# Near-Optimal SQ Lower Bounds for Agnostically Learning Halfspaces and ReLUs under Gaussian Marginals

NeurIPS 2020, 2020.

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Abstract:

We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. In the former problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \{ \pm 1\}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitra...More

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Introduction

- The authors study the fundamental problems of agnostically learning halfspaces and ReLU regression in the distribution-specific agnostic PAC model.
- In both of these problems, the authors are given i.i.d. samples from a joint distribution D on labeled examples (x, y), where x ∈ Rd is the example and y ∈ R is the corresponding label, and the goal is to compute a hypothesis that is competitive with the best-fitting halfspace or ReLU respectively.

Highlights

- 1.1 Background and Problem Motivation

We study the fundamental problems of agnostically learning halfspaces and Rectified Linear Unit (ReLU) regression in the distribution-specific agnostic PAC model - In both of these problems, we are given i.i.d. samples from a joint distribution D on labeled examples (x, y), where x ∈ Rd is the example and y ∈ R is the corresponding label, and the goal is to compute a hypothesis that is competitive with the best-fitting halfspace or ReLU respectively
- [GKK19] gave a qualitatively similar reduction implying a computational lower bound of dΩ(log(1/ǫ)) for Problem 1.2
- Our lower bounds suggest that the accuracy-runtime tradeoff of known polynomial time approximation schemes (PTAS) for these problems [Dan[15], DGK+20] that achieve error (1 + γ)OPT + ǫ, for all γ > 0, in time poly(dpoly(1/γ), 1/ǫ) is qualitatively best possible
- Consider the set of distributions {Pv}, where v is any unit vector, such that the projection of Pv in the v-direction is equal to A and in the orthogonal complement Pv is an independent standard Gaussian
- This set of distributions has Statistical Query (SQ) dimension dΩ(k). By known results this implies that distinguishing such a distribution from the standard Gaussian or learning a distribution with better than 1/poly(dk) correlation with such a distribution is hard in the SQ model

Results

**The authors' Results and Techniques**

The authors are ready to formally state the main results. For Problem 1.1 the authors prove: Theorem 1.4.- Let d ≥ 1 and ǫ ≥ d−c, for some sufficiently small constant c > 0.
- Any SQ algorithm that agnostically learns halfspaces on Rd under Gaussian marginals within additive error ǫ > 0 requires at least dc/ǫ many statistical queries to STAT(d−c/ǫ).
- The above statement says that any SQ algorithm for Problem 1.1 requires time at least dΩ(1/ǫ)
- This comes close to the known upper bound of dO(1/ǫ2) [KKMS08] and exponentially improves on the best known lower bound of dΩ(log(1/ǫ)) [KK14]

Conclusion

- The reduction-based hardness of [KK14, GKK19] imply SQ lower bounds of dΩ(log(1/ǫ)) for both problems.
- The authors' new SQ lower bounds are qualitatively optimal, nearly matching current algorithms.
- For both problems, the results show a sharp separation in the complexity of obtaining error O(OPT) + ǫ (which is poly(d/ǫ)) versus optimal error OPT + ǫ.
- Consider the set of distributions {Pv}, where v is any unit vector, such that the projection of Pv in the v-direction is equal to A and in the orthogonal complement Pv is an independent standard Gaussian.
- By known results this implies that distinguishing such a distribution from the standard Gaussian or learning a distribution with better than 1/poly(dk) correlation with such a distribution is hard in the SQ model

Summary

## Introduction:

The authors study the fundamental problems of agnostically learning halfspaces and ReLU regression in the distribution-specific agnostic PAC model.- In both of these problems, the authors are given i.i.d. samples from a joint distribution D on labeled examples (x, y), where x ∈ Rd is the example and y ∈ R is the corresponding label, and the goal is to compute a hypothesis that is competitive with the best-fitting halfspace or ReLU respectively.
## Results:

**The authors' Results and Techniques**

The authors are ready to formally state the main results. For Problem 1.1 the authors prove: Theorem 1.4.- Let d ≥ 1 and ǫ ≥ d−c, for some sufficiently small constant c > 0.
- Any SQ algorithm that agnostically learns halfspaces on Rd under Gaussian marginals within additive error ǫ > 0 requires at least dc/ǫ many statistical queries to STAT(d−c/ǫ).
- The above statement says that any SQ algorithm for Problem 1.1 requires time at least dΩ(1/ǫ)
- This comes close to the known upper bound of dO(1/ǫ2) [KKMS08] and exponentially improves on the best known lower bound of dΩ(log(1/ǫ)) [KK14]
## Conclusion:

The reduction-based hardness of [KK14, GKK19] imply SQ lower bounds of dΩ(log(1/ǫ)) for both problems.- The authors' new SQ lower bounds are qualitatively optimal, nearly matching current algorithms.
- For both problems, the results show a sharp separation in the complexity of obtaining error O(OPT) + ǫ (which is poly(d/ǫ)) versus optimal error OPT + ǫ.
- Consider the set of distributions {Pv}, where v is any unit vector, such that the projection of Pv in the v-direction is equal to A and in the orthogonal complement Pv is an independent standard Gaussian.
- By known results this implies that distinguishing such a distribution from the standard Gaussian or learning a distribution with better than 1/poly(dk) correlation with such a distribution is hard in the SQ model

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