Conditional Value At Risk

Definition

Conditional Value-at-Risk (CVaR) of a random cost $Z$ at confidence level $\alpha \in (0,1)$ is the
expected loss in the worst $(1-\alpha )$-tail of the cost distribution — the mean of $Z$ conditioned
on $Z$ exceeding its $(1-\alpha )$-quantile (the Value-at-Risk, VaR). It quantifies “how bad is bad”,
interpolating between the risk-neutral expectation ($\alpha \to 0$) and the worst-case essential
supremum ($\alpha \to 1$). Unlike VaR (a quantile, hence a chance constraint), CVaR is a coherent risk
measure — monotone, translation-invariant, positively homogeneous and sub-additive/convex
(coherent_risk_measures) — which makes the constraint
${\mathrm{CVaR}}_{\alpha }(Z)\le 0$ both convex and a conservative tightening of the chance constraint
$\mathbb{P}[Z>0]\le 1-\alpha$. CVaR is distribution-agnostic and independent of the free-flying /
free-floating regime; in all cited sources it is layered on top of the planning/control problem rather
than on the manipulator dynamics.

Key Equations

Symbols per notation.md.

The canonical Rockafellar–Uryasev optimization form (used directly by dixit2023risk),
with $(\cdot {)}^{+}=\max \{\cdot ,0\}$:

${\mathrm{CVaR}}_{1-\alpha }(Z):={\inf }_{z\in \mathbb{R}}\mathbb{E}\left[z+\frac{(Z-z{)}^{+}}{1-\alpha }\right],$

whose minimizer $z^{\star }={\mathrm{VaR}}_{1-\alpha }(Z)$, the $(1-\alpha )$-quantile. Equivalently, as the tail-average
of VaR (majumdar2017how, eq. for ${\mathrm{CVaR}}_{\alpha }$):

${\mathrm{CVaR}}_{\alpha }(Z)=\frac{1}{\alpha }{\int }_{1-\alpha }^1{\mathrm{VaR}}_{1-\tau }(Z)d\tau .$

Convention flag. $\alpha$ is the CVaR confidence level in notation.md (line: “CVaR confidence
level (risk layer)”), so the risk tail has mass $1-\alpha$ and the page writes ${\mathrm{CVaR}}_{1-\alpha }$, matching
dixit2023risk — i.e. the subscript is the tail mass ($\alpha \to 1$ = worst case).
majumdar2017how keeps the same confidence-level meaning for $\alpha$ but indexes the
subscript by the confidence level itself, writing ${\mathrm{CVaR}}_{\alpha }$ for the average of the upper $(1-\alpha )$-tail
(so again $\alpha \to 1$ = worst case); the tail-average formula below is quoted in that indexing.
ren2022chance uses $ϵ$ for the tail probability (subscript = tail mass, like dixit).
Same metric, different subscript-indexing — read the subscript convention before reusing a formula.

Source Support

• majumdar2017how — derives CVaR from an axiomatic (“coherent / distortion”) theory of
  risk for robotics; CVaR satisfies all six axioms (A1–A6) whereas VaR fails sub-additivity (A4). Primary for the
  definition and the coherence argument.
• dixit2023risk — gives the Rockafellar–Uryasev optimization form and its discrete/dual
  (risk-envelope) representation; embeds CVaR as the obstacle-avoidance constraint in risk-averse MPC and contrasts it
  with EVaR and total-variation-distance metrics.
• ren2022chance — uses CVaR as a conservative approximation of a chance constraint
  under a Gaussian-mixture uncertainty model: ${\mathrm{CVaR}}_{ϵ}\le 0\Rightarrow \mathbb{P}[\cdot ]\ge 1-ϵ$,
  with a closed-form second-order-cone reformulation; bounds the amount of violation, not just its probability.
• akella2024risk — situates CVaR within the worst-case / risk-neutral / risk-aware
  taxonomy of safety assessment; supporting context.

Related Topics

• coherent_risk_measures — CVaR is the canonical coherent (indeed distortion / spectral)
  risk measure; every distortion metric is a weighted integral of CVaR over confidence levels.
• chance_constraints — CVaR is the convex, sub-additive surrogate for a chance constraint;
  ${\mathrm{CVaR}}_{\alpha }(Z)\le 0$ implies $\mathbb{P}[Z>0]\le 1-\alpha$ and additionally caps the expected overshoot.
• risk_aware_mpc — CVaR (or a general coherent measure) is the risk functional optimized/constrained
  in the receding-horizon problems of dixit2023risk and ren2022chance.
• motion_planning_under_uncertainty — CVaR is the risk metric these planners
  impose on collision / constraint-violation cost to obtain safe, dynamically feasible trajectories.
• control_barrier_functions — CVaR has been combined with CBFs (dixit2023risk cites
  CVaR-CBF formulations) to enforce risk-bounded forward invariance of a safe set.

Open Questions

• All four sources apply CVaR to terrestrial/automotive systems (autonomous driving, bipeds, ground robots) with
  linear or linearized discrete-time dynamics; none treats a free-flying space manipulator. Does the convexity that makes
  ${\mathrm{CVaR}}_{\alpha }(Z)\le 0$ tractable survive when $Z$ is a collision/pointing cost coupled through the nonlinear
  circumcentroidal dynamics (ffsm_dynamics)?
• The natural risk variable for our inspection problem (versine pointing error, or singularity proximity ${\sigma }_G$)
  is not a financial loss — what is the right cost $Z$, and at what confidence level $\alpha$, for a CVaR constraint on
  proximity to a dynamic_singularity?
• majumdar2017how flags that CVaR (single-level, single-stage) can violate time consistency in multi-stage planning;
  does our receding-horizon inspection guidance need the nested/Markov-coherent form instead?