Alzheimer’s disease is a progressive neurodegenerative disorder, and identifying it in its early stages is essential for enabling timely therapeutic intervention. This study investigates the effectiveness of five state-of-the-art deep pretrained convolutional neural networks—VGG19, InceptionV3, ResNet50, InceptionResNetV2, and EfficientNetB0—in classifying Alzheimer’s disease using magnetic resonance imaging (MRI) data. Both shallow tuning and deep tuning of the models were explored for effective learning. Furthermore, we fine-tune VGG19 by unfreezing only Blocks 4 and 5, allowing adaptation of high-level features while preserving low-level representations. A cosine annealing learning rate scheduler with warm restarts dynamically adjusts the learning rate to accelerate convergence and mitigate overfitting. The five deep learning models and two machine learning models–SVM and LDA–were trained on brain MRI images from a subset of the ADNI dataset, available online on Kaggle. The dataset comprises three classes representing different stages of Alzheimer’s disease and a fourth class representing the control (healthy) group. The three disease-related classes, being minority classes, were merged into a single class, rendering the classification problem binary with a balanced class distribution (diseased versus healthy group). The results indicate that the selectively fine-tuned VGG19 model, trained using a cosine annealing learning rate schedule with warm restarts, achieved superior performance compared to state-of-the-art methods. It attained an accuracy of 97.35%, precision of 0.9751, recall of 0.9706, F1-score of 0.9729, and an AUC of 0.9734, outperforming existing approaches by a significant margin.