Margin-based learning, exemplified by linear and kernel methods, is one of the few classical settings where generalization guarantees are independent of the number of parameters. This makes it a central case study in modern highly over-parameterized learning. We ask what minimal mathematical structure underlies this phenomenon. We begin with a simple margin-based problem in arbitrary metric spaces: concepts are defined by a center point and classify points according to whether their distance lies below $r$ or above $R$. We show that whenever $R>3r$, this class is learnable in \emph{any} metric space. Thus, sufficiently large margins make learnability rely only on the triangle inequality, without any linear or analytic structure being necessary. Our first main result extends this phenomenon to concepts defined by bounded linear combinations of distance functions, and reveals a sharp threshold: there exists a universal constant such that whenever the margin is larger than this constant, the class is learnable in every metric space, while below it there exist metric spaces where it is not learnable at all. We then ask whether margin-based learnability can always be explained via an embedding into a linear space – that is, reduced to linear classification in some Banach space through a kernel-type construction. We answer this negatively by demonstrating a margin learnable class that cannot be embedded into any Banach space in which linear classification with margins is learnable.

Modern machine learning systems, such as generative models and recommendation systems, often evolve through a cycle of deploying a model, observing user interactions, and updating the model intermittently based on feedback. This mode of learning contrasts with common supervised learning frameworks, which focus on loss or regret minimization over a shared sequence of prediction tasks. Motivated by this deployment-driven learning cycle, we revisit the classical model of learning from equivalence queries, introduced by Angluin, which provides a simple abstraction of such interactions: a learner repeatedly proposes hypotheses and, whenever the deployed hypothesis is inadequate, receives a counterexample tailored to that hypothesis. Under fully adversarial counterexample generation, however, this model exhibits overly pessimistic worst-case behavior. Moreover, most existing work on learning from equivalence queries considers the \emph{full-information} setting, where the learner observes not only a counterexample but also its correct label. This is an assumption that does not always align with natural interactive settings. To address these considerations, we restrict the environment to generate counterexamples in a less adversarial manner by introducing a broad class of counterexample generators, which we call \emph{symmetric}. Informally, such symmetric counterexample generators select counterexamples based only on the symmetric difference between the hypothesis and the target, and encompass natural feedback mechanisms such as random counterexamples, as well as generators that select counterexamples minimizing a prescribed complexity measure over the instance space. Within this framework, we study learning from equivalence queries under both full-information and bandit feedback. We establish tight bounds on the number of learning rounds in both settings and outline directions for future research. Our techniques rely on a game-theoretic perspective on symmetric adversaries and combine adaptive weighting algorithms with minimax arguments.

We study the out-of-sample performance of multi-pass stochastic gradient descent (SGD) in the fundamental stochastic convex optimization (SCO) model. While one-pass SGD is known to achieve an optimal $\Theta(1/\sqrt{n})$ excess population loss given a sample of size $n$, much less is understood about the multi-pass version of the algorithm which is widely used in practice. Somewhat surprisingly, we show that in the general non-smooth case of SCO, just a few epochs of SGD can already hurt its out-of-sample performance significantly and lead to overfitting. In particular, using a step size $\eta = \Theta(1/\sqrt{n})$, which gives the optimal rate after one pass, can lead to population loss as large as $\Omega(1)$ after just one additional pass. More generally, we show that the population loss from the second pass onward is of the order $\Theta(1/(\eta T) + \eta \sqrt{T})$, where $T$ is the total number of steps. These results reveal a certain phase-transition in the out-of-sample behavior of SGD after the first epoch, as well as a sharp separation between the rates of overfitting in the smooth and non-smooth cases of SCO. Additionally, we extend our results to with-replacement SGD, proving that the same asymptotic bounds hold after $O(n \log n)$ steps. Finally, we also prove a lower bound of $\Omega(\eta \sqrt{n})$ on the generalization gap of one-pass SGD in dimension $d = {\widetilde O}(n)$, improving on recent results of Koren et al. (2022) and Schliserman et al. (2024).

