Information Complexity of Stochastic Convex Optimization: Applications to Generalization and Memorization (2402.09327v2)

Abstract: In this work, we investigate the interplay between memorization and learning in the context of \emph{stochastic convex optimization} (SCO). We define memorization via the information a learning algorithm reveals about its training data points. We then quantify this information using the framework of conditional mutual information (CMI) proposed by Steinke and Zakynthinou (2020). Our main result is a precise characterization of the tradeoff between the accuracy of a learning algorithm and its CMI, answering an open question posed by Livni (2023). We show that, in the $L^2$ Lipschitz--bounded setting and under strong convexity, every learner with an excess error $\varepsilon$ has CMI bounded below by $\Omega(1/\varepsilon^2)$ and $\Omega(1/\varepsilon)$, respectively. We further demonstrate the essential role of memorization in learning problems in SCO by designing an adversary capable of accurately identifying a significant fraction of the training samples in specific SCO problems. Finally, we enumerate several implications of our results, such as a limitation of generalization bounds based on CMI and the incompressibility of samples in SCO problems.

• The paper establishes a tradeoff between memorization and generalization in SCO using conditional mutual information to quantify the required data retention for ε-learners.
• It introduces a novel memorization measure inspired by membership inference attacks to precisely gauge how much training data is retained by learning algorithms.
• It demonstrates through adversarial constructions that any sample-efficient SCO learner must inherently memorize a significant fraction of its training data, challenging constant-sized compression schemes.

Exploring the Ties Between Memorization and Generalization in Stochastic Convex Optimization

Introduction

The paper presents a comprehensive paper on the intrinsic connection between memorization and generalization within the scope of Stochastic Convex Optimization (SCO). Memorization and generalization, often seen as counteracting aspects in machine learning, are revisited with insights emphasizing their indispensability for achieving optimal learning outcomes. The authors delve into the complexities surrounding Conditional Mutual Information (CMI) and its role in quantifying the extent of memorization required by learning algorithms to ensure strong generalization.

Main Contributions

The primary focus revolves around establishing a quantitative understanding of the tradeoff between a learning algorithm's accuracy and its memorization capacity, through the lens of CMI. Notable contributions include:

• A meticulous characterization of the CMI-accuracy tradeoff, shedding light on the essential amount of memorization inherent to ε-learners for various subclasses of SCO problems, notably the Lipschitz bounded (CLB) and the strongly convex (CSL) SCOs.
• Introduction of a novel memorization measure, inspiring from CMI and membership inference attacks, aimed at quantifying the extent of training data memorization by the learning algorithms.
• Through adversarial constructions, the paper underlines the necessity of memorization in learning, demonstrating that any sample-efficient learner must, to a significant extent, memorize its training dataset.
• It presents a rigorous argument against the existence of constant-sized (dimension-independent) sample compression schemes for SCO, further enriching the discourse on the characteristics and limitations of sample compression in learning algorithms.

Implications and Theoretical Contributions

The paper's exploration into the memorization-learning interplay via the CMI framework brings forth several critical insights and theoretical advancements:

• It reveals the limitations of conventional CMI-based generalization bounds in adequately capturing the optimal excess error across SCO settings. This is attributed to the identified lower bounds on CMI, which suggest that these bounds become vacuous for algorithms with optimal sample complexity.
• The work extends beyond mere characterization of CMI, offering a concrete adversary construction capable of distinguishing a significant fraction of training samples in specific SCO problems. This approach not only substantiates the theoretical underpinnings of memorization necessity but also delivers a practical scheme to evaluate memorization in learning algorithms.
• Discussions on the non-existence of dimension-independent sample compression schemes for SCO problems challenge prevailing assumptions in the machine learning community, highlighting the unique challenges posed by SCO in the context of data compression and memorization.

Future Directions

The findings and methodologies presented set a fertile ground for further exploration within the machine learning research community. Future works may extend beyond the confines of stochastic convex optimization, investigating the role of memorization in more complex and overparameterized models, such as deep neural networks. Additionally, the development of more refined measures for memorization, encompassing aspects of robustness and privacy, could pave the way for designing learning algorithms that balance the dual objectives of generalization and memorization more effectively.

Conclusion

This paper marks a significant step towards demystifying the complex relationship between memorization and generalization in the field of stochastic convex optimization. By rigorously analyzing the information complexities and necessitating memorization through conditional mutual information and adversarial constructs, this work delineates the intricate balance that learning algorithms must navigate to achieve optimality.