Optimal Multi-Distribution Learning (2312.05134v4)

Abstract: Multi-distribution learning (MDL), which seeks to learn a shared model that minimizes the worst-case risk across $k$ distinct data distributions, has emerged as a unified framework in response to the evolving demand for robustness, fairness, multi-group collaboration, etc. Achieving data-efficient MDL necessitates adaptive sampling, also called on-demand sampling, throughout the learning process. However, there exist substantial gaps between the state-of-the-art upper and lower bounds on the optimal sample complexity. Focusing on a hypothesis class of Vapnik-Chervonenkis (VC) dimension d, we propose a novel algorithm that yields an varepsilon-optimal randomized hypothesis with a sample complexity on the order of (d+k)/varepsilon^2 (modulo some logarithmic factor), matching the best-known lower bound. Our algorithmic ideas and theory are further extended to accommodate Rademacher classes. The proposed algorithms are oracle-efficient, which access the hypothesis class solely through an empirical risk minimization oracle. Additionally, we establish the necessity of randomization, revealing a large sample size barrier when only deterministic hypotheses are permitted. These findings resolve three open problems presented in COLT 2023 (i.e., citet[Problems 1, 3 and 4]{awasthi2023sample}).

• The paper presents a novel algorithm that dynamically adjusts weight distributions using a multiplicative update rule to minimize maximum loss across distributions.
• It provides a rigorous proof of convergence to an approximately optimal equilibrium with sample complexity scaling polynomially in dimension, task count, and precision.
• The algorithm enhances robustness and generalization in multi-task learning scenarios, efficiently handling adversarial and shifting data distributions.

Overview of the Minimax Optimizatiion Problem in Machine Learning

This paper addresses a minimax optimization problem in the context of machine learning, specifically focusing on a set of hypothesis functions with a VC-dimension denoted by $d$, and a set of distributions $D = \{D_1, D_2, \ldots, D_k\}$ over the feature set $\mathcal{X}$ and the label set $\mathcal{Y} = \{0,1\}$. The problem is framed as an optimization where the goal is to minimize the maximum expected loss across all distributions using a hypothesis from $\mathcal{H}$.

The authors present a novel algorithm designed to solve this optimization problem effectively. The primary contribution is an algorithm that dynamically adjusts the distribution of weights applied to various tasks, iteratively refining a model with a specific minimax approach.

Algorithmic Approach

The paper details the algorithmic framework as follows:

• Initialization: The algorithm begins by distributing weights equally across all tasks and initializes the hypothesis set.
• Sample Collection and Projection: Using a sample collection technique, the algorithm projects the hypothesis set according to the samples collected.
• Iterative Update Rule: The core loop employs an iterative strategy that involves updating weight distributions based on predefined neighborhood conditions. The estimation of the loss function for each hypothesis is computed utilizing collected data, and a multiplication-based updating procedure (termed as MWU or Multiplicative Weight Update) is employed.

Theoretical Analysis and Sample Complexity

The authors provide a rigorous proof of the algorithm's performance, demonstrating that it converges towards an $\tilde{O}(\epsilon)$-equilibrium. One key theoretical insight is that their algorithm guarantees that the maximization of the expected loss over the minimization strategy is kept within a tight error bound, characterized by sample complexity parameters.

An underlying assumption is that the solution's sample complexity scales with the dimensionality $d$, task number $k$, and the precision parameter $\epsilon$. Theoretical analysis ensures that the approach demands polynomial sample quantities relative to these dimensions, promising practical feasibility for certain large-scale applications.

Implications and Future Directions

The implications of this research extend across several critical aspects of machine learning:

1. Robustness in Adversarial Contexts: The minimax approach inherently enhances robustness, making it particularly applicable for scenarios where training data distributions can shift or be adversarially manipulated.
2. Generalization Across Tasks: By balancing distribution priorities, the proposed algorithm potentially generalizes effectively across heterogeneous data distributions common in multi-task learning spaces.
3. Efficient Resource Utilization: Ensuring efficiency in sample complexity is pertinent for resource-limited environments, suggesting usage in scenarios where data accessibility is a limiting factor.

Future research might explore extensions of this framework, including:

• Incorporating non-binary and multi-class label sets $\mathcal{Y}$.
• Applying this methodology to continuous domains where distributions are more complex and less discrete.
• Optimizing the computational efficiency of the iterative and sampling components through advancements in scalable algorithmic techniques.

In summary, this paper introduces a theoretically sound algorithm for tackling minimax problems in machine learning, offering valuable insights into distribution-robust learning strategies. The work promises potential advancements in developing robust predictive models, especially amidst varying real-world data distributional properties.