{"url":"/method/huber-loss","slug":"huber-loss","name":"Huber loss","full_name":"Huber loss","full_name_withheld":false,"description_markdown":"The Huber loss function describes the penalty incurred by an estimation procedure f. Huber (1964) defines the loss function piecewise by[1]\r\n\r\n    L δ ( a ) = { 1 2 a 2 for  | a | ≤ δ , δ ⋅ ( | a | − 1 2 δ ) , otherwise. {\\displaystyle L_{\\delta }(a)={\\begin{cases}{\\frac {1}{2}}{a^{2}}&{\\text{for }}|a|\\leq \\delta ,\\\\\\delta \\cdot \\left(|a|-{\\frac {1}{2}}\\delta \\right),&{\\text{otherwise.}}\\end{cases}}}\r\n\r\nThis function is quadratic for small values of a, and linear for large values, with equal values and slopes of the different sections at the two points where | a | = δ |a|=\\delta . The variable a often refers to the residuals, that is to the difference between the observed and predicted values a = y − f ( x ) a=y-f(x), so the former can be expanded to[2]\r\n\r\n    L δ ( y , f ( x ) ) = { 1 2 ( y − f ( x ) ) 2 for  | y − f ( x ) | ≤ δ , δ   ⋅ ( | y − f ( x ) | − 1 2 δ ) , otherwise. {\\displaystyle L_{\\delta }(y,f(x))={\\begin{cases}{\\frac {1}{2}}(y-f(x))^{2}&{\\text{for }}|y-f(x)|\\leq \\delta ,\\\\\\delta \\ \\cdot \\left(|y-f(x)|-{\\frac {1}{2}}\\delta \\right),&{\\text{otherwise.}}\\end{cases}}}\r\n\r\nThe Huber loss is the convolution of the absolute value function with the rectangular function, scaled and translated. Thus it \"smoothens out\" the former's corner at the origin. \r\n\r\n.. math::\r\n        \\ell(x, y) = L = \\{l_1, ..., l_N\\}^T\r\n\r\n    with\r\n\r\n    .. math::\r\n        l_n = \\begin{cases}\r\n        0.5 (x_n - y_n)^2, & \\text{if } |x_n - y_n| < delta \\\\\r\n        delta * (|x_n - y_n| - 0.5 * delta), & \\text{otherwise }\r\n        \\end{cases}","description_state":"present","introduced_year":null,"introduced_by":{"title":null,"paper":null,"first_author":null,"n_authors":0,"url_abs":null,"archive_paper_url":null},"source":{"url":null,"title":null,"url_on_a_paper_host":false},"code_snippet_url":null,"code_snippet_url_on_a_code_host":false,"categories":[{"area":"General","area_id":"general","collection":"Loss Functions","url":"/methods/category/loss-functions","pwc_aliases":[]}],"n_papers_tagged":77,"archive_num_papers":77,"papers_newest_first":[{"paper":null,"title":"SHORE: A Long-term User Lifetime Value Prediction Model in Digital 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