Multiaccuracy and multicalibration are multi-group fairness notions for prediction that have found numerous applications in learning and computational complexity [HKRR18]. They can be achieved from a single learning primitive: weak agnostic learning. A line of work starting from [GKR+22] has shown that multicalibration implies a very strong form of learning. Here we investigate the power of multiaccuracy as a learning primitive, both with and without the additional assumption of calibration. We find that multiaccuracy in itself is rather weak, but that the addition of global calibration (this notion is called calibrated multiaccuracy) boosts its power substantially, enough to recover implications that were previously known only assuming the stronger notion of multicalibration. We give evidence that multiaccuracy might not be as powerful as standard weak agnostic learning, by showing that there is no way to post-process a multiaccurate predictor to get a weak learner, even assuming the best hypothesis has correlation 1/2. Rather, we show that it yields a restricted form of weak agnostic learning, which requires some concept in the class to have correlation greater than 1/2 with the labels. However, by also requiring the predictor to be calibrated, we recover not just weak, but strong agnostic learning. A similar picture emerges when we consider the derivation of hardcore measures from predictors satisfying multigroup fairness notions [TTV09], [CDV24]. On the one hand, while multiaccuracy only yields hardcore measures of density half the optimal, we show that (a weighted version of) calibrated multiaccuracy achieves optimal density. Our results yield new insights into the complementary roles played by multiaccuracy and calibration in each setting. They shed light on why multiaccuracy and global calibration, although not particularly powerful by themselves, together yield considerably stronger notions.

We propose new learning algorithms for building selective classifiers, which are predictors that are allowed to abstain on some fraction of the domain. We study the model where a classifier may abstain from predicting at a fixed cost. Building on the recent framework on multigroup fairness and omniprediction, given a pre-specified class of loss functions, we provide an algorithm for building a single classifier that learns abstentions and predictions optimally for every loss in the entire class, where the abstentions are decided efficiently for each specific loss function by applying a fixed post-processing function. Our algorithm and theoretical guarantees generalize the previously-known algorithms for learning selective classifiers in formal learning-theoretic models. We then extend the traditional multigroup fairness algorithms to the selective classification setting and show that we can use a calibrated and multiaccurate predictor to efficiently build selective classifiers that abstain optimally not only globally but also locally within each of the groups in any pre-specified collection of possibly intersecting subgroups of the domain, and are also accurate when they do not abstain. We show how our abstention algorithms can be used as conformal prediction methods in the binary classification setting to achieve both marginal and group-conditional coverage guarantees for an intersecting collection of groups. We provide empirical evaluations for all of our theoretical results, demonstrating the practicality of our learning algorithms for abstaining optimally and fairly.

Pause tokens, simple filler symbols such as "...", consistently improve Transformer performance on both language and mathematical tasks, yet their theoretical effect remains unexplained. We provide the first formal separation result, proving that adding pause tokens to constant-depth, logarithmic-width Transformers strictly increases their computational expressivity. With bounded-precision activations, Transformers without pause tokens compute only a strict subset of $\mathsf{AC}^0$ functions, while adding a polynomial number of pause tokens enables expressing the complete class. For logarithmic-precision Transformers, we show that adding pause tokens achieves expressivity equivalent to $\mathsf{TC}^0$, matching known upper bounds. Empirically, we demonstrate that two‑layer causally masked Transformers can learn parity when supplied with pause tokens, a function that they appear unable to learn without them. Our results provide a rigorous theoretical explanation for prior empirical findings, clarify how pause tokens interact with width, depth, and numeric precision, and position them as a distinct mechanism, complementary to chain-of-thought prompting, for enhancing Transformer reasoning.

In order to understand the in-context learning phenomenon, recent works have adopted a stylized experimental framework and demonstrated that Transformers can match the performance of gradient-based learning algorithms for various classes of real-valued functions. However, the limitations of Transformers in implementing learning algorithms, and their ability to learn other forms of algorithms are not well understood. Additionally, the degree to which these capabilities are confined to attention-based models is unclear. Furthermore, it remains to be seen whether the insights derived from these stylized settings can be extrapolated to pretrained Large Language Models (LLMs). In this work, we take a step towards answering these questions by demonstrating the following: (a) On a test-bed with a variety of Boolean function classes, we find that Transformers can nearly match the optimal learning algorithm for 'simpler' tasks, while their performance deteriorates on more 'complex' tasks. Additionally, we find that certain attention-free models perform (almost) identically to Transformers on a range of tasks. (b) When provided a *teaching sequence*, i.e. a set of examples that uniquely identifies a function in a class, we show that Transformers learn more sample-efficiently. Interestingly, our results show that Transformers can learn to implement *two distinct* algorithms to solve a *single* task, and can adaptively select the more sample-efficient algorithm depending on the sequence of in-context examples. (c) Lastly, we show that extant LLMs, e.g. LLaMA-2, GPT-4, can compete with nearest-neighbor baselines on prediction tasks that are guaranteed to not be in their training set.

Transformer architectures have been widely adopted in foundation models. Due to their high inference costs, there is renewed interest in exploring the potential of efficient recurrent architectures (RNNs). In this paper, we analyze the differences in the representational capabilities of Transformers and RNNs across several tasks of practical relevance, including index lookup, nearest neighbor, recognizing bounded Dyck languages, and string equality. For the tasks considered, our results show separations based on the size of the model required for different architectures. For example, we show that a one-layer Transformer of logarithmic width can perform index lookup, whereas an RNN requires a hidden state of linear size. Conversely, while constant-size RNNs can recognize bounded Dyck languages, we show that one-layer Transformers require a linear size for this task. Furthermore, we show that two-layer Transformers of logarithmic size can perform decision tasks such as string equality or disjointness, whereas both one-layer Transformers and recurrent models require linear size for these tasks. We also show that a log-size two-layer Transformer can implement the nearest neighbor algorithm in its forward pass; on the other hand recurrent models require linear size. Our constructions are based on the existence of $N$ nearly orthogonal vectors in $O(\log N)$ dimensional space and our lower bounds are based on reductions from communication complexity problems. We supplement our theoretical results with experiments that highlight the differences in the performance of these architectures on practical-size sequences.

