Robust Learnability of Sample-Compressible Distributions under Noisy or Adversarial Perturbations

Abstract: Learning distribution families over $\mathbb{R}^d$ is a fundamental problem in unsupervised learning and statistics. A central question in this setting is whether a given family of distributions possesses sufficient structure to be (at least) information-theoretically learnable and, if so, to characterize its sample complexity. In 2018, Ashtiani et al. reframed \emph{sample compressibility}, originally due to Littlestone and Warmuth (1986), as a structural property of distribution classes, proving that it guarantees PAC-learnability. This discovery subsequently enabled a series of recent advancements in deriving nearly tight sample complexity bounds for various high-dimensional open problems. It has been further conjectured that the converse also holds: every learnable class admits a tight sample compression scheme. In this work, we establish that sample compressible families remain learnable even from perturbed samples, subject to a set of necessary and sufficient conditions. We analyze two models of data perturbation: (i) an additive independent noise model, and (ii) an adversarial corruption model, where an adversary manipulates a limited subset of the samples unknown to the learner. Our results are general and rely on as minimal assumptions as possible. We develop a perturbation-quantization framework that interfaces naturally with the compression scheme and leads to sample complexity bounds that scale gracefully with the noise level and corruption budget. As concrete applications, we establish new sample complexity bounds for learning finite mixtures of high-dimensional uniform distributions under both noise and adversarial perturbations, as well as for learning Gaussian mixture models from adversarially corrupted samples, resolving two open problems in the literature.

Robust Learnability of Sample-Compressible Distributions Under Noisy or Adversarial Perturbations

The paper "Robust Learnability of Sample-Compressible Distributions under Noisy or Adversarial Perturbations" by Arefe Boushehrian and Amir Najafi investigates the learnability of distribution classes under challenging conditions, specifically focusing on perturbations due to noise and adversarial attacks. This research builds on the notion of sample compressibility, reframing it as a critical structure within distribution classes to ensure learnability, even in adverse scenarios.

Overview

The central focus is to ascertain whether the concept of sample compressibility, which guarantees PAC-learnability, can sustain learnability amid perturbations. Two types of perturbations are addressed: additive noise, where samples are obscured by independent noise, and adversarial corruption, where a subset of samples is maliciously altered. Notably, the research posits conditions under which learnability is preserved, also delving into necessary and sufficient traits for recovering the original distribution from its perturbed version.

Key Contributions

1. Sample Compressibility Under Perturbations: The authors extend sample compressibility, traditionally a tool for clean samples, to environments with noise and adversarial attacks. For noisy situations, they suggest a perturbation-quantization framework that helps sustain learnability, handling noise levels and corruption budget.

2. Analytical Frameworks and Models: The analysis includes models where additive noise is applied globally to samples, alongside models where adversarial entities affect specific samples. For both cases, mathematical frameworks deliver sample complexity bounds, emphasizing minimal assumptions.

3. Applications and Implications: Concrete examples are explored, such as learning mixtures of Gaussian models or uniform distributions under perturbed conditions, to illustrate these robust learning frameworks.

4. Future Directions: Speculating on future pathways, the framework established can extend into more intricate distributions or potentially lead to computational efficiency techniques in high-dimensional models, particularly those suffering from data corruption or distortion.

Theoretical and Practical Implications

From a theoretical standpoint, the paper conjectures about the equivalence of PAC learnability and sample compression schemes, suggesting every learnable class should have a corresponding efficient compression model. Practically, the framework proposed is meaningful for real-world applications in data science and machine learning, where data corruption through noise or adversaries is a lurking threat.

Future Work and Open Questions

The exploration leaves several promising directions open, such as:
- Verifying the broader applicability of their assumptions across more diverse data scenarios.
- Developing efficient algorithms that apply these theoretical guarantees to computational practices.
- Investigating real-world adversarial learning scenarios to validate robustness beyond synthetic models.

Overall, the paper profoundly contributes to understanding robust distribution learning, offering a structured approach to tackling unsupervised learning challenges under corruptive influences. By framing the discussion around sample compressibility, the authors provide a unified approach potentially useful for a wide range of applications, ensuring that core distributional learning tasks remain resilient in data-sensitive environments.