Stochastic convex optimization is one of the most well-studied models for learning in modern machine learning. Nevertheless, a central fundamental question in this setup remained unresolved: how many data points must be observed so that any empirical risk minimizer (ERM) shows good performance on the true population? This question was proposed by Feldman who proved that $\Omega(\frac{d}{\epsilon} + \frac{1}{\epsilon^2} )$ data points are necessary (where $d$ is the dimension and $\epsilon > 0$ the accuracy parameter). Proving an $\omega(\frac{d}{\epsilon} + \frac{1}{\epsilon^2})$ lower bound was left as an open problem. In this work we show that in fact $\tilde{O}(\frac{d}{\epsilon} + \frac{1}{\epsilon^2})$ data points are also sufficient. This settles the question and yields a new separation between ERMs and uniform convergence. This sample complexity holds for the classical setup of learning bounded convex Lipschitz functions over the Euclidean unit ball. We further generalize the result and show that a similar upper bound holds for all symmetric convex bodies. The general bound is composed of two terms: (i) a term of the form $\tilde{O}(\frac{d}{\epsilon})$ with an inverse-linear dependence on the accuracy parameter, and (ii) a term that depends on the statistical complexity of the class of linear functions (captured by the Rademacher complexity). The proof builds a mechanism for controlling the behavior of stochastic convex optimization problems.

In this work, we investigate the interplay between memorization and learning in the context of stochastic convex optimization (SCO). We define memorization via the information a learning algorithm reveals about its training data points. We then quantify this information using the framework of conditional mutual information (CMI) proposed by Steinke and Zakynthinou (2020). Our main result is a precise characterization of the tradeoff between the accuracy of a learning algorithm and its CMI, answering an open question posed by Livni (2023). We show that, in the $L^2$ Lipschitz–bounded setting and under strong convexity, every learner with an excess error $\epsilon$ has CMI bounded below by $\Omega(1/\epsilon^2)$ and $\Omega(1/\epsilon)$, respectively. We further demonstrate the essential role of memorization in learning problems in SCO by designing an adversary capable of accurately identifying a significant fraction of the training samples in specific SCO problems. Finally, we enumerate several implications of our results, such as a limitation of generalization bounds based on CMI and the incompressibility of samples in SCO problems.

The study of adaptive data analysis examines how many statistical queries can be answered accurately using a fixed dataset while avoiding false discoveries (statistically inaccurate answers). In this paper, we tackle a question that precedes the field of study: Is data only valuable when it provides accurate answers to statistical queries? To answer this question, we use Stochastic Convex Optimization as a case study. In this model, algorithms are considered as analysts who query an estimate of the gradient of a noisy function at each iteration and move towards its minimizer. It is known that $O(1/ε^2)$ examples can be used to minimize the objective function, but none of the existing methods depend on the accuracy of the estimated gradients along the trajectory. Therefore, we ask: How many samples are needed to minimize a noisy convex function if we require $ε$-accurate estimates of $O(1/ε^2)$ gradients? Or, might it be that inaccurate gradient estimates are \emph{necessary} for finding the minimum of a stochastic convex function at an optimal statistical rate? We provide two partial answers to this question. First, we show that a general analyst (queries that may be maliciously chosen) requires $Ω(1/ε^3)$ samples, ruling out the possibility of a foolproof mechanism. Second, we show that, under certain assumptions on the oracle, $\tilde Ω(1/ε^{2.5})$ samples are necessary for gradient descent to interact with the oracle. Our results are in contrast to classical bounds that show that $O(1/ε^2)$ samples can optimize the population risk to an accuracy of $O(ε)$, but with spurious gradients.

We analyze the sample complexity of full-batch Gradient Descent (GD) in the setup of non-smooth Stochastic Convex Optimization. We show that the generalization error of GD, with common choice of hyper-parameters, can be $\tilde \Theta(d/m+1/\sqrt{m})$, where d is the dimension and m is the sample size. This matches the sample complexity of \emph{worst-case} empirical risk minimizers. That means that, in contrast with other algorithms, GD has no advantage over naive ERMs. Our bound follows from a new generalization bound that depends on both the dimension as well as the learning rate and number of iterations. Our bound also shows that, for general hyper-parameters, when the dimension is strictly larger than number of samples, $T=\Omega(1/\epsilon^4)$ iterations are necessary to avoid overfitting. This resolves an open problem by Schlisserman et al.23 and Amir er Al.21, and improves over previous lower bounds that demonstrated that the sample size must be at least square root of the dimension.