The local Rademacher complexity framework is one of the most successful general-purpose toolboxes for establishing sharp excess risk bounds for statistical estimators based on empirical risk minimization. However, applying this toolbox typically requires using the Bernstein condition, which often restricts the applicability domain to convex and proper settings. Recent years have witnessed several examples of problems where optimal statistical performance is only achievable via non-convex and improper estimators originating from aggregation theory, including the fundamental problem of model selection. These examples are currently outside the reach of the classical local Rademacher complexity theory. In this work, we build upon the recent approach to localization via offset Rademacher complexities, for which a general high-probability theory has yet to be established. Our main result is an exponential-tail offset Rademacher complexity excess risk upper bound that yields results at least as sharp as those obtainable via the classical theory. However, our bound applies under an estimator-dependent geometric condition (the “offset condition”) instead of the estimator-independent (but, in general, distribution-dependent) Bernstein condition on which the classical theory relies. Our results apply to improper prediction regimes not directly covered by the classical theory, such as optimal model selection aggregation for arbitrary classes (including infinite and non-convex classes), and early-stopping/iterative regularization; the Bernstein condition does not hold in both examples.

Despite the widespread success of Transformers on NLP tasks, recent works have found that they struggle to model several formal languages when compared to recurrent models.This raises the question of why Transformers perform well in practice and whether they have any properties that enable them to generalize better than recurrent models.In this work, we conduct an extensive empirical study on Boolean functions to demonstrate the following: (i) Random Transformers are relatively more biased towards functions of low sensitivity.(ii) When trained on Boolean functions, both Transformers and LSTMs prioritize learning functions of low sensitivity, with Transformers ultimately converging to functions of lower sensitivity.(iii) On sparse Boolean functions which have low sensitivity, we find that Transformers generalize near perfectly even in the presence of noisy labels whereas LSTMs overfit and achieve poor generalization accuracy.Overall, our results provide strong quantifiable evidence that suggests differences in the inductive biases of Transformers and recurrent models which may help explain Transformer's effective generalization performance despite relatively limited expressiveness.

The matrix completion problem involves reconstructing a low-rank matrix by using a given set of revealed (and potentially noisy) entries. Although existing methods address the completion of the entire matrix, the accuracy of the completed entries can vary significantly across the matrix, due to differences in the sampling distribution. For instance, users may rate movies primarily from their country or favorite genres, leading to inaccurate predictions for the majority of completed entries. We propose a novel formulation of the problem as Partial Matrix Completion, where the objective is to complete a substantial subset of the entries with high confidence. Our algorithm efficiently handles the unknown and arbitrarily complex nature of the sampling distribution, ensuring high accuracy for all completed entries and sufficient coverage across the matrix. Additionally, we introduce an online version of the problem and present a low-regret efficient algorithm based on iterative gradient updates. Finally, we conduct a preliminary empirical evaluation of our methods.

Coalescing random walks is a fundamental distributed process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a single random walk. The coalescence time is defined as the expected time until only one particle remains, starting from one particle at every node. Despite recent progress such as that of Cooper et al., the coalescence time for graphs, such as binary trees, d -dimensional tori, hypercubes, and, more generally, vertex-transitive graphs, remains unresolved. We provide a powerful toolkit that results in tight bounds for various topologies including the aforementioned ones. The meeting time is defined as the worst-case expected time required for two random walks to arrive at the same node at the same time. As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log 2 n ), the coalescence time of n random walks equals the meeting time up to constant factors. This upper bound is complemented by the construction of a graph family demonstrating that this result is the best possible up to constant factors. Finally, we prove a tight worst-case bound for the coalescence time of O(n 3 ) . By duality, our results yield identical bounds on the voter model. Our techniques also yield a new bound on the hitting time and cover time of regular graphs, improving and tightening previous results by Broder and Karlin, as well as those by Aldous and Fill.

A fundamental problem in adversarial machine learning is to quantify how much training data is needed in the presence of evasion attacks. In this paper we address this issue within the framework of PAC learning, focusing on the class of decision lists. Given that distributional assumptions are essential in the adversarial setting, we work with probability distributions on the input data that satisfy a Lipschitz condition: nearby points have similar probability. Our key results illustrate that the adversary's budget (that is, the number of bits it can perturb on each input) is a fundamental quantity in determining the sample complexity of robust learning. Our first main result is a sample-complexity lower bound: the class of monotone conjunctions (essentially the simplest non-trivial hypothesis class on the Boolean hypercube) and any superclass has sample complexity at least exponential in the adversary's budget. Our second main result is a corresponding upper bound: for every fixed k the class of k-decision lists has polynomial sample complexity against a log(n)-bounded adversary. This sheds further light on the question of whether an efficient PAC learning algorithm can always be used as an efficient log(n)-robust learning algorithm under the uniform distribution.

Distributional assumptions have been shown to be necessary for the robust learnability of concept classes when considering the exact-in-the-ball robust risk and access to random examples by Gourdeau et al. (2019). In this paper, we study learning models where the learner is given more power through the use of local queries, and give the first distribution-free algorithms that perform robust empirical risk minimization (ERM) for this notion of robustness. The first learning model we consider uses local membership queries (LMQ), where the learner can query the label of points near the training sample. We show that, under the uniform distribution, LMQs do not increase the robustness threshold of conjunctions and any superclass, e.g., decision lists and halfspaces. Faced with this negative result, we introduce the local equivalence query (LEQ) oracle, which returns whether the hypothesis and target concept agree in the perturbation region around a point in the training sample, as well as a counterexample if it exists. We show a separation result: on one hand, if the query radius $\lambda$ is strictly smaller than the adversary's perturbation budget $\rho$, then distribution-free robust learning is impossible for a wide variety of concept classes; on the other hand, the setting $\lambda=\rho$ allows us to develop robust ERM algorithms. We then bound the query complexity of these algorithms based on online learning guarantees and further improve these bounds for the special case of conjunctions. We finish by giving robust learning algorithms for halfspaces with margins on both $\{0,1\}^n$ and $\mathbb{R}^n$.

