SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\overline {{\mathcal {T}}} ^ {\mathrm {c s t n d}} \left(\mathcal {R} _ {\ell_ {0 - 1}} (h) - \mathcal {R} _ {\ell_ {0 - 1}} ^ {*} (\overline {{\mathcal {H}}}) + \mathcal {M} _ {\ell_ {0 - 1}} (\overline {{\mathcal {H}}})\right) \leq \mathcal {R} _ {\ell \mathrm {c s t n d}} (h) - \mathcal {R} _ {\ell \mathrm {c s t n d}} ^ {*} (\overline {{\mathcal {H}}}) + \mathcal {M} _ {\ell \mathrm {c s t n d}} (\overline {{\mathcal {H}}}). \tag {6}
$$
with  $\overline{\mathcal{T}}^{\mathrm{cstnd}}$  the  $\overline{\mathcal{H}}$ -estimation error transformation for constrained losses defined for all  $t \in [0,1]$  by  $\overline{\mathcal{T}}^{\mathrm{cstnd}}(t) =$
$$
\left\{ \begin{array}{l l} \inf  _ {\tau \geq 0} \sup  _ {\mu \in [ \tau - \Lambda_ {\min }, \tau + \Lambda_ {\min } ]} \big \{\frac {1 - t}{2} \big [ \Phi (\tau) - \Phi (- \tau + \mu) \big ] + \frac {1 + t}{2} \big [ \Phi (- \tau) - \Phi (\tau - \mu) \big ] \big \} & n = 2 \\ \inf  _ {P \in \big [ \frac {1}{n - 1}, 1 \big ]} \inf  _ {\tau_ {1} \geq \max  \{\tau_ {2}, 0 \}} \sup  _ {\mu \in C} \Big \{\frac {2 - P - t}{2} \big [ \Phi (- \tau_ {2}) - \Phi (- \tau_ {1} + \mu) \big ] + \frac {2 - P + t}{2} \big [ \Phi (- \tau_ {1}) - \Phi (- \tau_ {2} - \mu) \big ] \Big \} & n > 2, \end{array} \right.
$$
where  $C = \left[\max \{\tau_1, -\tau_2\} - \Lambda_{\min}, \min \{\tau_1, -\tau_2\} + \Lambda_{\min}\right]$  and  $\Lambda_{\min} = \inf_{x \in \mathcal{X}} \Lambda(x)$ . Furthermore, for any  $t \in [0,1]$ , there exist a distribution  $\mathcal{D}$  and a hypothesis  $h \in \mathcal{H}$  such that  $\mathcal{R}_{\ell_{0-1}}(h) - \mathcal{R}_{\ell_{0-1}}^*(\mathcal{H}) + \mathcal{M}_{\ell_{0-1}}(\mathcal{H}) = t$  and  $\mathcal{R}_{\ell \text{cstnd}}(h) - \mathcal{R}_{\ell \text{cstnd}}^*(\mathcal{H}) + \mathcal{M}_{\ell \text{cstnd}}(\mathcal{H}) = \mathcal{T}^{\text{cstnd}}(t)$ .
