Prostate cancer is the most prevalent cancer among men in Western countries, with 1.1 million new diagnoses every year. The gold standard for the diagnosis of prostate cancer is a pathologists' evaluation of prostate tissue. To potentially assist pathologists deep-learning-based cancer detection systems have been developed. Many of the state-of-the-art models are patch-based convolutional neural networks, as the use of entire scanned slides is hampered by memory limitations on accelerator cards. Patch-based systems typically require detailed, pixel-level annotations for effective training. However, such annotations are seldom readily available, in contrast to the clinical reports of pathologists, which contain slide-level labels. As such, developing algorithms which do not require manual pixel-wise annotations, but can learn using only the clinical report would be a significant advancement for the field. In this paper, we propose to use a streaming implementation of convolutional layers, to train a modern CNN (ResNet-34) with 21 million parameters end-to-end on 4712 prostate biopsies. The method enables the use of entire biopsy images at high-resolution directly by reducing the GPU memory requirements by 2.4 TB. We show that modern CNNs, trained using our streaming approach, can extract meaningful features from high-resolution images without additional heuristics, reaching similar performance as state-of-the-art patch-based and multiple-instance learning methods. By circumventing the need for manual annotations, this approach can function as a blueprint for other tasks in histopathological diagnosis. The source code to reproduce the streaming models is available at https://github.com/DIAGNijmegen/pathology-streaming-pipeline.