## 2 \(\varepsilon\)-In SVR and Its Dual Problem
### The Nonsmooth Version of \(\varepsilon\)-In SVR
: _The distribution defined in (15) becomes equivalent to the well-known distributions for some values of tunable parameters such that_  
\[\lim_{\eta_{1}=\sigma\eta_{2}\to\infty}p(x;\eta_{1},\eta_{2},\varepsilon)= \frac{1}{2\sigma+2\varepsilon}e^{-\frac{|x|_{\varepsilon}}{\sigma}} \tag{16}\]  
\[\lim_{\eta_{1}=\sigma^{2}\eta_{2}^{2}\to 0}p(x;\eta_{1},\eta_{2},\varepsilon)= \frac{1}{\sqrt{2\pi\sigma^{2}+2\varepsilon}}e^{-\frac{(x)^{2}_{\sigma}}{2 \sigma^{2}}} \tag{17}\]  
\[p(x;\eta_{1}=1,\eta_{2}=\frac{\pi}{2\sigma},\varepsilon)=\frac{1}{2\sigma+2 \varepsilon}sech_{\varepsilon}(x_{r},\frac{\pi}{2\sigma}) \tag{18}\]  
_where (16), (17) and (18) are equivalent to the Laplace, Gaussian and hyperbolic secant distributions for \(\varepsilon=0\), respectively._  
Proof. : Equation (15) can be rewritten as follows:  
\[p(x;\eta_{1},\eta_{2},\varepsilon)=\frac{\eta_{2}}{B(\frac{1}{2\eta_{1}}, \frac{1}{2})+2\varepsilon\eta_{2}}e^{-l_{\varepsilon}(x\eta_{1},\eta_{2})}. \tag{19}\]  
Substituting \(\lim_{\eta_{1}=\sigma\eta_{2}\to\infty}\frac{\eta_{2}}{B(\frac{1}{2\eta_{1}}, \frac{1}{2})+2\varepsilon\eta_{2}}=\frac{1}{2\sigma+2\varepsilon}\) and \(\lim_{\eta_{1}=\sigma\eta_{2}\to\infty}l_{\varepsilon}(x;\eta_{1},\eta_{2})= \frac{|x|_{\varepsilon}}{\sigma}\) into (19), it is obvious that (16) holds. Employing [PERSON]'s approximation of the Beta function, it follows that \(\lim_{\eta_{1}\to 0}B(\frac{1}{2\eta_{1}},\frac{1}{2})=\sqrt{2\pi\eta_{1}}\). Substituting this into (19) and considering \(\lim_{\eta_{1}=\sigma^{2}\eta_{2}^{2}\to 0}l_{\varepsilon}(x;\eta_{1},\eta_{2})= \frac{(x)^{2}_{\sigma}}{2\sigma^{2}}\), it is obvious that (17) holds. Equation (18) is obtained by substituting \(\eta_{1}=1\) and \(\eta_{2}=\frac{\pi}{2\sigma}\) into (15). Without loss of generality, \(\eta_{2}\) can be chosen as \(\eta_{2}=\sqrt{\frac{1}{2}\psi_{1}(\frac{1}{2\eta_{1}})}\), which results in the standardized PHS density functions described in [49], to take advantage of tuning only one parameter. Benefiting from this research, instead of tuning two parameters \(\eta_{1}\) and \(\eta_{2}\), it is reduced to tune only one parameter \(\eta_{1}\) which continues to cover the Vapnik's and \(\epsilon\)-insensitive \(l_{2}\) losses for the limit values such that  
\[\lim_{\eta_{1}\to\infty}l_{\epsilon}\Big{(}x;\eta_{1},\eta_{2}=\sqrt{\frac{1 }{2}\psi_{1}(\frac{1}{2\eta_{1}})}\Big{)}=\sqrt{2}|x|_{\epsilon} \tag{20}\]  
and  
\[\lim_{\eta_{1}\to 0}l_{\epsilon}(x;\eta_{1},\eta_{2}=\sqrt{\frac{1}{2}\psi_{1}( \frac{1}{2\eta_{1}})})=\frac{(x)_{\epsilon}^{2}}{2} \tag{21}\]  
[PERSON] et al. [50] states that, if a kernel function is bounded and its first derivative of the loss function is bounded, then the influence function is also bounded. This makes the Incosh loss function (9) attractive for building robust estimators. The derivative of the Incosh loss function is bounded by \(\frac{\eta_{2}}{\eta_{1}}\). \(\eta_{2}\) is chosen as \(\eta_{2}=\sqrt{\frac{1}{2}\psi_{1}(\frac{1}{2\eta_{1}})}\) where \(\psi_{1}(. )\) is the trigamma function. Since for \(x\geq 0\) it is known that \(\frac{1}{x}+\frac{1}{2x^{2}}\leq\psi_{1}(x)\leq\frac{1}{x}+\frac{1}{x^{2}}\), this bound becomes \(\frac{\eta_{2}}{\eta_{1}}\leq\sqrt{2+\frac{1}{\eta_{1}}}\). This reveals how the parameter \(\eta_{1}\) controls robustness: a small \(\eta_{1}\) can lead to a large influence function, making the estimator more susceptible to outliers. Conversely, a large \(\eta_{1}\) keeps the influence function small, leading to robustness. A bounded influence function signifies that there is a limit to how much a single outlier can affect the overall estimate derived by the model. This characteristic is directly linked to robustness. Since our loss function has a bounded influence function, it suggests the model is less susceptible to the negative effects of outliers, contributing to its overall robustness.