Credit attribution is crucial across various fields. In academic research, proper citation acknowledges prior work and establishes original contributions. Similarly, in generative models, such as those trained on existing artworks or music, it is important to ensure that any generated content influenced by these works appropriately credits the original creators. We study credit attribution by machine learning algorithms. We propose new definitions--relaxations of Differential Privacy--that weaken the stability guarantees for a designated subset of $k$ datapoints. These $k$ datapoints can be used non-stably with permission from their owners, potentially in exchange for compensation. Meanwhile, the remaining datapoints are guaranteed to have no significant influence on the algorithm's output. Our framework extends well-studied notions of stability, including Differential Privacy ($k = 0$), differentially private learning with public data (where the $k$ public datapoints are fixed in advance), and stable sample compression (where the $k$ datapoints are selected adaptively by the algorithm). We examine the expressive power of these stability notions within the PAC learning framework, provide a comprehensive characterization of learnability for algorithms adhering to these principles, and propose directions and questions for future research.

We examine the relationship between the mutual information between the output model and the empirical sample and the algorithm's generalization in the context of stochastic convex optimization. Despite increasing interest in information-theoretic generalization bounds, it is uncertain if these bounds can provide insight into the exceptional performance of various learning algorithms. Our study of stochastic convex optimization reveals that, for true risk minimization, dimension-dependent mutual information is necessary. This indicates that existing information-theoretic generalization bounds fall short in capturing the generalization capabilities of algorithms like SGD and regularized ERM, which have dimension-independent sample complexity.

We study best-of-both-worlds algorithms for bandits with switching cost, recently addressed by Rouyer et al., 2021. We introduce a surprisingly simple and effective algorithm that simultaneously achieves minimax optimal regret bound (up to logarithmic factors) of $\mathcal{O}(T^{2/3})$ in the oblivious adversarial setting and a bound of $\mathcal{O}(\min\{\log (T)/\Delta^2,T^{2/3}\})$ in the stochastically-constrained regime, both with (unit) switching costs, where $\Delta$ is the gap between the arms. In the stochastically constrained case, our bound improves over previous results due to Rouyer et al., 2021, that achieved regret of $\mathcal{O}(T^{1/3}/\Delta)$. We accompany our results with a lower bound showing that, in general, $\tilde{\mathcal{\Omega}}(\min\{1/\Delta^2,T^{2/3}\})$ switching cost regret is unavoidable in the stochastically-constrained case for algorithms with $\mathcal{O}(T^{2/3})$ worst-case switching cost regret.

We consider linear prediction with a convex Lipschitz loss, or more generally, stochastic convex optimization problems of generalized linear form, i.e.~where each instantaneous loss is a scalar convex function of a linear function. We show that in this setting, early stopped Gradient Descent (GD), without any explicit regularization or projection, ensures excess error at most $\varepsilon$ (compared to the best possible with unit Euclidean norm) with an optimal, up to logarithmic factors, sample complexity of $\tilde{O}(1/\varepsilon^2)$ and only $\tilde{O}(1/\varepsilon^2)$ iterations. This contrasts with general stochastic convex optimization, where $\Omega(1/\varepsilon^4)$ iterations are needed Amir et al. 2021. The lower iteration complexity is ensured by leveraging uniform convergence rather than stability. But instead of uniform convergence in a norm ball, which we show can guarantee suboptimal learning using $\Theta(1/\varepsilon^4)$ samples, we rely on uniform convergence in a distribution-dependent ball.

We study to what extent may stochastic gradient descent (SGD) be understood as a ``conventional'' learning rule that achieves generalization performance by obtaining a good fit to training data. We consider the fundamental stochastic convex optimization framework, where (one pass, $\textit{without}$-replacement) SGD is classically known to minimize the population risk at rate $O(1/\sqrt n)$, and prove that, surprisingly, there exist problem instances where the SGD solution exhibits both empirical risk and generalization gap of $\Omega(1)$. Consequently, it turns out that SGD is not algorithmically stable in $\textit{any}$ sense, and its generalization ability cannot be explained by uniform convergence or any other currently known generalization bound technique for that matter (other than that of its classical analysis). We then continue to analyze the closely related $\textit{with}$-replacement SGD, for which we show that an analogous phenomenon does not occur and prove that its population risk does in fact converge at the optimal rate. Finally, we interpret our main results in the context of without-replacement SGD for finite-sum convex optimization problems, and derive upper and lower bounds for the multi-epoch regime that significantly improve upon previously known results.

Let H be a binary-labeled concept class. We prove that H can be PAC learned by an (approximate) differentially private algorithm if and only if it has a finite Littlestone dimension. This implies a qualitative equivalence between online learnability and private PAC learnability.

