# Optimization Strategies in Quantum Machine Learning: A Performance Analysis
## 5 Comparative Analysis
### Performance Evaluation  
To comprehensively assess the effectiveness of predictive models optimized using different optimizers, a range of performance metrics--accuracy, precision, recall, and F1 score--were employed in this study. These metrics provide a quantitative evaluation of the model's classification performance and are derived from the confusion matrix, which serves as the foundation for their computation.  
The confusion matrix is a fundamental tool for evaluating classification models by organizing predictions into four categories:  
\[\begin{bmatrix}TP&FP\\ FN&TN\end{bmatrix} \tag{1}\]  
where  
True positive (TP): the number of instances correctly classified as positive;  
False positive (FP): the number of instances incorrectly classified as positive;  
True negative (TN): the number of instances correctly classified as negative;  
False negative (FN): the number of instances incorrectly classified as negative.  
Using these components, the following performance metrics are calculated.  
Accuracy measures the proportion of correctly classified instances (both positive and negative) out of the total number of instances. It is defined as follows:  
\[Accuracy=\frac{TP+TN}{\text{TP}+\text{TN}+\text{FP}+\text{FN}} \tag{2}\]  
While accuracy provides an overall sense of model performance, it may not be sufficient for imbalanced datasets where one class dominates the other.  
Precision evaluates the proportion of true positive predictions among all instances predicted as positive. It is particularly useful in scenarios where minimizing false positives is critical. Precision is calculated as follows:  
\[Precision=\frac{TP}{\text{TP}+\text{FP}} \tag{3}\]  
High precision indicates that the model makes accurate positive predictions with minimal false positives.  
Recall (Sensitivity)  
Recall, also known as sensitivity, measures the proportion of actual positive instances that are correctly identified by the model. It is especially important in applications where detecting all positive cases is prioritized. Recall is calculated as follows:  
\[Recall=\frac{TP}{\text{TP}+\text{FN}} \tag{4}\]  
A high recall value signifies that the model effectively captures most of the positive instances.  
The F1 score is the harmonic mean of precision and recall, providing a balanced measure that accounts for both false positives and false negatives. It is particularly valuable when there is an uneven class distribution. The F1 score is computed as follows:  
\[F1-Score=2\times\frac{Precision\times Recall}{Precision+Recall} \tag{5}\]  
The F1 score ranges between 0 and 1, with higher values indicating better overall performance [46].  
These metrics collectively offer a nuanced understanding of a model's predictive capabilities. By leveraging the confusion matrix as the basis for computation, they enable a detailed assessment of the trade-offs between correct and incorrect predictions across different classes. The subsequent sections present the results of these metrics for models optimized using various algorithms, providing insights into their relative strengths and limitations.  
This section evaluates the performance of three optimizers--COBYLA, L-BFGS-B, and ADAM--on the Cleveland dataset. The analysis is based on the confusion matrices derived from each optimizer's predictions, followed by the computation of key performance metrics: accuracy, precision, recall (sensitivity), and F1 score. Additionally, the training times are reported to assess the optimizers' computational efficiency.