We study the problem of learning mixtures of linear classifiers under Gaussian covariates.

We give superpolynomial statistical query (SQ) lower bounds for learning twohidden-layer ReLU networks with respect to Gaussian inputs in the standard (noise-free)model.

We study the complexity of Non-Gaussian Component Analysis (NGCA) in the Statistical Query (SQ) model. Prior work developed a general methodology to prove SQ lower bounds for this task that have been applicable to a wide range of contexts. In particular, it was known that for any univariate distribution A satisfying certain conditions, distinguishing between a standard multivariate Gaussian and a distribution that behaves like A in a random hidden direction and like a standard Gaussian in the orthogonal complement, is SQ-hard. The required conditions were that (1) A matches many low-order moments with the standard univariate Gaussian, and (2) the chi-squared norm of A with respect to the standard Gaussian is finite. While the moment-matching condition is necessary for hardness, the chi-squared condition was only required for technical reasons. In this work, we establish that the latter condition is indeed not necessary. In particular, we prove near-optimal SQ lower bounds for NGCA under the moment-matching condition only. Our result naturally generalizes to the setting of a hidden subspace. Leveraging our general SQ lower bound, we obtain near-optimal SQ lower bounds for a range of concrete estimation tasks where existing techniques provide sub-optimal or even vacuous guarantees.

The statistical-query model is a well-studied framework for analyzing the sample complexity of learning problems andcaptures mostknownlearning algorithms.

Our main result is a Statistical Query (SQ) lower bound suggesting that known algorithms for this problem are essentially best possible,even for the special case of uniform mixtures.In particular, we show that the complexity of any SQ algorithm for the problem is $n^{\mathrm{poly}(1/\Delta) \log(r)}$,where $\Delta$ is a lower bound on the pairwise $\ell_2$-separation between the $\mathbf{v}_{\ell}$'s.The key technical ingredient underlying our result is a new construction of spherical designs on the unit sphere that may be of independent interest.

We study the problem of PAC learning a single neuron in the presence of Massart noise. Specifically, for a known activation function $f: \mathbb{R}\to \mathbb{R}$, the learner is given access to labeled examples $(\mathbf{x}, y) \in \mathbb{R}^d \times \mathbb{R}$, where the marginal distribution of $\mathbf{x}$ is arbitrary and the corresponding label $y$ is a Massart corruption of $f(\langle \mathbf{w}, \mathbf{x} \rangle)$. The goal of the learner is to output a hypothesis $h: \mathbb{R}^d \to \mathbb{R}$ with small squared loss. For a range of activation functions, including ReLUs, we establish super-polynomial Statistical Query (SQ) lower bounds for this learning problem. In more detail, we prove that no efficient SQ algorithm can approximate the optimal error within any constant factor. Our main technical contribution is a novel SQ-hard construction for learning $\{ \pm 1\}$-weight Massart halfspaces on the Boolean hypercube that is interesting on its own right.

We give superpolynomial statistical query (SQ) lower bounds for learning two-hidden-layer ReLU networks with respect to Gaussian inputs in the standard (noise-free) model. No general SQ lower bounds were known for learning ReLU networks of any depth in this setting: previous SQ lower bounds held only for adversarial noise models (agnostic learning) (Kothari and Klivans 2014, Goel et al. 2020a, Diakonikolas et al. 2020a) or restricted models such as correlational SQ (Goel et al. 2020b, Diakonikolas et al. 2020b). Prior work hinted at the impossibility of our result: Vempala and Wilmes (2019) showed that general SQ lower bounds cannot apply to any real-valued family of functions that satisfies a simple non-degeneracy condition. To circumvent their result, we refine a lifting procedure due to Daniely and Vardi (2021) that reduces Boolean PAC learning problems to Gaussian ones. We show how to extend their technique to other learning models and, in many well-studied cases, obtain a more efficient reduction. As such, we also prove new cryptographic hardness results for PAC learning two-hidden-layer ReLU networks, as well as new lower bounds for learning constant-depth ReLU networks from membership queries.

We study the complexity of Non-Gaussian Component Analysis (NGCA) in the Statistical Query (SQ) model.Prior work developed a methodology to prove SQ lower bounds for NGCA that have been applicable to a wide range of contexts.In particular, it was known that for any univariate distribution $A$ satisfying certain conditions,distinguishing between a standard multivariate Gaussian and a distribution that behaves like $A$ in a random hidden direction and like a standard Gaussian in the orthogonal complement, is SQ-hard.The required conditions were that (1) $A$ matches many low-order moments with a standard Gaussian,and (2) the chi-squared norm of $A$ with respect to the standard Gaussian is finite.While the moment-matching condition is clearly necessary for hardness, the chi-squared condition was only required for technical reasons.In this work, we establish that the latter condition is indeed not necessary.In particular, we prove near-optimal SQ lower bounds for NGCA under the moment-matching condition only.