Complexity-Theoretic Implications of Multicalibration

TL;DR

This work reveals that multicalibration strengthens the complexity-theoretic Regularity Lemma by yielding low-complexity partitions that render complex objects locally indistinguishable from simple, constant-Bernoulli models. By leveraging these partitions, the authors derive IHCL++, PAME++, and DMT++—per-piece hardness, entropy, and dense-model results that hold without global hardness assumptions, and then show how to glue pieces to recover classical theorems. The approach unifies fairness definitions with foundational complexity concepts, enabling local-to-global constructions that improve density and hardness guarantees across a broad range of settings, including multiclass outputs. The results have potential implications for robust fairness, cryptography, information theory, and complexity, offering a framework to translate regularity phenomena into concrete hardness and entropy bounds with polynomial-partition complexity. Overall, the paper advances the toolkit for connecting algorithmic fairness with core complexity-theoretic phenomena, suggesting practical and theoretical avenues for further exploration.

Abstract

We present connections between the recent literature on multigroup fairness for prediction algorithms and classical results in computational complexity. Multiaccurate predictors are correct in expectation on each member of an arbitrary collection of pre-specified sets. Multicalibrated predictors satisfy a stronger condition: they are calibrated on each set in the collection. Multiaccuracy is equivalent to a regularity notion for functions defined by Trevisan, Tulsiani, and Vadhan (2009). They showed that, given a class $F$ of (possibly simple) functions, an arbitrarily complex function $g$ can be approximated by a low-complexity function $h$ that makes a small number of oracle calls to members of $F$, where the notion of approximation requires that $h$ cannot be distinguished from $g$ by members of $F$. This complexity-theoretic Regularity Lemma is known to have implications in different areas, including in complexity theory, additive number theory, information theory, graph theory, and cryptography. Starting from the stronger notion of multicalibration, we obtain stronger and more general versions of a number of applications of the Regularity Lemma, including the Hardcore Lemma, the Dense Model Theorem, and the equivalence of conditional pseudo-min-entropy and unpredictability. For example, we show that every boolean function (regardless of its hardness) has a small collection of disjoint hardcore sets, where the sizes of those hardcore sets are related to how balanced the function is on corresponding pieces of an efficient partition of the domain.

Figures (3)

• Figure 1: Illustration of the difference between IHCL and IHCL$++$, and how to recover IHCL from our ICHL$++$.
• Figure 2: Visual depiction of Theorem \ref{['thm:dmt++']} in the case where $\mathcal{S}$ and $\mathcal{V}$ correspond to the uniform distributions over $S$ and $V$, respectively. In this case, $\Pr_{x \sim \mathcal{S}}[x \in P] = |P \cap S|/|S|$ and $\Pr_{x \sim \mathcal{V}}[x \in P] = |P \cap V|/|V|$. The MC partition (indicated with black lines in the right figure) is such that, for each level set $P$ in the partition, the uniform distribution over $P \cap V$ is a model for the corresponding uniform distribution over $P \cap S$. One such level set is illustrated in purple in the right figure.
• Figure :

Theorems & Definitions (72)

• Theorem 1.1: Regularity Lemma ttv09, informally stated
• Theorem 1.2: Multicalibration Theorem hkrr18, informally stated
• Theorem 1.3: IHCL imp95hol05, informally stated
• Theorem 1.4: IHCL$++$, informal version
• Theorem 1.5: PAME vz12zhe14, informally stated
• Theorem 1.6: PAME$++$, informally stated
• Theorem 1.7: DMT gt08tz08rttv08imp08imp09, informally stated
• Theorem 1.8: DMT$++$, informally stated
• Lemma 1.9: Characterizing Indistinguishability from Constant-Bernoulli Functions, informally stated
• Definition 2.1: Multiaccuracy hkrr18kgz19