We study the problem of learning mixtures of $k$ Gaussians in $d$ dimensions. We make no separation assumptions on the underlying mixture components: we only require that the covariance matrices have bounded condition number and that the means and covariances lie in a ball of bounded radius. We give an algorithm that draws $d^{\textrm{poly}(k/\epsilon)}$ samples from the target mixture, runs in sample-polynomial time, and constructs a sampler whose output distribution is $\epsilon$-close from the unknown mixture in total variation. Prior works for this problem either (i) required exponential runtime in the dimension $d$, (ii) placed strong assumptions on the instance (e.g., spherical covariances or clusterability), or (iii) had doubly exponential dependence on the number of components $k$. Our approach departs from commonly used techniques for this problem like the method of moments. Instead, we leverage a recently developed reduction, based on diffusion models, from distribution learning to a supervised learning task called score matching. We give an algorithm for the latter by proving a structural result showing that the score function of a Gaussian mixture can be approximated by a piecewise-polynomial function, and there is an efficient algorithm for finding it. To our knowledge, this is the first example of diffusion models achieving a state-of-the-art theoretical guarantee for an unsupervised learning task.

We study the problem of learning from truncated samples: instead of observing samples from some underlying population $p^\ast$, we observe only the examples that fall in some survival set $S \subset \mathbb{R}^d$ whose probability mass (measured with respect to $p^\ast$) is at least $\alpha$. Assuming membership oracle access to the truncation set $S$, prior works obtained algorithms for the case where $p^\ast$ is Gaussian or more generally an exponential family with strongly convex likelihood --- albeit with a super-polynomial dependency on the (inverse) survival mass $1/\alpha$ both in terms of runtime and in number of oracle calls to the set $S$. In this work we design a new learning method with runtime and query complexity polynomial in $1/\alpha$. Our result significantly improves over the prior works by focusing on efficiently solving the underlying optimization problem using a general purpose optimization algorithm with minimal assumptions.

We study the task of online learning in the presence of Massart noise. Specifically, instead of assuming that the online adversary chooses an arbitrary sequence of labels, we assume that the context $\boldsymbol{x}$ is selected adversarially but the label $y$ presented to the learner disagrees with the ground-truth label of $\boldsymbol{x}$ with unknown probability {\em at most} $\eta$. We focus on the fundamental class of $\gamma$-margin linear classifiers and present the first computationally efficient algorithm that achieves mistake bound $\eta T + o(T)$. We point out that the mistake bound achieved by our algorithm is qualitatively tight for computationally efficient algorithms; this follows from the fact that, even in the offline setting, achieving 0-1 error better than $\eta$ requires super-polynomial time under standard complexity assumptions. We extend our online learning model to a $k$-arm contextual bandit setting where the rewards---instead of satisfying commonly used realizability assumptions---are consistent, in expectation, with some linear ranking function with weight vector $\boldsymbol{w}^\ast$. Given a list of contexts $\boldsymbol{x}_1,\ldots \boldsymbol{x}_k$, if $\boldsymbol{w}^*\cdot \boldsymbol{x}_i > \boldsymbol{w}^* \cdot \boldsymbol{x}_j$, the expected reward of action $i$ must be larger than that of $j$ by at least $\Delta$. We use our Massart online learner to design an efficient bandit algorithm that obtains expected reward at least $(1-1/k)~ \Delta T - o(T)$ bigger than choosing a random action at every round.

In recent years, masked diffusion models (MDMs) have emerged as a promising alternative approach for generative modeling over discrete domains. Compared to autoregressive models (ARMs), MDMs trade off complexity at training time with flexibility at inference time. At training time, they must learn to solve an exponentially large number of infilling problems, but at inference time, they can decode tokens in essentially arbitrary order. In this work we closely examine these two competing effects. On the training front, we theoretically and empirically demonstrate that MDMs indeed train on computationally intractable subproblems compared to their autoregressive counterparts. On the inference front, we show that a suitable strategy for adaptively choosing the token decoding order significantly enhances the capabilities of MDMs, allowing them to sidestep hard subproblems. On logic puzzles like Sudoku, we show that adaptive inference can boost solving accuracy in pretrained MDMs from $<7$\% to $\approx 90$\%, even outperforming ARMs that were explicitly trained via teacher forcing to learn the right order of decoding.

