When classical regularization methods like weight decay improve generalization in deep networks without the use of early stopping, it is likely not because they restrict the network's capacity in a meaningful way. Instead, these techniques are thought to encode specific inductive biases that happen to align well with the structural patterns present in the datasets of interest, functioning similarly to how architectural choices or distance metrics guide model preferences.

Claude

Given a finite training set, machine learning models must rely on certain assumptions to achieve human-level performance. These assumptions, known as inductive biases, encode preferences for solutions with specific properties that often reflect how humans think about the world. For example, a deep multilayer perceptron (MLP) has an inductive bias towards building up a complicated function through the composition of simpler functions. The necessity of these biases stems from the 'no free lunch' theorem, which dictates that algorithms must make assumptions to generalize effectively.

Inductive Bias in Machine Learning

Dive into Deep Learning

The approach to training machine learning models typically consists of two distinct phases. The first phase focuses on fitting the model to the training data. The second phase involves estimating the generalization error—which is defined as the model's true error on the underlying population—by evaluating its performance on a separate holdout dataset.

Phases of Machine Learning Training

In nonparametric methods such as the $$k$$-nearest neighbor algorithm, a distance function $$d$$ (or equivalently, a vector-valued basis function $$\phi(\mathbf{x})$$) must be specified to measure similarity between data points. This choice of distance metric is critical because it encodes the model's inductive bias. Even if any metric allows a model like $$1$$-nearest neighbor to achieve zero training error, different distance functions represent different underlying assumptions about the data patterns. Consequently, with finite data, these varying inductive biases will yield different predictors, and their generalization performance will depend on how compatible the chosen metric is with the true data distribution.

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