Numerous lines of work aim to control \emph{model disagreement} — the extent to which two machine learning models disagree in their predictions. We adopt a simple and standard notion of model disagreement in real-valued prediction problems, namely the expected squared difference in predictions between two models trained on independent samples, without any coordination of the training processes. We would like to be able to drive disagreement to zero with some natural parameter(s) of the training procedure using analyses that can be applied to existing training methodologies. We develop a simple general technique for proving bounds on independent model disagreement based on \emph{anchoring} to the average of two models within the analysis. We then apply this technique to prove disagreement bounds for four commonly used machine learning algorithms: (1) stacked aggregation over an arbitrary model class (where disagreement is driven to 0 with the number of models $k$ being stacked) (2) gradient boosting (where disagreement is driven to 0 with the number of iterations $k$) (3) neural network training with architecture search (where disagreement is driven to 0 with the size $n$ of the architecture being optimized over) and (4) regression tree training over all regression trees of fixed depth (where disagreement is driven to 0 with the depth $d$ of the tree architecture). For clarity, we work out our initial bounds in the setting of one-dimensional regression with squared error loss — but then show that all of our results generalize to multi-dimensional regression with any strongly convex loss.

We study a distributed learning problem in which learning agents are embedded in a directed acyclic graph (DAG). There is a fixed and arbitrary distribution over feature/label pairs, and each agent or vertex in the graph is able to directly observe only a subset of the features — potentially a different subset for every agent. The agents learn sequentially in some order consistent with a topological sort of the DAG, committing to a model mapping observations to predictions of the real-valued label. Each agent observes the predictions of their parents in the DAG, and trains their model using both the features of the instance that they directly observe, and the predictions of their parents as additional features. We ask when this process is sufficient to achieve information aggregation, in the sense that some agent in the DAG is able to learn a model whose error is competitive with the best model that could have been learned (in some hypothesis class) with direct access to all features, despite the fact that no single agent in the network has such access. We give upper and lower bounds for this problem for both linear and general hypothesis classes. Our results identify the depth of the DAG as the key parameter: information aggregation can occur over sufficiently long paths in the DAG, assuming that all of the relevant features are well represented along the path, and there are distributions over which information aggregation cannot occur even in the linear case, and even in arbitrarily large DAGs that do not have sufficient depth (such as a hub-and-spokes topology in which the spoke vertices collectively see all the features). We complement our theoretical results with a comprehensive set of experiments.

In traditional reinforcement learning (RL), the learner aims to solve a single objective optimization problem: find the policy that maximizes expected reward. However, in many real-world settings, it is important to optimize over multiple objectives simultaneously. For example, when we are interested in fairness, states might have feature annotations corresponding to multiple (intersecting) demographic groups to whom reward accrues, and our goal might be to maximize the reward of the group receiving the minimal reward. In this work, we consider a multi-objective optimization problem in which each objective is defined by a state-based reweighting of a single scalar reward function. This generalizes the problem of maximizing the reward of the minimum reward group. We provide oracle-efficient algorithms to solve these multi-objective RL problems even when the number of objectives is very large — for tabular MDPs, as well as for large MDPs when the group functions have additional structure. The contribution of this paper is that we are able to solve this class of multi-objective RL problems with a possibly exponentially large class of constraints over intersecting groups in both tabular and large state space MDPs in an oracle-efficient manner. Finally, we experimentally validate our theoretical results and demonstrate applications on a preferential attachment graph MDP.

Recently, diffusion models have become popular tools for image synthesis due to their high-quality outputs. However, like other large models, they may leak private information about their training data. Here, we demonstrate a privacy vulnerability of diffusion models through a membership inference (MI) attack, which aims to identify whether a target example belongs to the training set when given the trained diffusion model. Our proposed MI attack learns quantile regression models that predict (a quantile of) the distribution of reconstruction loss on examples not used in training. This allows us to define a granular hypothesis test for determining the membership of a point in the training set, based on thresholding the reconstruction loss of that point using a custom threshold tailored to the example. We also provide a simple bootstrap technique that takes a majority membership prediction over ”a bag of weak attackers” which improves the accuracy over individual quantile regression models. We show that our attack outperforms the prior state-of-the-art attack while being substantially less computationally expensive — prior attacks required training multiple ”shadow models” with the same architecture as the model under attack, whereas our attack requires training only much smaller models.

