A Model of Inductive Bias Learning (1106.0245v1)

Abstract: A major problem in machine learning is that of inductive bias: how to choose a learner's hypothesis space so that it is large enough to contain a solution to the problem being learnt, yet small enough to ensure reliable generalization from reasonably-sized training sets. Typically such bias is supplied by hand through the skill and insights of experts. In this paper a model for automatically learning bias is investigated. The central assumption of the model is that the learner is embedded within an environment of related learning tasks. Within such an environment the learner can sample from multiple tasks, and hence it can search for a hypothesis space that contains good solutions to many of the problems in the environment. Under certain restrictions on the set of all hypothesis spaces available to the learner, we show that a hypothesis space that performs well on a sufficiently large number of training tasks will also perform well when learning novel tasks in the same environment. Explicit bounds are also derived demonstrating that learning multiple tasks within an environment of related tasks can potentially give much better generalization than learning a single task.

• The paper establishes a theoretical framework that automates inductive bias learning across multiple tasks to reduce individual sampling requirements.
• It demonstrates that learning bias through task diversity leads to meta-generalization, where a well-chosen hypothesis space performs well on new tasks.
• The model underpins practical advances in feature learning and hierarchical systems, improving efficiency in applications like image recognition.

A Model of Inductive Bias Learning

Introduction

Jonathan Baxter's paper "A Model of Inductive Bias Learning" provides a comprehensive theoretical framework for addressing the pivotal problem of inductive bias in machine learning. Inductive bias concerns the choice of a hypothesis space that is sufficiently broad to contain solutions to specific learning problems while also being constrained enough to ensure reliable generalization from limited data. Traditional approaches often rely on expert knowledge to supply this bias, which has inherent limitations regarding accuracy and transferability. Baxter proposes a model for automating the learning of inductive bias when a learner is embedded within an environment consisting of multiple related tasks. This setting allows the learner to explore various tasks, thereby facilitating the discovery of a hypothesis space that is broadly effective across many problems.

The Model

Baxter’s model extends the PAC (Probably Approximately Correct) learning framework by considering a learner capable of sampling from multiple tasks within an environment. The environment is defined by a distribution over a set of learning tasks, and the learner's goal is to identify a hypothesis space that minimizes the average expected loss across this distribution. The paper proves that, under certain conditions, a hypothesis space performing well on a sufficiently large number of training tasks will likely perform well on novel tasks within the same environment.

Key Findings

The paper's central findings can be summarized as follows:

1. Reduction in Sampling Burden:
   • By learning multiple related tasks, the sampling burden (i.e., the number of examples required per task) for achieving good generalization is reduced.
2. Meta-Generalization:
   • The process of bias learning can be seen as achieving meta-generalization: a bias learner generalizes well if, after sufficient training on a variety of tasks, it produces a hypothesis space capable of containing good solutions for new, unseen tasks.

Baxter derives explicit bounds demonstrating that the number of examples required for each task decreases inversely with the number of tasks, under the condition that the potential complexity of the hypothesis spaces is controlled. This result implies that sharing information across related tasks can significantly improve learning efficiency.

Practical Implications

The practical implications of Baxter’s work are vast, especially in domains where related tasks are abundant:

• Feature Learning:
  • A notable application is in feature learning. Baxter formulates the problem of learning appropriate neural network features for related tasks as a bias learning problem. He proves upper bounds on the number of tasks and examples required for these features to generalize well to new tasks. This has practical consequences for tasks like handwritten character recognition or face recognition, where preprocessing steps (such as normalization) can be viewed as domain-level biases.
• Hierarchies in Learning:
  • The model paves the way for hierarchical learning systems where learning occurs at multiple levels. For instance, high-level features could be learned across an environment of tasks and subsequently leveraged to expedite learning in novel tasks.

Theoretical Implications and Future Directions

Baxter’s model prompts several theoretical questions and potential future research avenues:

1. Optimal Hypothesis Space Families:
   • Determining the optimal hypothesis space family that balances representational richness and practical learnability remains an open question. Further exploration is required to develop algorithms that automatically determine the structure of the hypothesis space family.
2. Task Relatedness:
   • Investigating the extent to which different tasks are related and developing methods to quantify and leverage this relatedness could further improve bias learning models.
3. Extended Hierarchies:
   • Expanding the two-level hierarchy of task and environment to deeper, more complex hierarchies could reveal additional efficiencies and capabilities for more sophisticated multi-level learning systems.

Conclusion

Baxter’s "A Model of Inductive Bias Learning" provides a foundational framework for understanding and automating the learning of inductive bias in environments with multiple related tasks. The model's implications are far-reaching, enabling significant improvements in both the sample and computational efficiency of learning systems. By offering a method to learn and generalize biases automatically, Baxter's work lays the groundwork for more adaptive and intelligent learning systems capable of efficiently tackling a broad range of tasks. Future research will undoubtedly build on these insights, seeking to refine and expand the capabilities of the model in practical and theoretical dimensions.