We give a new separation result between the generalization performance of stochastic gradient descent (SGD) and of full-batch gradient descent (GD) in the fundamental stochastic convex optimization model. While for SGD it is well-known that $O(1/\epsilon^2)$ iterations suffice for obtaining a solution with $\epsilon$ excess expected risk, we show that with the same number of steps GD may overfit and emit a solution with $\Omega(1)$ generalization error. Moreover, we show that in fact $\Omega(1/\epsilon^4)$ iterations are necessary for GD to match the generalization performance of SGD, which is also tight due to recent work by Bassily et al. (2020). We further discuss how regularizing the empirical risk minimized by GD essentially does not change the above result, and revisit the concepts of stability, implicit bias and the role of the learning algorithm in generalization.

Which classes can be learned properly in the online model? — that is, by an algorithm that on each round uses a predictor from the concept class. While there are simple and natural cases where improper learning is useful and even necessary, it is natural to ask how complex must the improper predictors be in such cases. Can one always achieve nearly optimal mistake/regret bounds using "simple" predictors? In this work, we give a complete characterization of when this is possible, thus settling an open problem which has been studied since the pioneering works of Angluin (1987) and Littlestone (1988). More precisely, given any concept class C and any hypothesis class H, we provide nearly tight bounds (up to a log factor) on the optimal mistake bounds for online learning C using predictors from H. Our bound yields an exponential improvement over the previously best known bound by Chase and Freitag (2020). As applications, we give constructive proofs showing that (i) in the realizable setting, a near-optimal mistake bound (up to a constant factor) can be attained by a sparse majority-vote of proper predictors, and (ii) in the agnostic setting, a near-optimal regret bound (up to a log factor) can be attained by a randomized proper algorithm. The latter was proven non-constructively by Rakhlin, Sridharan, and Tewari (2015). It was also achieved by constructive but improper algorithms proposed by Ben-David, Pal, and Shalev-Shwartz (2009) and Rakhlin, Shamir, and Sridharan (2012). A technical ingredient of our proof which may be of independent interest is a generalization of the celebrated Minimax Theorem (von Neumann, 1928) for binary zero-sum games with arbitrary action-sets: a simple game which fails to satisfy Minimax is "Guess the Larger Number". In this game, each player picks a natural number and the player who picked the larger number wins. Equivalently, the payoff matrix of this game is infinite triangular. We show that this is the only obstruction: if the payoff matrix does not contain triangular submatrices of unbounded sizes then the Minimax theorem is satisfied. This generalizes von Neumann’s Minimax Theorem by removing requirements of finiteness (or compactness) of the action-sets, and moreover it captures precisely the types of games of interest in online learning: namely, Littlestone games.

We study the generalization performance of $\text{\emph{full-batch}}$ optimization algorithms for stochastic convex optimization: these are first-order methods that only access the exact gradient of the empirical risk (rather than gradients with respect to individual data points), that include a wide range of algorithms such as gradient descent, mirror descent, and their regularized and/or accelerated variants. We provide a new separation result showing that, while algorithms such as stochastic gradient descent can generalize and optimize the population risk to within $\epsilon$ after $O(1/\epsilon^2)$ iterations, full-batch methods either need at least $\Omega(1/\epsilon^4)$ iterations or exhibit a dimension-dependent sample complexity.

We consider the problem of online classification under a privacy constraint. In this setting a learner observes sequentially a stream of labelled examples $(x_t, y_t)$, for $1 \leq t \leq T$, and returns at each iteration $t$ a hypothesis $h_t$ which is used to predict the label of each new example $x_t$. The learner's performance is measured by her regret against a known hypothesis class $\mathcal{H}$. We require that the algorithm satisfies the following privacy constraint: the sequence $h_1, \ldots, h_T$ of hypotheses output by the algorithm needs to be an $(\epsilon, \delta)$-differentially private function of the whole input sequence $(x_1, y_1), \ldots, (x_T, y_T)$.We provide the first non-trivial regret bound for the realizable setting. Specifically, we show that if the class $\mathcal{H}$ has constant Littlestone dimension then, given an oblivious sequence of labelled examples, there is a private learner that makes in expectation at most $O(\log T)$ mistakes -- comparable to the optimal mistake bound in the non-private case, up to a logarithmic factor. Moreover, for general values of the Littlestone dimension $d$, the same mistake bound holds but with a doubly-exponential in $d$ factor. A recent line of work has demonstrated a strong connection between classes that are online learnable and those that are differentially-private learnable. Our results strengthen this connection and show that an online learning algorithm can in fact be directly privatized (in the realizable setting).We also discuss an adaptive setting and provide a sublinear regret bound of $O(\sqrt{T})$.

