Generalization Error Bounds for Supervised Learning Algorithms via Auxiliary Distributions

Alapfogalmak

This work proposes a novel Auxiliary Distribution Method (ADM) to derive new upper bounds on the expected generalization error of supervised learning algorithms. The bounds are expressed in terms of various information-theoretic measures such as α-Jensen-Shannon divergence and α-Rényi divergence, which offer advantages over existing mutual information-based bounds.

Kivonat

The key highlights and insights of this work are:

1. The authors introduce a novel Auxiliary Distribution Method (ADM) to derive new upper bounds on the expected generalization error of supervised learning algorithms.

2. Using ADM, they derive bounds based on α-Jensen-Shannon (α-JS) divergence, which are always finite, unlike some existing mutual information-based bounds.

3. They also provide bounds based on α-Rényi divergence for 0 < α < 1, which can be finite even for deterministic supervised learning algorithms, in contrast to mutual information-based bounds.

4. The authors show how their bounds can be used to derive upper bounds on the excess risk of some learning algorithms and the generalization error under distribution mismatch between training and test data.

5. They outline conditions under which their proposed bounds might be tighter than earlier upper bounds.

The work provides a comprehensive analysis of generalization error bounds using various information-theoretic measures, offering new insights and tools for understanding the generalization properties of supervised learning algorithms.

Statisztikák

The loss function ℓ(w, z) is assumed to be σ-sub-Gaussian under certain distributions.
The hypothesis space W is assumed to be finite in some results.
The training data samples {Zi}ni=1 are assumed to be i.i.d.

Idézetek

"We suggest a novel method, i.e., the Auxiliary Distribution Method, that leads to new upper bounds on expected generalization errors that are appropriate for supervised learning scenarios."
"Our upper bounds based on α-Jensen-Shannon information are also finite."
"In contrast to mutual information-based bounds, our bounds—such as the α-Rényi information for 0 < α < 1—are finite for some deterministic supervised learning algorithms."

Főbb Kivonatok

Learning Algorithm Generalization Error Bounds via Auxiliary Distributions

by Gholamali Am... : arxiv.org 04-18-2024

https://arxiv.org/pdf/2210.00483.pdf

Mélyebb kérdések

How can the proposed bounds be extended to handle more complex loss functions or hypothesis spaces beyond the finite case

The proposed bounds can be extended to handle more complex loss functions or hypothesis spaces beyond the finite case by considering more general assumptions and properties of the loss functions and hypothesis spaces. For instance, instead of assuming sub-Gaussianity or boundedness of the loss functions, we can explore other properties such as sub-Exponentiality or sub-Gamma distributions. By incorporating these broader distributions, we can derive tighter and more general bounds that are applicable to a wider range of scenarios. Additionally, by relaxing the assumptions on the hypothesis space from being finite to more general spaces, such as infinite-dimensional function spaces, we can develop bounds that are suitable for more complex models and learning algorithms.

Can the ADM framework be applied to other machine learning problems beyond generalization error bounds, such as out-of-distribution generalization or domain adaptation

The ADM framework can indeed be applied to other machine learning problems beyond generalization error bounds. One such application is in the context of out-of-distribution generalization, where the distribution of the test data differs significantly from the training data. By utilizing auxiliary distributions and information-theoretic measures, the ADM framework can help derive upper bounds on the generalization error in scenarios with distribution mismatches. Furthermore, the framework can also be extended to address domain adaptation problems, where the goal is to adapt a model trained on one domain to perform well on a different but related domain. By incorporating auxiliary distributions and information measures specific to domain adaptation, the ADM framework can provide insights into the generalization performance of models in these challenging scenarios.

What are the potential applications and practical implications of the α-Jensen-Shannon and α-Rényi information measures in the context of supervised learning

The α-Jensen-Shannon and α-Rényi information measures have several potential applications and practical implications in the context of supervised learning. These measures offer a way to quantify the information content and relationships between the input data, output predictions, and the learning algorithm's parameters. In supervised learning, these measures can be used to assess the complexity of the learning problem, the dependencies between the training data and the hypothesis space, and the generalization performance of the model.
Specifically, the α-Jensen-Shannon information measure can be utilized as a loss function under label noise scenarios, in adversarial learning, and in active learning settings. It provides a way to capture the uncertainty and information content in the data, which can be crucial for robust and accurate model training. On the other hand, the α-Rényi information measure is valuable in feature extraction, image segmentation based on clustering, and other machine learning tasks where understanding the information content and relationships between variables is essential. By leveraging these information measures, researchers and practitioners can gain deeper insights into the learning process and improve the performance of supervised learning algorithms.