A Distributional-Lifting Theorem for PAC Learning (2506.16651v1)

Abstract: The apparent difficulty of efficient distribution-free PAC learning has led to a large body of work on distribution-specific learning. Distributional assumptions facilitate the design of efficient algorithms but also limit their reach and relevance. Towards addressing this, we prove a distributional-lifting theorem: This upgrades a learner that succeeds with respect to a limited distribution family $\mathcal{D}$ to one that succeeds with respect to any distribution $D^\star$, with an efficiency overhead that scales with the complexity of expressing $D^\star$ as a mixture of distributions in $\mathcal{D}$. Recent work of Blanc, Lange, Malik, and Tan considered the special case of lifting uniform-distribution learners and designed a lifter that uses a conditional sample oracle for $D^\star$, a strong form of access not afforded by the standard PAC model. Their approach, which draws on ideas from semi-supervised learning, first learns $D^\star$ and then uses this information to lift. We show that their approach is information-theoretically intractable with access only to random examples, thereby giving formal justification for their use of the conditional sample oracle. We then take a different approach that sidesteps the need to learn $D^\star$, yielding a lifter that works in the standard PAC model and enjoys additional advantages: it works for all base distribution families, preserves the noise tolerance of learners, has better sample complexity, and is simpler.

• The paper introduces the distributional-lifting theorem that extends efficient PAC learners from simple distributions to broader mixture models using decision tree and subcube decompositions.
• The work achieves sample complexity 2^(O(d))·poly(n) while preserving noise tolerance, highlighting both its theoretical and practical significance.
• The methodology sidesteps direct distribution learning by employing a direct search over decompositions, making the approach viable under standard PAC settings with random examples.

A Distributional-Lifting Theorem for PAC Learning

The paper "A Distributional-Lifting Theorem for PAC Learning" (2506.16651) advances the theoretical understanding of the distributional assumptions in PAC learning, providing a flexible framework that interpolates between distribution-specific and distribution-free learning. It offers a general method for converting efficient learners, originally designed for a restricted family of distributions, into learners that succeed on significantly broader distribution families, with the complexity of the lifting scaling with a formal measure of distributional complexity.

Background and Motivation

In the standard PAC model, the learner aims to approximate an unknown function $f$ given samples $(x, f(x))$ where $x$ is drawn from an unknown distribution $D$. Distribution-free PAC learning, with no assumptions on $D$, has produced predominantly negative results—even for simple concept classes—because $D$ can encode arbitrary computational difficulty. In contrast, distribution-specific PAC learning assumes a fixed, often simple, distribution (e.g., product measures), enabling a range of efficient algorithms, albeit under a strong and often unrealistic independence assumption.

The crucial question addressed is: Can the algorithmic successes of distribution-specific PAC learning be generalized to broader classes of data distributions without incurring the full hardness of the distribution-free model? The notion of distributional-lifting seeks to interpolate between these extremes, parameterizing distributional complexity and providing a way to leverage efficient learners under "simple" distributions as building blocks for more general settings.

Main Contribution: The Distributional-Lifting Theorem

The central result is a distributional-lifting theorem: if there exists a learner for a concept class $\mathcal{C}$ that is efficient for each distribution in a family $\mathcal{D}$ (closed under restrictions), then there exists a computationally efficient learner for any distribution $D^*$ that can be written as a mixture (decision tree or subcube partition) of distributions from $\mathcal{D}$. Specifically:

• Decision Tree Lifting: If $D^*$ admits a decomposition by a depth-$d$ decision tree whose leaves are distributions in $\mathcal{D}$, the lifted learner runs in $n^{O(d)}$ time with sample complexity $2^{O(d)} \cdot \text{poly}(n)$, matching the efficiency of the best-known approaches for constant $d$.
• Subcube Partition Lifting: The theorem further extends to cover subcube partitions with size $s$ and codimension $d$, capturing a broader class of distributions. The algorithm produces a "subcube list" hypothesis that is efficient to construct and applies to richer decompositions than those obtainable through decision trees alone.
• Noise Tolerance Preservation: The distributional-lifting procedure preserves the noise-robustness of the base learner. If the original learner is robust to label noise or contamination (in the sense of total variation), the lifted learner maintains this property, an advance over prior approaches which did not preserve robustness.

