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Scaling laws of optimization – Machine Learning Research Blog

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Scaling laws have been one of the key achievements of theoretical analysis in various fields of applied mathematics and computer science, answering the following key question: How fast does my method or my algorithm converge as a function of (potentially partially) observable problem parameters. For supervised machine learning and statistics, probably the simplest and oldest of all such scaling laws is the classical rate σ2d/n for ordinary least-squares (where d is the number of features, n the number of observations, and σ2 the noise variance). This has been extended over the years in the statistics community for various non-linear function estimation, with excess risks for regression essentially proportional to n−2s/(2s+d), where s is now the order of derivatives that exist for the optimal prediction function (see, e.g., these two nice books [1, 2]). Such scaling laws also have a long history in the part of statistical physics that looks into high-dimensional statistical estimation p

Scaling laws have been one of the key achievements of theoretical analysis in various fields of applied mathematics and computer science, answering the following key question: How fast does my method or my algorithm converge as a function of (potentially partially) observable problem parameters. For supervised machine learning and statistics, probably the simplest and oldest of all such scaling laws is the classical rate \(\sigma^2 d / n\) for ordinary least-squares (where \(d\) is the number of features, \(n\) the number of observations, and \(\sigma^2\) the noise variance). This has been ext

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