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Let $x_1,\ldots,x_n \in \mathbb R^d$ and $y_1,\ldots,y_n \in \{\pm 1\}$, and $\epsilon, h \gt 0$. Define $\theta(t) := Q((t-\epsilon)/h)$, where $Q(z) := \int_{z}^\infty \phi (z)\mathrm{d}z$ is the Ga …

Let $V$ be a finite-dimensional real vector space (e.g space of $m \times n$ real matrices equiped with Hilbert-Schmidt inner product $(A,B) \to \mathrm{tr}(AB^\top)$, and let $f:V^2 \to \mathbb R$, $ …

Let $C$ be a (nonempty) compact subset of euclidean $\mathbb R^n$, and consider the set-valued map $p_C:\mathbb R^n \to 2^C$ defined by $$ p_C(x) = \{c \in C \mid \|x-c\| = \mbox{dist}(x,C)\}, $$ wher …

Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time understanding a cer …

Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation $$ \begin{split} \mathbb E_{z …

Let $\Phi$ be the Gaussian CDF and for $\gamma\ge 0$ and $h>0$, define a loss function $\ell_h:\{\pm 1\} \times \mathbb R$ by $$ \ell_{\gamma,h}(y,y') := \phi_{\gamma,h}(yy') := \Phi((yy'-\gamma)/h). …

Diclaimer. I'm not sure this is the right venue for this question, but I'll give it a try Definition [Strong / metric slope]. Given a complete metric space $(M,d)$ and a function $f:M \to (-\infty,+ …

Disclaimer. Not sure this is MO-level but would really appreciate some help with this. Thanks in advance. Moved from SE. Let $a,b,c \ge 0$, and define a function $g:\mathbb R \to \mathbb R$ by $g(t) : …