Sample Compression Scheme Reductions

TL;DR

This paper introduces reductions from multiclass, regression, and robust learning sample compression schemes to binary schemes, highlighting implications for the sample compression conjecture and demonstrating learnability without bounded-size compression schemes.

Contribution

It provides novel reductions connecting various learning settings to binary compression schemes, extending the scope of the sample compression conjecture and exploring learnability limitations.

Findings

Reductions from multiclass and regression schemes to binary schemes.

Implications for the sample compression conjecture if resolved.

Existence of learnable classes without bounded-size compression schemes.

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).