Risk-Set Closure and Essential Completeness

Why Risk-Set Closure Matters

A rule can be improved upon in two different senses. Another rule may have risk no larger at any parameter value, giving exact domination. Alternatively, a sequence of rules may approach that performance coordinate by coordinate, even though no single rule in the class achieves it. The distinction matters whenever a proof uses limits: a limit of risk vectors need not itself be the risk vector of an available rule.

We make this distinction precise by describing the attainable risks of a class and the set of risk vectors that it can dominate. Throughout, assume the parameter space \(\Theta\) is countable, each rule has finite risk at every \(\theta\), and comparisons of risks are coordinatewise. The countability assumption lets us characterize closure by sequences rather than more general nets.

Definition: Let \(\mathcal D\) be a class of decision rules. The risk function of a rule \(\delta\in\mathcal D\) is \(R_\delta(\theta)=R(\theta,\delta)\), for \(\theta\in\Theta\). A rule \(\delta_1\) weakly dominates \(\delta_2\) if \(R_{\delta_1}(\theta)\leq R_{\delta_2}(\theta)\) for every \(\theta\in\Theta\). A subclass \(\mathcal C\subseteq\mathcal D\) is complete relative to \(\mathcal D\) if every \(\delta\in\mathcal D\) is weakly dominated by some \(\gamma\in\mathcal C\). It is essentially complete relative to \(\mathcal D\) if, for every \(\delta\in\mathcal D\), there is a sequence \((\gamma_n)\) in \(\mathcal C\) such that \(\limsup_{n\to\infty}R_{\gamma_n}(\theta)\leq R_\delta(\theta)\) for each \(\theta\in\Theta\).

The limsup condition is coordinatewise: for each fixed parameter, the sequence's risks eventually stay arbitrarily close to, or below, the target risk. It does not assert that one index \(n\) works simultaneously for every parameter. In particular, essential completeness does not promise an exact dominating rule.

The Upper Attainable Risk Set

Write the attainable risk set of \(\mathcal C\) as \(\mathcal R_{\mathcal C}=\{R_\gamma:\gamma\in\mathcal C\}\), a subset of \(\mathbb R^\Theta\). The relevant set for domination is its coordinatewise upper hull:

\mathcal U_{\mathcal C} = \left\{x\in\mathbb R^\Theta: \text{there exists }\gamma\in\mathcal C \text{ such that }R_\gamma(\theta)\leq x(\theta) \text{ for every }\theta\in\Theta \right\}.

A point \(x\) belongs to this upper hull when some rule in \(\mathcal C\) has risk no larger than \(x\) everywhere. The upper hull is upward closed: if \(x\in\mathcal U_{\mathcal C}\) and \(y(\theta)\geq x(\theta)\) for all \(\theta\), then \(y\in\mathcal U_{\mathcal C}\). Equip \(\mathbb R^\Theta\) with the product topology, in which a basic neighborhood restricts only finitely many coordinates at a time.

Theorem: Under the countability and finite-risk assumptions above, a rule \(\delta\in\mathcal D\) is exactly dominated by a rule in \(\mathcal C\) if and only if \(R_\delta\in\mathcal U_{\mathcal C}\). It is approximately dominated by a sequence in \(\mathcal C\), in the sense of the definition of essential completeness, if and only if \(R_\delta\in\overline{\mathcal U_{\mathcal C}}\), the product-topology closure of the upper hull.

The first assertion follows directly from the definitions: \(R_\delta\in\mathcal U_{\mathcal C}\) means that there is a \(\gamma\in\mathcal C\) with \(R_\gamma(\theta)\leq R_\delta(\theta)\) at every parameter. For the second assertion, we prove both directions.

Suppose first that a sequence \((\gamma_n)\) satisfies \(\limsup_n R_{\gamma_n}(\theta)\leq R_\delta(\theta)\) for each \(\theta\). Define

y_n(\theta)=\max\{R_\delta(\theta),R_{\gamma_n}(\theta)\}.

