On the Computational Landscape of Replicable Learning (2405.15599v2)

Abstract: We study computational aspects of algorithmic replicability, a notion of stability introduced by Impagliazzo, Lei, Pitassi, and Sorrell [2022]. Motivated by a recent line of work that established strong statistical connections between replicability and other notions of learnability such as online learning, private learning, and SQ learning, we aim to understand better the computational connections between replicability and these learning paradigms. Our first result shows that there is a concept class that is efficiently replicably PAC learnable, but, under standard cryptographic assumptions, no efficient online learner exists for this class. Subsequently, we design an efficient replicable learner for PAC learning parities when the marginal distribution is far from uniform, making progress on a question posed by Impagliazzo et al. [2022]. To obtain this result, we design a replicable lifting framework inspired by Blanc, Lange, Malik, and Tan [2023] that transforms in a black-box manner efficient replicable PAC learners under the uniform marginal distribution over the Boolean hypercube to replicable PAC learners under any marginal distribution, with sample and time complexity that depends on a certain measure of the complexity of the distribution. Finally, we show that any pure DP learner can be transformed to a replicable one in time polynomial in the accuracy, confidence parameters and exponential in the representation dimension of the underlying hypothesis class.

• The paper establishes a computational separation by showing a concept class that is efficiently PAC learnable and replicable but not efficiently online learnable under standard cryptographic assumptions.
• It introduces a replicable lifting framework that efficiently transforms learners from uniform to arbitrary marginal distributions using decision tree complexity.
• It demonstrates that pure differentially private learners can be converted into replicable learners, albeit with exponential computational overhead relative to the concept class's representation dimension.

Computational Aspects of Algorithmic Replicability

Overview

This paper addresses the computational dimensions of algorithmic replicability, a stability notion introduced by Impagliazzo et al. The primary focus is to deepen our understanding of the computational connections between replicability and other learning paradigms, like online learning, private learning, and Statistical Query (SQ) learning. Significant theoretical and practical implications are drawn through both negative and positive results.

Key Contributions

Computational Separation from Online Learning

One major contribution of this work is to explore whether efficient replicability is computationally separable from efficient online learning. A pivotal result shows that under standard cryptographic assumptions, there exists a concept class that is efficiently PAC learnable and replicable but not efficiently learnable online.

Replicable Lifting Framework

The paper introduces an efficient framework for transforming replicable PAC learners under the uniform distribution to replicable PAC learners under any marginal distribution. The transformation's complexity depends on the decision tree complexity of the distribution. An essential implication here is the ability to learn parities efficiently and replicably under unknown distributions.

Transformation of Pure Differentially Private Learners

Another significant contribution is demonstrating that any pure differentially private (DP) learner can be transformed into a replicable learner. While the sample complexity remains polynomial in error, confidence, and replicability parameters, the computational complexity is exponential concerning the representation dimension of the concept class.

Detailed Analysis

Enhanced Understanding of Online Learning Separation

The computational separation between replicability and online learning is cemented through the concept class $\Sigma$, constructed using pseudorandom function generators. This concept class exhibits the characteristic that, while efficiently PAC learnable and replicable, it resists efficient online learning under the assumption of one-way functions.

Efficient Replicable Learning of Parities

A robust replicable PAC learning process is designed for learning affine parities under non-uniform distributions. The transformation leverages decision tree complexity to ensure computational efficiency. The process includes partitioning the instance space based on a tree representation and using replicable learners for uniform marginals on these partitions.

The authors not only answer the posed question regarding learning affine parities under general distributions but extend their framework to demonstrate broader applicability.

Computational Transformations across Learning Paradigms

Attention is also given to the computational relationships between replicability and various stability notions such as DP and online learning. The authors confirm that there exists a concept class that is efficiently learnable under approximate DP but not under replicability, contingent on OWF assumptions. This result adds a nuanced layer to the overall understanding of the computational landscape of stable learning.

Computational Landscape of Stability

The results of this paper allow for a nuanced landscape of computational stability in learning frameworks to be drawn.

• Black-Box Transformations: It is established that pure DP can be transformed to online learning; replicability can be transformed to approximate DP and so can SQ learning.
• Separations: Notably, the paper delineates multiple separations:
  • Efficient learning through replicability but not online learning.
  • Approximate DP learning but not online learning.
  • Inefficient pure DP to replicable transformation compared to the more efficient approximate DP to replicable transformation under the same assumptions.

Future Directions and Implications

This paper sets a foundational ground for several future research directions.

1. Complexity Measures: Further exploration is needed into alternative complexity measures beyond decision tree complexity. Understanding the trade-offs involved in these transformations can lead to more optimized algorithms.
2. Extensions in Learning Paradigms: Extending the results to a broader range of concept classes and distributions would solidify these theoretical constructs further.
3. Practical Implementations: While primarily theoretical, the results bear potential implications for practical applications, particularly in making machine learning experiments more replicable and reliable.
4. Security Assumptions: The security assumptions (like OWF) underpin many results. Investigating weaker assumptions or different cryptographic bases can yield new insights.

Conclusion

By bridging gaps in understanding across replicability, online learning, SQ learning, and differential privacy, this paper sets an advanced discourse on the computational aspects of algorithmic replicability. The exploration not only allows for refined theoretical models but drives forward the practical dialogue on making machine learning outcomes more dependable in the real world.