$$
\begin{array}{l} \mathcal {C} _ {\ell \text {c s t n d}} (h, x) = \sum_ {y \in \mathcal {Y}} p (x, y) \ell^ {\text {c s t n d}} (h, x, y) \\ = \sum_ {y \in \mathcal {Y}} p (x, y) \sum_ {y ^ {\prime} \neq y} \Phi (- h (x, y ^ {\prime})) \\ = \sum_ {y \in \mathcal {Y}} \Phi (- h (x, y)) \sum_ {y ^ {\prime} \neq y} p (x, y ^ {\prime}) \\ = \sum_ {y \in \mathcal {Y}} \Phi (- h (x, y)) (1 - p (x, y)) \\ = \Phi (- h (x, y _ {\max }) \left(1 - p (x, y _ {\max })\right) + \Phi (- h (x, \mathsf {h} (x))) \left(1 - p (x, \mathsf {h} (x))\right) \\ + \sum_ {y \notin \{y _ {\max }, h (x) \}} \Phi (- h (x, y)) (1 - p (x, y)). \\ \end{array}
$$
For any  $h \in \overline{\mathcal{H}}$  and  $x \in \mathfrak{X}$ , by the definition of  $\overline{\mathcal{H}}$ , we can always find a family of hypotheses  $\{h_\mu\} \subset \mathcal{H}$  such that  $h_\mu(x, \cdot)$  take the following values:
Since  $h_\mu(x,y) \in [-\Lambda(x), \Lambda(x)]$ , we have the following constraints on  $\mu$ :
$$
\begin{array}{l} - \Lambda (x) - h (x, y _ {\max }) \leq \mu \leq \Lambda (x) - h (x, y _ {\max }) \\ - \Lambda (x) + h (x, \mathsf {h} (x)) \leq \mu \leq \Lambda (x) + h (x, \mathsf {h} (x). \\ \end{array}
$$
Let  $p_1 = p(x, y_{\max})$ ,  $p_2 = p(x, \mathsf{h}(x))$ ,  $\tau_1 = h(x, \mathsf{h}(x))$  and  $\tau_2 = h(x, y_{\max})$  to simplify the notation. Then, the constraint on  $\mu$  can be expressed as
$$
\mu \in \overline {{C}}, \quad \overline {{C}} = \left[ \max  \left\{\tau_ {1}, - \tau_ {2} \right\} - \Lambda (x), \min  \left\{\tau_ {1}, - \tau_ {2} \right\} + \Lambda (x) \right]
$$
Since  $\max \{\tau_1, -\tau_2\} - \min \{\tau_1, -\tau_2\} = |\tau_1 + \tau_2| \leq |\tau_1| + |\tau_2| \leq 2\Lambda(x)$ ,  $C$  is not an empty set. By the definition of  $h_\mu$ , we have for any  $h \in \mathcal{H}$  and  $x \in \mathcal{X}$ ,
$$
\begin{array}{l} \mathcal {C} _ {\ell^ {\text {c s t n d}}} (h, x) - \inf  _ {\mu \in \overline {{C}}} \mathcal {C} _ {\ell^ {\text {c s t n d}}} (h _ {\mu}, x) \\ = \sup  _ {\mu \in \overline {{C}}} \left\{\left(1 - p _ {1}\right) \left[ \Phi (- \tau_ {2}) - \Phi (- \tau_ {1} + \mu) \right] + \left(1 - p _ {2}\right) \left[ \Phi (- \tau_ {1}) - \Phi (- \tau_ {2} - \mu) \right] \right\} \\ = \sup  _ {\mu \in \overline {{C}}} \left\{\frac {2 - P - p _ {1} + p _ {2}}{2} \left[ \Phi (- \tau_ {2}) - \Phi (- \tau_ {1} + \mu) \right] + \frac {2 - P + p _ {1} - p _ {2}}{2} \left[ \Phi (- \tau_ {1}) - \Phi (- \tau_ {2} - \mu) \right] \right\} \\ (P = p _ {1} + p _ {2} \in [ \frac {1}{n - 1}, 1 ]) \\ = \inf  _ {P \in \left[ \frac {1}{n - 1}, 1 \right]} \inf  _ {\tau_ {1} \geq \max  \left\{\tau_ {2}, 0 \right\}} \sup  _ {\mu \in \overline {{C}}} \left\{\frac {2 - P - p _ {1} + p _ {2}}{2} \left[ \Phi (- \tau_ {2}) - \Phi (- \tau_ {1} + \mu) \right] \right. \\ \left. + \frac {2 - P + p _ {1} - p _ {2}}{2} \left[ \Phi (- \tau_ {1}) - \Phi (- \tau_ {2} - \mu) \right] \right\} \quad (\tau_ {1} \geq 0, \tau_ {2} \leq \tau_ {1}) \\ \geq \inf  _ {P \in \left[ \frac {1}{n - 1}, 1 \right]} \inf  _ {\tau_ {1} \geq \max  \left\{\tau_ {2}, 0 \right\}} \sup  _ {\mu \in C} \left\{\frac {2 - P - p _ {1} + p _ {2}}{2} \left[ \Phi (- \tau_ {2}) - \Phi (- \tau_ {1} + \mu) \right] \right. \\ \left. + \frac {2 - P + p _ {1} - p _ {2}}{2} \left[ \Phi (- \tau_ {1}) - \Phi (- \tau_ {2} - \mu) \right] \right\} \\ (C = \left[ \max  \left\{\tau_ {1}, - \tau_ {2} \right\} - \Lambda_ {\min }, \min  \left\{\tau_ {1}, - \tau_ {2} \right\} + \Lambda_ {\min } \right] \subset \overline {{C}} \text {s i n c e} \Lambda_ {\min } \leq \Lambda (x)) \\ = \inf  _ {P \in \left[ \frac {1}{n - 1}, 1 \right]} \inf  _ {\tau_ {1} \geq \max  \{\tau_ {2}, 0 \}} \left\{\frac {2 - P - p _ {1} + p _ {2}}{2} \Phi (- \tau_ {2}) + \frac {2 - P + p _ {1} - p _ {2}}{2} \Phi (- \tau_ {1}) \right. \\ \left. - \inf  _ {\mu \in C} \left\{\frac {2 - P - p _ {1} + p _ {2}}{2} \Phi (- \tau_ {1} + \mu) + \frac {2 - P + p _ {1} - p _ {2}}{2} \Phi (- \tau_ {2} - \mu) \right\} \right\} \\ = \mathbb {T} ^ {\text {c s t n d}} \left(p _ {1} - p _ {2}\right) \\ = \mathcal {T} ^ {\text {c s t n d}} \left(\Delta \mathcal {C} _ {\ell_ {0 - 1}, \mathcal {H}} (h, x)\right). \tag {by Lemma 1} \\ \end{array}
$$
Note that for  $n = 2$ , an additional constraint  $\tau_{1} + \tau_{2} = 1$  is imposed and the expression can be simplified as
$$
\begin{array}{l} \mathcal {C} _ {\ell \text {c s t n d}} (h, x) - \inf  _ {\mu \in \overline {{C}}} \mathcal {C} _ {\ell \text {c s t n d}} (h _ {\mu}, x) \\ \geq \inf  _ {\tau \geq 0} \sup  _ {\mu \in [ \tau - \Lambda_ {\min }, \tau + \Lambda_ {\min } ]} \left\{\frac {1 - p _ {1} + p _ {2}}{2} \left[ \Phi (\tau) - \Phi (- \tau + \mu) \right] + \frac {1 + p _ {1} - p _ {2}}{2} \left[ \Phi (- \tau) - \Phi (\tau - \mu) \right] \right\} \\ = \mathfrak {T} ^ {\operatorname {c s t n d}} \left(p _ {1} - p _ {2}\right) \\ = \mathcal {T} ^ {\text {c s t n d}} \left(\Delta \mathcal {C} _ {\ell_ {0 - 1}, \mathcal {H}} (h, x)\right). \tag {by Lemma 1} \\ \end{array}