Notable Theoretical and Numerical Results

• Sample Complexity: The new lifter achieves $2^{O(d)} \cdot \text{poly}(n)$ sample complexity, improving upon prior work (which required $n^{O(d)}$ samples for the lifted learner).
• Broader Class of Distributions: By handling general subcube partitions (not just decision trees), the lifter supports a strictly larger set of distributions for finite $d$ and $s$, broadening its applicability in practice.
• Computational Complexity: For practical values (e.g., $d = O(\log n)$), the algorithm remains polynomial in $n$.

Methodological Innovations

Previous distributional-lifting approaches, such as Blanc, Lange, Malik, and Tan (2023), relied on conditional sample oracles to efficiently learn the underlying distributional structure (e.g., decision tree decomposition), which is not available in the standard PAC model. This paper proves an impossibility result: with only random samples, it is information-theoretically infeasible to reconstruct the exact decomposition, justifying the hardness of the previous approach.

The key methodological shift is to sidestep explicit distribution learning and, instead, build a hypothesis space via direct search over decompositions, selecting the best according to empirical test error on a validation set. This makes the algorithm viable in the standard PAC setting (random examples only), simplifies implementation, and maintains robust generalization guarantees—despite the inability to learn the underlying distribution structure.

Algorithmic Overview

For Decision Tree Decompositions

1. Training Phase: For each restriction (up to depth $d$), run the base learner on the subset of the data consistent with the restriction.
2. Test Phase: For all possible depth-$d$ trees, combine the associated hypotheses to produce a composite hypothesis; select the one with lowest empirical error on a held-out test set.
3. Efficiency: Utilizing recursive structure, the enumeration over trees is rendered feasible for moderate $d$ ($n^{O(d)}$ time).

For Subcube Partitions

1. Train on All Restrictions: Learn a hypothesis for every possible subcube (restriction) up to a given codimension.
2. Greedy Cover Algorithm: Sequentially select subcubes and associated hypotheses that cover large portions of the remaining dataset with low error, in the spirit of greedy set cover.
3. Hypothesis Construction: Form an ordered "subcube list," mapping points to the first covering subcube, and assign the corresponding hypothesis.
4. Efficiency: Achievable because the number of candidate subcubes grows only moderately for constant or logarithmic codimension.

Implications and Theoretical Significance

This work addresses several key open issues:

• Elimination of the Conditional Oracle Assumption: The lifter is practical under the standard PAC model, not requiring strong oracle access.
• Generality and Extensibility: It lifts arbitrary base learners (not only those for the uniform distribution) and applies to any family of distributions closed under restriction.
• Robustness: Preservation of robustness under label or example contamination broadens the practical utility of the technique to real-world imperfect data.

Limitations and Contrasts with Previous Work

• Optimality Boundaries: The sample and computational complexity scale exponentially with the decomposition depth; extreme values of $d$ or $s$ render the method less practical.
• Lower Bound Justification: Impossibility results established in the paper formally demonstrate that no method relying solely on random samples can efficiently learn the underlying distributional decomposition, nor can the conditional oracle approach be replaced via sampling.

Broader Impacts and Future Directions

The distributional-lifting framework clarifies the interaction between distributional complexity and the computational feasibility of learning. By parameterizing the "hardness" of a distribution through tree or subcube decomposition depth, the results enable a gradual relaxation of distributional assumptions, giving finer-grained control over the balance between generality and efficiency.

Potential future directions:

• Lower Bound Analyses for Specific Concept Classes: Establishing computational lower bounds parameterized by both concept and distributional complexity.
• Empirical Validation: Implementing the lifting algorithm on practical concept classes (e.g., DNF, decision trees) with structured but correlated distributions commonly observed in real-world data.
• Algorithmic Optimization: Developing heuristics or approximate search methods for decompositions to mitigate the exponential dependence on decomposition parameters.
• Extensions to Unlabeled Data or Semi-supervised Frameworks: Exploring the synergy between the distributional-lifting methodology and semi-supervised learning paradigms.

Conclusion

The paper provides a significant advancement in the theory of PAC learning, formally establishing a means to leverage efficient learning algorithms (originally designed for simple distributions) in considerably more general distributional regimes. By encoding distributional complexity via mixture decompositions and providing robust, efficient lifting mechanisms, this work bridges previously disconnected islands in computational learning theory, and points towards a spectrum of learning guarantees that interpolate between severe and null distributional assumptions. The distributional-lifting paradigm is likely to inform both future theoretical research and the design of practical learning algorithms accommodating structured, but nontrivially correlated, real-world data.