We study the expressive power of kernel methods and the algorithmic feasibility of multiple kernel learning for a special rich class of kernels. Specifically, we define Euclidean kernels, a diverse class that includes most, if not all, families of kernels studied in literature such as polynomial kernels and radial basis functions. We then describe the geometric and spectral structure of this family of kernels over the hypercube (and to some extent for any compact domain). Our structural results allow us to prove meaningful limitations on the expressive power of the class as well as derive several efficient algorithms for learning kernels over different domains.

We prove that every concept class with finite Littlestone dimension can be learned by an (approximate) differentially-private algorithm. This answers an open question of Alon et al. (STOC 2019) who proved the converse statement (this question was also asked by Neel et al. (FOCS 2019)). Together these two results yield an equivalence between online learnability and private PAC learnability. We introduce a new notion of algorithmic stability called “global stability” which is essential to our proof and may be of independent interest. We also discuss an application of our results to boosting the privacy and accuracy parameters of differentially-private learners.

We revisit the fundamental problem of prediction with expert advice, in a setting where the environment is benign and generates losses stochastically, but the feedback observed by the learner is subject to a moderate adversarial corruption. We prove that a variant of the classical Multiplicative Weights algorithm with decreasing step sizes achieves constant regret in this setting and performs optimally in a wide range of environments, regardless of the magnitude of the injected corruption. Our results reveal a surprising disparity between the often comparable Follow the Regularized Leader (FTRL) and Online Mirror Descent (OMD) frameworks: we show that for experts in the corrupted stochastic regime, the regret performance of OMD is in fact strictly inferior to that of FTRL.

We study the sample complexity of private synthetic data generation over an unbounded sized class of statistical queries, and show that any class that is privately proper PAC learnable admits a private synthetic data generator (perhaps non-efficient). A differentially private synthetic generator is an algorithm that receives an IID data and publishes synthetic data that is indistinguishable from the true data w.r.t a given fixed class of statistical queries. The synthetic data set can then be used by a data scientist without compromising the privacy of the original data set. Previous work on synthetic data generators focused on the case that the query class $\D$ is finite and obtained sample complexity bounds that scale logarithmically with the size $|\D|$. Here we construct a private synthetic data generator whose sample complexity is independent of the domain size, and we replace finiteness with the assumption that $\D$ is privately PAC learnable (a formally weaker task, hence we obtain equivalence between the two tasks). Our proof relies on a new type of synthetic data generator, Sequential Synthetic Data Generators, which we believe may be of interest of their own right. A sequential SDG is defined by a sequential game between a generator that proposes synthetic distributions and a discriminator that tries to distinguish between real and fake distributions. We characterize the classes that admit a sequential-SDG and show that they are exactly Littlestone classes. Given the online nature of the sequential setting, it is natural that Littlestone classes arise in this context. Nevertheless, the characterization of sequential--SDGs by Littlestone classes turns out to be technically challenging, and to the best of the author's knowledge, does not follow via simple reductions to online prediction.

The notion of implicit bias, or implicit regularization, has been suggested as a means to explain the surprising generalization ability of modern-days overparameterized learning algorithms. This notion refers to the tendency of the optimization algorithm towards a certain structured solution that often generalizes well. Recently, several papers have studied implicit regularization and were able to identify this phenomenon in various scenarios. We revisit this paradigm in arguably the simplest non-trivial setup, and study the implicit bias of Stochastic Gradient Descent (SGD) in the context of Stochastic Convex Optimization. As a first step, we provide a simple construction that rules out the existence of a \emph{distribution-independent} implicit regularizer that governs the generalization ability of SGD. We then demonstrate a learning problem that rules out a very general class of \emph{distribution-dependent} implicit regularizers from explaining generalization, which includes strongly convex regularizers as well as non-degenerate norm-based regularizations. Certain aspects of our constructions point out to significant difficulties in providing a comprehensive explanation of an algorithm's generalization performance by solely arguing about its implicit regularization properties.