$$
Since  $\mathfrak{T}^{\mathrm{cstnd}}$  is convex, by Jensen's inequality, we obtain for any hypothesis  $h\in \mathcal{H}$  and any distribution,
$$
\begin{array}{l} \mathfrak {T} ^ {\text {c s t n d}} \left(\mathcal {R} _ {\ell_ {0 - 1}} (h) - \mathcal {R} _ {\ell_ {0 - 1}} ^ {*} (\mathcal {H}) + \mathcal {M} _ {\ell_ {0 - 1}} (\mathcal {H})\right) \\ = \mathcal {T} ^ {\text {c s t n d}} \left(\underset {X} {\mathbb {E}} \left[ \Delta \mathcal {C} _ {\ell_ {0 - 1}, \mathcal {H}} (h, x) \right]\right) \\ \leq \mathbb {E} _ {X} \left[ \mathcal {T} ^ {\text {c s t n d}} \left(\Delta \mathcal {C} _ {\ell_ {0 - 1}, \mathcal {H}} (h, x)\right) \right] \\ \leq \mathbb {E} _ {X} \left[ \Delta \mathcal {C} _ {\ell^ {\text {c s t n d}}, \mathcal {H}} (h, x) \right] \\ = \mathcal {R} _ {\ell \mathrm {c s t n d}} (h) - \mathcal {R} _ {\ell \mathrm {c s t n d}} ^ {*} (\mathcal {H}) + \mathcal {M} _ {\ell \mathrm {c s t n d}} (\mathcal {H}). \\ \end{array}
$$
Let  $n = 2$ . For any  $t \in [0,1]$ , we consider the distribution that concentrates on a singleton  $\{x\}$  and satisfies  $p(x,1) = \frac{1 + t}{2}$ ,  $p(x,2) = \frac{1 - t}{2}$ . For any  $\epsilon > 0$ , by the definition of infimum, we can take  $h \in \mathcal{H}$  such that  $h(x,2) = \tau_{\epsilon} \geq 0$  and satisfies
$$
\sup _ {\mu \in [ \tau_ {\epsilon} - \Lambda_ {\min }, \tau_ {\epsilon} + \Lambda_ {\min } ]} \left\{\frac {1 - t}{2} [ \Phi (\tau_ {\epsilon}) - \Phi (- \tau_ {\epsilon} + \mu) ] + \frac {1 + t}{2} [ \Phi (- \tau_ {\epsilon}) - \Phi (\tau_ {\epsilon} - \mu) ] \right\} <   \mathcal {T} ^ {\mathrm {c s t n d}} (t) + \epsilon .
$$
$$
\begin{array}{l} \mathcal {T} ^ {\text {c s t n d}} (t) \leq \mathcal {R} _ {\ell^ {\text {c s t n d}}} (h) - \mathcal {R} _ {\ell^ {\text {c s t n d}}} ^ {*} (\mathcal {H}) + \mathcal {M} _ {\ell^ {\text {c s t n d}}} (\mathcal {H}) \\ = \mathcal {R} _ {\ell \text {c s t n d}} (h) - \mathbb {E} _ {X} \left[ \mathcal {C} _ {\ell \text {c s t n d}} ^ {*} (\mathcal {H}, x) \right] \\ = \mathcal {C} _ {\ell \text {c s t n d}} (h, x) - \mathcal {C} _ {\ell \text {c s t n d}} ^ {*} (\mathcal {H}, x) \\ = \sup  _ {\mu \in \left[ \tau_ {\epsilon} - \Lambda_ {\min }, \tau_ {\epsilon} + \Lambda_ {\min } \right]} \left\{\frac {1 - t}{2} \left[ \Phi \left(\tau_ {\epsilon}\right) - \Phi (- \tau_ {\epsilon} + \mu) \right] + \frac {1 + t}{2} \left[ \Phi (- \tau_ {\epsilon}) - \Phi (\tau_ {\epsilon} - \mu) \right] \right\} \\ <   \mathfrak {T} ^ {\mathrm {c s t n d}} (t) + \epsilon . \\ \end{array}
