Achieving reliable observations of avalanche debris is crucial for many applications including avalanche forecasting. The ability to continuously monitor the avalanche activity, in space and time, would provide indicators on the potential instability of the snowpack and would allow a better characterization of avalanche risk periods and zones. In this work, we use Sentinel-1 SAR (synthetic aperture radar) data and an independent in-situ avalanche inventory (ground truth) to automatically detect avalanche debris in the French Alps during the remarkable winter season 2017-18. Convolutional neural networks are applied on SAR image patches to locate avalanche debris signatures. We are able to successfully locate new avalanche deposits with as much as 77% confidence on the most susceptible mountain zone (compared to 53% with a baseline method). One of the challenges of this study is to make an efficient use of remote sensing measurements on a complex terrain. It explores the following questions: to what extent can deep learning methods improve the detection of avalanche deposits and help us to derive relevant avalanche activity statistics at different scales (in time and space) that could be useful for a large number of users (researchers, forecasters, government operators)?