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 consider an active learning setting where a learner is presented with a pool $S$ of $n$ unlabeled examples belonging to a domain $\mathcal X$ and asks queries to find the underlying labeling that agrees with a target concept $h^\ast \in \mathcal H$. In contrast to traditional active learning that queries a single example for its label, we study more general \emph{region queries} that allow the learner to pick a subset of the domain $T \subset \mathcal X$ and a target label $y$ and ask a labeler whether $h^\ast(x) = y $ for every example in the set $T \cap S$. Such more powerful queries allow us to bypass the limitations of traditional active learning and use significantly fewer rounds of interactions to learn but can potentially lead to a significantly more complex query language. Our main contribution is quantifying the trade-off between the number of queries and the complexity of the query language used by the learner. We measure the complexity of the region queries via the VC dimension of the family of regions. We show that given any hypothesis class $\H$ with VC dimension $d$, one can design a region query family $Q$ with VC dimension $6d$ such that for every set of $n$ examples $S \subset \X$ and every $h^* \in \H$, a learner can submit $O(d\log n)$ queries from $Q$ to a labeler and perfectly label $S$. We show a matching lower bound by designing a hypothesis class $\H$ with VC dimension $d$ and a dataset $S \subset \X$ of size $n$ such that any learning algorithm using any query class with VC dimension $(d-2)/3$ must make $\poly(n)$ queries to label $S$ perfectly. Finally, we focus on well-studied hypothesis classes including unions of intervals, high-dimensional boxes, and $d$-dimensional halfspaces, and obtain stronger results. In particular, we design learning algorithms that (i) are computationally efficient and (ii) work even when the queries are not answered based on the learner’s pool of examples $S$ but on some unknown superset $L$ of $S$.

We study the power of query access for the fundamental task of agnostic learning under the Gaussian distribution. In the agnostic model, no assumptions are made on the labels of the examples and the goal is to compute a hypothesis that is competitive with the best-fit function in a known class, i.e., it achieves error opt$+\epsilon$, where opt is the error of the best function in the class. We focus on a general family of Multi-Index Models (MIMs), which are d-variate functions that depend only on few relevant directions, i.e., have the form$g$(Wx) for an unknown link function$g$and a$k\times d$matrix W. Multi-index models cover a wide range of commonly studied function classes, including real-valued function classes such as constant-depth neural networks with ReLU activations, and Boolean concept classes such as intersections of halfspaces. Our main result shows that query access gives significant runtime improvements over random examples for agnostically learning both real-valued and Boolean-valued MIMs. Under standard regularity assumptions for the link function (namely, bounded variation or surface area), we give an agnostic query learner for MIMs with running time$O(k)^{\text{poly}(1/\epsilon}$) poly$(d)$. In contrast, algorithms that rely only on random labeled examples inherently require$d^{\text{poly}(1/\epsilon}$samples and runtime, even for the basic problem of agnostically learning a single ReLU or a halfspace. As special cases of our general approach, we obtain the following results: •For the class of depth-ℓ, width-S ReLU networks on$\mathbb{R}^{d}$, our agnostic query learner runs in time poly$(d)2^{\text{poly}(\ell S/\epsilon)}$. This bound qualitatively matches the runtime of an algorithm by [1] for the realizable PAC setting with random examples. •For the class of arbitrary intersections of$k$halfspaces on$\mathbb{R}^{d}$, our agnostic query learner runs in time poly$(d)2^{\text{poly}(\log(k)/\epsilon)}$. Prior to our work, no improvement over the agnostic PAC model complexity (without queries) was known, even for the case of a single halfspace. In both these settings, we provide evidence that the$2^{\text{poly}(1/\epsilon)}$runtime dependence is required for proper query learners, even for agnosticallylearning a single ReL U or halfspace. Our algorithmic result establishes a strong computational separation between the agnostic PAC and the agnostic PAC+Query models under the Gaussian distribution for a range of natural function classes. Prior to our work, no such separation was known for any natural concept class - even for the case of a single halfspace, for which it was an open problem posed by Feldman [2]. Our results are enabled by a general dimension-reduction technique that leverages query access to estimate gradients of (a smoothed version of) the underlying label function.

