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Machine learning models are powerful tools for solving complex problems, but they can easily become overly complex themselves, leading to overfitting. Regularization techniques help prevent overfitting by imposing constraints on the model’s parameters. One common regularization technique is L2 regularization, also known as weight decay. In this blog post, we’ll explore the big idea behind L2 regularization and weight decay, their equivalence in stochastic gradient descent (SGD), and why weight decay is preferred over L2 regularization in more advanced optimizers like Adam.

The big idea behind L2 regularization and weight decay is straightforward: networks with smaller weights tend to overfit less and generalize better. In other words, by reducing the magnitude of the model’s parameters, we can make it less prone to fitting the noise in the training data and improve its ability to make accurate predictions on unseen data.

Weight decay is a regularization technique that operates by subtracting a fraction of the previous weights when updating the weights during training, effectively making the weights smaller over time. Unlike L2 regularization which adds a penalty terms to the loss function (see below), weight decay directly influences the weight update step itself. This subtraction of a portion of the existing weights ensures that during each iteration of training, the model’s parameters are nudged towards smaller values. By gradually diminishing the magnitude of the weights, weight decay helps prevent overfitting and encourages the model to generalize better to unseen data. In mathematical terms, weight decay can be represented as a weight update step that subtracts a scaled version of the current weights, where the scaling factor is controlled by a small regularization parameter.

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