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A centered Gaussian minimizes bounded symmetric quasi-convex loss

Statement

Assume AC. Let $X \sim N_{d} (m, Σ)$ with positive-definite covariance $Σ$ . Let $L : R^{d} \to [0, M]$ be Borel measurable and symmetric, $L (- x) = L (x)$ , and quasi-convex, meaning each sublevel set ${x : L (x) \leq t}$ is convex. Then for every $a \in R^{d}$ , $E L (X - a) \geq E L (X - m) .$ Thus the Gaussian center $m$ minimizes expected loss. The result includes quasi-convex losses whose sublevel sets are not closed.

Facts & Assumptions

Given: AC; a finite-dimensional nondegenerate Gaussian vector; and a bounded Borel measurable symmetric loss with convex sublevel sets.

[A1]

The Axiom of Choice: AC is assumed for the finite-dimensional multivariate normal construction and characteristic laws used here.

[F1]

Brunn–Minkowski inequality for compact convex sets: For nonempty compact convex $U, V \subset R^{d}$ , $λ_{d} (U + V)^{1 / d} \geq λ_{d} (U)^{1 / d} + λ_{d} (V)^{1 / d} .$

[F2]

The boundary of a convex set is Lebesgue null: A convex set and its closure differ only by a Lebesgue-null boundary; lower-dimensional convex sets are null.

[F3]

Multivariate normal law, including singular covariance, Standard normal and normal laws, and The standard normal density has total mass one: A nondegenerate Gaussian is an invertible affine image of a standard Gaussian vector with density $(2 π)^{- d / 2} e^{- ∥ z ∥_{2}^{2} / 2}$ . The Euclidean norm is the one in The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ .

[F4]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: Nonnegative measurable integrands may be integrated in either order.

[F5]

A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not: An invertible linear map scales Lebesgue measure by the absolute determinant. In particular, reflection $x \mapsto - x$ preserves volume and scaling by $1 / 2$ multiplies it by $2^{- d}$ . Thus the unweighted inequality in [F1], applied to $U, V$ , gives $λ_{d} ((U + V) / 2)^{1 / d} \geq \frac{λ_{d} (U)^{1 / d} + λ_{d} (V)^{1 / d}}{2} .$

[F6]

Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation: Translation preserves Lebesgue measure; accordingly, translating the variable in a Lebesgue integral does not change the integral after the integrand is translated.

[F7]

Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: Closed bounded subsets of $R^{d}$ are compact. Thus intersections of the closed set $C$ with the closed balls used below are compact and measurable.

Proof

Proof technique: prove the Gaussian shift inequality for convex level sets by Brunn–Minkowski, then integrate their indicators.

1.1F5F7givenalgebra

First let $C \subseteq R^{d}$ be closed, symmetric, and convex. For each radius $r > 0$ and shift $v$ , put $U = C \cap (v + \overline{B} (0, r)), V = C \cap (- v + \overline{B} (0, r)) .$ They are compact convex sets by [F7], and $V = - U$ , so they have equal volume by [F5]. For $u \in U$ and $w \in V$ , symmetry and convexity put $(u + w) / 2$ in $C$ , while $u - v$ and $w + v$ lie in the radius- $r$ ball, so $∥ (u + w) / 2 ∥_{2} = ∥ (u - v + w + v) / 2 ∥_{2} \leq r$ . Thus $(U + V) / 2 \subseteq C \cap \overline{B} (0, r)$ . This sets up the Brunn–Minkowski comparison.

2.1F1F5step 1.1algebra

If $U$ has positive volume, [F1] and its derived weighted form [F5] give $λ_{d} (C \cap \overline{B} (0, r)) \geq λ_{d} ((U + V) / 2) \geq λ_{d} (U),$ since $λ_{d} (V) = λ_{d} (U)$ . If $U$ has zero volume the same inequality follows from nonnegativity. Therefore $λ_{d} (C \cap (v + \overline{B} (0, r))) \leq λ_{d} (C \cap \overline{B} (0, r)) .$ This is the required shifted-volume inequality.

3.1F3F4F5F6step 2.1algebra

Let $Z \sim N_{d} (0, I_{d})$ . Its density is a positive constant times $e^{- ∥ z ∥_{2}^{2} / 2}$ . For any shift $v$ , translation invariance [F6] and symmetry of $C$ (using the reflection in [F5]) give $P (Z \in C + v) = c_{d} \int_{C} e^{- ∥ z + v ∥_{2}^{2} / 2} d z = c_{d} \int_{C} e^{- ∥ z - v ∥_{2}^{2} / 2} d z$ , where $c_{d} > 0$ is the normalizing constant. The identity $e^{- ∥ z - v ∥_{2}^{2} / 2} = \int_{0}^{\infty} e^{- t} 1_{{∥ z - v ∥_{2}^{2} / 2 \leq t}} d t$ and [F4] express the latter integral as $c_{d} \int_{0}^{\infty} e^{- t} λ_{d} (C \cap (v + \overline{B} (0, \sqrt{2 t}))) d t .$ By step 2.1, this is at most the same expression at shift zero. Hence $P (Z \in C + v) \leq P (Z \in C)$ . This proves the Gaussian shift inequality.

4.1F2F6step 3.1given

If $C$ is convex, symmetric, but not closed, replace it by $\overline{C}$ . The difference is contained in $\partial C$ , which is null by [F2]; its translate is also null by [F6]. Thus the probability inequality in step 3.1 holds for every convex symmetric Borel set. If $C$ has empty interior, [F2] makes both probabilities zero, which also covers this boundary case.

5.1A1F2F3step 4.1algebra

Write $X = m + A Z$ with $A = Σ^{1 / 2}$ invertible. For any symmetric convex Borel set $C$ , the event $X - a \in C$ is $Z \in A^{- 1} C + A^{- 1} (a - m)$ . The set $A^{- 1} C$ is again symmetric convex, so step 4.1 gives $P (X - a \in C) \leq P (X - m \in C) .$ Apply this to every sublevel set $C_{t} = {x : L (x) \leq t}$ . This gives the comparison for each loss sublevel.

6.1F4step 5.1givenalgebra∎

For any random variable $Y$ with $0 \leq Y \leq M$ , $E Y = \int_{0}^{M} P (Y > t) d t$ , by [F4]. Step 5.1 says $P (L (X - a) \leq t) \leq P (L (X - m) \leq t)$ for each $t$ , hence the corresponding tail probabilities are ordered in the reverse direction. Integrating them gives $E L (X - a) \geq E L (X - m)$ . This proves the statement without requiring closed loss sublevels.

Depends on

Used by

Bounded bowl-shaped losses have Gaussian-centered Lipschitz minorants Lemma

Dependency tree · two levels

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Sources

John Duchi, A few notes on contiguity, asymptotics, and local asymptotic normality, §4, PDF pp. 16–17 (standard reference, not scraped)
Jan van Neerven, γ-Radonifying operators: A Survey, §8, Lemma 8.1 and Theorem 8.2 (Anderson), printed p. 29 (standard reference, not scraped)