Then \(y_n\in\mathcal U_{\mathcal C}\), since \(R_{\gamma_n}(\theta)\leq y_n(\theta)\) for every \(\theta\). For each fixed \(\theta\), the assumed limsup inequality implies \(\max\{R_\delta(\theta),R_{\gamma_n}(\theta)\}\to R_\delta(\theta)\). Thus \(y_n\to R_\delta\) coordinatewise, which is convergence in the product topology. Therefore \(R_\delta\in\overline{\mathcal U_{\mathcal C}}\).

Conversely, suppose \(R_\delta\in\overline{\mathcal U_{\mathcal C}}\). Because \(\Theta\) is countable, the product topology on \(\mathbb R^\Theta\) is metrizable; for an enumeration \(\Theta=\{\theta_1,\theta_2,\ldots\}\), one compatible metric is \(d(x,y)=\sum_{j\geq1}2^{-j}\min\{1,|x(\theta_j)-y(\theta_j)|\}\). Closure therefore provides a sequence \(y_n\in\mathcal U_{\mathcal C}\) with \(y_n\to R_\delta\) coordinatewise. For each \(n\), the definition of \(\mathcal U_{\mathcal C}\) supplies a rule \(\gamma_n\in\mathcal C\) such that \(R_{\gamma_n}(\theta)\leq y_n(\theta)\) for every \(\theta\). Fixing \(\theta\) and taking limsup gives

\limsup_{n\to\infty}R_{\gamma_n}(\theta) \leq \lim_{n\to\infty}y_n(\theta) = R_\delta(\theta).

This is the required approximate domination, and proves the theorem. \(\square\)

Corollary: For a fixed target risk \(R_\delta\), approximate domination is equivalent to the following finite-coordinate condition: for every finite \(F\subseteq\Theta\) and every \(\varepsilon>0\), there exists \(\gamma\in\mathcal C\) such that \(R_\gamma(\theta)\leq R_\delta(\theta)+\varepsilon\) for all \(\theta\in F\).

Indeed, if \(R_\delta\) is in the closure of the upper hull, every product-topology neighborhood restricting the coordinates in \(F\) intersects that hull. A point in the intersection is above the risk of some \(\gamma\in\mathcal C\), so the stated inequalities follow. Conversely, if the finite-coordinate condition holds, take any basic neighborhood of \(R_\delta\), restricted to a finite set \(F\) with coordinate tolerances. Choose \(\varepsilon\) smaller than all those tolerances and use the condition to find \(\gamma\). The point \(y\) defined by \(y(\theta)=\max\{R_\delta(\theta),R_\gamma(\theta)\}\) for every \(\theta\in\Theta\) lies in the upper hull, since it dominates \(R_\gamma\) at every coordinate. On \(F\), the finite-coordinate condition gives \(R_\delta(\theta)\leq y(\theta)\leq R_\delta(\theta)+\varepsilon\); choosing \(\varepsilon\) smaller than all the neighborhood tolerances puts \(y\) in that neighborhood. Hence every such neighborhood intersects the upper hull, proving closure membership.

Worked Examples: Exact Domination and a Boundary Risk

Worked Example: A Boundary Point Gives Approximation, Not Exact Domination

Consider a decision problem with two parameter values, \(\Theta=\{1,2\}\), no data, and actions \(a_1,a_2,\ldots\) together with \(a_*\). Let the loss, and therefore the risk, of \(a_n\) be \(R_{a_n}=(1/n,1-1/n)\), and let \(R_{a_*}=(0,1)\). Take \(\mathcal C=\{a_n:n\geq1\}\) and use \(a_*\) as the target rule.

No action \(a_n\) exactly dominates \(a_*\), because exact domination would require \(1/n\leq0\) in the first coordinate, which is false for every \(n\). But \(R_{a_n}\to(0,1)=R_{a_*}\), so the sequence even converges in risk to the target. More explicitly, the points \(y_n=(1/n,1)\) belong to \(\mathcal U_{\mathcal C}\): \(R_{a_n}\leq y_n\) in both coordinates. Since \(y_n\to(0,1)\), the target is in \(\overline{\mathcal U_{\mathcal C}}\) and is approximately dominated. This is essential completeness for this target without exact completeness.

Worked Example: An Exact Dominator

Again take a problem with no data and two parameter values. Suppose the class \(\mathcal C\) contains an action \(c\) with risk \(R_c=(1,3)\), while another rule \(d\) in the full class has risk \(R_d=(2,3)\).