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 pool-based active learning, where a learner has a large pool $S$ of unlabeled examples and can adaptively ask a labeler questions to learn these labels. The goal of the learner is to output a labeling for $S$ that can compete with the best hypothesis from a given hypothesis class $\mathcal{H}$. We focus on halfspace learning, one of the most important problems in active learning. It is well known that in the standard active learning model, learning the labels of an arbitrary pool of examples labeled by some halfspace up to error $\epsilon$ requires at least $\Omega(1/\epsilon)$ queries. To overcome this difficulty, previous work designs simple but powerful query languages to achieve $O(\log(1/\epsilon))$ query complexity, but only focuses on the realizable setting where data are perfectly labeled by some halfspace. However, when labels are noisy, such queries are too fragile and lead to high query complexity even under the simple random classification noise model. In this work, we propose a new query language called threshold statistical queries and study their power for learning under various noise models. Our main algorithmic result is the first query-efficient algorithm for learning halfspaces under the popular Massart noise model. With an arbitrary dataset corrupted with Massart noise at noise rate $\eta$, our algorithm uses only $\mathrm{polylog(1/\epsilon)}$ threshold statistical queries and computes an $(\eta + \epsilon)$-accurate labeling in polynomial time. For the harder case of agnostic noise, we show that it is impossible to beat $O(1/\epsilon)$ query complexity even for the much simpler problem of learning singleton functions (and thus for learning halfspaces) using a reduction from agnostic distributed learning.

We study the efficient learnability of low-degree polynomial threshold functions (PTFs) in the presence of a constant fraction of adversarial corruptions. Our main algorithmic result is a polynomial-time PAC learning algorithm for this concept class in the strong contamination model under the Gaussian distribution with error guarantee Od, c(opt1−c), for any desired constant c>0, where opt is the fraction of corruptions. In the strong contamination model, an omniscient adversary can arbitrarily corrupt an opt-fraction of the data points and their labels. This model generalizes the malicious noise model and the adversarial label noise model. Prior to our work, known polynomial-time algorithms in this corruption model (or even in the weaker adversarial label noise model) achieved error Õd(opt1/(d+1)), which deteriorates significantly as a function of the degree d. Our algorithm employs an iterative approach inspired by localization techniques previously used in the context of learning linear threshold functions. Specifically, we use a robust perceptron algorithm to compute a good partial classifier and then iterate on the unclassified points. In order to achieve this, we need to take a set defined by a number of polynomial inequalities and partition it into several well-behaved subsets. To this end, we develop new polynomial decomposition techniques that may be of independent interest.

In online classification, a learner is presented with a sequence of examples and aims to predict their labels in an online fashion so as to minimize the total number of mistakes. In the self-directed variant, the learner knows in advance the pool of examples and can adaptively choose the order in which predictions are made. Here we study the power of choosing the prediction order and establish the first strong separation between worst-order and random-order learning for the fundamental task of linear classification. Prior to our work, such a separation was known only for very restricted concept classes, e.g., one-dimensional thresholds or axis-aligned rectangles.We present two main results.If $X$ is a dataset of $n$ points drawn uniformly at random from the $d$-dimensional unit sphere, we design an efficient self-directed learner thatmakes $O(d \log \log(n))$ mistakes and classifies the entire dataset.If $X$ is an arbitrary $d$-dimensional dataset of size $n$, we design an efficient self-directed learner that predicts the labels of $99%$ of the points in $X$ with mistake bound independent of $n$. In contrast, under a worst- or random-ordering, the number of mistakes must be at least $\Omega(d \log n)$, even when the points are drawn uniformly from the unit sphere and the learner only needs to predict the labels for $1%$ of them.

Deep Neural Networks and Reinforcement Learning methods have empirically shown great promise in tackling challenging combinatorial problems. In those methods a deep neural network is used as a solution generator which is then trained by gradient-based methods (e.g., policy gradient) to successively obtain better solution distributions. In this work we introduce a novel theoretical framework for analyzing the effectiveness of such methods. We ask whether there exist generative models that (i) are expressive enough to generate approximately optimal solutions; (ii) have a tractable, i.e, polynomial in the size of the input, number of parameters; (iii) their optimization landscape is benign in the sense that it does not contain sub-optimal stationary points. Our main contribution is a positive answer to this question. Our result holds for a broad class of combinatorial problems including Max- and Min-Cut, Max-$k$-CSP, Maximum-Weight-Bipartite-Matching, and the Traveling Salesman Problem. As a byproduct of our analysis we introduce a novel regularization process over vanilla gradient descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.

