continuous spaces and active information gathering remains future work.

Abstract

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An agent still learning its environment should be cautious while ignorant and bold once confident. The entropic value-at-risk captures this through a robust-optimization identity---a confidence level fixes the radius of a relative-entropy ball of alternative models---but that ball cannot reach catastrophes the nominal deems impossible, precisely what a safe agent must hedge. We instead use an optimal-transport ball and study the coherent risk measure it induces, the Wasserstein entropic value-at-risk. It has a variational dual mirroring the entropic formula (an inverse temperature becomes a transport price), occupies a definite place in the risk hierarchy, and provably accounts for the reachable catastrophes the entropic measure ignores; we verify both dualities numerically. Driving the transport radius by belief entropy then yields a closed-form robust dynamic-programming operator whose caution contracts as the belief sharpens, with a certified safety sandwich and a sharp safety switch.

1 RISK UNDER EVOLVING UNCERTAINTY

The section motivates risk that adapts to evidence: worst-case control remains overly conservative, while expected-value planning can catastrophically misjudge hidden regimes. It introduces WEVaR as an optimal-transport replacement for entropic risk, preserving robust guarantees while representing reachable catastrophes excluded by relative-entropy balls.

  • Motivation: A drone facing an unobserved wind regime needs caution that is maximal under ignorance and vanishes as evidence sharpens.Worst-case control is safe but excessively conservative, whereas expected-value or Bayesian planning can commit to a fatal wrong guess.
  • Entropic baseline: Entropic value-at-risk sweeps from the mean to the essential supremum and equals optimization over a relative-entropy ball of radius −ln α.Its induced optimization is convex and admits well-posed, convergent estimation.
  • Entropic limitation: Relative-entropy ambiguity cannot assign positive probability to catastrophes that the nominal law assigns zero probability.As the learned nominal kernel concentrates away from rare disasters, the entropic adversary loses the ability to represent them.
  • Contributions: The paper replaces the entropic ambiguity ball with an optimal-transport ball while retaining the entropic value-at-risk’s robust formulation and guarantees.The transport geometry makes reachable but nominally impossible outcomes representable, with cost proportional to their distance.
  • Contributions: WEVaR has a Kantorovich–Rubinstein variational dual mirroring the entropic formula, is convex and well-posed, and admits a closed mean-plus-Lipschitz form.The paper also proves coherence and places WEVaR in the risk hierarchy, including a transport–entropy sandwich and strict accounting for zero-nominal-probability catastrophes.

2 THE ENTROPIC VALUE-AT-RISK AND ITS ROBUST FORMULATION

The entropic value-at-risk is a canonical coherent risk measure with a dual/primal robust formulation over relative-entropy alternatives. Its exponential-tilt optimizer is well defined but remains supported on nominal support, making EVaR blind to losses deemed impossible by the reference law.

  • Risk-measure foundations: EVaR is presented alongside conditional value-at-risk as a canonical instance of a coherent risk measure.Coherence comprises monotonicity, translation invariance, positive homogeneity, and subadditivity.
  • Robust formulation: The inverse temperature t is the multiplier on the relative-entropy constraint, and the worst-case law is an exponential tilt Q⋆.
  • Robust formulation: The reparametrized EVaR objective is convex, yielding a well-posed one-dimensional convex program with a unique solution.
  • Support limitation: EVaRα(X) is independent of X on every zero-nominal-probability state because its worst-case tilt is supported on supp(P).This support restriction is identified as EVaR’s blind spot for losses the nominal law deems impossible.

3 SWAPPING THE BALL: A WASSERSTEIN ROBUST RISK

This section replaces the relative-entropy ambiguity ball with a 1-Wasserstein ball, yielding a tractable variational risk measure with a transport-price dual. WEVaR is coherent, spans the risk hierarchy, and captures reachable catastrophes that entropic risk ignores.