Machine unlearning is motivated by principles of data autonomy. The premise is that a person can request to have their data's influence removed from deployed models, and those models should be updated as if they were retrained without the person's data. We show that these updates expose individuals to high-accuracy reconstruction attacks which allow the attacker to recover their data in its entirety, even when the original models are so simple that privacy risk might not otherwise have been a concern. We show how to mount a near-perfect attack on the deleted data point from linear regression models. We then generalize our attack to other loss functions and architectures, and empirically demonstrate the effectiveness of our attacks across a wide range of datasets (capturing both tabular and image data). Our work highlights that privacy risk is significant even for extremely simple model classes when individuals can request deletion of their data from the model.

Reinforcement learning (RL) in large or infinite state spaces is notoriously challenging, both theoretically (where worst-case sample and computational complexities must scale with state space cardinality) and experimentally (where function approximation and policy gradient techniques often scale poorly and suffer from instability and high variance). One line of research attempting to address these difficulties makes the natural assumption that we are given a collection of base or *constituent* policies (possibly heuristic) upon which we would like to improve in a scalable manner. In this work we aim to compete with the *max-following policy*, which at each state follows the action of whichever constituent policy has the highest value. The max-following policy is always at least as good as the best constituent policy, and may be considerably better. Our main result is an efficient algorithm that learns to compete with the max-following policy, given only access to the constituent policies (but not their value functions). In contrast to prior work in similar settings, our theoretical results require only the minimal assumption of an ERM oracle for value function approximation for the constituent policies (and not the global optimal policy or the max-following policy itself) on samplable distributions. We illustrate our algorithm's experimental effectiveness and behavior on several robotic simulation testbeds.

We show how to take a regression function that is appropriately multicalibrated and efficiently post-process it into an approximately error minimizing classifier satisfying a large variety of fairness constraints. The post-processing requires no labeled data, and only a modest amount of unlabeled data and computation. The computational and sample complexity requirements of computing are comparable to the requirements for solving a single fair learning task optimally, but it can in fact be used to solve many different downstream fairness-constrained learning problems efficiently. Our post-processing method easily handles intersecting groups, generalizing prior work on post-processing regression functions to satisfy fairness constraints that only applied to disjoint groups. Our work extends recent work showing that multicalibrated regression functions are omnipredictors (i.e. can be post-processed to optimally solve unconstrained ERM problems) to constrained optimization problems.

We study the connection between multicalibration and boosting for squared error regression. First we prove a useful characterization of multicalibration in terms of a “swap regret” like condition on squared error. Using this characterization, we give an exceedingly simple algorithm that can be analyzed both as a boosting algorithm for regression and as a multicalibration algorithm for a class $\mathcal{H}$ that makes use only of a standard squared error regression oracle for $\mathcal{H}$. We give a weak learning assumption on $\mathcal{H}$ that ensures convergence to Bayes optimality without the need to make any realizability assumptions — giving us an agnostic boosting algorithm for regression. We then show that our weak learning assumption on $\mathcal{H}$ is both necessary and sufficient for multicalibration with respect to $\mathcal{H}$ to imply Bayes optimality, answering an open question. We also show that if $\mathcal{H}$ satisfies our weak learning condition relative to another class $\mathcal{C}$ then multicalibration with respect to $\mathcal{H}$ implies multicalibration with respect to $\mathcal{C}$. Finally we investigate the empirical performance of our algorithm experimentally.

Membership inference attacks are designed to determine, using black box access to trained models, whether a particular example was used in training or not. Membership inference can be formalized as a hypothesis testing problem. The most effective existing attacks estimate the distribution of some test statistic (usually the model's confidence on the true label) on points that were (and were not) used in training by training many \emph{shadow models}---i.e. models of the same architecture as the model being attacked, trained on a random subsample of data. While effective, these attacks are extremely computationally expensive, especially when the model under attack is large. \footnotetext[0]{ Martin and Shuai are the lead authors, and other authors are ordered alphabetically. \{maberlop,shuat\}@amazon.com} We introduce a new class of attacks based on performing quantile regression on the distribution of confidence scores induced by the model under attack on points that are not used in training. We show that our method is competitive with state-of-the-art shadow model attacks, while requiring substantially less compute because our attack requires training only a single model. Moreover, unlike shadow model attacks, our proposed attack does not require any knowledge of the architecture of the model under attack and is therefore truly ``black-box". We show the efficacy of this approach in an extensive series of experiments on various datasets and model architectures. Our code is available at \href{https://github.com/amazon-science/quantile-mia}{github.com/amazon-science/quantile-mia.}