$$
By letting  $\epsilon \to 0$ , we conclude the proof. The proof for  $n > 2$  directly extends from the case when  $n = 2$ . Indeed, for any  $t \in [0,1]$ , we consider the distribution that concentrates on a singleton  $\{x\}$  and satisfies  $p(x,1) = \frac{1 + t}{2}$ ,  $p(x,2) = \frac{1 - t}{2}$ ,  $p(x,y) = 0, 3 \leq y \leq n$ . For any  $\epsilon > 0$ , by the definition of infimum, we can take  $h \in \mathcal{H}$  such that  $h(x,1) = \tau_{1,\epsilon}$ ,  $h(x,2) = \tau_{2,\epsilon}$ ,  $h(x,3) = 0$ ,  $3 \leq y \leq n$  and satisfies  $\tau_{1,\epsilon} + \tau_{2,\epsilon} = 0$ , and
$$
\begin{array}{l} \inf_{P\in \left[\frac{1}{n - 1},1\right]}\sup_{\mu \in C}\Bigg\{\frac{2 - P - t}{2}\bigl[\Phi \bigl( - \tau_{2,\epsilon}\bigr) - \Phi \bigl( - \tau_{1,\epsilon} + \mu \bigr)\bigr ] + \frac{2 - P + t}{2}\bigl[\Phi \bigl( - \tau_{1,\epsilon}\bigr) - \Phi \bigl( - \tau_{2,\epsilon} - \mu \bigr)\bigr ]\Bigg\} \\ = \sup  _ {\mu \in C} \left\{\frac {1 - t}{2} \left[ \Phi (- \tau_ {2, \epsilon}) - \Phi (- \tau_ {1, \epsilon} + \mu) \right] + \frac {1 + t}{2} \left[ \Phi (- \tau_ {1, \epsilon}) - \Phi (- \tau_ {2, \epsilon} - \mu) \right] \right\} \\ <   \mathfrak {T} ^ {\mathrm {c s t n d}} (t) + \epsilon . \\ \end{array}
$$
$$
\mathfrak {T} ^ {\text {c s t n d}} (t) \leq \mathcal {R} _ {\ell^ {\text {c s t n d}}} (h) - \mathcal {R} _ {\ell^ {\text {c s t n d}}} ^ {*} (\mathcal {H}) + \mathcal {M} _ {\ell^ {\text {c s t n d}}} (\mathcal {H}) <   \mathfrak {T} ^ {\text {c s t n d}} (t) + \epsilon .
$$
F.2 Constrained exponential loss
$$
\mathcal {R} _ {\ell_ {0 - 1}} (h) - \mathcal {R} _ {\ell_ {0 - 1}} ^ {*} (\overline {{\mathcal {H}}}) + \mathcal {M} _ {\ell_ {0 - 1}} (\overline {{\mathcal {H}}}) \leq \Psi^ {- 1} \big (\mathcal {R} _ {\ell \mathrm {c s t n d}} (h) - \mathcal {R} _ {\ell \mathrm {c s t n d}} ^ {*} (\overline {{\mathcal {H}}}) + \mathcal {M} _ {\ell \mathrm {c s t n d}} (\overline {{\mathcal {H}}}) \big)
$$
Proof. For  $n = 2$ , plugging in  $\Phi(t) = e^{-t}$  in Theorem 12, gives
$$
\overline {{\mathcal {T}}} ^ {\mathrm {c s t n d}} (t) = \inf  _ {\tau \geq 0} \sup  _ {\mu \in [ \tau - \Lambda_ {\min }, \tau + \Lambda_ {\min } ]} \left\{\frac {1 - t}{2} [ e ^ {- \tau} - e ^ {\tau - \mu} ] + \frac {1 + t}{2} [ e ^ {\tau} - e ^ {- \tau + \mu} ] \right\}.