  • Variational dual: The Wasserstein robust risk admits a one-dimensional convex variational dual in which transport price λ replaces entropic inverse temperature.The optimal price satisfies λ⋆≤Lip_d(X), and ε 7→WEVaR_ε(X) is concave and nondecreasing.
  • Variational dual: For ε below a saturation threshold, the dual objective reaches its transport-price bound at λ = Lip_d(X), after which WEVaR rises concavely toward max_s X(s).The closed form is exact for two-point supports.
  • Coherence: WEVaR_ε is a coherent risk measure for every ε ≥0.It is the support function of the convex compact Wasserstein ball, which yields positive homogeneity, subadditivity, monotonicity, and translation invariance.
  • Hierarchy and catastrophe: WEVaR_0(X) = E_P[X], and WEVaR_ε(X) ↑ ess sup(X) as ε ↑D, so Wasserstein risk sweeps the full hierarchy.Here D is the diameter of the finite metric space.
  • Hierarchy and catastrophe: If P(s⋆) = 0 and X(s⋆) > E_P[X], EVaR_α(X) is independent of X(s⋆), while WEVaR_ε(X) strictly increases once ε > dist_d(s⋆, supp P).The Wasserstein ball therefore accounts for reachable but zero-probability catastrophes excluded by the relative-entropy ball.

4 EMPIRICAL GUARANTEES

Numerical experiments verify the robust programs and their dual formulations to solver tolerance, while confirming the closed form and the Wasserstein measure’s ability to reach catastrophes excluded by relative entropy. Across radii, both measures move from the mean toward the worst case, with a quantified Wasserstein gap at fixed radius.

  • Numerical verification: Gurobi 12 verifies both robust programs and their duals on a five-state metric space, including the entropic program–primal and WEVaR transport–dual pairings.The space is X = [0, 2, 4, 8, 100] with d(s, s′) = |s − s′|.
  • Numerical verification: 1.2 × 10−4 and 9 × 10−7 are the respective agreement tolerances for the entropic primal–program and transport linear program–Kantorovich–Rubinstein dual.The closed form (5) is exact for ε ≤ 0.1 and otherwise upper-bounded by the exact one-dimensional dual.
  • Risk comparison: Both measures rise from the mean toward the worst case across radius sweeps, with EVaRα ≤ WEVaR at the Pinsker-matched radius.At fixed confidence, WEVaR at a fixed radius changes by 807.5.
  • Geometric catastrophe reachability: The relative-entropy ball remains trapped on the nominal support edge when FAIL has zero mass, whereas the 1-Wasserstein ball reaches the FAIL vertex.The worst-case adversary lies on the Wasserstein ball’s boundary.

5 EVOLVING UNCERTAINTY: BELIEF ENTROPY AS THE RADIUS

The transport radius is set to belief entropy, so ambiguity contracts as observations sharpen and the robust controller moves toward risk neutrality. The resulting closed-form dynamic-programming operator is contractive, has a unique fixed point, and supports safety guarantees throughout learning.

  • Belief-entropy radius: Setting ε(b) = βH(b) makes the ambiguity ball contract toward the nominal kernel as belief entropy decreases, shifting risk attitude from worst-case toward risk-neutral.The controller’s closed-form inner functional is E_P̄b[V] − βH(b) Lip_d(V).
  • Closed-form robust update: The closed-form robust update costs O(|S|^2), is a γ-contraction for γ < 1 and on proper MDPs for γ = 1, and therefore has a unique fixed point.The update avoids a coupling linear program.
  • Safety guarantees: The fixed point satisfies a safety sandwich between the always-maximally-cautious value and the type-aware oracle, converging to the oracle as H(b) → 0.This establishes adaptive safety while uncertainty resolves.
  • Evolving safety switch: 1100: On Ambiguous Bridge, Lip_d(V) = 1100, yielding closed-form SPRINT and CRAWL action values that a Gurobi transport solve reproduces to 10^-13.The example contrasts a risky action that can achieve +100 or −1000 with an always-safe but costly action.
  • Guarantees, illustrated: 8.4 to 4.4: The corridor’s ceiling gap shrinks from the uniform belief to b(calm) = 0.95 and reaches zero at identification; value iteration converges at empirical rate 0.90 = γ.The closed-form update matches Gurobi at every sampled belief, preserving a valid safety bound at every iterate.