The replicability crisis in the social, behavioral, and data sciences has led to the formulation of algorithm frameworks for replicability --- i.e., a requirement that an algorithm produce identical outputs (with high probability) when run on two different samples from the same underlying distribution. While still in its infancy, provably replicable algorithms have been developed for many fundamental tasks in machine learning and statistics, including statistical query learning, the heavy hitters problem, and distribution testing. In this work we initiate the study of replicable reinforcement learning, providing a provably replicable algorithm for parallel value iteration, and a provably replicable version of R-Max in the episodic setting. These are the first formal replicability results for control problems, which present different challenges for replication than batch learning settings.

We introduce AdaMix, an adaptive differentially private algorithm for training deep neural network classifiers using both private and public image data. While pre-training language models on large public datasets has enabled strong differential privacy (DP) guarantees with minor loss of accuracy, a similar practice yields punishing trade-offs in vision tasks. A few-shot or even zero-shot learning baseline that ignores private data can outperform fine-tuning on a large private dataset. AdaMix incorporates few-shot training, or cross-modal zero-shot learning, on public data prior to private fine-tuning, to improve the trade-off. AdaMix reduces the error increase from the non-private upper bound from the 167–311% of the baseline, on average across 6 datasets, to 68-92% depending on the desired privacy level selected by the user. AdaMix tackles the trade-off arising in visual classification, whereby the most privacy sensitive data, corresponding to isolated points in representation space, are also critical for high classification accuracy. In addition, AdaMix comes with strong theoretical privacy guarantees and convergence analysis.

We provide a differentially private algorithm for producing synthetic data simultaneously useful for multiple tasks: marginal queries and multitask machine learning (ML). A key innovation in our algorithm is the ability to directly handle numerical features, in contrast to a number of related prior approaches which require numerical features to be first converted into {high cardinality} categorical features via {a binning strategy}. Higher binning granularity is required for better accuracy, but this negatively impacts scalability. Eliminating the need for binning allows us to produce synthetic data preserving large numbers of statistical queries such as marginals on numerical features, and class conditional linear threshold queries. Preserving the latter means that the fraction of points of each class label above a particular half-space is roughly the same in both the real and synthetic data. This is the property that is needed to train a linear classifier in a multitask setting. Our algorithm also allows us to produce high quality synthetic data for mixed marginal queries, that combine both categorical and numerical features. Our method consistently runs 2-5x faster than the best comparable techniques, and provides significant accuracy improvements in both marginal queries and linear prediction tasks for mixed-type datasets.

We consider a recently introduced framework in which fairness is measured by worst-case outcomes across groups, rather than by the more standard differences between group outcomes. In this framework we provide provably convergent oracle-efficient learning algorithms (or equivalently, reductions to non-fair learning) for minimax group fairness. Here the goal is that of minimizing the maximum loss across all groups, rather than equalizing group losses. Our algorithms apply to both regression and classification settings and support both overall error and false positive or false negative rates as the fairness measure of interest. They also support relaxations of the fairness constraints, thus permitting study of the tradeoff between overall accuracy and minimax fairness. We compare the experimental behavior and performance of our algorithms across a variety of fairness-sensitive data sets and show empirical cases in which minimax fairness is strictly and strongly preferable to equal outcome notions.

We propose, implement, and evaluate a new algo-rithm for releasing answers to very large numbersof statistical queries likek-way marginals, sub-ject to differential privacy. Our algorithm makesadaptive use of a continuous relaxation of thePro-jection Mechanism, which answers queries on theprivate dataset using simple perturbation, and thenattempts to find the synthetic dataset that mostclosely matches the noisy answers. We use a con-tinuous relaxation of the synthetic dataset domainwhich makes the projection loss differentiable,and allows us to use efficient ML optimizationtechniques and tooling. Rather than answering allqueries up front, we make judicious use of ourprivacy budget by iteratively finding queries forwhich our (relaxed) synthetic data has high error,and then repeating the projection. Randomizedrounding allows us to obtain synthetic data in theoriginal schema. We perform experimental evalu-ations across a range of parameters and datasets,and find that our method outperforms existingalgorithms on large query classes.

