Generalization in Overparameterized Linear Models
When you train a linear model with more parameters than data points — a situation called overparameterization — the model can fit the training data perfectly, achieving zero error. This seems, at first glance, to contradict classical generalization theory, which suggests that models with too many parameters are likely to overfit, memorizing the training data and failing to generalize to new examples. Yet, in practice, overparameterized linear models often generalize surprisingly well. To understand this, recall the concepts of minimum-norm and maximum-margin solutions discussed previously. When fitting linear models with more parameters than constraints, there are infinitely many solutions that fit the data exactly. However, standard training algorithms like gradient descent tend to select particular solutions — such as the one with the smallest Euclidean norm — without any explicit regularization term. This selection is an example of implicit bias: the algorithm's preference for certain solutions, which turns out to have a profound impact on generalization.
When there are more parameters than data points, a linear model can fit the training data in infinitely many ways. However, not all solutions are equally simple. Algorithms like gradient descent tend to find the simplest solution that still fits the data — often the one with the smallest weights (minimum norm). This simplicity acts like an invisible form of regularization, favoring solutions that are less likely to overfit and more likely to generalize to new data.
In mathematical terms, if you use gradient descent to minimize the squared loss in an overparameterized linear model, the algorithm converges to the minimum-norm solution among all possible interpolating solutions. This minimum-norm solution often has desirable generalization properties, especially when the data is not too noisy and the true relationship is close to linear. The implicit bias of the algorithm, therefore, guides the model toward solutions that generalize well, even in the absence of explicit regularization.
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When you train a linear model with more parameters than data points — a situation called overparameterization — the model can fit the training data perfectly, achieving zero error. This seems, at first glance, to contradict classical generalization theory, which suggests that models with too many parameters are likely to overfit, memorizing the training data and failing to generalize to new examples. Yet, in practice, overparameterized linear models often generalize surprisingly well. To understand this, recall the concepts of minimum-norm and maximum-margin solutions discussed previously. When fitting linear models with more parameters than constraints, there are infinitely many solutions that fit the data exactly. However, standard training algorithms like gradient descent tend to select particular solutions — such as the one with the smallest Euclidean norm — without any explicit regularization term. This selection is an example of implicit bias: the algorithm's preference for certain solutions, which turns out to have a profound impact on generalization.
When there are more parameters than data points, a linear model can fit the training data in infinitely many ways. However, not all solutions are equally simple. Algorithms like gradient descent tend to find the simplest solution that still fits the data — often the one with the smallest weights (minimum norm). This simplicity acts like an invisible form of regularization, favoring solutions that are less likely to overfit and more likely to generalize to new data.
In mathematical terms, if you use gradient descent to minimize the squared loss in an overparameterized linear model, the algorithm converges to the minimum-norm solution among all possible interpolating solutions. This minimum-norm solution often has desirable generalization properties, especially when the data is not too noisy and the true relationship is close to linear. The implicit bias of the algorithm, therefore, guides the model toward solutions that generalize well, even in the absence of explicit regularization.
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