## 3 Proposed Methodology
Similarly, the \(\sigma\) establishes our function's total difference from its mean value. In short,\(\sigma\) and \(\ell\) define the function's horizontal and vertical domains. \[ \begin{split} tpe(\eta\|\varkappa)&=\frac{p( \varkappa\|\eta)p(\eta)}{p(\varkappa)}\\ tpe(\varkappa\|(\eta))&=\begin{Bmatrix}l(\varkappa) &if\eta<\eta^{*}\\ g(\varkappa)&if\eta\geq\eta^{*}\end{Bmatrix}\\ EI_{\eta^{*}}(\varkappa)&=\int_{-\infty}^{\eta^{*}}(\eta^{*}-\eta)p( \eta\mid\varkappa)\\ d(\eta)&=\int_{-\infty}^{\eta^{*}}(\eta^{*}-\eta)\frac{p( \varkappa\mid\eta)p(\eta)}{p(\varkappa)}d(\eta)\\ \tau&=p(\eta<\eta^{*})\\ p(\varkappa)&=\int_{F}p(\varkappa\mid\eta)p(\eta)\\ d(\eta)&=\tau l(\varkappa)+(1-\tau)g(\varkappa)\\ \alpha&=\frac{\tau\eta^{*}l(\varkappa)-l(\varkappa)\int_{-\infty}^{\eta^{*} }p(\eta)d(\eta)}{\tau l(\varkappa)+(1-\tau)g(\varkappa)}\\ \beta&=\tau+\frac{g(\varkappa)}{l(\varkappa)}(1-\tau)\\ El_{\eta^{*}}(\varkappa)&=\alpha\propto\beta^{-1}\end{split}  \tag{6}\]  
Upper Confidence Bound (UCB) comprises explicit exploitation \(\mu(\varkappa)\) and exploration \(\sigma(\varkappa)\) components and is perhaps as straightforward as an acquisition function can acquire (7). \[a(\varkappa;\lambda)=\mu(\varkappa)+\lambda\sigma(\varkappa) \tag{7}\]  
The exploitation vs. exploration decision is simple and easier to tweak with upper confidence bound via the hyperparameter\(\lambda\). The upper confidence bound is a weighted total of the anticipated efficiency recorded by \(\mu(\varkappa)\) the Gaussian Process and the ambiguity represented by the gaussian process's standard error\(\sigma(\varkappa)\). When \(\lambda\) is modest, AHT will favor alternatives that are likely to function well, i.e., have a large \(\mu(\varkappa)\). On the opposite, when \(\lambda\) is big, AHT encourages the discovery of previously unexplored portions of the solution space. The Surrogate Model (SM) and the Acquisition Function (AF) are critical components of an Adaptive Hyperparameter Tuning (AHT) proposed algorithm. Surrogate models are frequently Gaussian Processes (GP) that can accommodate recorded datasets while quantifying the ambiguity of unseen regions. As a result, the surrogate model is to attempt to estimate the mysterious black-box function \(\mathbb{F}(\varkappa)\). The AF then \"reviews\" the surrogate model to identify which regions of the scope of \(\mathbb{F}(\varkappa)\) are worthy of utilizing and which regions are worth investigating. As a result, the acquisition function has a great premium in locations in which \(\mathbb{F}(\varkappa)\) is ideal, or in regions where we have still not examined. On the opposite, the acquisition function expected score is modest in places where \(\mathbb{F}(\varkappa)\) is unsatisfactory or where we have recently tested. We determine the next wild estimate \(\mathbb{F}(\varkappa)\) to attempt by identifying the \(\varkappa\) that maximizes the AF instead of explicitly maximizing \(\mathbb{F}(\varkappa)\), whose analytical version we do not know, we instead maximize the acquisition function which is considerably simpler to perform and significantly less costly. From the above mentioned algorithms, we apply the Tree Parzen Estimator, and the output of the algorithm is forward to the selection function. In this step, three common algorithms (Probability of Improvement (PI), Lower/Upper Confidence Bound (L/UCB),and Expected Improvement (EI)) are used for calculating expected improvements. The mathematical calculation of the most commonly used algorithm is depicted in Equation (8). We want to maximize f(\(\varkappa\)), and the best option we have so far is \(\varkappa\). Then we can describe \"improvement,\" I(\(\varkappa\)) in Equation (8). As a result, if the incoming \(\varkappa\) we are considering has an underlying quantity f(\(\varkappa\)) that is smaller than f(\(\varkappa^{*}\)), f(\(\varkappa\)) - f(\(\varkappa*\)) is negligible. Therefore, we are not getting any better, because the above calculation produces 0 because the largest amount among any negative integer and 0 is 0.