We pose the following question in computational learning theory: \textit{can we efficiently test whether a training set satisfies the assumptions of a given noise model?} This question has remained unaddressed despite decades of research on learning in the presence of noise. In this work, we show that this task is tractable and present the first efficient algorithm to test various noise assumptions on the training data. To model this question, we extend the recently proposed testable learning framework of Rubinfeld and Vasilyan (2023) and require a learner to run an associated test that satisfies the following two conditions: (1) whenever the test accepts, the learner outputs a classifier along with a \textit{certificate of optimality}, and (2) the test must pass for any dataset drawn according to a specified modeling assumption on both the marginal distribution and the noise model. We then consider the problem of learning halfspaces over Gaussian marginals with Massart noise (where each label can be flipped with probability less than $1/2$ depending on the input features), and give a fully-polynomial time testable learning algorithm. We also show a separation between the classical setting of learning in the presence of structured noise and testable learning. In fact, for the simple case of random classification noise (where each label is flipped with fixed probability $\eta = 1/2$), we show that testable learning requires super-polynomial time while classical learning is trivial.

Recent work has shown the surprising power of low-degree {\em sandwiching} polynomial approximators in the context of challenging learning settings such as learning with distribution shift, testable learning, and learning with contamination. A pair of sandwiching polynomials approximate a target function in expectation while also providing \emph{pointwise} upper and lower bounds on the function’s values. In this paper, we give a new method for constructing low-degree sandwiching polynomials that yield greatly improved degree bounds for several fundamental function classes and marginal distributions. In particular, we obtain degree $\mathrm{poly}(k)$ sandwiching polynomials for functions of $k$ halfspaces under the Gaussian distribution, improving exponentially over the prior $2^{O(k)}$ bound. More broadly, our approach applies to function classes that are low-dimensional and have smooth boundary. In contrast to prior work, our proof is relatively simple and directly uses the smoothness of the target function’s boundary to construct sandwiching Lipschitz functions, which are amenable to results from high-dimensional approximation theory. For low-dimensional polynomial threshold functions (PTFs) with respect to Gaussians, we obtain doubly exponential improvements without applying the FT-mollification method of Kane used in the best previous result.

Recent work on provably efficient algorithms for learning with distribution shift has focused on two models: PQ learning (Goldwasser et al., 2020) and TDS learning (Klivans et al., 2024). Algorithms for TDS learning are allowed to reject a test set entirely if distribution shift is detected. In contrast, PQ learners may only reject points that are deemed out-of-distribution on an individual basis. Our main result is a surprising equivalence between these two models in the distribution-free setting. In particular, we give an efficient black-box reduction from PQ learning to TDS learning for any Boolean concept class. This equivalence implies the first hardness results for distribution-free TDS learning of basic concept classes such as halfspaces. The main technical contribution underlying our equivalence is a method for boosting, via branching programs, the weak distinguishing power of TDS learners that have rejected the target domain. We also show that giving a learner access to {\em membership queries} sidesteps these hardness results and allows for efficient, distribution-free PQ learnability of halfspaces. Our algorithm iteratively recovers large-margin separators obtained by applying successive Forster transforms on the training data.

We give the first fully polynomial-time algorithm for learning halfspaces with respect to the uniform distribution on the hypercube in the presence of contamination, where an adversary may corrupt some fraction of examples and labels arbitrarily. We achieve an error guarantee of ηO(1)+є where η is the noise rate. Such a result was not known even in the agnostic setting, where only labels can be adversarially corrupted. All prior work over the last two decades has a superpolynomial dependence in 1/є or succeeds only with respect to continuous marginals (such as log-concave densities).

Sparse linear regression is one of the most basic questions in machine learning and statistics. Here, we are given as input a design matrix X ∈ ℝN × d and measurements or labels y ∈ ℝN where y = X w* + ξ, and ξ is the noise in the measurements. Importantly, we have the additional constraint that the unknown signal vector w* is sparse: it has k non-zero entries where k is much smaller than the ambient dimension. Our goal is to output a prediction vector w that has small prediction error: 1/N· ||X w* − X w||22. Information-theoretically, we know what is best possible in terms of measurements: under most natural noise distributions, we can get prediction error at most є with roughly N = O(k logd/є) samples. Computationally, this currently needs dΩ(k) run-time. Alternately, with N = O(d), we can get polynomial-time. Thus, there is an exponential gap (in the dependence on d) between the two and we do not know if it is possible to get do(k) run-time and o(d) samples. We give the first generic positive result for worst-case design matrices X: For any X, we show that if the support of w* is chosen at random, we can get prediction error є with N = poly(k, logd, 1/є) samples and run-time poly(d,N). This run-time holds for any design matrix X with condition number up to 2poly(d). Previously, such results were known for worst-case w*, but only for random design matrices from well-behaved families, matrices that have a very low condition number (poly(logd); e.g., as studied in compressed sensing), or those with special structural properties.

