Minimum-Norm Solutions in Linear Models
When you work with linear models, you often encounter systems of equations that do not have a unique solution. This happens in underdetermined settings, where the number of unknowns (parameters) exceeds the number of equations (data points). For example, if you have a data matrix X with shape (n,d) where n<d, and you want to solve Xw=y for the parameter vector w, there are infinitely many possible w that fit the data exactly because the system does not constrain all degrees of freedom. This raises a fundamental question: if there are many solutions, which one will your learning algorithm find?
A key result is that certain algorithms, such as gradient descent applied to underdetermined linear systems, always converge to the solution with the smallest Euclidean norm (the minimum-norm solution). This minimum-norm solution is unique among all solutions that fit the data exactly, and the algorithm's implicit bias leads it to select this particular solution without any explicit regularization.
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Can you explain how regularization helps in underdetermined systems?
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When you work with linear models, you often encounter systems of equations that do not have a unique solution. This happens in underdetermined settings, where the number of unknowns (parameters) exceeds the number of equations (data points). For example, if you have a data matrix X with shape (n,d) where n<d, and you want to solve Xw=y for the parameter vector w, there are infinitely many possible w that fit the data exactly because the system does not constrain all degrees of freedom. This raises a fundamental question: if there are many solutions, which one will your learning algorithm find?
A key result is that certain algorithms, such as gradient descent applied to underdetermined linear systems, always converge to the solution with the smallest Euclidean norm (the minimum-norm solution). This minimum-norm solution is unique among all solutions that fit the data exactly, and the algorithm's implicit bias leads it to select this particular solution without any explicit regularization.
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