We study the problem of learning a clustering of an online set of points. The specific formulation we use is the k-means objective: At each time step the algorithm has to maintain a set of k candidate centers and the loss incurred by the algorithm is the squared distance between the new point and the closest center. The goal is to minimize regret with respect to the best solution to the k-means objective in hindsight. We show that provided the data lies in a bounded region, learning is possible, namely an implementation of the Multiplicative Weights Update Algorithm (MWUA) using a discretized grid achieves a regret bound of $\tilde{O}(\sqrt{T})$ in expectation. We also present an online-to-offline reduction that shows that an efficient no-regret online algorithm (despite being allowed to choose a different set of candidate centres at each round) implies an offline efficient algorithm for the k-means problem, which is known to be NP-hard. In light of this hardness, we consider the slightly weaker requirement of comparing regret with respect to $(1 + \epsilon)OPT$ and present a no-regret algorithm with runtime $O\left(T \mathrm{poly}(\log(T),k,d,1/\epsilon)^{O(kd)}\right)$. Our algorithm is based on maintaining a set of points of bounded size which is a coreset that helps identifying the \emph{relevant} regions of the space for running an adaptive, more efficient, variant of the MWUA. We show that simpler online algorithms, such as \emph{Follow The Leader} (FTL), fail to produce sublinear regret in the worst case. We also report preliminary experiments with synthetic and real-world data. Our theoretical results answer an open question of Dasgupta (2008).

We give an efficient algorithm for learning a binary function in a given class $C$ of bounded VC dimension, with training data distributed according to $P$ and test data according to $Q$, where $P$ and $Q$ may be arbitrary distributions over $X$. This is the generic form of what is called \textit{covariate shift}, which is impossible in general as arbitrary $P$ and $Q$ may not even overlap. However, recently guarantees were given in a model called PQ-learning (Goldwasser et al., 2020) where the learner has: (a) access to unlabeled test examples from $Q$ (in addition to labeled samples from $P$, i.e., semi-supervised learning); and (b) the option to \textit{reject} any example and abstain from classifying it (i.e., selective classification). The algorithm of Goldwasser et al. (2020) requires an (agnostic) noise-tolerant learner for $C$. The present work gives a polynomial-time PQ-learning algorithm, called \textit{Slice-and-Dice}, that uses an oracle to a “reliable” learner for $C$, where reliable learning (Kalai et al., 2012) is a model of learning with one-sided noise. Furthermore, this reduction is optimal in the sense that we show the equivalence of reliable and PQ learning.

A common challenge across all areas of machine learning is that training data is not distributed like test data, due to natural shifts or adversarial examples; such examples are referred to as out-of-distribution (OOD) test examples. We consider a model where one may abstain from predicting, at a fixed cost. In particular, our transductive abstention algorithm takes labeled training examples and unlabeled test examples as input, and provides predictions with optimal prediction loss guarantees. The loss bounds match standard generalization bounds when test examples are i.i.d. from the training distribution, but add an additional term that is the cost of abstaining times the statistical distance between the train and test distribution (or the fraction of adversarial examples). For linear regression, we give a polynomial-time algorithm based on Celis-Dennis-Tapia optimization algorithms. For binary classification, we show how to efficiently implement it using a proper agnostic learner (i.e., an Empirical Risk Minimizer) for the class of interest. Our work builds on recent work of Goldwasser, Kalais, and Montasser (2020) who gave error and abstention guarantees for transductive binary classification.

It is becoming increasingly important to understand the vulnerability of machine learning models to adversarial attacks. In this paper we study the feasibility of adversarially robust learning from the perspective of computational learning theory, considering both sample and computational complexity. In particular, our definition of robust learnability requires polynomial sample complexity. We start with two negative results. We show that no non-trivial concept class can be robustly learned in the distribution-free setting against an adversary who can perturb just a single input bit. We show, moreover, that the class of monotone conjunctions cannot be robustly learned under the uniform distribution against an adversary who can perturb $\omega(\log n)$ input bits. However, we also show that if the adversary is restricted to perturbing $O(\log n)$ bits, then one can robustly learn the class of $1$-decision lists (which subsumes monotone conjunctions) with respect to the class of log-Lipschitz distributions. We then extend this result to show learnability of 2-decision lists and monotone $k$-decision lists in the same distributional and adversarial setting. Finally, we provide a simple proof of the computational hardness of robust learning on the boolean hypercube. Unlike previous results of this nature, our result does not rely on a more restricted model of learning, such as the statistical query model, nor on any hardness assumption other than the existence of an (average-case) hard learning problem in the PAC framework; this allows us to have a clean proof of the reduction, and the assumption is no stronger than assumptions that are used to build cryptographic primitives.

Discovering the causal effect of a decision is critical to nearly all forms of decision-making. In particular, it is a key quantity in drug development, in crafting government policy, and when implementing a real-world machine learning system. Given only observational data, confounders often obscure the true causal effect. Luckily, in some cases, it is possible to recover the causal effect by using certain observed variables to adjust for the effects of confounders. However, without access to the true causal model, finding this adjustment requires brute-force search. In this work, we present an algorithm that exploits auxiliary variables, similar to instruments, in order to find an appropriate adjustment by a gradient-based optimization method. We demonstrate that it outperforms practical alternatives in estimating the true causal effect, without knowledge of the full causal graph.

We study the low rank regression problem y = Mx + ε, where x and y are d1 and d2 dimensional vectors respectively. We consider the extreme high-dimensional setting where the number of observations n is less than d1 + d2. Existing algorithms are designed for settings where n is typically as large as rank(M)(d1+d2). This work provides an efficient algorithm which only involves two SVD, and establishes statistical guarantees on its performance. The algorithm decouples the problem by first estimating the precision matrix of the features, and then solving the matrix denoising problem. To complement the upper bound, we introduce new techniques for establishing lower bounds on the performance of any algorithm for this problem. Our preliminary experiments confirm that our algorithm often out-performs existing baseline, and is always at least competitive.

