Complexity-Theoretic Implications of Multicalibration (2312.17223v2)

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.

• The paper generalizes the Hardcore Lemma into IHCL++ by identifying multiple local hardcore sets that enhance density parameters across multicalibrated partitions.
• The paper characterizes pseudo-entropy by developing local versions of average min-entropy through multicalibration, deepening the understanding of entropy in computational contexts.
• The paper extends the Dense Model Theorem to DMT++, offering new insights into pseudodense set modeling and the interplay between fairness and computational complexity.

Complexity-Theoretic Implications of Multicalibration

The paper entitled "Complexity-Theoretic Implications of Multicalibration" explores the relationships between fairness in prediction algorithms and foundational results in computational complexity theory. It emphasizes the concepts of multiaccuracy and multicalibration, which are integral to understanding fair predictions across multiple subpopulations.

Multicalibration and Algorithmic Fairness

Multicalibration is a method ensuring that predictors do not just achieve mean accuracy over a global population but are also calibrated for specific, possibly overlapping, subgroups. This notion arises from the desire to bridge both individual and group fairness. A predictor is termed multicalibrated when it provides calibrated predictions for every predetermined subgroup within a population. This property ensures the predictor's approximation behaves consistently with actual subgroup distributions. Multicalibration proves particularly useful in sensitive applications where fairness across various demographic categories is crucial.

Connection to Computational Complexity

One of the paper's significant claims is the equivalence between multiaccuracy and a regularity notion for functions formulated by Trevisan, Tulsiani, and Vadhan. Their Regularity Lemma suggests that any complex function can be approximated by a simpler function in a manner indistinguishable by a specified class of functions. This result links with various computational theories and entices connections between algorithmic fairness and core complexity principles.

The Regularity Lemma's implications in additive number theory, information theory, and cryptography are leveraged in the paper to derive stronger applications such as the Hardcore Lemma and the Dense Model Theorem. This forms the crux of how the fairness in multiaccurate and multicalibrated predictors can be harnessed further into theoretical computer science.

Key Results and Contributions

1. Hardcore Lemma Generalization: The paper extends the traditional Hardcore Lemma into what is dubbed IHCL++. It discovers multiple "local" hardcore sets within a multicalibrated partition, offering better density parameters relative to the data balanced on each partition piece.
2. Characterizing Pseudo-Entropy: The work develops stronger characterizations of pseudo-average min-entropy by introducing local versions inferred from multicalibration, thus providing a broader understanding of entropy in computational contexts.
3. Dense Model Theorem Extensions: By applying multicalibration principles, the Dense Model Theorem is extended into DMT++. The authors generate 'local models' for pseudodense sets by sectioning dense distributions on partitions resulting from multicalibration.

Potential and Future Directions

The implications of this research stretch into practical applications for AI and machine learning, particularly in scenarios necessitating predictive fairness across demographic groups. Moreover, the theoretical avenues imply potential novel approaches in cryptography and complexity theory application fields.

Future research could explore refining multicalibration algorithms to be more computation-efficient and examining their application beyond existing domains. Additionally, the uniform complexity implications for learning multicalibrated predictors present an intriguing area for further exploration.

In summary, this paper extends the dialogue between fairness in prediction algorithms and computational complexity, offering novel insights and practical advancements in both fields. The exploration provides a path for utilizing multicalibration not only for rigorous fairness in algorithms but also for deepening theoretical computer science frameworks.