$$
By differentiating with respect to  $\tau$ , we can see that the infimum is achieved when  $\tau = 0$  modulo some elementary analysis. Thus,  $\overline{\mathcal{T}}^{\mathrm{cstnd}}$  can be reformulated as
$$
\begin{array}{l} \overline {{\mathcal {T}}} ^ {\text {c s t n d}} = \sup  _ {\mu \in [ - \Lambda_ {\min }, \Lambda_ {\min } ]} \left\{\frac {1 - t}{2} \left[ 1 - e ^ {- \mu} \right] + \frac {1 + t}{2} \left[ 1 - e ^ {\mu} \right] \right\} \\ = 1 + \sup  _ {\mu \in \left[ - \Lambda_ {\min }, \Lambda_ {\min } \right]} g (\mu). \\ \end{array}
$$
where  $g(\mu) = -\frac{1 - t}{2} e^{-\mu} - \frac{1 + t}{2} e^{\mu}$ . Since  $g$  is continuous, it attains its supremum over a compact set. Note that  $g$  is concave and differentiable. In view of that, the maximum over the open set  $(-\infty, +\infty)$  can be obtained by setting its gradient to zero. Differentiate  $g(\mu)$  to optimize, we obtain
$$
g \left(\mu^ {*}\right) = 0, \quad \mu^ {*} = \frac {1}{2} \log \frac {1 - t}{1 + t}
$$
Moreover, by the concavity,  $g(\mu)$  is non-increasing when  $\mu \geq \mu^{*}$ . Since  $\mu^{*} \leq 0$  and  $\Lambda_{\min} \geq 0$ , we have
$$
\mu^ {*} \leq 0 \leq \Lambda_ {\min }
$$
In view of the constraint, if  $\mu^{*} \geq -\Lambda_{\mathrm{min}}$ , the maximum is achieved by  $\mu = \mu^{*}$ . Otherwise, if  $\mu^{*} < -\Lambda_{\mathrm{min}}$ , since  $g(\mu)$  is non-increasing when  $\mu \geq \mu^{*}$ , the maximum is achieved by  $\mu = -\Lambda_{\mathrm{min}}$ . Since  $\mu^{*} \geq -\Lambda_{\mathrm{min}}$  is equivalent to  $t \leq \frac{e^{2\Lambda_{\mathrm{min}} - 1}}{e^{2\Lambda_{\mathrm{min}}} + 1}$ , the maximum can be expressed as
$$
\max  _ {\mu \in [ - \Lambda_ {\min }, \Lambda_ {\min } ]} g (\mu) = \left\{ \begin{array}{l l} g (\mu^ {*}) & t \leq \frac {e ^ {2 \Lambda_ {\min}} - 1}{e ^ {2 \Lambda_ {\min}} + 1} \\ g (- \Lambda_ {\min }) & \text {o t h e r w i s e} \end{array} \right.
$$
$$
g \left(\mu^ {*}\right) = - \sqrt {1 - t ^ {2}}
$$
$$
g \left(- \Lambda_ {\min }\right) = - \frac {1 - t}{2} e ^ {\Lambda_ {\min }} - \frac {1 + t}{2} e ^ {- \Lambda_ {\min }}.
$$
Then, if  $t \leq \frac{e^{2\Lambda_{\min}} - 1}{e^{2\Lambda_{\min}} + 1}$ , we obtain
$$
\overline {{\mathcal {T}}} ^ {\mathrm {c s t n d}} = 1 - \sqrt {1 - t ^ {2}}.
$$
$$
\begin{array}{l} \overline {{\mathcal {T}}} ^ {\mathrm {c s t n d}} = 1 - \frac {1 - t}{2} e ^ {\Lambda_ {\min }} - \frac {1 + t}{2} e ^ {- \Lambda_ {\min }} \\ = \frac {t}{2} \left(e ^ {\Lambda_ {\min}} - e ^ {- \Lambda_ {\min}}\right) + \frac {2 - e ^ {\Lambda_ {\min}} - e ^ {- \Lambda_ {\min}}}{2}. \\ \end{array}
$$
For  $n > 2$ , plugging in  $\Phi(t) = e^{-t}$  in Theorem 12, gives
$$
\overline {{\mathcal {T}}} ^ {\mathrm {c s t n d}} (t) = \inf  _ {P \in \left[ \frac {1}{n - 1}, 1 \right]} \inf  _ {\tau_ {1} \geq \max  \{\tau_ {2}, 0 \}} \sup  _ {\mu \in C} \Bigg \{\frac {2 - P - t}{2} \big [ e ^ {\tau_ {2}} - e ^ {\tau_ {1} - \mu} \big ] + \frac {2 - P + t}{2} \big [ e ^ {\tau_ {1}} - e ^ {\tau_ {2} + \mu} \big ] \Bigg \}.
