{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-z-loss-a-shift-and-scale-invariant","title":"The Z-loss: a shift and scale invariant classification loss belonging to the Spherical Family","arxiv_id":"1604.08859","date":"2016-04-29","proceeding":null,"authors":["Alexandre de Brébisson","Pascal Vincent"],"abstract":"Despite being the standard loss function to train multi-class neural\nnetworks, the log-softmax has two potential limitations. First, it involves\ncomputations that scale linearly with the number of output classes, which can\nrestrict the size of problems we are able to tackle with current hardware.\nSecond, it remains unclear how close it matches the task loss such as the top-k\nerror rate or other non-differentiable evaluation metrics which we aim to\noptimize ultimately. In this paper, we introduce an alternative classification\nloss function, the Z-loss, which is designed to address these two issues.\nUnlike the log-softmax, it has the desirable property of belonging to the\nspherical loss family (Vincent et al., 2015), a class of loss functions for\nwhich training can be performed very efficiently with a complexity independent\nof the number of output classes. We show experimentally that it significantly\noutperforms the other spherical loss functions previously investigated.\nFurthermore, we show on a word language modeling task that it also outperforms\nthe log-softmax with respect to certain ranking scores, such as top-k scores,\nsuggesting that the Z-loss has the flexibility to better match the task loss.\nThese qualities thus makes the Z-loss an appealing candidate to train very\nefficiently large output networks such as word-language models or other extreme\nclassification problems. On the One Billion Word (Chelba et al., 2014) dataset,\nwe are able to train a model with the Z-loss 40 times faster than the\nlog-softmax and more than 4 times faster than the hierarchical softmax.","url_abs":"http://arxiv.org/abs/1604.08859v2","url_pdf":"http://arxiv.org/pdf/1604.08859v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"the-z-loss-a-shift-and-scale-invariant","repo_url":"https://github.com/pascal20100/factored_output_layer","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"classification","task_name":"General Classification"},{"task_slug":"language-modeling","task_name":"Language Modeling"},{"task_slug":"language-modelling","task_name":"Language Modelling"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1604.08859","atlas_url":"https://app.syntology.ai/?focus=1604.08859","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}