PAC-Bayes is a useful framework for deriving generalization bounds which was introduced by McAllester ('98). This framework has the flexibility of deriving distribution- and algorithm-dependent bounds, which are often tighter than VC-related uniform convergence bounds. In this manuscript we present a limitation for the PAC-Bayes framework. We demonstrate an easy learning task which is not amenable to a PAC-Bayes analysis. Specifically, we consider the task of linear classification in 1D; it is well-known that this task is learnable using just $O(\log(1/\delta)/\epsilon)$ examples. On the other hand, we show that this fact can not be proved using a PAC-Bayes analysis: for any algorithm that learns 1-dimensional linear classifiers there exists a (realizable) distribution for which the PAC-Bayes bound is arbitrarily large.

We present provable algorithms for learning linear representations which are trained in a supervised fashion across a number of tasks. Furthermore, whereas previous methods in the context of multitask learning only allow for generalization within tasks that have already been observed, our representations are both efficiently learnable and accompanied by generalization guarantees to unseen tasks. Our method relies on a certain convex relaxation of a non-convex problem, making it amenable to online learning procedures. We further ensure that a low-rank representation is maintained, and we allow for various trade-offs between sample complexity and per-iteration cost, depending on the choice of algorithm.

This work studies distributed learning in the spirit of Yao’s model of communication complexity: consider a two-party setting, where each of the players gets a list of labelled examples and they communicate in order to jointly perform some learning task. To naturally fit into the framework of learning theory, the players can send each other examples (as well as bits) where each example/bit costs one unit of communication. This enables a uniform treatment of infinite classes such as half-spaces in $\R^d$, which are ubiquitous in machine learning. We study several fundamental questions in this model. For example, we provide combinatorial characterizations of the classes that can be learned with efficient communication in the proper-case as well as in the improper-case. These findings imply unconditional separations in this context between various learning tasks, e.g. realizable versus agnostic learning, proper versus improper learning, etcetera. %They also imply lower bounds that match the performance %of algorithm from previous works. The derivation of these results hinges on a type of decision problems we term “{\it realizability problems}” where the goal is deciding whether a distributed input sample is consistent with an hypothesis from a pre-specified class. From a technical perspective, the protocols we devise (i.e. the upper bounds) are based on ideas from machine learning and the impossibility results (i.e. the lower bounds) are based on ideas from communication complexity.

A basic question in learning theory is to identify if two distributions are identical when we have access only to examples sampled from the distributions. This basic task is considered, for example, in the context of Generative Adversarial Networks (GANs), where a discriminator is trained to distinguish between a real-life distribution and a synthetic distribution. Classically, we use a hypothesis class $H$ and claim that the two distributions are distinct if for some $h\in H$ the expected value on the two distributions is (significantly) different. Our starting point is the following fundamental problem: "is having the hypothesis dependent on more than a single random example beneficial". To address this challenge we define $k$-ary based discriminators, which have a family of Boolean $k$-ary functions $\G$. Each function $g\in \G$ naturally defines a hyper-graph, indicating whether a given hyper-edge exists. A function $g\in \G$ distinguishes between two distributions, if the expected value of $g$, on a $k$-tuple of i.i.d examples, on the two distributions is (significantly) different. We study the expressiveness of families of $k$-ary functions, compared to the classical hypothesis class $H$, which is $k=1$. We show a separation in expressiveness of $k+1$-ary versus $k$-ary functions. This demonstrate the great benefit of having $k\geq 2$ as distinguishers. For $k\geq 2$ we introduce a notion similar to the VC-dimension, and show that it controls the sample complexity. We proceed and provide upper and lower bounds as a function of our extended notion of VC-dimension.

We show that every approximately differentially private learning algorithm (possibly improper) for a class H with Littlestone dimension d requires Ω(log*(d)) examples. As a corollary it follows that the class of thresholds over ℕ can not be learned in a private manner; this resolves open questions due to [Bun et al. 2015] and [Feldman and Xiao, 2015]. We leave as an open question whether every class with a finite Littlestone dimension can be learned by an approximately differentially private algorithm.

We ask whether there exists an efficient unsupervised learning algorithm for mixture of Gaussians in the over-complete case (number of mixtures is larger than the dimension). The notion of learning is taken to be worst-case compression-based, to allow for improper learning.