$$
where  $C = \left[\max \{\tau_1, -\tau_2\} - \Lambda_{\min}, \min \{\tau_1, -\tau_2\} + \Lambda_{\min}\right]$ . By differentiating with respect to  $\tau_2$  and  $P$ , we can see that the infimum is achieved when  $\tau_2 = \tau_1 = 0$  and  $P = 1$  modulo some elementary analysis. Thus,  $\overline{\mathcal{T}}^{\mathrm{cstnd}}$  can be reformulated as
$$
\begin{array}{l} \overline {{\mathcal {T}}} ^ {\mathrm {c s t n d}} = \sup  _ {\mu \in C} \left\{\frac {1 - t}{2} \left[ 1 - e ^ {- \mu} \right] + \frac {1 + t}{2} \left[ 1 - e ^ {\mu} \right] \right\} \\ = 1 + \sup_{\mu \in C}g(\mu). \\ \end{array}
$$
where  $C = \left[-\Lambda_{\min}, \Lambda_{\min}\right]$  and  $g(\mu) = -\frac{1 - t}{2} e^{-\mu} - \frac{1 + t}{2} e^{\mu}$ . Since  $g$  is continuous, it attains its supremum over a compact set. Note that  $g$  is concave and differentiable. In view of that, the maximum over the open set  $(-\infty, +\infty)$  can be obtained by setting its gradient to zero. Differentiate  $g(\mu)$  to optimize, we obtain
$$
g (\mu^ {*}) = 0, \quad \mu^ {*} = \frac {1}{2} \log \frac {1 - t}{1 + t}
$$
$$
\mu^ {*} \leq 0 \leq \Lambda_ {\mathrm {m i n}}
$$
In view of the constraint, if  $\mu^{*} \geq -\Lambda_{\mathrm{min}}$ , the maximum is achieved by  $\mu = \mu^{*}$ . Otherwise, if  $\mu^{*} < -\Lambda_{\mathrm{min}}$ , since  $g(\mu)$  is non-increasing when  $\mu \geq \mu^{*}$ , the maximum is achieved by  $\mu = -\Lambda_{\mathrm{min}}$ . Since  $\mu^{*} \geq -\Lambda_{\mathrm{min}}$  is equivalent to  $t \leq \frac{e^{2\Lambda_{\mathrm{min}}} - 1}{e^{2\Lambda_{\mathrm{min}}} + 1}$ , the maximum can be expressed as
$$
\max  _ {\mu \in [ - \Lambda_ {\min }, \Lambda_ {\min } ]} g (\mu) = \left\{ \begin{array}{l l} g (\mu^ {*}) & t \leq \frac {e ^ {2 \Lambda_ {\min } - 1}}{e ^ {2 \Lambda_ {\min } + 1}} \\ g (- \Lambda_ {\min }) & \text {o t h e r w i s e} \end{array} \right.
$$
$$
\begin{array}{l} \overline {{\mathcal {T}}} ^ {\mathrm {c s t n d}} = 1 - \frac {1 - t}{2} e ^ {\Lambda_ {\mathrm {m i n}}} - \frac {1 + t}{2} e ^ {- \Lambda_ {\mathrm {m i n}}} \\ = \frac {t}{2} \left(e ^ {\Lambda_ {\min }} - e ^ {- \Lambda_ {\min }}\right) + \frac {2 - e ^ {\Lambda_ {\min }} - e ^ {- \Lambda_ {\min }}}{2}. \\ \end{array}
$$
Since  $\overline{\mathfrak{T}}^{\mathrm{cstnd}}$  is convex, by Theorem 12, for any  $h\in \overline{\mathcal{H}}$  and any distribution,
$$
\Psi \big (t \big) = \left\{ \begin{array}{l l} 1 - \sqrt {1 - t ^ {2}} & t \leq \frac {e ^ {2 \Lambda_ {\operatorname* {m i n}}} - 1}{e ^ {2 \Lambda_ {\operatorname* {m i n}}} + 1} \\ \frac {t}{2} \big (e ^ {\Lambda_ {\operatorname* {m i n}}} - e ^ {- \Lambda_ {\operatorname* {m i n}}} \big) + \frac {2 - e ^ {\Lambda_ {\operatorname* {m i n}}} - e ^ {- \Lambda_ {\operatorname* {m i n}}}}{2} & \text {o t h e r w i s e .} \end{array} \right.
$$