Computational-Statistical Gaps in Gaussian Single-Index Models (2403.05529v2)

Abstract: Single-Index Models are high-dimensional regression problems with planted structure, whereby labels depend on an unknown one-dimensional projection of the input via a generic, non-linear, and potentially non-deterministic transformation. As such, they encompass a broad class of statistical inference tasks, and provide a rich template to study statistical and computational trade-offs in the high-dimensional regime. While the information-theoretic sample complexity to recover the hidden direction is linear in the dimension $d$, we show that computationally efficient algorithms, both within the Statistical Query (SQ) and the Low-Degree Polynomial (LDP) framework, necessarily require $\Omega(d^{k^\star/2})$ samples, where $k^\star$ is a "generative" exponent associated with the model that we explicitly characterize. Moreover, we show that this sample complexity is also sufficient, by establishing matching upper bounds using a partial-trace algorithm. Therefore, our results provide evidence of a sharp computational-to-statistical gap (under both the SQ and LDP class) whenever $k^\star>2$. To complete the study, we provide examples of smooth and Lipschitz deterministic target functions with arbitrarily large generative exponents $k^\star$.

• The paper establishes that efficient SQ and LDP algorithms need Ω(d^(κ/2)) samples, highlighting a critical computational-statistical gap in high-dimensional inference.
• It introduces the generative exponent κ, demonstrating its pivotal role in dictating the minimum sample complexity for accurate model recovery.
• The work bridges theoretical and practical insights, providing a foundation for developing more efficient algorithms in Single-Index Models with refined statistical guarantees.

Computational-Statistical Gaps in Gaussian Single-Index Models

The paper "Computational-Statistical Gaps in Gaussian Single-Index Models" by Damian, Pillaud-Vivien, Lee, and Bruna addresses performance trade-offs between computational complexity and statistical efficiency in solving high-dimensional inference tasks using Single-Index Models (SIMs). These represent a crucial family of high-dimensional regression problems where observations depend on a one-dimensional parametric projection via a nonlinear transformation.

Key Results

The major finding in the paper revolves around defining the sample complexity required to detect the underlying one-dimensional structure using common algorithmic frameworks. The authors establish that while the information-theoretic sample complexity yields a linear relationship concerning the dimension, sophisticated computational methods need significantly more samples to achieve similar performance. Specifically, they demonstrate that efficient algorithms operating within the Statistical Query (SQ) and Low-Degree Polynomial (LDP) frameworks necessitate $\Omega(d^{\kappa/2})$ samples, unveiling a notable computational-statistical gap when $\kappa > 2$.

• Sample Complexity Lower Bounds:
  • A key theoretical contribution is the characterization of the SQ and LDP lower bounds, identifying the minimum sample requirements compatible with specific algorithmic frameworks. Leveraging the orthogonality of Hermite polynomials, the authors show that achieving efficient estimation in SIMs isn't feasible with polynomial time unless the dataset size scales as $n = \Theta(d^{\kappa/2})$, thereby providing evidence of a computational-statistical barrier under these frameworks.
• Generative Exponent and its Implications:
  • A crucial outcome of the paper is the introduction and application of the generative exponent $\kappa$ that signifies the smallest degree at which non-zero Hermite coefficients provide useful statistical information. The authors show that this parameter governs the SQ complexity, and they achieve exact consistency by determining its precise role in inferential efficiency through meticulous analysis of $\chi^2$-divergence measures between model spaces.

Theoretical Contributions

The paper provides multiple theoretical insights, particularly highlighting the lower bounds in SQ and LDP methodologies. The construction and deployment of a partial-trace algorithm also contribute to bridging the theoretical gap by demonstrating an upper bound capable of matching the lower computational-statistical horizon within the SQ framework, showcasing polynomial time recovery with order-optimal samples for cases where $\kappa \leq 2$.

The role of generative exponents and their resulting impact on SIM complexity is monumental as it dictates whether computational lower bounds can be overcome. The distinction between the generative and information exponents, with $\kappa \leq k$, underscores their influence in defining the challenge's statistical substratum.

Implications & Future Directions

The findings have several theoretical and practical implications, particularly affirming that a stark computational-to-statistical gap exists for determination in high-dimensional SIMs, conditional to $\kappa > 2$. The integrative approach provides a template for evaluating other statistical inference models subjected to similar constraints, such as Non-Gaussian Component Analysis.

Future developments could aim at refining the partial trace approach to extend its practicality and scope, particularly under conditions where $\kappa > 2$. Research could also explore connections between the generative exponent and alternative metrics for inference tasks to deepen understanding of computational barriers across other statistical frameworks.

Additionally, the paper hints at the potential of variational perspectives by inferring that the generative exponent can be affected by label transformations, motivating continued exploration of operational mathematic methods to transform learning intricacies into tractable forecasts.

In conclusion, this work makes a significant contribution to the understanding of computational-statistics trade-offs in single-index models, providing a foundational template for analyzing such intersections in broader contexts. This research underscores the nuanced complexity of balancing algorithmic efficiency against statistical precision and sets the stage for further explorations into the inherent challenges in high-dimensional statistical inference.