Recently there has been a surge of interest in understanding implicit regularization properties of iterative gradient-based optimization algorithms. In this paper, we study the statistical guarantees on the excess risk achieved by early-stopped unconstrained mirror descent algorithms applied to the unregularized empirical risk with the squared loss for linear models and kernel methods. By completing an inequality that characterizes convexity for the squared loss, we identify an intrinsic link between offset Rademacher complexities and potential-based convergence analysis of mirror descent methods. Our observation immediately yields excess risk guarantees for the path traced by the iterates of mirror descent in terms of offset complexities of certain function classes depending only on the choice of the mirror map, initialization point, step-size, and the number of iterations. We apply our theory to recover, in a rather clean and elegant manner via rather short proofs, some of the recent results in the implicit regularization literature, while also showing how to improve upon them in some settings.

We study the problem of hypothesis testing between two discrete distributions, where we only have access to samples after the action of a known reversible Markov chain, playing the role of noise. We derive instance-dependent minimax rates for the sample complexity of this problem, and show how its dependence in time is related to the spectral properties of the Markov chain. We show that there exists a wide statistical window, in terms of sample complexity for hypothesis testing between different pairs of initial distributions. We illustrate these results in several concrete examples.

It is becoming increasingly important to understand the vulnerability of machine learning models to adversarial attacks. In this paper we study the feasibility of robust learning from the perspective of computational learning theory, considering both sample and computational complexity. In particular, our definition of robust learnability requires polynomial sample complexity. We start with two negative results. We show that no non-trivial concept class can be robustly learned in the distribution-free setting against an adversary who can perturb just a single input bit. We show moreover that the class of monotone conjunctions cannot be robustly learned under the uniform distribution against an adversary who can perturb $\omega(\log n)$ input bits. However if the adversary is restricted to perturbing $O(\log n)$ bits, then the class of monotone conjunctions can be robustly learned with respect to a general class of distributions (that includes the uniform distribution). Finally, we provide a simple proof of the computational hardness of robust learning on the boolean hypercube. Unlike previous results of this nature, our result does not rely on another computational model (e.g. the statistical query model) nor on any hardness assumption other than the existence of a hard learning problem in the PAC framework.

We study a decentralized cooperative stochastic multi-armed bandit problem with K arms on a network of N agents. In our model, the reward distribution of each arm is the same for each agent and rewards are drawn independently across agents and time steps. In each round, each agent chooses an arm to play and subsequently sends a message to her neighbors. The goal is to minimize the overall regret of the entire network. We design a fully decentralized algorithm that uses an accelerated consensus procedure to compute (delayed) estimates of the average of rewards obtained by all the agents for each arm, and then uses an upper confidence bound (UCB) algorithm that accounts for the delay and error of the estimates. We analyze the regret of our algorithm and also provide a lower bound. The regret is bounded by the optimal centralized regret plus a natural and simple term depending on the spectral gap of the communication matrix. Our algorithm is simpler to analyze than those proposed in prior work and it achieves better regret bounds, while requiring less information about the underlying network. It also performs better empirically.

We investigate implicit regularization schemes for gradient descent methods applied to unpenalized least squares regression to solve the problem of reconstructing a sparse signal from an underdetermined system of linear measurements under the restricted isometry assumption. For a given parametrization yielding a non-convex optimization problem, we show that prescribed choices of initialization, step size and stopping time yield a statistically and computationally optimal algorithm that achieves the minimax rate with the same cost required to read the data up to poly-logarithmic factors. Beyond minimax optimality, we show that our algorithm adapts to instance difficulty and yields a dimension-independent rate when the signal-to-noise ratio is high enough. Key to the computational efficiency of our method is an increasing step size scheme that adapts to refined estimates of the true solution. We validate our findings with numerical experiments and compare our algorithm against explicit $\ell_{1}$ penalization. Going from hard instances to easy ones, our algorithm is seen to undergo a phase transition, eventually matching least squares with an oracle knowledge of the true support.

Coalescing random walks is a fundamental stochastic process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a single random walk. The coalescence time is defined as the expected time until only one particle remains, starting from one particle at every node. Despite recent progress such as by Cooper, Elsässer, Ono, Radzik [13] and Cooper, Frieze and Radzik [12], the coalescence time for graphs such as binary trees, d-dimensional tori, hypercubes and more generally, vertex-transitive graphs, remains unresolved. We provide a powerful toolkit that results in tight bounds for various topologies including the aforementioned ones. The meeting time is defined as the worst-case expected time required for two random walks to arrive at the same node at the same time. As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log2 n), the coalescence time of n random walks equals the meeting time up to constant factors. This upper bound is complemented by the construction of a graph family demonstrating that this result is the best possible up to constant factors. For almost-regular graphs, we bound the coalescence time by the hitting time, resolving the discrete-time variant of a conjecture by Aldous for this class of graphs. Finally, we prove that for any graph the coalescence time is bounded by O(n3) (which is tight for the Barbell graph); surprisingly even such a basic question about the coalescing time was not answered before this work. By duality, our results give bounds on the voter model and therefore give bounds on the consensus time in arbitrary undirected graphs. We also establish a new bound on the hitting time and cover time of regular graphs, improving and tightening previous results by Broder and Karlin [10], as well as those by Aldous and Fill [1].