A decision-theoretic characterization of perfect calibration is that an agent seeking to minimize a proper loss in expectation cannot improve their outcome by post-processing a perfectly calibrated predictor. Hu and Wu (FOCS’24) use this to define an approximate calibration measure called calibration decision loss (CDL), which measures the maximal improvement achievable by any post-processing over any proper loss. Unfortunately, CDL turns out to be intractable to even weakly approximate in the offline setting, given black-box access to the predictions and labels.

The seminal work of Linial, Mansour, and Nisan gave a quasipolynomial-time algorithm for learning constant-depth circuits ($\mathsf{AC}^0$) with respect to the uniform distribution on the hypercube. Extending their algorithm to the setting of malicious noise, where both covariates and labels can be adversarially corrupted, has remained open. Here we achieve such a result, inspired by recent work on learning with distribution shift. Our running time essentially matches their algorithm, which is known to be optimal assuming various cryptographic primitives. Our proof uses a simple outlier-removal method combined with Braverman’s theorem for fooling constant-depth circuits. We attain the best possible dependence on the noise rate and succeed in the harshest possible noise model (i.e., contamination or so-called “nasty noise").

We give the first provably efficient algorithms for learning neural networks with respect to distribution shift. We work in the Testable Learning with Distribution Shift framework (TDS learning) of Klivans et al. (2024), where the learner receives labeled examples from a training distribution and unlabeled examples from a test distribution and must either output a hypothesis with low test error or reject if distribution shift is detected. No assumptions are made on the test distribution. All prior work in TDS learning focuses on classification, while here we must handle the setting of nonconvex regression. Our results apply to real-valued networks with arbitrary Lipschitz activations and work whenever the training distribution has strictly sub-exponential tails. For training distributions that are bounded and hypercontractive, we give a fully polynomial-time algorithm for TDS learning one hidden-layer networks with sigmoid activations. We achieve this by importing classical kernel methods into the TDS framework using data-dependent feature maps and a type of kernel matrix that couples samples from both train and test distributions.

Inspired by recent work on learning with distribution shift, we give a general outlier removal algorithm called *iterative polynomial filtering* and show a number of striking applications for supervised learning with contamination: (1) We show that any function class that can be approximated by low-degree polynomials with respect to a hypercontractive distribution can be efficiently learned under bounded contamination (also known as *nasty noise*). This is a surprising resolution to a longstanding gap between the complexity of agnostic learning and learning with contamination, as it was widely believed that low-degree approximators only implied tolerance to label noise. (2) For any function class that admits the (stronger) notion of sandwiching approximators, we obtain near-optimal learning guarantees even with respect to heavy additive contamination, where far more than $1/2$ of the training set may be added adversarially. Prior related work held only for regression and in a list-decodable setting. (3) We obtain the first efficient algorithms for tolerant testable learning of functions of halfspaces with respect to any fixed log-concave distribution. Even the non-tolerant case for a single halfspace in this setting had remained open. These results significantly advance our understanding of efficient supervised learning under contamination, a setting that has been much less studied than its unsupervised counterpart.

In the well-studied agnostic model of learning, the goal of a learner– given examples from an arbitrary joint distribution on $\mathbb{R}^d \times \{\pm 1\}$– is to output a hypothesis that is competitive (to within $\epsilon$) of the best fitting concept from some class. In order to escape strong hardness results for learning even simple concept classes in this model, we introduce a smoothed analysis framework where we require a learner to compete only with the best classifier that is robust to small random Gaussian perturbation. This subtle change allows us to give a wide array of learning results for any concept that (1) depends on a low-dimensional subspace (aka multi-index model) and (2) has a bounded Gaussian surface area. This class includes functions of halfspaces and (low-dimensional) convex sets, cases that are only known to be learnable in non-smoothed settings with respect to highly structured distributions such as Gaussians. Perhaps surprisingly, our analysis also yields new results for traditional non-smoothed frameworks such as learning with margin. In particular, we obtain the first algorithm for agnostically learning intersections of $k$-halfspaces in time $k^{\poly(\frac{\log k}{\epsilon \gamma}) }$ where $\gamma$ is the margin parameter. Before our work, the best-known runtime was exponential in $k$ (Arriaga and Vempala, 1999).

