Computational-Statistical Gaps for Improper Learning in Sparse Linear Regression
(2402.14103)Abstract
We study computational-statistical gaps for improper learning in sparse linear regression. More specifically, given $n$ samples from a $k$-sparse linear model in dimension $d$, we ask what is the minimum sample complexity to efficiently (in time polynomial in $d$, $k$, and $n$) find a potentially dense estimate for the regression vector that achieves non-trivial prediction error on the $n$ samples. Information-theoretically this can be achieved using $\Theta(k \log (d/k))$ samples. Yet, despite its prominence in the literature, there is no polynomial-time algorithm known to achieve the same guarantees using less than $\Theta(d)$ samples without additional restrictions on the model. Similarly, existing hardness results are either restricted to the proper setting, in which the estimate must be sparse as well, or only apply to specific algorithms. We give evidence that efficient algorithms for this task require at least (roughly) $\Omega(k2)$ samples. In particular, we show that an improper learning algorithm for sparse linear regression can be used to solve sparse PCA problems (with a negative spike) in their Wishart form, in regimes in which efficient algorithms are widely believed to require at least $\Omega(k2)$ samples. We complement our reduction with low-degree and statistical query lower bounds for the sparse PCA problems from which we reduce. Our hardness results apply to the (correlated) random design setting in which the covariates are drawn i.i.d. from a mean-zero Gaussian distribution with unknown covariance.
Overview
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This paper investigates the computational and statistical challenges in achieving optimal prediction errors with minimal samples in sparse linear regression, especially under improper learning settings.
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It highlights the significant gap between computational feasibility and statistical optimality, indicating that efficient polynomial-time algorithms require substantially more samples than the theoretically minimal number.
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The study provides evidence of a lower sample complexity bound for efficient algorithms, suggesting a profound computational-statistical gap that challenges current algorithmic approaches.
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The implications of these findings are discussed in terms of theoretical importance, practical relevance, and the need for novel algorithmic strategies to bridge this gap in future machine learning research.
Exploring the Computational-Statistical Gaps in Sparse Linear Regression
Introduction
In the domain of machine learning, sparse linear regression models have garnered significant attention due to their relevance in various applications where the true model is believed to be sparse. Such models are particularly notable in scenarios characterized by high dimensionality and sparsity, where the goal is to predict an output variable as a linear combination of a small subset of the predictor variables. This paper explore the computational-statistical gaps associated with improper learning in sparse linear regression, focusing on the (correlated) random design setting.
Computational-Statistical Gaps
At the heart of this analysis is the exploration of sample complexity—the minimum number of samples required to achieve non-trivial prediction error—while considering computational efficiency. Information-theoretically, achieving prediction error with a minimal number of samples, specifically Θ(k log(d/k)), is well-understood; however, the computational complexity of achieving this with efficient (polynomial-time) algorithms remains less clear, particularly without additional model restrictions. Existing algorithms operating under polynomial time constraints typically require a substantially higher sample complexity, often Ω(d), illuminating a gap between what is computationally feasible and statistically optimal.
Hardness in Improper Learning
This work puts forward evidence that suggests a lower bound on sample complexity for efficient algorithms in improper learning settings of sparse linear regression. It demonstrates that achieving prediction error akin to the information-theoretical minimum likely necessitates Ω(k2) samples—a significant departure from the optimistic k log(d/k) bound, revealing a profound computational-statistical gap. This assertion is substantiated through a reduction from sparse PCA problems with a negative spike, which are widely regarded as computationally intractable under certain sample regimes.
Theoretical Implications and Practical Relevance
The findings present a nuanced understanding of the limitations inherent in current algorithmic approaches for sparse linear regression under improper learning settings. From a theoretical perspective, they align with the broader narrative within statistical learning that identifies stark differences in the sample complexities necessary for statistical consistency versus those required for computational feasibility. Practically, these results serve as a critical consideration for researchers and practitioners working on high-dimensional data, advising caution against over-reliance on computational shortcuts which may not provide statistically robust estimates.
Future Directions
The conjectured computational-statistical gap opens several avenues for future research. A pivotal question is whether novel algorithmic frameworks or learning regimes could narrow this gap. Furthermore, alternative models that go beyond the Gaussian assumptions or explore structured sparsity could offer new insights, potentially leading to more efficient algorithms that do not compromise on the optimal sample complexity. Lastly, a deeper exploration into the nature of improper learning, in different statistical models, could unearth broader principles governing the interplay between computational efficiency and statistical rigour.
Conclusion
By articulating the apparent computational limitations in achieving statistically optimal sample complexity for sparse linear regression in high-dimensional settings, this paper adds a critical dimension to the discourse on efficient learning algorithms. It underscores the necessity for the continued development of algorithmic techniques that better bridge the computational-statistical divide, an endeavor that remains paramount in the advancement of machine learning and data science.
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