Binary surrogate classification risk

Let $Y\in \{-1,+1\}$ be the label in a supervised prediction problem with the full power-set sigma-algebra on this binary label set, and choose the finite-label conditional probability function ordered as $(-1,+1)$; write $\eta (x)={\eta }_{+1}(x)$. Let $\mathrm{\Phi }:\mathbb{R}\to [0,\mathrm{\infty })$ be finite, convex and nondecreasing. It is continuous on $\mathbb{R}$; extend it to $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$ by its one-sided monotone limits, which may equal $+\mathrm{\infty }$. The resulting extended function is Borel measurable. For $u\in \bar{\mathbb{R}}$ define the conditional margin risk $L_{\mathrm{\Phi }}(x,u)=\eta (x)\mathrm{\Phi }(-u)+(1-\eta (x))\mathrm{\Phi }(u),$ using $0\cdot (+\mathrm{\infty })=0$ at endpoints. A measurable extended-real conditional minimizer is a measurable $h_{\mathrm{\Phi }}^{\ast }:\mathsf{X}\to \bar{\mathbb{R}}$ satisfying $L_{\mathrm{\Phi }}(x,h_{\mathrm{\Phi }}^{\ast }(x))\le L_{\mathrm{\Phi }}(x,u)$ for every $x$ and every $u\in \bar{\mathbb{R}}$. For a measurable real-valued score $h$, its margin loss is ${\mathrm{\ell }}_{\mathrm{\Phi }}(h(x),y)=\mathrm{\Phi }(-yh(x))$, its surrogate risk is $L_{\mathrm{\Phi }}(h)=\mathbb{E}[\mathrm{\Phi }(-Yh(X))]$, and its conditional-minimum risk is $L_{\mathrm{\Phi }}^{\ast }=\mathbb{E}[L_{\mathrm{\Phi }}(X,h_{\mathrm{\Phi }}^{\ast }(X))]$. Under the zero-score-is-error convention, its classification risk is $R_{01}(h)=P(Yh(X)\le 0)$, and the Bayes classification risk is $R_{01}^{\ast }=\mathbb{E}[\min \{\eta (X),1-\eta (X)\}]$. Write $h^{\ast }(x)=\eta (x)-\frac{1}{2}$ for the centered conditional class probability.

Supervised prediction problems: fixes the measurable instance and label spaces and joint law.

A finite label has a conditional probability function: the chosen $\eta$ is a measurable version whose event integrals give the conditional label probabilities.

Population and empirical risk: population risk is an extended nonnegative integral of the loss under the joint law.

Loss functions and risk: fixes the loss and risk conventions used for the margin loss.

A measurable function between measurable spaces: a function is measurable when preimages of measurable sets are measurable.

A convex function on an open convex set is continuous: every finite convex function on the open convex set $\mathbb{R}$ is continuous.