Smoothed Analysis for Learning Concepts with Low Intrinsic Dimension (2407.00966v1)

Abstract: In traditional models of supervised learning, the goal of a learner -- given examples from an arbitrary joint distribution on $\mathbb{R}^d \times {\pm 1}$ -- is to output a hypothesis that is competitive (to within $\epsilon$) of the best fitting concept from some class. In order to escape strong hardness results for learning even simple concept classes, we introduce a smoothed-analysis framework that requires a learner to compete only with the best classifier that is robust to small random Gaussian perturbation. This subtle change allows us to give a wide array of learning results for any concept that (1) depends on a low-dimensional subspace (aka multi-index model) and (2) has a bounded Gaussian surface area. This class includes functions of halfspaces and (low-dimensional) convex sets, cases that are only known to be learnable in non-smoothed settings with respect to highly structured distributions such as Gaussians. Surprisingly, our analysis also yields new results for traditional non-smoothed frameworks such as learning with margin. In particular, we obtain the first algorithm for agnostically learning intersections of $k$-halfspaces in time $k^{poly(\frac{\log k}{\epsilon \gamma}) }$ where $\gamma$ is the margin parameter. Before our work, the best-known runtime was exponential in $k$ (Arriaga and Vempala, 1999).

• The paper introduces a smoothed analysis framework that relaxes optimality by comparing learners to classifiers robust to Gaussian perturbations.
• The method leverages low-degree polynomial approximations and L1-regression to efficiently approximate classifiers in low intrinsic dimension settings.
• Results show improved computational bounds and sample complexities under sub-Gaussian and bounded conditions, extending margin-based agnostic learning.

Smoothed Analysis for Learning Concepts with Low Intrinsic Dimension

The paper "Smoothed Analysis for Learning Concepts with Low Intrinsic Dimension" presents a novel framework aimed at addressing the computational hardness inherent in traditional models of supervised learning. The authors propose a smoothed-analysis framework which necessitates that a learner competes only with the best classifier robust to minor random Gaussian perturbations. This nuanced alteration facilitates a broad array of learning results for concepts dependent on low-dimensional subspaces and possessing bounded Gaussian surface area.

Conceptual Framework and Primary Contributions

In the standard PAC and agnostic learning paradigms, achieving a classifier that approximates the optimal one in a target concept class, especially under arbitrary joint distributions, is computationally prohibitive. To circumvent such intractability, the authors introduce a relaxed optimality notion where the learner's goal is to match the performance of the best classifier subject to Gaussian perturbations.

Key Definitions and Model

The authors define their smoothed learning model formally. For a concept class $\mathcal{F}$ and a distribution $D$ over $\mathbb{R}^d \times \{\pm 1\}$, the optimal error involving Gaussian perturbation ($\sigma$-smoothed setting) is given by:

$opt_\sigma = \inf_{f \in \mathcal{F}} \mathbb{E}_{\mathbf{z} \sim \mathcal{N}} \left[ \pr_{(x, y) \sim D}[f(x + \sigma \mathbf{z}) \neq y] \right]$

Results and Implications

The results demonstrate significant improvements over existing methods in handling various concept classes under weaker assumptions such as sub-Gaussian marginals. The reliance on Gaussian surface area (GSA) as a complexity measure, and the focus on concepts with low intrinsic dimension, like intersections of halfspaces, lead to the development of efficient learning algorithms. These algorithms either improve upon the existing computational bounds or provide new feasible methods for otherwise intractable problems.

Learning under Sub-Gaussian and Bounded Distributions

For distributions with sub-Gaussian tails, the authors show an algorithm with: $$N = d^{poly\left(\frac{k\Gamma}{\sigma \epsilon}\right)} \log\left(\frac{1}{\delta}\right) \text{ samples and } poly(d, N) \text{ runtime}$$

When the marginal distribution is bounded, dramatic improvements are presented with: $$N = k^{poly\left(\frac{\Gamma}{\epsilon \sigma}\right)} \log\left(\frac{1}{\delta}\right) \text{ samples and } poly(d, N) \text{ runtime}$$

Connections to Existing Models

1. Agnostic Learning with Margin: The smoothed learning model shows equivalence and improvement over margin-based learning by translating the geometric property of margins into the probabilistic setting of Gaussian perturbations. For intersections of $k$-halfspaces, this yields quasi-polynomial time complexities.
2. Learning under Smoothed Distributions: It extends the framework to scenarios where the $x$-marginal itself is smoothed. Under this setting, significant runtime improvements are demonstrated, utilizing the new smoothed learning framework.
3. Agnostic Learning with Anti-concentration: By leveraging anti-concentration properties, the authors generalize their results and remove strong dependencies on sample complexity, extending them to broader class functions and distributions.

Technical Foundation and Polynomial Regression

The methodology rests heavily on using low-degree polynomial approximations and leveraging $L_1$-regression techniques:

1. Polynomial Approximation: Through the introduction of Ornstein-Uhlenbeck noise operators, the authors approximate the given function $f$ through polynomials parameterized by Gaussian perturbations. They provide rigorous proofs for bounded and sub-Gaussian distributions ensuring the degree of approximating polynomial scales polynomially with the inverse error $\epsilon$.
2. Dimensionality Reduction: For bounded distributions, the authors apply random projections to navigate from high-dimensional spaces to manageable subspaces, significantly optimizing the polynomial regression step.

Conclusion

This paper contributes a substantial advance in learning theory by introducing a smoothed analysis framework that mitigates traditional computational hardness via Gaussian perturbations. The theoretical insights and algorithmic solutions presented provide strong foundations and pathways for further research in efficiently learning complex concepts under various realistic distributional assumptions. Future studies can investigate extending these results to other complexity measures and broader distribution classes, furthering the impact of the smoothed learning approach on both theoretical and practical aspects of machine learning.