We revisit the fundamental problem of learning with distribution shift, in which a learner is given labeled samples from training distribution D, unlabeled samples from test distribution D’ and is asked to output a classifier with low test error. The standard approach in this setting is to bound the loss of a classifier in terms of some notion of distance between D and D’. These distances, however, seem difficult to compute and do not lead to efficient algorithms. We depart from this paradigm and define a new model called testable learning with distribution shift, where we can obtain provably efficient algorithms for certifying the performance of a classifier on a test distribution. In this model, a learner outputs a classifier with low test error whenever samples from D and D’ pass an associated test; moreover, the test must accept (with high probability) if the marginal of D equals the marginal of D’. We give several positive results for learning well-studied concept classes such as halfspaces, intersections of halfspaces, and decision trees when the marginal of D is Gaussian or uniform on the hypercube. Prior to our work, no efficient algorithms for these basic cases were known without strong assumptions on D’. For halfspaces in the realizable case (where there exists a halfspace consistent with both D and D’), we combine a moment-matching approach with ideas from active learning to simulate an efficient oracle for estimating disagreement regions. To extend to the non-realizable setting, we apply recent work from testable (agnostic) learning. More generally, we prove that any function class with low-degree $\mathcal{L}_2$-sandwiching polynomial approximators can be learned in our model. Since we require $\mathcal{L}_2$- sandwiching (instead of the usual $\mathcal{L}_1$ loss), we cannot directly appeal to convex duality and instead apply constructions from the pseudorandomness literature to obtain the required approximators. We also provide lower bounds to show that the guarantees we obtain on the performance of our output hypotheses are best possible up to constant factors, as well as a separation showing that realizable learning in our model is incomparable to (ordinary) agnostic learning.

Recent work of Klivans, Stavropoulos, and Vasilyan initiated the study of testable learning with distribution shift (TDS learning), where a learner is given labeled samples from training distribution $\mathcal{D}$, unlabeled samples from test distribution $\mathcal{D}’$, and the goal is to output a classifier with low error on $\mathcal{D}’$ whenever the training samples pass a corresponding test. Their model deviates from all prior work in that no assumptions are made on $\mathcal{D}’$. Instead, the test must accept (with high probability) when the marginals of the training and test distributions are equal. Here we focus on the fundamental case of intersections of halfspaces with respect to Gaussian training distributions and prove a variety of new upper bounds including a $2^{(k/\epsilon)^{O(1)}} \mathsf{poly}(d)$-time algorithm for TDS learning intersections of $k$ homogeneous halfspaces to accuracy $\epsilon$ (prior work achieved $d^{(k/\epsilon)^{O(1)}}$). We work under the mild assumption that the Gaussian training distribution contains at least an $\epsilon$ fraction of both positive and negative examples ($\epsilon$-balanced). We also prove the first set of SQ lower-bounds for any TDS learning problem and show (1) the $\epsilon$-balanced assumption is necessary for $\mathsf{poly}(d,1/\epsilon)$-time TDS learning for a single halfspace and (2) a $d^{\tilde{\Omega}(\log 1/\epsilon)}$ lower bound for the intersection of two general halfspaces, even with the $\epsilon$-balanced assumption. Our techniques significantly expand the toolkit for TDS learning. We use dimension reduction and coverings to give efficient algorithms for computing a localized version of discrepancy distance, a key metric from the domain adaptation literature.

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 $d$ 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 $\mathrm{opt}+\epsilon$ (and extends to any fixed strongly log-concave target distribution). For adversarial noise, our tester-learner obtains error $O(\mathrm{opt})+\epsilon$ 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 study the problem of PAC learning $\gamma$-margin halfspaces with Massart noise. We propose a simple proper learning algorithm, the Perspectron, that has sample complexity $\widetilde{O}((\epsilon\gamma)^{-2})$ and achieves classification error at most $\eta+\epsilon$ where $\eta$ is the Massart noise rate. Prior works (DGT19, CKMY20) came with worse sample complexity guarantees (in both $\epsilon$ and $\gamma$) or could only handle random classification noise (DDKWZ23,KITBMV23)--- a much milder noise assumption. We also show that our results extend to the more challenging setting of learning generalized linear models with a known link function under Massart noise, achieving a similar sample complexity to the halfspace case. This significantly improves upon the prior state-of-the-art in this setting due to CKMY20, who introduced this model.