Hierarchical clustering is a recursive partitioning of a dataset into clusters at an increasingly finer granularity. Motivated by the fact that most work on hierarchical clustering was based on providing algorithms, rather than optimizing a specific objective, Dasgupta framed similarity-based hierarchical clustering as a combinatorial optimization problem, where a “good” hierarchical clustering is one that minimizes a particular cost function [23]. He showed that this cost function has certain desirable properties: To achieve optimal cost, disconnected components (namely, dissimilar elements) must be separated at higher levels of the hierarchy, and when the similarity between data elements is identical, all clusterings achieve the same cost. We take an axiomatic approach to defining “good” objective functions for both similarity- and dissimilarity-based hierarchical clustering. We characterize a set of admissible objective functions having the property that when the input admits a “natural” ground-truth hierarchical clustering, the ground-truth clustering has an optimal value. We show that this set includes the objective function introduced by Dasgupta. Equipped with a suitable objective function, we analyze the performance of practical algorithms, as well as develop better and faster algorithms for hierarchical clustering. We also initiate a beyond worst-case analysis of the complexity of the problem and design algorithms for this scenario.

Machine learning methods are widely used for a variety of prediction problems. Prediction as a service is a paradigm in which service providers with technological expertise and computational resources may perform predictions for clients. However, data privacy severely restricts the applicability of such services, unless measures to keep client data private (even from the service provider) are designed. Equally important is to minimize the nature of computation and amount of communication required between client and server. Fully homomorphic encryption offers a way out, whereby clients may encrypt their data, and on which the server may perform arithmetic computations. The one drawback of using fully homomorphic encryption is the amount of time required to evaluate large machine learning models on encrypted data. We combine several ideas from the machine learning literature, particularly work on quantization and sparsification of neural networks, together with algorithmic tools to speed-up and parallelize computation using encrypted data.

Kleinberg (2002) stated three axioms that any clustering procedure should satisfy and showed there is no clustering procedure that simultaneously satisfies all three. One of these, called the consistency axiom, requires that when the data is modified in a helpful way, i.e. if points in the same cluster are made more similar and those in different ones made less similar, the algorithm should output the same clustering. To circumvent this impossibility result, research has focused on considering clustering procedures that have a clustering quality measure (or a cost) and showing that a modification of Kleinberg’s axioms that takes cost into account lead to feasible clustering procedures. In this work, we take a different approach, based on the observation that the consistency axiom fails to be satisfied when the “correct” number of clusters changes. We modify this axiom by making use of cost functions to determine the correct number of clusters, and require that consistency holds only if the number of clusters remains unchanged. We show that single linkage satisfies the modified axioms, and if the input is well-clusterable, some popular procedures such as k-means also satisfy the axioms, taking a step towards explaining the success of these objective functions for guiding the design of algorithms.

Hierarchical clustering is a recursive partitioning of a dataset into clusters at an increasingly finer granularity. Motivated by the fact that most work on hierarchical clustering was based on providing algorithms, rather than optimizing a specific objective, [19] framed similarity-based hierarchical clustering as a combinatorial optimization problem, where a ‘good’ hierarchical clustering is one that minimizes some cost function. He showed that this cost function has certain desirable properties, such as in order to achieve optimal cost, disconnected components must be separated first and that in ‘structureless’ graphs, i.e., cliques, all clusterings achieve the same cost. We take an axiomatic approach to defining ‘good’ objective functions for both similarity and dissimilarity-based hierarchical clustering. We characterize a set of admissible objective functions (that includes the one introduced by Dasgupta) that have the property that when the input admits a ‘natural’ ground-truth hierarchical clustering, the ground-truth clustering has an optimal value. Equipped with a suitable objective function, we analyze the performance of practical algorithms, as well as develop better and faster algorithms for hierarchical clustering. For similarity-based hierarchical clustering, [19] showed that a simple recursive sparsest-cut based approach achieves an O(log3/2 n)-approximation on worst-case inputs. We give a more refined analysis of the algorithm and show that it in fact achieves an -approximation1. This improves upon the LP-based O(log n)-approximation of [33]. For dissimilarity-based hierarchical clustering, we show that the classic average-linkage algorithm gives a factor 2 approximation, and provide a simple and better algorithm that gives a factor 3/2 approximation. This aims at explaining the success of these heuristics in practice. Finally, we consider a ‘beyond-worst-case’ scenario through a generalisation of the stochastic block model for hierarchical clustering. We show that Dasgupta's cost function also has desirable properties for these inputs and we provide a simple algorithm that for graphs generated according to this model yields a 1 + o(1) factor approximation.

We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipschitz-continuous functions or functions with bounded gradients. This is with good reason-any learning algorithm suffers linear regret even against piecewise constant functions that are chosen adversarially, arguably the simplest of non-Lipschitz continuous functions. The smoothed setting we consider is inspired by the seminal work of Spielman and Teng (2004) and the recent work of Gupta and Roughgarden (2016)-in this setting, the sequence of functions may be chosen by an adversary, however, with some uncertainty in the location of discontinuities. We give algorithms that achieve sublinear regret in the full information and bandit settings.

We give the first dimension-efficient algorithms for learning Rectified Linear Units (ReLUs), which are functions of the form $\mathbf{x} \mapsto \mathsf{max}(0, \mathbf{w} ⋅\mathbf{x})$ with $\mathbf{w} ∈\mathbb{S}^n-1$. Our algorithm works in the challenging Reliable Agnostic learning model of Kalai, Kanade and Mansour (2012) where the learner is given access to a distribution $\mathcal{D}$ on labeled examples but the labeling may be arbitrary. We construct a hypothesis that simultaneously minimizes the false-positive rate and the loss on inputs given positive labels by $\mathcal{D}$, for any convex, bounded, and Lipschitz loss function. The algorithm runs in polynomial-time (in $n$) with respect to \em any distribution on $\mathbb{S}^n-1$ (the unit sphere in $n$ dimensions) and for any error parameter $ε= Ω(1 / \log n)$ (this yields a PTAS for a question raised by F. Bach on the complexity of maximizing ReLUs). These results are in contrast to known efficient algorithms for reliably learning linear threshold functions, where $ε$ must be $Ω(1)$ and strong assumptions are required on the marginal distribution. We can compose our results to obtain the first set of efficient algorithms for learning constant-depth networks of ReLU with fixed polynomial-dependence in the dimension. For depth-2 networks of sigmoids, we obtain the first algorithms that have a polynomial dependency in \em all parameters. Our techniques combine kernel methods and polynomial approximations with a “dual-loss” approach to convex programming. As a byproduct we obtain a number of applications including the first set of efficient algorithms for “convex piecewise-linear fitting” and the first efficient algorithms for noisy polynomial reconstruction of low-weight polynomials on the unit sphere.