We give the first polynomial-time algorithm for the testable learning of halfspaces in the presence of adversarial label noise under the Gaussian distribution. In the recently introduced testable learning model, one is required to produce a tester-learner such that if the data passes the tester, then one can trust the output of the robust learner on the data. Our tester-learner runs in time $\text{poly}(d/\epsilon)$ and outputs a halfspace with misclassification error $O(\text{opt})+\epsilon$, where $\text{opt}$ is the 0-1 error of the best fitting halfspace. At a technical level, our algorithm employs an iterative soft localization technique enhanced with appropriate testers to ensure that the data distribution is sufficiently similar to a Gaussian. Finally, our algorithm can be readily adapted to yield an efficient and testable active learner requiring only $d ~ \text{polylog}(1/\epsilon)$ labeled examples.

Knowledge distillation with unlabeled examples is a powerful training paradigm for generating compact and lightweight student models in applications where the amount of labeled data is limited but one has access to a large pool of unlabeled data. In this setting, a large teacher model generates "soft" pseudo-labels for the unlabeled dataset which are then used for training the student model. Despite its success in a wide variety of applications, a shortcoming of this approach is that the teacher's pseudo-labels are often noisy, leading to impaired student performance. In this paper, we present a principled method for knowledge distillation with unlabeled examples that we call Student-Label Mixing (SLaM) and we show that it consistently improves over prior approaches by evaluating it on several standard benchmarks. Finally, we show that SLaM comes with theoretical guarantees; along the way we give an algorithm improving the best-known sample complexity for learning halfspaces with margin under random classification noise, and provide the first convergence analysis for so-called ``forward loss-adjustment" methods.

We study fixed-design online learning where the learner is allowed to choose the order of the datapoints in order to minimize their regret (aka self-directed online learning). We focus on the fundamental task of online linear regression: the learner is given a dataset $X$ with $n$ examples in $d$ dimensions and at step $t$ they select a point $x_t \in X$, predict a value $\widetilde y_t$, and suffer loss $(\widetilde y_t - w^\ast \cdot x_t)^2$. The goal is to design algorithms that order the examples and achieve better regret than random- or worst-order online algorithms. For an arbitrary dataset $X$, we show that, under the Exponential Time Hypothesis, no efficient algorithm can approximate the optimal (best-order) regret within a factor of $d^{1/\poly(\log \log d)}$. We then show that, for structured datasets, we can bypass the above hardness result and achieve nearly optimal regret. When the examples of $X$ are drawn i.i.d.\ from the uniform distribution on the sphere, we present an algorithm based on the greedy heuristic of selecting ``easiest'' examples first that achieves a $\log d$-approximation of the optimal regret.

Abstract We study opinion formation games based on the famous model proposed by Friedkin and Johsen (FJ model). In today’s huge social networks the assumption that in each round agents update their opinions by taking into account the opinions of all their friends is unrealistic. So, we are interested in the convergence properties of simple and natural variants of the FJ model that use limited information exchange in each round and converge to the same stable point. As in the FJ model, we assume that each agent i has an intrinsic opinion $$s_i \in [0,1]$$ s i ∈ [ 0 , 1 ] and maintains an expressed opinion $$x_i(t) \in [0,1]$$ x i ( t ) ∈ [ 0 , 1 ] in each round t. To model limited information exchange, we consider an opinion formation process where each agent i meets with one random friend j at each round t and learns only her current opinion $$x_j(t)$$ x j ( t ) . The amount of influence j imposes on i is reflected by the probability $$p_{ij}$$ p ij with which i meets j. Then, agent i suffers a disagreement cost that is a convex combination of $$(x_i(t) - s_i)^2$$ ( x i ( t ) - s i ) 2 and $$(x_i(t) - x_j(t))^2$$ ( x i ( t ) - x j ( t ) ) 2 . An important class of dynamics in this setting are no regret dynamics, i.e. dynamics that ensure vanishing regret against the experienced disagreement cost to the agents. We show an exponential gap between the convergence rate of no regret dynamics and of more general dynamics that do not ensure no regret. We prove that no regret dynamics require roughly $$\varOmega (1/\varepsilon )$$ Ω ( 1 / ε ) rounds to be within distance $$\varepsilon $$ ε from the stable point of the FJ model. On the other hand, we provide an opinion update rule that does not ensure no regret and converges to $$x^*$$ x ∗ in $$\tilde{O}(\log ^2(1/\varepsilon ))$$ O ~ ( log 2 ( 1 / ε ) ) rounds. Finally, in our variant of the FJ model, we show that the agents can adopt a simple opinion update rule that ensures no regret to the experienced disagreement cost and results in an opinion vector that converges to the stable point $$x^*$$ x ∗ of the FJ model within distance $$\varepsilon $$ ε in $$\textrm{poly}(1/\varepsilon )$$ poly (

