Information-Theoretic Foundations for Machine Learning (2407.12288v4)

Abstract: The progress of machine learning over the past decade is undeniable. In retrospect, it is both remarkable and unsettling that this progress was achievable with little to no rigorous theory to guide experimentation. Despite this fact, practitioners have been able to guide their future experimentation via observations from previous large-scale empirical investigations. In this work, we propose a theoretical framework which attempts to provide rigor to existing practices in machine learning. To the theorist, we provide a framework which is mathematically rigorous and leaves open many interesting ideas for future exploration. To the practitioner, we provide a framework whose results are simple, and provide intuition to guide future investigations across a wide range of learning paradigms. Concretely, we provide a theoretical framework rooted in Bayesian statistics and Shannon's information theory which is general enough to unify the analysis of many phenomena in machine learning. Our framework characterizes the performance of an optimal Bayesian learner as it learns from a stream of experience. Unlike existing analyses that weaken with increasing data complexity, our theoretical tools provide accurate insights across diverse machine learning settings. Throughout this work, we derive theoretical results and demonstrate their generality by apply them to derive insights specific to settings. These settings range from learning from data which is independently and identically distributed under an unknown distribution, to data which is sequential, to data which exhibits hierarchical structure amenable to meta-learning, and finally to data which is not fully explainable under the learner's beliefs (misspecification). These results are particularly relevant as we strive to understand and overcome increasingly difficult machine learning challenges in this endlessly complex world.

• The paper provides a rigorous theoretical framework using Bayesian statistics and Shannon’s information theory to analyze machine learning phenomena.
• It bridges empirical practices with theory by linking estimation error to mutual information and employing rate-distortion concepts for continuous latent variables.
• The framework is applied across models—including linear regression, logistic regression, deep neural networks, and nonparametric methods—highlighting its wide-ranging impact.

Information-Theoretic Foundations for Machine Learning

The paper "Information-Theoretic Foundations for Machine Learning" provides a rigorous theoretical framework which leverages Bayesian statistics and Shannon's Information Theory to analyze machine learning phenomena. This framework addresses the need for a unifying theoretical basis in the field, which has largely relied on empirical observations and heuristic methods. The authors, Hong Jun Jeon and Benjamin Van Roy, propose a comprehensive approach to characterize the performance of an optimal Bayesian learner and provide insights applicable across various data regimes, including iid, sequential, and hierarchical data.

Theoretical Framework and Key Concepts

The core of the paper is the formulation of machine learning as a process of reducing uncertainty about an unknown latent variable, $\theta$, which determines the probabilistic relationship between inputs and outputs. This is modeled within a Bayesian framework where $\theta$ is treated as a random variable with a known prior distribution.

The performance of learning algorithms is measured using the cumulative expected log-loss, with optimal performance characterized by the Bayesian posterior distribution $P(Y_{t+1}\in\cdot|H_t)$, where $H_t$ is the history of observed data up to time $t$. The central result, Theorem 1, elegantly connects the estimation error of an optimal Bayesian algorithm to the mutual information between the history of observations and the latent variable $\theta$: $L_T = \frac{I(H_T;\theta)}{T}.$ This theorem implies that the estimation error is fundamentally a measure of the information gained about $\theta$ from the observed data, and thus decays as more data is acquired.

Rate-Distortion Theory

The authors extend the analysis by incorporating rate-distortion theory to handle continuous latent variables. They define a rate-distortion function $H_{\epsilon, T}(\theta)$ which quantifies the trade-off between the number of nats retained about $\theta$ and the distortion tolerable in predictions. The rate-distortion function offers a bound on the estimation error: $$\sup_{\epsilon \geq 0}\ \min\left\{\frac{H_{\epsilon, T}(\theta)}{T}, \epsilon\right\}\ \leq\ L_T\ \leq\ \inf_{\epsilon \geq 0}\ \frac{H_{\epsilon, T}(\theta)}{T} + \epsilon.$$

Application to Concrete Problems

The paper applies this theoretical framework to several machine learning scenarios:

1. Linear Regression: For a data generating process where outputs are linear combinations of inputs with Gaussian noise, the authors derive bounds on estimation error that depend on the dimensionality of inputs and the sample size. They demonstrate that the error scales linearly with the parameter dimension and inversely with the number of observations.
2. Logistic Regression: For binary classification using logistic regression, they establish error bounds that mirror those of linear regression, extending the insights to classification tasks.
3. Deep Neural Networks: They explore more complex models like deep neural networks, showing that the estimation error scales linearly with the number of parameters. This is a significant improvement over previous bounds that scale with the product of parameters and depth.
4. Nonparametric Learning: The framework extends to nonparametric settings where the hypothesis class is infinite-dimensional. By leveraging concentration properties of the latent variables, they provide insights into the sample efficiency of learning under such complex models.

Sequential and Meta-Learning

Theoretical results for learning from sequential data are also discussed. The paper addresses the limitations of previous models which required mixing time assumptions, and demonstrates that their framework can handle autoregressive models, including transformers with self-attention mechanisms, without such constraints. Additionally, meta-learning, where the learning process itself adapts across tasks, is explored using a hierarchical Bayesian model.

Misspecified Models

The paper tackles the practical scenario of model misspecification, providing bounds on the additional error incurred when the assumed prior distribution deviates from the true data generating process. This is crucial for understanding the robustness of learning algorithms in real-world applications where the underlying models are often complex and not fully known.

Implications and Future Directions

The theoretical insights presented in this paper provide a robust foundation for understanding and guiding the development of machine learning algorithms. The results emphasize the crucial role of information theory in analyzing the limits of learning and point towards new methods for designing algorithms that balance computational constraints and learning efficacy. This rigorous approach can significantly impact the development of scalable and robust machine learning systems, potentially informing the design of algorithms that make optimal use of available data and computational resources.

Future research can explore more sophisticated models and learning settings, such as reinforcement learning or lifelong learning, where the information-theoretic principles discussed can provide further enhancements. The integration of rate-distortion theory with practical algorithm design remains an open and promising area for exploration.

Overall, this paper lays a solid theoretical groundwork that bridges gaps between empirical machine learning practices and formal statistical analysis, promising more coherent and theoretically sound advancements in the field.