In this paper we initiate the study of financial asset design with fairness as an explicit goal. We consider a variation on the classical problem of optimal portfolio design. In our setting, an individual consumer is specified by her risk tolerance, which corresponds to the variance in returns she is willing to accept in exchange for higher expected returns. We must design a (small) collection of portfolios and assign each consumer to a portfolio at lower or approximately equal risk than her tolerance. Fairness is imposed by demanding that the portfolios designed do not discriminate (in terms of expected returns) against less wealthy clients (or other specified protected groups).

We propose and analyze differentially private (DP) mechanisms for call auctions as an alternative to the complex and ad-hoc privacy efforts that are common in modern electronic markets. We prove that the number of shares cleared in the DP mechanisms compares favorably to the non-private optimal and provide a matching lower bound. We analyze the incentive properties of our mechanisms and their behavior under natural no-regret learning dynamics by market participants. We include simulation results and connections to the finance literature on market impact.

Motivated by settings in which predictive models may be required to be non-discriminatory with respect to certain attributes (such as race), but even collecting the sensitive attribute may be forbidden or restricted, we initiate the study of fair learning under the constraint of differential privacy. Our first algorithm is a private implementation of the equalized odds post-processing approach of (Hardt et al., 2016). This algorithm is appealingly simple, but must be able to use protected group membership explicitly at test time, which can be viewed as a form of “disparate treatment”. Our second algorithm is a differentially private version of the oracle-efficient in-processing approach of (Agarwal et al., 2018) which is more complex but need not have access to protected group membership at test time. We identify new tradeoffs between fairness, accuracy, and privacy that emerge only when requiring all three properties, and show that these tradeoffs can be milder if group membership may be used at test time. We conclude with a brief experimental evaluation.

We study a network formation game where agents receive benefits by forming connections to other agents but also incur both direct and indirect costs from the formed connections. Specifically, once the agents have purchased their connections, an attack starts at a randomly chosen vertex in the network and spreads according to the independent cascade model with a fixed probability, destroying any infected agents. The utility or welfare of an agent in our game is defined to be the expected size of the agent's connected component post-attack minus her expenditure in forming connections. Our goal is to understand the properties of the equilibrium networks formed in this game. Our first result concerns the edge density of equilibrium networks. A network connection increases both the likelihood of remaining connected to other agents after an attack as well the likelihood of getting infected by a cascading spread of infection. We show that the latter concern primarily prevails and any equilibrium network in our game contains only $O(n\log n)$ edges where $n$ denotes the number of agents. On the other hand, there are equilibrium networks that contain $\Omega(n)$ edges showing that our edge density bound is tight up to a logarithmic factor. Our second result shows that the presence of attack and its spread through a cascade does not significantly lower social welfare as long as the network is not too dense. We show that any non-trivial equilibrium network with $O(n)$ edges has $\Theta(n^2)$ social welfare, asymptotically similar to the social welfare guarantee in the game without any attacks.

We study a game between two firms which each provide a service based on machine learning. The firms are presented with the opportunity to purchase a new corpus of data, which will allow them to potentially improve the quality of their products. The firms can decide whether or not they want to buy the data, as well as which learning model to build on that data. We demonstrate a reduction from this potentially complicated action space to a one-shot, two-action game in which each firm only decides whether or not to buy the data. The game admits several regimes which depend on the relative strength of the two firms at the outset and the price at which the data is being offered. We analyze the game's Nash equilibria in all parameter regimes and demonstrate that, in expectation, the outcome of the game is that the initially stronger firm's market position weakens whereas the initially weaker firm's market position becomes stronger. Finally, we consider the perspective of the users of the service and demonstrate that the expected outcome at equilibrium is not the one which maximizes the welfare of the consumers.

We propose a new family of fairness definitions for classification problems that combine some of the best properties of both statistical and individual notions of fairness. We posit not only a distribution over individuals, but also a distribution over (or collection of) classification tasks. We then ask that standard statistics (such as error or false positive/negative rates) be (approximately) equalized across individuals, where the rate is defined as an expectation over the classification tasks. Because we are no longer averaging over coarse groups (such as race or gender), this is a semantically meaningful individual-level constraint. Given a sample of individuals and problems, we design an oracle-efficient algorithm (i.e. one that is given access to any standard, fairness-free learning heuristic) for the fair empirical risk minimization task. We also show that given sufficiently many samples, the ERM solution generalizes in two directions: both to new individuals, and to new classification tasks, drawn from their corresponding distributions. Finally we implement our algorithm and empirically verify its effectiveness.

