Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mathcal{P}^{a}_{\rho}$ be the set of The Measures Noise-Connected to the Reference Measure §space, let $\varphi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ and let $\mu\in\mathcal{P}^{a}_{\rho}$. Let $L^{2}(\mu;X^{a})$ be the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields. For $\nu\in\mathcal{P}^{a}_{\rho}$, $\Pi^{a}(\mu,\nu)$ is the set of Couplings of Finite Noise Cost and Their Noise Cost §couplings, and for $\pi\in\Pi^{a}(\mu,\nu)$ the noise cost $I^{a}(\pi)$ is a nonnegative real number by Couplings of Finite Noise Cost and Their Noise Cost §cost and $\mathcal{J}^{a}(\eta,\pi)$ is the noise displacement pairing of $\eta\in L^{2}(\mu;X^{a})$ along $\pi$. Square roots are those of Existence and Uniqueness of the Nonnegative Square Root.

1. (Differentiability along noise couplings)¶ Let $\eta\in L^{2}(\mu;X^{a})$. The function $\varphi$ is differentiable along noise couplings at $\mu$ with gradient $\eta$ if for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\theta\in\mathbb{R}$ with $0<\theta$ such that

$$\bigl|\varphi(\nu)-\varphi(\mu)-\mathcal{J}^{a}(\eta,\pi)\bigr|\le\varepsilon\,\sqrt{I^{a}(\pi)}$$

for every $\nu\in\mathcal{P}^{a}_{\rho}$ and every $\pi\in\Pi^{a}(\mu,\nu)$ with $I^{a}(\pi)<\theta^{2}$. The function $\varphi$ is differentiable along noise couplings at $\mu$ if there is an $\eta\in L^{2}(\mu;X^{a})$ with this property.

2. (The gradient along noise couplings)¶ Suppose that $\varphi$ is differentiable along noise couplings at $\mu$. Then exactly one $\eta\in L^{2}(\mu;X^{a})$ has the property of clause 1. Indeed, one exists by that hypothesis. Suppose $\eta$ and $\eta'$ both have it and let $\varepsilon\in\mathbb{R}$ be positive, so that $\varepsilon\cdot2^{-1}$ is positive by Elementary Order Arithmetic in an Ordered Field §halving. Let $\theta$ and $\theta'$ be as in clause 1 for $\eta$ and for $\eta'$ respectively, with $\varepsilon\cdot2^{-1}$ there in place of $\varepsilon$, and let $\theta''=\min\{\theta,\theta'\}$ be their minimum, which is positive because it equals $\theta$ or $\theta'$ by claim 2 of Elementary Properties of the Minimum of Two Elements. Let $\nu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\mu,\nu)$ satisfy $I^{a}(\pi)<(\theta'')^{2}$. Since $\theta''\le\theta$ and $\theta''\le\theta'$ by claim 1 of that lemma, and both sides of each of these inequalities are nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $(\theta'')^{2}\le\theta^{2}$ and $(\theta'')^{2}\le(\theta')^{2}$, whence $I^{a}(\pi)<\theta^{2}$ and $I^{a}(\pi)<(\theta')^{2}$ by Elementary Order Arithmetic in an Ordered Field §mixed-transitivity. Both estimates of clause 1 are therefore available at $\pi$, and, by the linearity of the noise displacement pairing (Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §linear, applied with $s=1$ and $t=-1$),

$$\mathcal{J}^{a}(\eta-\eta',\pi)=\bigl(\varphi(\nu)-\varphi(\mu)-\mathcal{J}^{a}(\eta',\pi)\bigr)-\bigl(\varphi(\nu)-\varphi(\mu)-\mathcal{J}^{a}(\eta,\pi)\bigr),$$

so that the triangle inequality Properties of the Absolute Value in an Ordered Field §triangle, together with the symmetry Properties of the Absolute Value in an Ordered Field §symmetry, bounds $|\mathcal{J}^{a}(\eta-\eta',\pi)|$ by $\varepsilon\cdot2^{-1}\sqrt{I^{a}(\pi)}+\varepsilon\cdot2^{-1}\sqrt{I^{a}(\pi)}$, which equals $\varepsilon\sqrt{I^{a}(\pi)}$ by the distributive law of Field and the halving identity of Elementary Order Arithmetic in an Ordered Field §halving. Thus

$$\bigl|\mathcal{J}^{a}(\eta-\eta',\pi)\bigr|\le\varepsilon\,\sqrt{I^{a}(\pi)} .$$

So $\eta-\eta'$ satisfies the hypothesis of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §vanishing, whence $\eta-\eta'$ is the zero element of $L^{2}(\mu;X^{a})$ by that claim and $\eta=\eta'$. This unique $\eta$ is written $\nabla\varphi(\mu)$ and called the gradient along noise couplings of $\varphi$ at $\mu$. This notion is not to be confused with the noise gradient $\nabla_{a}\psi$ of a bounded $C^{1}$ cylindrical function $\psi$ on $X$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient), which is a map on $X$.