The sample complexity of learning a Boolean-valued function class is precisely characterized by its Rademacher complexity. This has little bearing, however, on the sample complexity of efficient agnostic learning. We introduce refutation complexity, a natural computational analog of Rademacher complexity of a Boolean concept class and show that it exactly characterizes the sample complexity of efficient agnostic learning. Informally, refutation complexity of a class C is the minimum number of example-label pairs required to efficiently distinguish between the case that the labels correlate with the evaluation of some member of C (structure) and the case where the labels are i.i.d. Rademacher random variables (noise). The easy direction of this relationship was implicitly used in the recent framework for improper PAC learning lower bounds of Daniely and co-authors via connections to the hardness of refuting random constraint satisfaction problems. Our work can be seen as making the relationship between agnostic learning and refutation implicit in their work into an explicit equivalence. In a recent, independent work, Salil Vadhan discovered a similar relationship between refutation and PAC-learning in the realizable (i.e. noiseless) case.

The abundance of unlabeled data makes semi-supervised learning (SSL) an attractive approach for improving the accuracy of learning systems. However, we are still far from a complete theoretical understanding of the benefits of this learning scenario in terms of sample complexity. In particular, for many natural learning settings it can in fact be shown that SSL does not improve sample complexity. Thus far, the only case where SSL provably helps, without compatibility assumptions, is a recent combinatorial construction of Darnstadt et al. Deriving similar theoretical guarantees for more commonly used approaches to SSL remains a challenge. Here, we provide the first analysis of manifold based SSL, where there is a provable gap between supervised learning and SSL, and this gap can be arbitrarily large. Proving the required lower bound is a technical challenge, involving tools from geometric measure theory. The algorithm we analyse is similar to subspace clustering, and thus our results demonstrate that this method can be used to improve sample complexity.

We extend the model of Multi-Armed Bandit with unit switching cost to incorporate a metric between the actions. We consider the case where the metric over the actions can be modeled by a complete binary tree, and the distance between two leaves is the size of the subtree of their least common ancestor, which abstracts the case that the actions are points on the continuous interval $[0,1]$ and the switching cost is their distance. In this setting, we give a new algorithm that establishes a regret of $\widetilde{O}(\sqrt{k}T + T/k)$, where $k$ is the number of actions and $T$ is the time horizon. When the set of actions corresponds to whole $[0,1]$ interval we can exploit our method for the task of bandit learning with Lipschitz loss functions, where our algorithm achieves an optimal regret rate of $\widetilde{Θ}(T^2/3)$, which is the same rate one obtains when there is no penalty for movements. As our main application, we use our new algorithm to solve an adaptive pricing problem. Specifically, we consider the case of a single seller faced with a stream of patient buyers. Each buyer has a private value and a window of time in which they are interested in buying, and they buy at the lowest price in the window, if it is below their value. We show that with an appropriate discretization of the prices, the seller can achieve a regret of $\widetilde{O}(T^2/3)$ compared to the best fixed price in hindsight, which outperform the previous regret bound of $\widetilde{O}(T^3/4)$ for the problem.

Infinite Layer Networks (ILN) have been proposed as an architecture that mimics neural networks while enjoying some of the advantages of kernel methods. ILN are networks that integrate over infinitely many nodes within a single hidden layer. It has been demonstrated by several authors that the problem of learning ILN can be reduced to the kernel trick, implying that whenever a certain integral can be computed analytically they are efficiently learnable. In this work we give an online algorithm for ILN, which avoids the kernel trick assumption. More generally and of independent interest, we show that kernel methods in general can be exploited even when the kernel cannot be efficiently computed but can only be estimated via sampling. We provide a regret analysis for our algorithm, showing that it matches the sample complexity of methods which have access to kernel values. Thus, our method is the first to demonstrate that the kernel trick is not necessary, as such, and random features suffice to obtain comparable performance.

We present a new affine-invariant optimization algorithm called Online Lazy Newton. The regret of Online Lazy Newton is independent of conditioning: the algorithm's performance depends on the best possible preconditioning of the problem in retrospect and on its \emph{intrinsic} dimensionality. As an application, we show how Online Lazy Newton can be used to achieve an optimal regret of order $\sqrt{rT}$ for the low-rank experts problem, improving by a $\sqrt{r}$ factor over the previously best known bound and resolving an open problem posed by Hazan et al (2016).