We study fairness in linear bandit problems. Starting from the notion of meritocratic fairness introduced in~\citeJKMR16, we carry out a more refined analysis of a more general problem, achieving better performance guarantees with fewer modelling assumptions on the number and structure of available choices as well as the number selected. We also analyze the previously-unstudied question of fairness in infinite linear bandit problems, obtaining instance-dependent regret upper bounds as well as lower bounds demonstrating that this instance-dependence is necessary. The result is a framework for meritocratic fairness in an online linear setting that is substantially more powerful, general, and realistic than the current state of the art.

The most prevalent notions of fairness in machine learning fix a small collection of pre-defined groups (such as race or gender), and then ask for approximate parity of some statistic of the classifier (such as false positive rate) across these groups. Constraints of this form are susceptible to fairness gerrymandering, in which a classifier is fair on each individual group, but badly violates the fairness constraint on structured subgroups, such as certain combinations of protected attribute values. We thus consider fairness across exponentially or infinitely many subgroups, defined by a structured class of functions over the protected attributes. We first prove that the problem of auditing subgroup fairness for both equality of false positive rates and statistical parity is computationally equivalent to the problem of weak agnostic learning — which means it is hard in the worst case, even for simple structured subclasses. However, it also suggests that common heuristics for learning can be applied to successfully solve the auditing problem in practice. We then derive an algorithm that provably converges in a polynomial number of steps to the best subgroup-fair distribution over classifiers, given access to an oracle which can solve the agnostic learning problem. The algorithm is based on a formulation of subgroup fairness as a zero-sum game between a Learner (the primal player) and an Auditor (the dual player). We implement a variant of this algorithm using heuristic oracles, and show that we can effectively both audit and learn fair classifiers on a real dataset.

We consider the problem of online learning in the linear contextual bandits setting, but in which there are also strong individual fairness constraints governed by an unknown similarity metric. These constraints demand that we select similar actions or individuals with approximately equal probability DHPRZ12, which may be at odds with optimizing reward, thus modeling settings where profit and social policy are in tension. We assume we learn about an unknown Mahalanobis similarity metric from only weak feedback that identifies fairness violations, but does not quantify their extent. This is intended to represent the interventions of a regulator who "knows unfairness when he sees it" but nevertheless cannot enunciate a quantitative fairness metric over individuals. Our main result is an algorithm in the adversarial context setting that has a number of fairness violations that depends only logarithmically on T, while obtaining an optimal O(sqrt(T)) regret bound to the best fair policy.

We consider a new learning model in which a joint distribution over vector pairs $(x,y)$ is determined by an unknown function $c(x)$ that maps input vectors $x$ not to individual outputs, but to entire \em distributions\/ over output vectors $y$. Our main results take the form of rather general reductions from our model to algorithms for PAC learning the function class and the distribution class separately, and show that virtually every such combination yields an efficient algorithm in our model. Our methods include a randomized reduction to classification noise and an application of Le Cam’s method to obtain robust learning algorithms.

We initiate the study of fairness in reinforcement learning, where the actions of a learning algorithm may affect its environment and future rewards. Our fairness constraint requires that an algorithm never prefers one action over another if the long-term (discounted) reward of choosing the latter action is higher. Our first result is negative: despite the fact that fairness is consistent with the optimal policy, any learning algorithm satisfying fairness must take time exponential in the number of states to achieve non-trivial approximation to the optimal policy. We then provide a provably fair polynomial time algorithm under an approximate notion of fairness, thus establishing an exponential gap between exact and approximate fairness.

We consider the problem of selecting a strong pool of individuals from several populations with incomparable skills (e.g. soccer players, mathematicians, and singers) in a fair manner. The quality of an individual is defined to be their relative rank (by cumulative distribution value) within their own population, which permits cross-population comparisons. We study algorithms which attempt to select the highest quality subset despite the fact that true CDF values are not known, and can only be estimated from the finite pool of candidates. Specifically, we quantify the regret in quality imposed by “meritocratic” notions of fairness, which require that individuals are selected with probability that is monotonically increasing in their true quality. We give algorithms with provable fairness and regret guarantees, as well as lower bounds, and provide empirical results which suggest that our algorithms perform better than the theory suggests. We extend our results to a sequential batch setting, in which an algorithm must repeatedly select subsets of individuals from new pools of applicants, but has the benefit of being able to compare them to the accumulated data from previous rounds.

