Derandomizing Multi-Distribution Learning (2409.17567v1)

Abstract: Multi-distribution or collaborative learning involves learning a single predictor that works well across multiple data distributions, using samples from each during training. Recent research on multi-distribution learning, focusing on binary loss and finite VC dimension classes, has shown near-optimal sample complexity that is achieved with oracle efficient algorithms. That is, these algorithms are computationally efficient given an efficient ERM for the class. Unlike in classical PAC learning, where the optimal sample complexity is achieved with deterministic predictors, current multi-distribution learning algorithms output randomized predictors. This raises the question: can these algorithms be derandomized to produce a deterministic predictor for multiple distributions? Through a reduction to discrepancy minimization, we show that derandomizing multi-distribution learning is computationally hard, even when ERM is computationally efficient. On the positive side, we identify a structural condition enabling an efficient black-box reduction, converting existing randomized multi-distribution predictors into deterministic ones.

• The paper shows that converting randomized predictors to deterministic ones in multi-distribution learning incurs computational hardness under standard complexity assumptions.
• The study demonstrates an efficient derandomization algorithm under label-consistent conditions that slightly increases sample complexity and training time.
• The findings highlight the trade-off between algorithmic determinism and computational efficiency in collaborative learning scenarios.

Derandomizing Multi-Distribution Learning: An Analytical Overview

The paper "Derandomizing Multi-Distribution Learning" by Kasper Green Larsen, Omar Montasser, and Nikita Zhivotovskiy addresses a pivotal concern in the field of collaborative learning: the transition from randomized to deterministic predictors in multi-distribution scenarios. Collaborative learning aims to develop a unified predictor that performs well across multiple data distributions, leveraging samples from each during training. While existing algorithms have achieved near-optimal sample complexities with oracle efficiency, these methods typically generate randomized predictors, raising questions about the feasibility of deriving deterministic solutions. The authors present both the theoretical challenges and potential methodologies for derandomizing such algorithms, alongside the implications and intricacies of this transition.

Problem Statement and Framework

The paper explores the context of multi-distribution learning where the input consists of multiple unknown data distributions $D_1, \dots, D_k$ over $X \times \{-1,1\}$. The objective is to learn a classifier $f : X \to \{-1,1\}$ that minimizes the maximum error across any individual distribution: $\er_P(f) := \max_i \er_{D_i}(f) \leq \min_{h \in H} \max_i \er_{D_i}(h) + \epsilon,$ where $H$ is the hypothesis class with finite VC dimension $d$. The challenge in extending this framework from a single to multiple distributions reveals that while deterministic classifiers are feasible under classical PAC learning, achieving the optimal performance in a multi-distribution setup typically necessitates randomized predictors.

Computational Hardness and Derandomization

A significant contribution of the paper is the demonstration of computational hardness in derandomizing multi-distribution learning. Through a reduction to discrepancy minimization, the authors show that derandomizing the output of multi-distribution learning algorithms is computationally hard if $\text{BPP} \neq \text{NP}$. Specifically, for any hypothesis class $H$ with VC-dimension $d$ that can shatter a set of points in polynomial time, any multi-distribution learning algorithm that guarantees a deterministic predictor with high probability must incur either super-polynomial training time or evaluation time, assuming $n = \min\{d, k, 1/\epsilon\}$ tends to infinity.

Positive Results and Algorithmic Derandomization

Despite the computational barriers, the paper identifies a structural condition under which derandomization is feasible. Particularly in scenarios where the label distributions $D_i(y \mid x)$ are consistent across distributions (label-consistent distributions), the authors present an efficient algorithm to convert a randomized multi-distribution learner into a deterministic one. This algorithm leverages a black-box approach, using existing randomized predictors to produce a deterministic classifier with a manageable increase in sample complexity and training time. Formally, given a multi-distribution learning algorithm that uses $m(k,d,OPT,\epsilon,\delta)$ samples, the deterministic learner's sample complexity stands at: $$m(k,d,OPT,\epsilon/2,\delta/2) + O(k\ln^2(k/\delta) / \epsilon^2),$$ and the training time is incremented to: $$t(k,d,OPT,\epsilon/2,\delta/2) + \tilde{O}(k/\epsilon^2 + \ln(|X|/\delta)).$$

Implications and Future Directions

From a theoretical standpoint, the findings illustrate that while multi-distribution learning inherently benefits from randomized algorithms, specific structural properties can mitigate the barriers to derandomization. The distinction between lightly and heavily biased instances within the input domain plays a crucial role in achieving efficient deterministic predictors.

Practically, the results underscore the importance of structural insights in designing learning algorithms that balance computational efficiency and determinism. The complexity bounds provided suggest that future research can explore more refined conditions or alternative structures where derandomization might be efficiently achievable. Moreover, extending these results to infinite input domains offers a fertile ground for advancing the theoretical underpinnings and practical implementations of multi-distribution learning.

Conclusion

The paper "Derandomizing Multi-Distribution Learning" significantly advances our understanding of the computational limits and potential methodologies for transitioning from randomized to deterministic predictors in collaborative learning scenarios. By elucidating the inherent hardness and identifying feasible paths forward under specific conditions, the authors pave the way for further exploration and innovation in the field of AI and machine learning. The balance between sample efficiency, computational feasibility, and predictor determinism remains a pivotal consideration for ongoing and future research.