We consider the non-stochastic Multi-Armed Bandit problem in a setting where there is a fixed and known metric on the action space that determines a cost for switching between any pair of actions. The loss of the online learner has two components: the first is the usual loss of the selected actions, and the second is an additional loss due to switching between actions. Our main contribution gives a tight characterization of the expected minimax regret in this setting, in terms of a complexity measure $\mathcal{C}$ of the underlying metric which depends on its covering numbers. In finite metric spaces with $k$ actions, we give an efficient algorithm that achieves regret of the form $\widetilde(\max\set{\mathcal{C}^{1/3}T^{2/3},\sqrt{kT}})$, and show that this is the best possible. Our regret bound generalizes previous known regret bounds for some special cases: (i) the unit-switching cost regret $\widetilde{\Theta}(\max\set{k^{1/3}T^{2/3},\sqrt{kT}})$ where $\mathcal{C}=\Theta(k)$, and (ii) the interval metric with regret $\widetilde{\Theta}(\max\set{T^{2/3},\sqrt{kT}})$ where $\mathcal{C}=\Theta(1)$. For infinite metrics spaces with Lipschitz loss functions, we derive a tight regret bound of $\widetilde{\Theta}(T^{\frac{d+1}{d+2}})$ where $d \ge 1$ is the Minkowski dimension of the space, which is known to be tight even when there are no switching costs.

Neural networks have recently re-emerged as a powerful hypothesis class, yielding impressive classification accuracy in multiple domains. However, their training is a non convex optimization problem. Here we address this difficulty by turning to "improper learning" of neural nets. In other words, we learn a classifier that is not a neural net but is competitive with the best neural net model given a sufficient number of training examples. Our approach relies on a novel kernel which integrates over the set of possible neural models. It turns out that the corresponding integral can be evaluated in closed form via a simple recursion. The learning problem is then an SVM with this kernel, and a global optimum can thus be found efficiently. We also provide sample complexity results which depend on the stability of the optimal neural net.

We consider the problem of prediction with expert advice when the losses of the experts have low-dimensional structure: they are restricted to an unknown d-dimensional subspace. We devise algorithms with regret bounds that are independent of the number of experts and depend only on the rank d. For the stochastic model we show a tight bound of Θ(\sqrtdT), and extend it to a setting of an approximate d subspace. For the adversarial model we show an upper bound of O(d\sqrtT) and a lower bound of Ω(\sqrtdT).

We consider a seller with an unlimited supply of a single good, who is faced with a stream of $T$ buyers. Each buyer has a window of time in which she would like to purchase, and would buy at the lowest price in that window, provided that this price is lower than her private value (and otherwise, would not buy at all). In this setting, we give an algorithm that attains $O(T^{2/3})$ regret over any sequence of $T$ buyers with respect to the best fixed price in hindsight, and prove that no algorithm can perform better in the worst case.

We consider classification and regression tasks where we have missing data and assume that the (clean) data resides in a low rank subspace. Finding a hidden subspace is known to be computationally hard. Nevertheless, using a non-proper formulation we give an efficient agnostic algorithm that classifies as good as the best linear classifier coupled with the best low-dimensional subspace in which the data resides. A direct implication is that our algorithm can linearly (and non-linearly through kernels) classify provably as well as the best classifier that has access to the full data.

The existence of a compression scheme for every concept class with bounded VC-dimension is one of the oldest open problems in statistical learning theory. Here we demonstrate the existence of such compression schemes under stronger assumptions than finite VC-dimension. Specifically, for each concept class we associate a family of concept classes that we call the alternating concept classes. Under the assumption that these concept classes have bounded VC dimension, we prove existence of a compression scheme. This result is motivated by recent progress in the field of model theory with respect to an analogues problem. In fact, our proof can be considered as a constructive proof of these advancements. This means that we describe the reconstruction function explicitly. Not less important, the theorems and proofs we present are in purely combinatorial terms and are available to the reader who is unfamiliar with model theory. Also, using tools from model theory, we apply our results and prove existence of compression schemes in interesting cases, such as concept classes defined by hyperplanes, polynomials, exponentials, restricted analytic functions and compositions, additions and multiplications of all of the above.

The vanishing ideal of a set of n points S, is the set of all polynomials that attain the value of zero on all the points in S. Such ideals can be compactly represented using a small set of polynomials known as generators of the ideal. Here we describe and analyze an efficient procedure that constructs a set of generators of a vanishing ideal. Our procedure is numerically stable, and can be used to find approximately vanishing polynomials. The resulting polynomials capture nonlinear structure in data, and can for example be used within supervised learning. Empirical comparison with kernel methods show that our method constructs more compact classifiers with comparable accuracy.