We study the fundamental problem of learning a single neuron, i.e., a function of the form $\x \mapsto \sigma(\vec w \cdot \x)$ for monotone activations $\sigma:\R \mapsto \R$, with respect to the $L_2^2$-loss in the presence of adversarial label noise. Specifically, we are given labeled examples from a distribution $D$ on $(\x{}, y) \in \R^d \times \R$ such that there exists $\vec w^\ast \in \R^d$ achieving $F(\vec w^\ast) = \opt$, where $F(\vec w) = \E_{(\x{},y) \sim D}[(\sigma(\vec w\cdot \x) - y)^2]$. The goal of the learner is to output a hypothesis vector $\wt{\vec w}$ such that $F(\wt{\vec w}) = C \, \opt+\eps$ with high probability, where $C$ is a universal constant. As our main contribution, we give efficient constant-factor approximate learners for a broad class of distributions (including log-concave distributions) and activation functions (including ReLUs and sigmoids). Concretely, for the class of isotropic log-concave distributions, we obtain the following important corollaries: \begin{itemize}[leftmargin=3pc, rightmargin = 1.5pc] \item For the logistic activation, i.e., $\sigma(t) = 1/(1+e^{-t})$, we obtain the first polynomial-time constant factor approximation, even under the Gaussian distribution. Moreover, our algorithm has sample complexity $\wt{O}(d/\eps)$, which is tight within polylogarithmic factors. \item For the ReLU activation, i.e., $\sigma(t) = \max(0,t)$, we give an efficient algorithm with sample complexity $\wt{O}(d \, \polylog(1/\eps))$. Prior to our work, the best known constant-factor approximate learner had sample complexity $\Omega(d/\eps)$. \end{itemize} In both settings, our algorithms are simple, performing gradient-descent on the (regularized) $L_2^2$-loss. The correctness of our algorithms relies on novel structural results that we establish, showing that (essentially all) stationary points of the underlying non-convex loss are approximately optimal.

We study the problem of learning general {—} i.e., not necessarily homogeneous {—} halfspaces with adversarial label noise under the Gaussian distribution. Prior work has provided a sophisticated polynomial-time algorithm for this problem. In this work, we show that the problem can be solved directly via online gradient descent applied to a sequence of natural non-convex surrogates. This approach yields a simple iterative learning algorithm for general halfspaces with near-optimal sample complexity, runtime, and error guarantee. At the conceptual level, our work establishes an intriguing connection between learning halfspaces with adversarial noise and online optimization that may find other applications.

