Noncompact Actions and Escaping Minimizers

Why Noncompact Action Spaces Need Attention

In a decision problem, an action space need not be compact. For example, an action may be any real number, so \(A=\mathbb{R}\). When an objective is minimized over a compact action space, lower semicontinuity ensures that a minimizing sequence has a convergent subsequence whose limit is an optimizer. On a noncompact space, that sequence may instead move indefinitely far away. A finite infimum can then fail to be attained.

This is one reason compactness appears in compactness-based complete-class arguments: it helps prevent limits of nearly optimal choices from disappearing outside the available space. But compactness is not the only possible route to existence. For an action space contained in Euclidean space, a sufficiently coercive objective keeps all near-minimizers in a bounded region. Closedness then gives the compactness needed to extract a convergent subsequence.

An infimum is a lower bound approached arbitrarily closely; it need not be the value of the objective at any action. It is important to distinguish a finite infimum that is not attained from an objective that is unbounded below. In the latter case, the infimum is \(-\infty\), not a finite number.

Compactness and Lower Semicontinuity

The compact-space existence result is the basic comparison point. A function \(g\) is lower semicontinuous (l.s.c.) on a metric space if, whenever \(a_n\to a\), it satisfies \(g(a)\leq\liminf_{n\to\infty}g(a_n)\). Intuitively, the value at a limit cannot exceed the limiting lower value of nearby objective values. This condition allows the objective value at a limit point to be no worse than the values along a minimizing sequence.

Proof. First, \(g\) is bounded below. Otherwise, for each positive integer \(n\) one could choose \(a_n\in K\) with \(g(a_n)<-n\). Compactness gives a subsequence \(a_{n_j}\to a^*\in K\). Since \(n_j\to\infty\), \(\liminf_j g(a_{n_j})=-\infty\). Lower semicontinuity would imply \(g(a^*)\leq-\infty\), which is impossible because \(g\) is real-valued. Thus \(g\) has a real lower bound. It also has a finite upper bound on its infimum, since \(K\) is nonempty and the value at any chosen point is finite. Therefore \(m=\inf_{a\in K}g(a)\) is finite.

Choose a minimizing sequence with \(g(a_n)\to m\). By compactness, some subsequence \(a_{n_j}\) converges to \(a^*\in K\). Lower semicontinuity gives

Since \(m\) is the infimum, \(m\leq g(a^*)\). Therefore \(g(a^*)=m\), so \(a^*\) attains the infimum. \(\square\)

Coercivity as an Alternative to Compactness

On a noncompact subset of Euclidean space, coercivity prevents near-minimizers from escaping. The relevant condition is about the objective at large action values, not about the parameter space of the statistical model.

Proof. Choose \(a_0\in A\). By coercivity, there is \(R<\infty\) such that \(g(a)\geq g(a_0)\) whenever \(a\in A\) and \(\|a\|\geq R\). The set \(K=A\cap\{a:\|a\|\leq R\}\) is compact: it is closed and bounded in \(\mathbb{R}^k\). If \(K\) is nonempty, the compact-space result just proved shows that \(g\) is bounded below on \(K\). If \(K\) is empty, there are no points of \(A\) to consider inside that ball. Outside the ball, \(g(a)\geq g(a_0)\). In either case, \(g\) is bounded below on \(A\); since \(A\) is nonempty and \(g(a_0)\) is finite, \(m=\inf_{a\in A}g(a)\) is finite.

Take a minimizing sequence \(a_n\in A\) with \(g(a_n)\to m\). Eventually \(g(a_n)\leq g(a_0)+1\). Coercivity, applied with \(M=g(a_0)+1\), gives a radius outside which \(g(a)\geq g(a_0)+1\). Thus all sufficiently late terms of the minimizing sequence lie inside a bounded ball. Since \(A\) is closed, a subsequence converges to some \(a^*\in A\). Lower semicontinuity gives \(g(a^*)\leq m\); the definition of \(m\) gives \(m\leq g(a^*)\). Hence \(g(a^*)=m\), and the infimum is attained. \(\square\)

The proof identifies the role of each assumption. Coercivity rules out escape by a minimizing sequence. Closedness ensures that a subsequential limit remains an available action. Lower semicontinuity ensures that the limit is at least as good as the limiting objective values. In particular, every sublevel set \(\{a\in A:g(a)\leq c\}\), for finite \(c\), is bounded by coercivity and closed by lower semicontinuity; it is therefore compact.

What Coercivity Does—and Does Not—Replace

Coercivity is a condition on a particular objective function. It can replace compactness of the action space when the question is whether that objective has a minimizer, provided the space is closed and the objective is lower semicontinuous. It does not, by itself, establish all the compactness properties needed in a complete-class theorem. Such arguments may involve sequences of decision rules, risk functions, or probability kernels, and each requires its own hypotheses and limit argument.

The closedness condition also matters. Coercivity at infinity controls sequences whose norms grow without bound; it does not prevent a sequence from approaching a missing boundary point at finite distance.

Common Mistakes

• Calling every escaping sequence evidence of an unattained finite optimum. For \(g(a)=-a\), the objective values tend to \(-\infty\), so there is no finite infimum. Check whether the infimum is finite before describing a sequence as approaching it.
• Assuming a finite infimum must be attained. For \(g(a)=e^{-a}\), the infimum is zero, but every available action has strictly positive objective value.
• Using coercivity without checking the action space. A closed subset of Euclidean space is needed for the theorem above. Coercivity at infinity alone does not prevent a minimizing sequence from approaching an excluded finite boundary point.
• Applying objective-level existence to an entire decision-theoretic limit argument. Coercivity can yield an optimizer for a specified objective; it does not automatically supply compactness or risk-limit properties for a class of decision rules.