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We give the first efficient algorithm for learning halfspaces in the testable learning model recently defined by Rubinfeld and Vasilyan [2022]. In this model, a learner certifies that the accuracy of its output hypothesis is near optimal whenever the training set passes an associated test, and training sets drawn from some target distribution must pass the test. This model is more challenging than distribution-specific agnostic or Massart noise models where the learner is allowed to fail arbitrarily if the distributional assumption does not hold. We consider the setting where the target distribution is the standard Gaussian in dimensions and the label noise is either Massart or adversarial (agnostic). For Massart noise, our tester-learner runs in polynomial time and outputs a hypothesis with (information-theoretically optimal) error (and extends to any fixed strongly log-concave target distribution). For adversarial noise, our tester-learner obtains error in polynomial time. Prior work on testable learning ignores the labels in the training set and checks that the empirical moments of the covariates are close to the moments of the base distribution. Here we develop new tests of independent interest that make critical use of the labels and combine them with the moment-matching approach of Gollakota et al. [2022]. This enables us to implement a testable variant of the algorithm of Diakonikolas et al. [2020a, 2020b] for learning noisy halfspaces using nonconvex SGD. 

We give the first result for agnostically learning Single-Index Models (SIMs) with arbitrary monotone and Lipschitz activations. All prior work either held only in the realizable setting or required the activation to be known. Moreover, we only require the marginal to have bounded second moments, whereas all prior work required stronger distributional assumptions (such as anticoncentration or boundedness). Our algorithm is based on recent work by [GHK+23] on omniprediction using predictors satisfying calibrated multiaccuracy. Our analysis is simple and relies on the relationship between Bregman divergences (or matching losses) and ℓp distances. We also provide new guarantees for standard algorithms like GLMtron and logistic regression in the agnostic setting. 

We give the first tester-learner for halfspaces that succeeds universally over a wide class of structured distributions. Our universal tester-learner runs in fully polynomial time and has the following guarantee: the learner achieves error O(opt)+ϵ on any labeled distribution that the tester accepts, and moreover, the tester accepts whenever the marginal is any distribution that satisfies a Poincare inequality. In contrast to prior work on testable learning, our tester is not tailored to any single target distribution but rather succeeds for an entire target class of distributions. The class of Poincare distributions includes all strongly log-concave distributions, and, assuming the Kannan--Lovasz--Simonovits (KLS) conjecture, includes all log-concave distributions. In the special case where the label noise is known to be Massart, our tester-learner achieves error opt+ϵ while accepting all log-concave distributions unconditionally (without assuming KLS).Our tests rely on checking hypercontractivity of the unknown distribution using a sum-of-squares (SOS) program, and crucially make use of the fact that Poincare distributions are certifiably hypercontractive in the SOS framework. 

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) or restricted models such as correlational SQ. Prior work hinted at the impossibility of our result: Vempala and Wilmes 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 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 label queries. 

We prove the first superpolynomial lower bounds for learning one-layer neural networks with respect to the Gaussian distribution using gradient descent. We show that any classifier trained using gradient descent with respect to square-loss will fail to achieve small test error in polynomial time given access to samples labeled by a one-layer neural network. For classification, we give a stronger result, namely that any statistical query (SQ) algorithm (including gradient descent) will fail to achieve small test error in polynomial time. Prior work held only for gradient descent run with small batch sizes, required sharp activations, and applied to specific classes of queries. Our lower bounds hold for broad classes of activations including ReLU and sigmoid. The core of our result relies on a novel construction of a simple family of neural networks that are exactly orthogonal with respect to all spherically symmetric distributions.