Label Ranking (LR) is the supervised task of learning a sorting function that maps feature vectors $x \in \mathbb{R}^d$ to rankings $\sigma(x) \in \mathbb S_k$ over a finite set of $k$ labels. We focus on the fundamental case of learning linear sorting functions (LSFs) under Gaussian marginals: $x$ is sampled from the $d$-dimensional standard normal and the ground truth ranking $\sigma^\star(x)$ is the ordering induced by sorting the coordinates of the vector $W^\star x$, where $W^\star \in \mathbb{R}^{k \times d}$ is unknown. We consider learning LSFs in the presence of bounded noise: assuming that a noiseless example is of the form $(x, \sigma^\star(x))$, we observe $(x, \pi)$, where for any pair of elements $i \neq j$, the probability that the order of $i, j$ is different in $\pi$ than in $\sigma^\star(x)$ is at most $\eta < 1/2$. We design efficient non-proper and proper learning algorithms that learn hypotheses within normalized Kendall's Tau distance $\epsilon$ from the ground truth with $N= \widetilde{O}(d\log(k)/\epsilon)$ labeled examples and runtime $\mathrm{poly}(N, k)$. For the more challenging top-$r$ disagreement loss, we give an efficient proper learning algorithm that achieves $\epsilon$ top-$r$ disagreement with the ground truth with $N = \widetilde{O}(d k r /\epsilon)$ samples and $\mathrm{poly}(N)$ runtime.

Distillation with unlabeled examples is a popular and powerful method for training deep neural networks in settings where the amount of labeled data is limited: A large “teacher” neural network is trained on the labeled data available, and then it is used to generate labels on an unlabeled dataset (typically much larger in size). These labels are then utilized to train the smaller “student” model which will actually be deployed. Naturally, the success of the approach depends on the quality of the teacher’s labels, since the student could be confused if trained on inaccurate data. This paper proposes a principled approach for addressing this issue based on a “debiasing" reweighting of the student’s loss function tailored to the distillation training paradigm. Our method is hyper-parameter free, data-agnostic, and simple to implement. We demonstrate significant improvements on popular academic datasets and we accompany our results with a theoretical analysis which rigorously justifies the performance of our method in certain settings.

We study the problem of PAC learning halfspaces on ℝd with Massart noise under the Gaussian distribution. In the Massart model, an adversary is allowed to flip the label of each point x with unknown probability η(x) ≤ η, for some parameter η ∈ [0,1/2]. The goal is to find a hypothesis with misclassification error of OPT + є, where OPT is the error of the target halfspace. This problem had been previously studied under two assumptions: (i) the target halfspace is homogeneous (i.e., the separating hyperplane goes through the origin), and (ii) the parameter η is strictly smaller than 1/2. Prior to this work, no nontrivial bounds were known when either of these assumptions is removed. We study the general problem and establish the following:

We show a statistical version of Taylor’s theorem and apply this result to non-parametric density estimation from truncated samples, which is a classical challenge in Statistics [Woodroofe 1985, Stute 1993]. The single-dimensional version of our theorem has the following implication: "For any distribution P on [0, 1] with a smooth log-density function, given samples from the conditional distribution of P on [a, a + \varepsilon] \subset [0, 1], we can efficiently identify an approximation to P over the whole interval [0, 1], with quality of approximation that improves with the smoothness of P". To the best of knowledge, our result is the first in the area of non-parametric density estimation from truncated samples, which works under the hard truncation model, where the samples outside some survival set S are never observed, and applies to multiple dimensions. In contrast, previous works assume single dimensional data where each sample has a different survival set $S$ so that samples from the whole support will ultimately be collected.

We study the problem of agnostically learning halfspaces under the Gaussian distribution. Our main result is the {\em first proper} learning algorithm for this problem whose running time qualitatively matches that of the best known improper agnostic learner. Building on this result, we also obtain the first proper polynomial time approximation scheme (PTAS) for agnostically learning homogeneous halfspaces. Our techniques naturally extend to agnostically learning linear models with respect to other activation functions, yielding the first proper agnostic algorithm for ReLU regression.

For many learning problems one may not have access to fine grained label information; e.g., an image can be labeled as husky, dog, or even animal depending on the expertise of the annotator. In this work, we formalize these settings and study the problem of learning from such coarse data. Instead of observing the actual labels from a set $\mathcal{Z}$, we observe coarse labels corresponding to a partition of $\mathcal{Z}$ (or a mixture of partitions). Our main algorithmic result is that essentially any problem learnable from fine grained labels can also be learned efficiently when the coarse data are sufficiently informative. We obtain our result through a generic reduction for answering Statistical Queries (SQ) over fine grained labels given only coarse labels. The number of coarse labels required depends polynomially on the information distortion due to coarsening and the number of fine labels $|\mathcal{Z}|$. We also investigate the case of (infinitely many) real valued labels focusing on a central problem in censored and truncated statistics: Gaussian mean estimation from coarse data. We provide an efficient algorithm when the sets in the partition are convex and establish that the problem is NP-hard even for very simple non-convex sets.