Hiererachical clustering, that is computing a recursive partitioning of a dataset to obtain clusters at increasingly finer granularity is a fundamental problem in data analysis. Although hierarchical clustering has mostly been studied through procedures such as linkage algorithms, or top-down heuristics, rather than as optimization problems, recently Dasgupta [1] proposed an objective function for hierarchical clustering and initiated a line of work developing algorithms that explicitly optimize an objective (see also [2, 3, 4]). In this paper, we consider a fairly general random graph model for hierarchical clustering, called the hierarchical stochastic blockmodel (HSBM), and show that in certain regimes the SVD approach of McSherry [5] combined with specific linkage methods results in a clustering that give an O(1)-approximation to Dasgupta’s cost function. We also show that an approach based on SDP relaxations for balanced cuts based on the work of Makarychev et al. [6], combined with the recursive sparsest cut algorithm of Dasgupta, yields an O(1) approximation in slightly larger regimes and also in the semi-random setting, where an adversary may remove edges from the random graph generated according to an HSBM. Finally, we report empirical evaluation on synthetic and real-world data showing that our proposed SVD-based method does indeed achieve a better cost than other widely-used heurstics and also results in a better classification accuracy when the underlying problem was that of multi-class classification.

Discovering statistical structure from links is a fundamental problem in the analysis of social networks. Choosing a misspecified model, or equivalently, an incorrect inference algorithm will result in an invalid analysis or even falsely uncover patterns that are in fact artifacts of the model. This work focuses on unifying two of the most widely used link-formation models: the stochastic block model (SBM) and the small world (or latent space) model (SWM). Integrating techniques from kernel learning, spectral graph theory, and nonlinear dimensionality reduction, we develop the first statistically sound polynomial-time algorithm to discover latent patterns in sparse graphs for both models. When the network comes from an SBM, the algorithm outputs a block structure. When it is from an SWM, the algorithm outputs estimates of each node's latent position.

We consider the problem of estimating the graph size, where one is given only local access to the graph. We formally define a query model in which one starts with a seed node and is allowed to make queries about neighbours of nodes that have already been seen. In the case of undirected graphs, an estimator of Katzir et al. (2014) based on a sample from the stationary distribution π uses O(1/(||π||2) + davg) queries; we prove that this is tight. In addition, we establish this as a lower bound even when the algorithm is allowed to crawl the graph arbitrarily; the results of Katzir et al. give an upper bound that is worse by a multiplicative factor tmix · log (n).

We consider the problem of estimating the graph size, where one is given only local access to the graph. We formally define a query model in which one starts with a seed node and is allowed to make queries about neighbours of nodes that have already been seen. In the case of undirected graphs, an estimator of Katzir et al. (2014) based on a sample from the stationary distribution pi uses O(1/||pi||_2 + d_avg) queries; we prove that this is tight. In addition, we establish this as a lower bound even when the algorithm is allowed to crawl the graph arbitrarily; the results of Katzir et al. give an upper bound that is worse by a multiplicative factor t_mix(1/n^4). The picture becomes significantly different in the case of directed graphs. We show that without strong assumptions on the graph structure, the number of nodes cannot be predicted to within a constant multiplicative factor without using a number of queries that are at least linear in the number of nodes; in particular, rapid mixing and small diameter, properties that most real-world networks exhibit, do not suffice. The question of interest is whether any algorithm can beat breadth-first search. We introduce a new parameter, generalising the well-studied conductance, such that if a suitable bound on it exists and is known to the algorithm, the number of queries required is sublinear in the number of edges; we show that this is tight.

We consider the problem of stable matching with dynamic preference lists. At each time-step, the preference list of some player may change by swapping random adjacent members. The goal of a central agency (algorithm) is to maintain an approximately stable matching, in terms of number of blocking pairs, at all time-steps. The changes in the preference lists are not reported to the algorithm, but must instead be probed explicitly. We design an algorithm that in expectation and with high probability maintains a matching that has at most O((log n)^2 blocking pairs.

Leskovec, Kleinberg and Faloutsos (2005) observed that many social networks exhibit properties such as shrinking (i.e. bounded) diameter, densification, and (power-law) heavy tail degree distributions. To explain these phenomena, they introduced a generative model, called the Forest Fire model, and using simulations showed that this model indeed exhibited these properties; however, proving this rigorously was left as an open problem. In this paper, we analyse one of these properties, shrinking diameter. We define a restricted version of their model that incorporates the main features that seem to contribute towards this property, and prove that the graphs generated by this model exhibit shrinking distance to the seed graph. We prove that an even simpler model, the random walk model, already exhibits this phenomenon.

The stochastic block model is a classical cluster exhibiting random graph model that has been widely studied in statistics, physics, and computer science. In its simplest form, the model is a random graph with two equal-sized clusters, with intracluster edge probability p, and intercluster edge probability q. We focus on the sparse case, i.e., p, q = O(1/n), which is practically more relevant and also mathematically more challenging. A conjecture of Decelle, Krzakala, Moore, and Zdeborová, based on ideas from statistical physics, predicted a specific threshold for clustering. The negative direction of the conjecture was proved by Mossel, Neeman, and Sly (2012), and more recently, the positive direction was independently proved by Massoulié and Mossel, Neeman, and Sly. In many real network clustering problems, nodes contain information as well. We study the interplay between node and network information in clustering by studying a labeled block model, where in addition to the edge information, the true cluster labels of a small fraction of the nodes are revealed. In the case of two clusters, we show that below the threshold, a small amount of node information does not affect recovery. On the other hand, we show that for any small amount of information, efficient local clustering is achievable as long as the number of clusters is sufficiently large (as a function of the amount of revealed information).

