TL;DR: Generalization is a model’s ability to make accurate predictions on new, unseen data after training. This post explains probabilistic and robustness-based views of generalization, including empirical risk minimization (ERM) and perturbation tests.

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Generalization refers to the ability of a machine learning (ML) method trained on a training set to make accurate predictions on new, unseen data. This is a central goal of ML and AI: to learn patterns that extend beyond the data available during training.

Most ML systems use empirical risk minimization (ERM) to learn a hypothesis $\hat{h} \in \mathcal{H}$ by minimizing the average loss over a training set of data points ${\bf z}^{(1)}, \ldots, {\bf z}^{(m)}$, denoted as $\mathcal{D}^{(\rm train)}$. However, success on the training set does not guarantee success on unseen data—this discrepancy is the challenge of generalization.

To study generalization mathematically, we often assume a probabilistic model for data generation, such as the i.i.d. assumption. That is, we interpret data points as independent random variables (RVs) with an identical probability distribution $p({\bf z})$. This distribution, while unknown, allows us to define the risk of a trained model $\hat{h}$ as the expected loss:

\[\bar{L} \big( \hat{h} \big) = \mathbb{E}_{{\bf z} \sim p({\bf z})} \left\{ L(\hat{h}, {\bf z}) \right\}.\]

The difference between risk $\bar{L}(\hat{h})$ and empirical risk $\widehat{L}(\hat{h}|\mathcal{D}^{(\rm train)})$ is called the generalization gap. Tools from probability theory—such as concentration inequalities and uniform convergence—can be used to bound this gap under specific conditions (Shalev-Shwartz and Ben-David 2014).

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Generalization Without Probability

Probability theory is one way to study generalization, but not the only one. Another approach uses perturbations to the training data. The idea is simple: a good model $\hat{h}$ should be robust. Its prediction $\hat{h}({\bf x})$ should not change much if we slightly modify the input ${\bf x}$ of a data point ${\bf z}$.

For example, an object detector trained on smartphone photos should still detect the object if a few random pixels are masked (Su, Vargas, and Sakurai 2019). Similarly, it should produce consistent predictions if the object is rotated (Mallat 2016).

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❓ Frequently Asked Questions

What is generalization in machine learning?
Generalization is a model’s ability to make accurate predictions on new, unseen data after training on a known dataset.

What causes poor generalization?
Overfitting, data imbalance, and overly complex models often lead to poor generalization.

Can we study generalization without probability theory?
Yes. Deterministic approaches like robustness testing with perturbations offer a complementary perspective.

What is the generalization gap?
It’s the difference between the empirical loss on training data and the expected loss on unseen data.

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📚 References

Mallat, Stéphane. 2016. “Understanding Deep Convolutional Networks.”
Philosophical Transactions of the Royal Society A 374 (2065): 20150203.
https://doi.org/10.1098/rsta.2015.0203

Shalev-Shwartz, S., and S. Ben-David. 2014. Understanding Machine Learning: From Theory to Algorithms.
Cambridge University Press.

Su, J., D. V. Vargas, and K. Sakurai. 2019. “One Pixel Attack for Fooling Deep Neural Networks.”
IEEE Transactions on Evolutionary Computation 23 (5): 828–841.
https://doi.org/10.1109/TEVC.2019.2890858

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📚 This explanation is part of the Aalto Dictionary of Machine Learning —
an open-access multilingual glossary developed at Aalto University to support accessible and precise communication in ML.