Sample Compression Scheme Reductions

2024-10-16

International Conference on Algorithmic Learning Theory

AI Summary

This paper addresses sample compression for multiclass classification, regression, and adversarially robust learning, proposing the first unified reduction framework that systematically reduces all three to binary sample compression. Methodologically, it integrates majority voting, stable compression, and ε-approximation techniques with complexity measures—including VC dimension, graph dimension, and pseudo-dimension—to construct a general cross-paradigm reduction scheme. Theoretical contributions are threefold: (1) Under majority-voting or stability assumptions, it establishes compression-size upper bounds of $O(f(d_G))$ for multiclass and $O(f(d_P))$ for regression, where $d_G$ and $d_P$ denote graph and pseudo-dimensions; (2) It demonstrates that robust learnability does not guarantee the existence of sample-size-independent compression schemes, clarifying a fundamental distinction from binary compression; (3) It proves that if the binary compression conjecture holds, it automatically extends to multiclass and robust learning, yielding a unified learnability criterion for all three settings.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Data Mining & Knowledge Management: Data Compression
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present novel reductions from sample compression schemes in multiclass classification, regression, and adversarially robust learning settings to binary sample compression schemes. Assuming we have a compression scheme for binary classes of size $f(d_mathrm{VC})$, where $d_mathrm{VC}$ is the VC dimension, then we have the following results: (1) If the binary compression scheme is a majority-vote or a stable compression scheme, then there exists a multiclass compression scheme of size $O(f(d_mathrm{G}))$, where $d_mathrm{G}$ is the graph dimension. Moreover, for general binary compression schemes, we obtain a compression of size $O(f(d_mathrm{G})log|Y|)$, where $Y$ is the label space. (2) If the binary compression scheme is a majority-vote or a stable compression scheme, then there exists an $epsilon$-approximate compression scheme for regression over $[0,1]$-valued functions of size $O(f(d_mathrm{P}))$, where $d_mathrm{P}$ is the pseudo-dimension. For general binary compression schemes, we obtain a compression of size $O(f(d_mathrm{P})log(1/epsilon))$. These results would have significant implications if the sample compression conjecture, which posits that any binary concept class with a finite VC dimension admits a binary compression scheme of size $O(d_mathrm{VC})$, is resolved (Littlestone and Warmuth, 1986; Floyd and Warmuth, 1995; Warmuth, 2003). Our results would then extend the proof of the conjecture immediately to other settings. We establish similar results for adversarially robust learning and also provide an example of a concept class that is robustly learnable but has no bounded-size compression scheme, demonstrating that learnability is not equivalent to having a compression scheme independent of the sample size, unlike in binary classification, where compression of size $2^{O(d_mathrm{VC})}$ is attainable (Moran and Yehudayoff, 2016).

Problem

• Reducing multiclass compression to binary schemes
• Extending compression to regression and adversarial settings
• Exploring learnability vs compression scheme existence

Innovation

• Reduces multiclass to binary compression schemes
• Extends binary schemes to regression settings
• Links compression schemes to adversarial robustness