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

Deep Kumar Ganguly; Jan Křetínský

cs.AIAug 19, 2026

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

Authors: Deep Kumar Ganguly, Jan Křetínský

Organizations: Technical University of Munich, Germany · Masaryk University, Brno, Czech Republic

Abstract

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.

Figures & tables

Figure 1: Why Wasserstein, geometrically. On the next-state simplex at a confident belief—the nominal

\bar{P}_{b}

places zero mass on Fail —the relative-entropy ball (red) is trapped on the support edge, blind to the catastrophe, while the

1

-Wasserstein ball (blue) reaches the Fail vertex; the worst-case adversary (

\times

) sits on its boundary. This is Thm. 2 (iii) drawn on

\Delta(\mathcal{S})

Figure 2: The entropy-modulated ambiguity set

\mathcal{U}(b)

on the next-state simplex. As the belief sharpens the radius

\varepsilon=\beta\mathcal{H}(b)

contracts (

1.04\!\to\!0.20

) and the nominal

\bar{P}_{b}

drifts toward the goal; the worst-case adversary (

\times

) loses its reach toward the failure vertex—robustness that vanishes with uncertainty (the geometric content of Thm. 3 ).

Figure 3: Computed guarantees on a five-state corridor with a hidden slip type (

\gamma=0.9

, Gurobi-checked). (a) The adaptive value

V^{\star}(s_{0},b)

stays inside the safety sandwich between the always-maximally-cautious floor

V_{\mathrm{wc}}

and the type-aware oracle ceiling; the price of robustness shrinks as the belief sharpens. (b) From arbitrary initializations, value iteration with the closed-form operator collapses geometrically onto

V^{\star}

along the

\gamma^{k}

envelope, certifying the bound at every iterate.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Gurobi-verified comparison on a five-state metric space. (a) hierarchy and the Pinsker sandwich; (b) the zero-probability catastrophe: the entropic value-at-risk is invariant to the disaster’s magnitude,

\mathrm{WEVaR}

is not.

Figure 5: Manifolds over the belief simplex

\Delta(\mathcal{Z})

|\mathcal{Z}|=3

(calm/breezy/storm): ambiguity radius, robust value, and policy region (cruise only near the calm vertex).

Figure 6: Robust action values vs. belief. Left: the Ambiguous Bridge ( Sprint vs. Crawl , switch at

\alpha^{\star}\approx 0.983

). Right: the stormy-drone canyon ( Cruise vs. Creep , switch at

b(\mathrm{calm})\approx 0.63

Figure 7: Catastrophe rate vs. time-to-goal on the stormy-drone canyon (

4000

episodes).

\mathrm{WEVaR}

dominates the static-robust Maximin baseline; the Bayesian planner is already cautious here (see caveat).

Risk-Sensitive RL

Risk-sensitive reinforcement learning optimizes policies for downside risk, tail outcomes, uncertainty, or safety constraints beyond expected reward.

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Quantifying Risk Under Evolving Uncertainty: Belief-Dependent Robustness for Safe Sequential Decision Making

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Technical University of Munich, Germany · Masaryk University, Brno, Czech Republic

Risk-Sensitive Planning Robust RL Risk-Sensitive RL Safe RL Distributionally Robust RL Bayesian RL

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How cautious should an agent be while it is still learning its environment? We propose RATTL (Risk-Adversarial Total-Reward Learning), which ties caution to epistemic uncertainty: the agent holds a Bayesian posterior over unknown dynamics and plans against a Wasserstein ambiguity set whose radius is a monotone function of that posterior. The radius contracts with evidence, so behaviour interpolates continuously between worst-case robustness and risk-neutral total-reward maximization. The design follows the duality underlying the Entropic Value-at-Risk, which converts the choice of a risk level into the choice of an ambiguity radius. We show the resulting planning problem is well posed under transience and compactness conditions, and prove a Safety Sandwich: the RATTL value lies between the uninformed robust value and the full- knowledge optimum, with a gap that vanishes as the posterior concentrates. In a canonical binary-hazard instance, the induced criterion reduces to Conditional Value-at-Risk at a level set by the posterior entropy. A worked example shows the agent deferring the efficient action until a sharp identification threshold. RATTL targets runtime safety for agents, including LLM-based systems, acting under uncertainty.

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