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Breast cancer is one of the most common cancer in women, with more than 1,300,000 cases and 450,000 deaths each year worldwide. Recent studies show that early breast cancer detection, along with suitable treatment, could significantly reduce breast cancer death rates in the long-term. While the consequences of a false positive diagnosis can be psychologically and socioeconomically burdensome, the result of a false negative diagnosis can be devastating, especially in terms of health detriment. In this context, the false positive and false negative rates commonly achieved by radiologists are extremely arduous to estimate and control, and some authors have estimated figures of up to 20% of total diagnoses or more. Novel ideas in computer-assisted diagnosis have been prompted by the introduction of deep learning techniques in general and of convolutional neural networks (CNN) in particular. In this paper, we design and validate an ad-hoc CNN architecture specialized in breast lesion classification and heuristically explore possible parameter combinations and architecture styles in order to propose a model selection criterion which can pose the emphasis on reducing false negatives while still retaining acceptable accuracy. We achieve good classification performance on the validation and test set, demonstrating how an ad-hoc, random initialization CNN architecture can provide practical aid in the classification and staging of breast cancer.