We study the problem of designing online algorithms given advice about the input. While prior work had focused on deterministic advice, we only assume distributional access to the instances of interest, and the goal is to learn a competitive algorithm given access to i.i.d. samples. We aim to be competitive against an adversary with prior knowledge of the distribution, while also performing well against worst-case inputs. We focus on the classical online problems of ski-rental and prophet-inequalities, and provide sample complexity bounds for the underlying learning tasks. First, we point out that for general distributions it is information-theoretically impossible to beat the worst-case competitive-ratio with any finite sample size. As our main contribution, we establish strong positive results for well-behaved distributions. Specifically, for the broad class of log-concave distributions, we show that $\mathrm{poly}(1/\epsilon)$ samples suffice to obtain $(1+\epsilon)$-competitive ratio. Finally, we show that this sample upper bound is close to best possible, even for very simple classes of distributions.

We study the problem of PAC learning homogeneous halfspaces with Tsybakov noise. In the Tsybakov noise model, the label of every example is independently flipped with an adversarially controlled probability that can be arbitrarily close to 1/2 for a fraction of the examples. We give the first polynomial-time algorithm for this fundamental learning problem. Our algorithm learns the true halfspace within any desired accuracy and succeeds under a broad family of well-behaved distributions including log-concave distributions.

We study the problem of PAC learning one-hidden-layer ReLU networks with $k$ hidden units on $\mathbb{R}^d$ under Gaussian marginals in the presence of additive label noise. For the case of positive coefficients, we give the first polynomial-time algorithm for this learning problem for $k$ up to $\tilde{O}(\sqrt{\log d})$. Previously, no polynomial time algorithm was known, even for $k=3$. This answers an open question posed by Klivans (2017). Importantly, our algorithm does not require any assumptions about the rank of the weight matrix and its complexity is independent of its condition number. On the negative side, for the more general task of PAC learning one-hidden-layer ReLU networks with arbitrary real coefficients, we prove a Statistical Query lower bound of $d^{\Omega(k)}$. Thus, we provide a separation between the two classes in terms of efficient learnability. Our upper and lower bounds are general, extending to broader families of activation functions.

We study the problem of learning halfspaces with Massart noise in the distribution-specific PAC model. We give the first computationally efficient algorithm for this problem with respect to a broad family of distributions, including log-concave distributions. This resolves an open question posed in a number of prior works. Our approach is extremely simple: We identify a smooth {\em non-convex} surrogate loss with the property that any approximate stationary point of this loss defines a halfspace that is close to the target halfspace. Given this structural result, we can use SGD to solve the underlying learning problem.

We study the problem of agnostically learning homogeneous halfspaces in the distribution-specific PAC model. For a broad family of structured distributions, including log-concave distributions, we show that non-convex SGD efficiently converges to a solution with misclassification error $O(\opt)+\eps$, where $\opt$ is the misclassification error of the best-fitting halfspace. In sharp contrast, we show that optimizing any convex surrogate inherently leads to misclassification error of $\omega(\opt)$, even under Gaussian marginals.

We study the problem of estimating the parameters of a Gaussian distribution when samples are only shown if they fall in some (unknown) set. This core problem in truncated statistics has long history going back to Galton, Lee, Pearson and Fisher. Recent work by Daskalakis et al. (FOCS'18), provides the first efficient algorithm that works for arbitrary sets in high dimension when the set is known, but leaves as an open problem the more challenging and relevant case of unknown truncation set. Our main result is a computationally and sample efficient algorithm for estimating the parameters of the Gaussian under arbitrary unknown truncation sets whose performance decays with a natural measure of complexity of the set, namely its Gaussian surface area. Notably, this algorithm works for large families of sets including intersections of halfspaces, polynomial threshold functions and general convex sets. We show that our algorithm closely captures the tradeoff between the complexity of the set and the number of samples needed to learn the parameters by exhibiting a set with small Gaussian surface area for which it is information theoretically impossible to learn the true Gaussian with few samples.