The theory of learning under the uniform distribution is rich and deep, with connections to cryptography, computational complexity, and the analysis of boolean functions to name a few areas. This theory however is very limited due to the fact that the uniform distribution and the corresponding Fourier basis are rarely encountered as a statistical model. A family of distributions that vastly generalizes the uniform distribution on the Boolean cube is that of distributions represented by Markov Random Fields (MRF). Markov Random Fields are one of the main tools for modeling high dimensional data in many areas of statistics and machine learning. In this paper we initiate the investigation of extending central ideas, methods and algorithms from the theory of learning under the uniform distribution to the setup of learning concepts given examples from MRF distributions. In particular, our results establish a novel connection between properties of MCMC sampling of MRFs and learning under the MRF distribution.

The stochastic block model is a classical cluster-exhibiting random graph model that has been widely studied in statistics, physics and computer science. In its simplest form, the model is a random graph with two equal-sized clusters, with intra-cluster edge probability p, and inter-cluster edge probability q. We focus on the sparse case, i.e. p, q = O(1/n), which is practically more relevant and also mathematically more challenging. A conjecture of Decelle, Krzakala, Moore and Zdeborova, based on ideas from statistical physics, predicted a specific threshold for clustering. The negative direction of the conjecture was proved by Mossel, Neeman and Sly (2012), and more recently the positive direction was proven independently by Massoulie and Mossel, Neeman, and Sly. In many real network clustering problems, nodes contain information as well. We study the interplay between node and network information in clustering by studying a labeled block model, where in addition to the edge information, the true cluster labels of a small fraction of the nodes are revealed. In the case of two clusters, we show that below the threshold, a small amount of node information does not affect recovery. On the other hand, we show that for any small amount of information efficient local clustering is achievable as long as the number of clusters is sufficiently large (as a function of the amount of revealed information).

We study several questions in the \emphreliable agnostic learning framework of Kalai et al. (2009), which captures learning tasks in which one type of error is costlier than other types. A positive reliable classifier is one that makes no false positive errors. The goal in the \emphpositive reliable agnostic framework is to output a hypothesis with the following properties: (i) its false positive error rate is at most ε, (ii) its false negative error rate is at most εmore than that of the best positive reliable classifier from the class. A closely related notion is \emphfully reliable agnostic learning, which considers \emphpartial classifiers that are allowed to predict “unknown” on some inputs. The best fully reliable partial classifier is one that makes no errors and minimizes the probability of predicting “unknown”, and the goal in fully reliable learning is to output a hypothesis that is almost as good as the best fully reliable partial classifier from a class. For distribution-independent learning, the best known algorithms for PAC learning typically utilize polynomial threshold representations, while the state of the art agnostic learning algorithms use point-wise polynomial approximations. We show that \emphone-sided polynomial approximations, an intermediate notion between polynomial threshold representations and point-wise polynomial approximations, suffice for learning in the reliable agnostic settings. We then show that majorities can be fully reliably learned and disjunctions of majorities can be positive reliably learned, through constructions of appropriate one-sided polynomial approximations. Our fully reliable algorithm for majorities provides the first evidence that fully reliable learning may be strictly easier than agnostic learning. Our algorithms also satisfy strong attribute-efficiency properties, and in many cases they provide smooth tradeoffs between sample complexity and running time.

We study linear control problems with quadratic losses and adversarially chosen tracking targets. We present an efficient algorithm for this problem and show that, under standard conditions on the linear system, its regret with respect to an optimal linear policy grows as O(\log^2 T), where T is the number of rounds of the game. We also study a problem with adversarially chosen transition dynamics; we present an exponentially-weighted average algorithm for this problem, and we give regret bounds that grow as O(\sqrt T).

In a seminal paper, Valiant (2006) introduced a computational model for evolution to address the question of complexity that can arise through Darwinian mechanisms. Valiant views evolution as a restricted form of computational learning, where the goal is to evolve a hypothesis that is close to the ideal function. Feldman (2008) showed that (correlational) statistical query learning algorithms could be framed as evolutionary mechanisms in Valiant's model. P. Valiant (2012) considered evolvability of real-valued functions and also showed that weak-optimization algorithms that use weak-evaluation oracles could be converted to evolutionary mechanisms.

We introduce a new model of membership query (MQ) learning, where the learning algorithm is restricted to query points that are close to random examples drawn from the underlying distribution. The learning model is intermediate between the PAC model (Valiant,1984) and the PAC+MQ model (where the queries are allowed to be arbitrary points). Membership query algorithms are not popular among machine learning practitioners. Apart from the obvious difficulty of adaptively querying labellers, it has also been observed that querying unnatural points leads to increased noise from human labellers (Lang and Baum, 1992). This motivates our study of learning algorithms that make queries that are close to examples generated from the data distribution. We restrict our attention to functions defined on the n-dimensional Boolean hypercube and say that a membership query is local if its Hamming distance from some example in the (random) training data is at most O(log(n)). We show the following results in this model: \beginenumerate \item The class of sparse polynomials (with coefficients in R) over {0, 1}^n is polynomial time learnable under a large class of locally smooth distributions using O(log(n))-local queries. This class also includes the class of O(log(n))-depth decision trees. \item The class of polynomial-sized decision trees is polynomial time learnable under product distributions using O(log(n))-local queries. \item The class of polynomial size DNF formulas is learnable under the uniform distribution using O(log(n))-local queries in time n^O(log(log(n))). \item In addition we prove a number of results relating the proposed model to the traditional PAC model and the PAC+MQ model. \endenumerate

We study the problem of online learning Markov Decision Processes (MDPs) when both the transition distributions and loss functions are chosen by an adversary. We present an algorithm that, under a mixing assumption, achieves $O(\sqrt{T\log|\Pi|}+\log|\Pi|)$ regret with respect to a comparison set of policies $\Pi$. The regret is independent of the size of the state and action spaces. When expectations over sample paths can be computed efficiently and the comparison set $\Pi$ has polynomial size, this algorithm is efficient. We also consider the episodic adversarial online shortest path problem. Here, in each episode an adversary may choose a weighted directed acyclic graph with an identified start and finish node. The goal of the learning algorithm is to choose a path that minimizes the loss while traversing from the start to finish node. At the end of each episode the loss function (given by weights on the edges) is revealed to the learning algorithm. The goal is to minimize regret with respect to a fixed policy for selecting paths. This problem is a special case of the online MDP problem. For randomly chosen graphs and adversarial losses, this problem can be efficiently solved. We show that it also can be efficiently solved for adversarial graphs and randomly chosen losses. When both graphs and losses are adversarially chosen, we present an efficient algorithm whose regret scales linearly with the number of distinct graphs. Finally, we show that designing efficient algorithms for the adversarial online shortest path problem (and hence for the adversarial MDP problem) is as hard as learning parity with noise, a notoriously difficult problem that has been used to design efficient cryptographic schemes.