A fundamental notion of distance between train and test distributions from the field of domain adaptation is discrepancy distance. While in general hard to compute, here we provide the first set of provably efficient algorithms for testing *localized* discrepancy distance, where discrepancy is computed with respect to a fixed output classifier. These results imply a broad set of new, efficient learning algorithms in the recently introduced model of Testable Learning with Distribution Shift (TDS learning) due to Klivans et al. (2023). Our approach generalizes and improves all prior work on TDS learning: (1) we obtain *universal* learners that succeed simultaneously for large classes of test distributions, (2) achieve near-optimal error rates, and (3) give exponential improvements for constant depth circuits. Our methods further extend to semi-parametric settings and imply the first positive results for low-dimensional convex sets. Additionally, we separate learning and testing phases and obtain algorithms that run in fully polynomial time at test time.

We study the problem of learning under arbitrary distribution shift, where the learner is trained on a labeled set from one distribution but evaluated on a different, potentially adversarially generated test distribution. We focus on two frameworks: *PQ learning* [GKKM'20], allowing abstention on adversarially generated parts of the test distribution, and *TDS learning* [KSV'23], permitting abstention on the entire test distribution if distribution shift is detected. All prior known algorithms either rely on learning primitives that are computationally hard even for simple function classes, or end up abstaining entirely even in the presence of a tiny amount of distribution shift. We address both these challenges for natural function classes, including intersections of halfspaces and decision trees, and standard training distributions, including Gaussians. For PQ learning, we give efficient learning algorithms, while for TDS learning, our algorithms can tolerate moderate amounts of distribution shift. At the core of our approach is an improved analysis of spectral outlier-removal techniques from learning with nasty noise. Our analysis can (1) handle arbitrarily large fraction of outliers, which is crucial for handling arbitrary distribution shifts, and (2) obtain stronger bounds on polynomial moments of the distribution after outlier removal, yielding new insights into polynomial regression under distribution shifts. Lastly, our techniques lead to novel results for tolerant *testable learning* [RV'23], and learning with nasty noise.

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 Gopalan et al. [2023] on Omniprediction using predictors satisfying calibrated multiaccuracy. Our analysis is simple and relies on the relationship between Bregman divergences (or matching losses) and $\ell_p$ 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 $O(\mathrm{opt}) + \epsilon$ 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 $\mathrm{opt} + \epsilon$ 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 study the problem of covering and learning sums $X = X_1 + \cdots + X_n$ of independent integer-valued random variables $X_i$ (SIIRVs) with infinite support. De et al. at FOCS 2018, showed that even when the collective support of $X_i$'s is of size $4$, the maximum value of the support necessarily appears in the sample complexity of learning $X$. In this work, we address two questions: (i) Are there general families of SIIRVs with infinite support that can be learned with sample complexity independent of both $n$ and the maximal element of the support? (ii) Are there general families of SIIRVs with infinite support that admit proper sparse covers in total variation distance? As for question (i), we provide a set of simple conditions that allow the infinitely supported SIIRV to be learned with complexity $ \text{poly}(1/\epsilon)$ bypassing the aforementioned lower bound. We further address question (ii) in the general setting where each variable $X_i$ has unimodal probability mass function and is a different member of some, possibly multi-parameter, exponential family $\mathcal{E}$ that satisfies some structural properties. These properties allow $\mathcal{E}$ to contain heavy tailed and non log-concave distributions. Moreover, we show that for every $\epsilon > 0$, and every $k$-parameter family $\mathcal{E}$ that satisfies some structural assumptions, there exists an algorithm with $\widetilde{O}(k) \cdot \text{poly}(1/\epsilon)$ samples that learns a sum of $n$ arbitrary members of $\mathcal{E}$ within $\epsilon$ in TV distance. The output of the learning algorithm is also a sum of random variables within the family $\mathcal{E}$. En route, we prove that any discrete unimodal exponential family with bounded constant-degree central moments can be approximated by the family corresponding to a bounded subset of the initial (unbounded) parameter space.

We consider the problem of learning the true ordering of a set of alternatives from largely incomplete and noisy rankings. We introduce a natural generalization of both the Mallows model, a popular model of ranking distributions, and the extensively studied model of ranking from pairwise comparisons. Our selective Mallows model outputs a noisy ranking on any given subset of alternatives, based on an underlying Mallows distribution. Assuming a sequence of subsets where each pair of alternatives appears frequently enough, we obtain strong asymptotically tight upper and lower bounds on the sample complexity of learning the underlying complete central ranking and the (identities and the) ranking of the top k alternatives from selective Mallows rankings. Moreover, building on the work of (Braverman and Mossel, 2009), we show how to efficiently compute the maximum likelihood complete ranking from selective Mallows rankings.