Is Efficient PAC Learning Possible with an Oracle That Responds 'Yes' or 'No'?
(2406.11667)Abstract
The empirical risk minimization (ERM) principle has been highly impactful in machine learning, leading both to near-optimal theoretical guarantees for ERM-based learning algorithms as well as driving many of the recent empirical successes in deep learning. In this paper, we investigate the question of whether the ability to perform ERM, which computes a hypothesis minimizing empirical risk on a given dataset, is necessary for efficient learning: in particular, is there a weaker oracle than ERM which can nevertheless enable learnability? We answer this question affirmatively, showing that in the realizable setting of PAC learning for binary classification, a concept class can be learned using an oracle which only returns a single bit indicating whether a given dataset is realizable by some concept in the class. The sample complexity and oracle complexity of our algorithm depend polynomially on the VC dimension of the hypothesis class, thus showing that there is only a polynomial price to pay for use of our weaker oracle. Our results extend to the agnostic learning setting with a slight strengthening of the oracle, as well as to the partial concept, multiclass and real-valued learning settings. In the setting of partial concept classes, prior to our work no oracle-efficient algorithms were known, even with a standard ERM oracle. Thus, our results address a question of Alon et al. (2021) who asked whether there are algorithmic principles which enable efficient learnability in this setting.
Overview
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The paper explores whether efficient Probably Approximately Correct (PAC) learning can be achieved using weaker oracles than empirical risk minimization (ERM), addressing various settings such as binary and multiclass classification, and regression.
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Key results demonstrate efficient PAC learning with a weak consistency oracle that only responds 'yes' or 'no' and a weak ERM oracle that provides empirical risk without the hypothesis, with complexity bounds related to the VC dimension.
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The research has practical implications for distributed and adversarial learning environments and theoretically challenges the traditional reliance on ERM oracles, suggesting new directions for investigating alternative oracle types.
Is Efficient PAC Learning Possible with an Oracle That Responds "Yes" or "No"?
The paper explores the fundamental question of whether efficient Probably Approximately Correct (PAC) learning can be achieved using a significantly weaker oracle than the commonly employed empirical risk minimization (ERM) oracle. This question is addressed under different settings including binary and multiclass classification, as well as regression. The authors provide affirmative answers, demonstrating that efficient learning is possible with a weak consistency oracle and a weak ERM oracle.
Key Results and Contributions
Realizable PAC Learning with Weak Consistency Oracle:
- The paper introduces a weak consistency oracle, which only responds "yes" or "no" to whether a given dataset is realizable by some hypothesis in the class.
- The authors develop an algorithm that relies on this weak oracle and prove that it can PAC learn any partial concept class with finite VC dimension.
- Sample and oracle complexity is shown to be polynomial in the VC dimension.
Agnostic PAC Learning with Weak ERM Oracle:
- Extending to the agnostic setting, a weak ERM oracle returns the empirical risk of the best hypothesis but not the hypothesis itself.
- An algorithm is presented that uses this weak ERM oracle to perform agnostic PAC learning.
- The sample complexity for the agnostic setting is only polynomially worse than that of the realizable setting, dependent on the VC dimension.
Extensions to Multiclass and Regression:
- Multiclass Classification: The weak consistency and weak ERM oracles are adapted to handle multiclass concept classes. The algorithms maintain polynomial sample complexity in terms of the Natarajan dimension and log of the number of classes.
- Regression: For regression tasks, the authors use a range consistency oracle and extend the weak ERM oracle concept. They show PAC learnability with polynomial bounds related to the fat-shattering dimension.
Lower Bounds and Oracle-Efficiency:
- The paper establishes that for multiclass concept classes and real-valued functions with specific complexity measures, a polynomial number of queries to even a strong ERM oracle does not suffice.
- This underscores that the algorithms provided for using weak oracles are nearly optimal in terms of complexity.
Implications and Theoretical Impact
The implications of this research are profound both practically and theoretically.
Practical Implications:
- The realization that weaker oracles can suffice for efficient learning tasks suggests more feasible implementations in distributed learning environments where query costs might be a critical factor.
- The algorithms developed might be more robust in adversarial settings where full responses from an ERM oracle might be prohibitive due to privacy or computational concerns.
Theoretical Implications:
- This work challenges the traditional reliance on ERM oracles and opens new avenues for investigating alternative oracle types in learning theory.
- By demonstrating that efficient learning does not necessitate full access to an ERM oracle, it provides a new perspective on the necessary and sufficient conditions for PAC learning.
Future Directions
The findings suggest several intriguing directions for future research:
Sample Complexity Improvements:
- Investigating whether the polynomial gap in sample complexity between using weak and strong oracles can be closed further with refined algorithms or new oracles.
Other Learning Paradigms:
- Extending the concept of weak oracles to more complex learning settings such as online learning, reinforcement learning, and bandits.
Practical Implementations & Empirical Validation:
- Developing practical implementations of the proposed algorithms and empirically validating their performance in real-world scenarios.
Oracle Complexity in Other Domains:
- Exploring weak oracle concepts in other domains (e.g., unsupervised learning, semi-supervised learning) to establish generalizable principles.
Conclusion
The paper makes a significant contribution by showing that PAC learning can be efficiently achieved with weaker oracles than previously thought necessary. Through rigorous theoretical development and proofs, it sets the stage for more efficient and potentially more robust learning paradigms that extend beyond the traditional reliance on ERM. This research paves the way for novel learning strategies that are both effective and computationally conservative, presenting a landmark shift in understanding the capabilities of learning algorithms under constrained oracle access.
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