We study the online decision problem where the set of available actions varies over time, also called the sleeping experts problem. We consider the setting where the performance comparison is made with respect to the best ordering of actions in hindsight. In this paper, both the payoff function and the availability of actions is adversarial. Kleinberg et al. (2008) gave a computationally efficient no-regret algorithm in the setting where payoffs are stochastic. Kanade et al. (2009) gave an efficient no-regret algorithm in the setting where action availability is stochastic.

We consider the online distributed non-stochastic experts problem, where the distributed system consists of one coordinator node that is connected to k sites, and the sites are required to communicate with each other via the coordinator. At each time-step t, one of the k site nodes has to pick an expert from the set {1, . . . , n}, and the same site receives information about payoffs of all experts for that round. The goal of the distributed system is to minimize regret at time horizon T, while simultaneously keeping communication to a minimum. The two extreme solutions to this problem are: (i) Full communication: This essentially simulates the non-distributed setting to obtain the optimal O(\sqrt{log(n)T}) regret bound at the cost of T communication. (ii) No communication: Each site runs an independent copy – the regret is O(\sqrt{log(n)kT}) and the communication is 0. This paper shows the difficulty of simultaneously achieving regret asymptotically better than \sqrt{kT} and communication better than T. We give a novel algorithm that for an oblivious adversary achieves a non-trivial trade-off: regret O(\sqrt{k^{5(1+\epsilon)/6} T}) and communication O(T/k^\epsilon), for any value of \epsilon in (0, 1/5). We also consider a variant of the model, where the coordinator picks the expert. In this model, we show that the label-efficient forecaster of Cesa-Bianchi et al. (2005) already gives us strategy that is near optimal in regret vs communication trade-off.

Valiant (2007) introduced a computational model of evolution and suggested that Darwinian evolution be studied in the framework of computational learning theory. Valiant describes evolution as a restricted form of learning where exploration is limited to a set of possible mutations and feedback is received through the survival of the fittest mutation. In subsequent work Feldman (2008) showed that evolvability in Valiant's model is equivalent to learning in the correlational statistical query (CSQ) model. We extend Valiant's model to include genetic recombination and show that in certain cases, recombination can significantly speed-up the process of evolution in terms of the number of generations, though at the expense of population size. This follows via a reduction from parallel-CSQ algorithms to evolution with recombination. This gives an exponential speed-up (in terms of the number of generations) over previous known results for evolving conjunctions and half spaces with respect to restricted distributions.

Generalized Linear Models (GLMs) and Single Index Models (SIMs) provide powerful generalizations of linear regression, where the target variable is assumed to be a (possibly unknown) 1-dimensional function of a linear predictor. In general, these problems entail non-convex estimation procedures, and, in practice, iterative local search heuristics are often used. Kalai and Sastry (2009) provided the first provably efficient method, the \emph{Isotron} algorithm, for learning SIMs and GLMs, under the assumption that the data is in fact generated under a GLM and under certain monotonicity and Lipschitz (bounded slope) constraints. The Isotron algorithm interleaves steps of perceptron-like updates with isotonic regression (fitting a one-dimensional non-decreasing function). However, to obtain provable performance, the method requires a fresh sample every iteration. In this paper, we provide algorithms for learning GLMs and SIMs, which are both computationally and statistically efficient. We modify the isotonic regression step in Isotron to fit a Lipschitz monotonic function, and also provide an efficient $O(n \log(n))$ algorithm for this step, improving upon the previous $O(n^2)$ algorithm. We provide a brief empirical study, demonstrating the feasibility of our algorithms in practice.

We prove strong noise-tolerance properties of a potential-based boosting algorithm, similar to MadaBoost (Domingo and Watanabe, 2000) and SmoothBoost (Servedio, 2003). Our analysis is in the agnostic framework of Kearns, Schapire and Sellie (1994), giving polynomial-time guarantees in presence of arbitrary noise. A remarkable feature of our algorithm is that it can be implemented without reweighting examples, by randomly relabeling them instead. Our boosting theorem gives, as easy corollaries, alternative derivations of two recent non-trivial results in computational learning theory: agnostically learning decision trees (Gopalan et al, 2008) and agnostically learning halfspaces (Kalai et al, 2005). Experiments suggest that the algorithm performs similarly to Madaboost.

Due to fundamental physical limitations and power constraints, we are witnessing a radical change in commodity microprocessor architectures to multicore designs. Continued performance on multicore processors now requires the exploitation of concurrency at the algorithmic level. In this paper, we identify key issues in algorithm design for multicore processors and propose a computational model for these systems. We introduce SWARM (software and algorithms for running on multi-core), a portable open-source parallel library of basic primitives that fully exploit multicore processors. Using this framework, we have implemented efficient parallel algorithms for important primitive operations such as prefix-sums, pointer-jumping, symmetry breaking, and list ranking; for combinatorial problems such as sorting and selection; for parallel graph theoretic algorithms such as spanning tree, minimum spanning tree, graph decomposition, and tree contraction; and for computational genomics applications such as maximum parsimony. The main contributions of this paper are the design of the SWARM multicore framework, the presentation of a multicore algorithmic model, and validation results for this model. SWARM is freely available as open-source from http://multicore-swarm.sourceforge.net/.