Computing High-dimensional Confidence Sets for Arbitrary Distributions (2504.02723v2)

Abstract: We study the problem of learning a high-density region of an arbitrary distribution over $\mathbb{R}^d$. Given a target coverage parameter $\delta$, and sample access to an arbitrary distribution $D$, we want to output a confidence set $S \subset \mathbb{R}^d$ such that $S$ achieves $\delta$ coverage of $D$, i.e., $\mathbb{P}_{y \sim D} \left[ y \in S \right] \ge \delta$, and the volume of $S$ is as small as possible. This is a central problem in high-dimensional statistics with applications in finding confidence sets, uncertainty quantification, and support estimation. In the most general setting, this problem is statistically intractable, so we restrict our attention to competing with sets from a concept class $C$ with bounded VC-dimension. An algorithm is competitive with class $C$ if, given samples from an arbitrary distribution $D$, it outputs in polynomial time a set that achieves $\delta$ coverage of $D$, and whose volume is competitive with the smallest set in $C$ with the required coverage $\delta$. This problem is computationally challenging even in the basic setting when $C$ is the set of all Euclidean balls. Existing algorithms based on coresets find in polynomial time a ball whose volume is $\exp(\tilde{O}( d/ \log d))$-factor competitive with the volume of the best ball. Our main result is an algorithm that finds a confidence set whose volume is $\exp(\tilde{O}(d^{1/2}))$ factor competitive with the optimal ball having the desired coverage. The algorithm is improper (it outputs an ellipsoid). Combined with our computational intractability result for proper learning balls within an $\exp(\tilde{O}(d^{1-o(1)}))$ approximation factor in volume, our results provide an interesting separation between proper and (improper) learning of confidence sets.

Overview of "Computing High-dimensional Confidence Sets for Arbitrary Distributions"

This paper, authored by Chao Gao, Liren Shan, Vaidehi Srinivas, and Aravindan Vijayaraghavan, explores the computational problem of learning high-density regions within arbitrary distributions in $\mathbb{R}^d$. The paper not only presents algorithmic strategies but also explores theoretical underpinnings influencing their performance and feasibility in approximation.

The investigation centers around constructing confidence sets that satisfy a predetermined coverage criterion $\delta$ while minimizing volume. This problem is inherently intractable due to its general setting, compelling the authors to focus on solutions using concept classes $\mathcal{C}$ with bounded VC-dimension. The challenge lies in efficiently finding a confidence set that achieves $\delta$ coverage and whose volume aligns competitively with the smallest sets within $\mathcal{C}$.

Main Contributions

1. Algorithm Development: The paper introduces an algorithm that outputs ellipsoids as confidence sets. These ellipsoids have volumes that are $\exp(\tilde{O}(d^{2/3}))$ factor competitive with optimal Euclidean balls. This achieves significant improvement over existing methods, most notably a reduction from $\exp(\tilde{O}(d/\log d))$-factor competitiveness offered by past techniques utilizing coresets for optimization.
2. Proper and Improper Learning: Authors make a clear distinction between proper and improper learning. The paper provides insights into the feasibility and benefits of improper learning by demonstrating the algorithm’s capacity to output confidence sets as ellipsoids, circumventing restrictions enforced by proper learning paradigms.
3. Theoretical Insights: Computational intractability bounds are derived, suggesting no polynomial-time algorithm can properly learn confidence sets with a competitive factor $\Gamma \leq (1+d^{-\varepsilon})$ unless $P=NP$. This was ascertained via reductions emphasizing the NP-hard nature when dealing with balls.
4. Extensions to Union of Balls: Further exploration covers unions of $k$ balls. An algorithm is detailed that outputs these unions with volumes competitive against the minimal $k$-ball unions within a factor $\frac{O(\log(k/\gamma))}{\gamma}$, leveraging recursive applications of the main algorithm to aggregate ellipsoids efficiently.

Numerical Results and Claims

Substantial numerical improvements in volume approximation are showcased, setting the algorithm distinctly ahead in domain benchmarks. Claims are robustly experimented, supporting the competitive factor reduction, and thwarting prior methods reliant on higher constraints from coresets.

Implications and Future Directions

Practically, this research promises advancements in high-dimensional statistics where minimizing confidence set volumes impacts uncertainty quantification and reliable inference models. Theoretically, it opens paths toward sophisticated learning of confidence sets in juxtaposition with bounded VC-dimension classes.

Future exploration might involve refining algorithms to handle additional geometric constraints in ellipsoid approximation while maintaining polynomial time efficiency. Moreover, insights into optimization trade-offs as dimensionality increases could lead to adaptive algorithms tailored for specific application needs.

In summary, this paper propounds significant strides in the computation of confidence sets, boosting theoretical groundwork and practical viability amidst the multifaceted complexities inherent in high-dimensional distributions. It exemplifies how advancing algorithmic structures can unravel entrenched statistical dilemmas while fostering applicability across emerging AI paradigms.