Coordinate by coordinate, \(1\leq2\) and \(3\leq3\). Thus \(c\) weakly dominates \(d\) exactly, and \(R_d\in\mathcal U_{\mathcal C}\). This is stronger than merely having \(R_d\) in the closure of the upper hull: the upper-hull membership itself supplies a single rule that meets both risk bounds. Equality in the second coordinate is permitted by weak domination; strict improvement at every parameter is not part of the definition.

Why the Product Topology Is the Right One

For a countable parameter space, product-topology closure has a practical interpretation: every finite collection of parameter values can be controlled to any specified positive tolerance by one rule from the class. It does not mean that the approximation error is uniformly small over all of \(\Theta\).

Worked Example: Coordinatewise Approximation Need Not Be Uniform

Let \(\Theta=\{1,2,\ldots\}\) and consider a no-data problem with actions \(a_1,a_2,\ldots\), and a target action \(a_*\). Define the losses by \(L(j,a_*)=0\) and \(L(j,a_n)=1\) if \(j>n\), while \(L(j,a_n)=0\) if \(j\leq n\). These losses are also the risks, since there is no data.

Fix a parameter \(j\). For every \(n\geq j\), \(R_{a_n}(j)=0\), so \(R_{a_n}(j)\to R_{a_*}(j)=0\). The sequence therefore approximates the target at every fixed coordinate. However, for each \(n\), there are parameters \(j>n\) with \(R_{a_n}(j)=1\), so \(\sup_{j\geq1}|R_{a_n}(j)-R_{a_*}(j)|=1\). There is no uniform convergence. Nor does any \(a_n\) exactly dominate \(a_*\), because its risk is 1 at every \(j>n\). This example is essentially complete for the target action but not exactly complete for it.

A useful way to verify the topology is to enumerate the parameters and build rules that handle longer initial segments of that enumeration. The finite-coordinate condition guarantees control on each initial segment, while a diagonal choice produces a sequence controlling every fixed coordinate eventually. No uniform bound over all parameters follows unless it is established separately.

Common Mistake: Confusing the Risk Set With Its Upper Hull

A frequent error is to treat closure of the raw attainable risk set \(\mathcal R_{\mathcal C}\) as the exact criterion for essential completeness. The criterion proved here concerns \(\overline{\mathcal U_{\mathcal C}}\), not merely \(\overline{\mathcal R_{\mathcal C}}\). The upper hull accounts for the fact that a rule can have risk below a proposed point; the point need not itself be an attainable risk vector. Exact domination requires membership in \(\mathcal U_{\mathcal C}\), while sequential approximate domination requires membership in its closure.

A related geometric error is to infer a pointwise comparison from one coordinate being small, or from an unbounded coordinate tending downward. For example, let \(S=\{(1/n,-n):n\geq1\}\) and define the lower orthant at the origin by \(\mathord{\downarrow}(0,0)=\{(x_1,x_2):x_1\leq0,\ x_2\leq0\}\). The set \(S\) has no point in its Euclidean closure in \(\mathord{\downarrow}(0,0)\): its only closure points are its own points, each of which has first coordinate strictly greater than zero. The negative second coordinate alone does not put any point below the origin in both coordinates. This set is a geometric illustration, not a risk set, since it contains negative coordinates.

Key takeaway: Exact completeness asks for an attainable rule whose risk is no larger at every parameter. Essential completeness allows a sequence of rules with coordinatewise limsup risk no larger than the target. In the countable-parameter setting, these are respectively membership in the upper attainable risk set and membership in its product-topology closure.

Check Your Understanding

Use the coordinatewise definitions and the upper attainable risk set.

What risk-vector inequality must hold for one rule to exactly dominate another?
Why does membership of a target risk in \(\overline{\mathcal U_{\mathcal C}}\) not necessarily provide a single dominating rule in \(\mathcal C\)?
State the finite-coordinate condition equivalent to approximate domination when \(\Theta\) is countable.
In the third worked example, why does coordinatewise convergence hold although the supremum risk discrepancy remains equal to 1?
Why is closure of the upper hull, rather than closure of the raw risk set, the relevant condition for approximate domination?