6 DISCUSSION

The discussion unifies entropic and Wasserstein robust risks as relative-entropy and optimal-transport realizations of a coherent risk whose ambiguity radius represents epistemic uncertainty. It identifies entropic optimal transport as the bridge and highlights extensions to continuous spaces and active information gathering.

  • Unifying perspective: Entropic and Wasserstein robust risks are presented as relative-entropy and optimal-transport faces of one coherent-risk framework.The framework interprets the ambiguity radius as the agent’s epistemic uncertainty.
  • Unifying perspective: Entropic optimal transport bridges the relative-entropy and optimal-transport formulations.The discussion connects this bridge to Cuturi (2013) and Peyré and Cuturi (2019).
  • Future directions: Natural next steps are scaling the closed-form operator to continuous spaces via Lipschitz critics and extending it to active information gathering.These directions are identified as future extensions of the presented framework.

APPENDIX: EXTENDED PROOFS

The appendix fixes the finite-state setting, bounded-loss assumptions, and notation for the reference law and Wasserstein-1 distance. It defines W1 through couplings between alternative and reference laws.

  • Standing notation: The setup assumes a finite state space S with |S| = n, metric diameter D, reference law P, and bounded loss X : S → R.These objects provide the standing notation for the extended proofs.
  • Wasserstein distance: W1(Q, P) is defined as the minimum transport cost over couplings Γ(Q, P), using the metric d on S.The transport cost sums d(s, s′)γ(s, s′) over state pairs.

A. THEOREM 1 · B. PROPOSITION 1 (COHERENCE)

Theorem 1 derives a one-dimensional convex dual for the Wasserstein entropic value-at-risk and characterizes its dependence on the transport radius. Proposition 1 shows that this risk measure is coherent through its Wasserstein risk envelope.

  • A. THEOREM 1: Theorem 1 reduces the worst-case expectation over a Wasserstein ball to a single linear program over couplings with fixed first marginal P.The program is feasible at zero transport cost and bounded because X is bounded, so linear-program strong duality applies.
  • A. THEOREM 1: The dual objective is a well-posed one-dimensional convex program because the pointwise continuation value is convex in λ and λε + E_P[X^c_λ] is convex on [0, ∞).For general Polish spaces, the identity is identified with Kantorovich–Rubinstein/Wasserstein-DRO strong duality.
  • A. THEOREM 1: The minimizing transport price satisfies λ⋆ ≤ Lip_d(X), since beyond the Lipschitz constant the dual objective is strictly increasing.For λ ≥ Lip_d(X), the continuation value equals X pointwise and the objective becomes λε + E_P[X].
  • A. THEOREM 1: WEVaR_ε(X) is nondecreasing and concave in ε because its Wasserstein feasible set expands with ε and its dual is an infimum of affine functions.The derivative analysis identifies a threshold through ε and the active transport distance, while the two-state formula is exact up to exhaustion at a vertex.
  • B. PROPOSITION 1 (COHERENCE): Proposition 1 represents ρ(X) = WEVaR_ε(X) as the supremum of expected payoffs over the nonempty, convex, compact Wasserstein risk envelope B.The nominal distribution P belongs to B, ensuring the envelope is nonempty.
  • B. PROPOSITION 1 (COHERENCE): The induced risk measure satisfies monotonicity, translation invariance, positive homogeneity, and subadditivity, hence is coherent.These properties follow directly by taking suprema of expectations over the same envelope B and match the axioms of Artzner et al. (1999).

C. THEOREM 2

Theorem 2 places WEVaR between nominal expectation and worst-case risk, while showing that transport ambiguity can account for catastrophes outside the nominal support. In particular, WEVaR reaches worst-case risk at the transport diameter and responds strictly to sufficiently reachable catastrophic outcomes.

  • (i) Sweep from mean to worst case: WEVaR_0(X) equals the nominal expectation, and WEVaR_ε(X) increases to max_s X(s) as ε increases to the metric diameter D.The worst-case value is attained using a point mass at an outcome maximizing X.
  • (iii) Catastrophe domination: If P(s⋆) = 0 and X(s⋆) exceeds the nominal expectation, EVaR_α(X) is independent of X(s⋆) because finite-KL alternatives remain absolutely continuous with respect to P.The entropic measure therefore assigns zero mass to the nominally impossible catastrophe.
  • (iii) Catastrophe domination: For ε > δ, where δ is the distance from s⋆ to supp P, WEVaR_ε(X) is strictly increasing in the catastrophic payoff X(s⋆).A feasible transport law moves positive mass from a nearest supported state to s⋆, and sufficiently large X(s⋆) makes every optimizer assign positive mass there.
  • (iii) Catastrophe domination: The Wasserstein ball can include distributions outside the nominal support, unlike the finite-relative-entropy ball used by EVaR.Transporting η mass from a nearest supported state to s⋆ costs at most ηδ, making the catastrophe feasible when ε > δ.

D. THEOREM 3

Theorem 3 establishes a scalar dual for the Wasserstein robust dynamic-programming operator, alongside contraction and a unique fixed point. It also bounds the robust value between worst-case-entropy and Bayesian benchmarks, with convergence to the known-type model as beliefs sharpen.

  • Closed form of the inner minimization: The inner minimization reduces to a one-dimensional concave maximization with unsaturated value Ē_Pb[W] − βH(b)Lip_d(W), replacing an O(n^2)-variable transportation LP.Evaluating Lip_d(W) costs O(n^2), while the dual is a scalar program.
  • Contraction: For γ < 1, monotonicity and the scaled constant shift make T a contraction; for γ = 1, the shift is exact and T is nonexpansive.Under properness, T is also contractive in a weighted supremum norm, with ∥T^mV − T^mV′∥_w ≤ κ∥V − V′∥_w for some m ≥ 1 and κ ∈ [0, 1).
  • Contraction: Either contraction route yields a unique fixed point V⋆ for the robust operator T.The weighted-norm result applies under properness, meaning every policy and kernel selection reaches the terminal set in uniformly bounded expected time.
  • Safety sandwich: The robust fixed point satisfies V⋆(s,b) ≤ VBayes(s,b) ≤ Ez∼b[V⋆opt(s,z)], so robustness is bounded above by the Bayesian value and the value of knowing the type.The first inequality follows because the nominal kernel belongs to the ambiguity set; the second is the nonnegative value-of-information bound.
  • Safety sandwich: Replacing βH(b) by β ln |Z| defines V⋆wc and gives V⋆(s,b) ≥ V⋆wc(s,b), while b → δz⋆ implies V⋆(s,b) → V⋆(s,δz⋆).The lower bound requires no nested-set condition, and convergence follows from H(b) → 0, ε(b) → 0, and the Lipschitz dependence of value on radius.

E. SUPPLEMENTARY EXPERIMENTS AND FIGURES

Supplementary experiments verify the entropic and Wasserstein dualities, including WEVaR’s response to catastrophes assigned zero nominal probability. Belief-dependent ambiguity produces interpretable value and policy manifolds, safety switches, and a planner comparison with a stated caveat.

  • Quantitative verification of the dualities: WEVaR rises toward the worst case and exceeds EVaRα at the Pinsker-matched radius, while responding to zero-probability disasters that leave EVaRα unchanged.Figure 4 provides a Gurobi-verified comparison across radii on a five-state metric space.
  • Manifolds over the belief simplex: The three-type belief simplex exposes the ambiguity radius ε(b) = βH(b), robust value V ⋆(s0, b), and policy regions, with cruising only near the calm vertex.The entropy radius is maximal at the centroid and zero at the vertices.
  • Safety switches: α⋆≈0.983 on the Ambiguous Bridge and b(calm) ≈0.63 in the stormy-drone canyon mark computable switches from safe to optimal actions.The Ambiguous Bridge threshold is set by reward asymmetry and is independent of β.
  • Safety-vs-efficiency rollouts, and an honest caveat: Across 4000 episodes, WEVaR dominates static-robust MAXIMIN with comparable safety and higher return, while Bayesian and WEVaR separation is small in this calibrated-belief canyon.The paper attributes the small separation to the Bayesian planner already being near-optimally cautious under a severe catastrophe.
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