We consider settings in which we wish to incentivize myopic agents (such as Airbnb landlords, who may emphasize short-term profits and property safety) to treat arriving clients fairly, in order to prevent overall discrimination against individuals or groups. We model such settings in both classical and contextual bandit models in which the myopic agents maximize rewards according to current empirical averages, but are also amenable to exogenous payments that may cause them to alter their choices. Our notion of fairness asks that more qualified individuals are never (probabilistically) preferred over less qualifie ones [8].

The widespread use of machine learning to make consequential decisions about individual citizens (such as those involving credit, employment, insurance, and education) has been accompanied by rising alarm over instances of bias or discrimination in the algorithms and models used. While legal, regulatory and watchdog challenges to discriminatory algorithms will play an important role, it is also crucial to examine and quantify the extent to which social norms such as fairness can be "endogenized" into the learning process itself. Can we develop a rigorous and useful science of fair machine learning?

We introduce the study of fairness in multi-armed bandit problems. Our fairness definition demands that, given a pool of applicants, a worse applicant is never favored over a better one, despite a learning algorithm’s uncertainty over the true payoffs. In the classic stochastic bandits problem we provide a provably fair algorithm based on “chained” confidence intervals, and prove a cumulative regret bound with a cubic dependence on the number of arms. We further show that any fair algorithm must have such a dependence, providing a strong separation between fair and unfair learning that extends to the general contextual case. In the general contextual case, we prove a tight connection between fairness and the KWIK (Knows What It Knows) learning model: a KWIK algorithm for a class of functions can be transformed into a provably fair contextual bandit algorithm and vice versa. This tight connection allows us to provide a provably fair algorithm for the linear contextual bandit problem with a polynomial dependence on the dimension, and to show (for a different class of functions) a worst-case exponential gap in regret between fair and non-fair learning algorithms.

We consider the problem of learning from revealed preferences in an online setting. In our framework, each period a consumer buys an optimal bundle of goods from a merchant according to her (linear) utility function and current prices, subject to a budget constraint. The merchant observes only the purchased goods, and seeks to adapt prices to optimize his profits. We give an efficient algorithm for the merchant's problem that consists of a learning phase in which the consumer's utility function is (perhaps partially) inferred, followed by a price optimization step. We also give an alternative online learning algorithm for the setting where prices are set exogenously, but the merchant would still like to predict the bundle that will be bought by the consumer, for purposes of inventory or supply chain management. In contrast with most prior work on the revealed preferences problem, we demonstrate that by making stronger assumptions on the form of utility functions, efficient algorithms for both learning and profit maximization are possible, even in adaptive, online settings.

We report on a series of behavioral experiments in human computation on three different tasks over networks: graph coloring, community detection (or graph clustering), and competitive contagion. While these tasks share similar action spaces and interfaces, they capture a diversity of computational challenges: graph coloring is a search problem, clustering is an optimization problem, and competitive contagion is a game-theoretic problem. In contrast with most of the prior literature on human-subject experiments in networks, in which collectives of subjects are embedded "in" the network, and have only local information and interactions, here individual subjects have a global (or "over") view and must solve "whole network" problems alone. Our primary findings are that subject performance is impressive across all three problem types; that subjects find diverse and novel strategies for solving each task; and that collective performance can often be strongly correlated with known algorithms.

We study a very general class of games — multi-dimensional aggregative games — which in particular generalize both anonymous games and weighted congestion games. For any such game that is also large , we solve the equilibrium selection problem in a strong sense. In particular, we give an efficient weak mediator : a mechanism which has only the power to listen to reported types and provide non-binding suggested actions, such that (a) it is an asymptotic Nash equilibrium for every player to truthfully report their type to the mediator, and then follow its suggested action; and (b) that when players do so, they end up coordinating on a particular asymptotic pure strategy Nash equilibrium of the induced complete information game. In fact, truthful reporting is an ex-post Nash equilibrium of the mediated game, so our solution applies even in settings of incomplete information, and even when player types are arbitrary or worst-case (i.e. not drawn from a common prior). We achieve this by giving an efficient differentially private algorithm for computing a Nash equilibrium in such games. The rates of convergence to equilibrium in all of our results are inverse polynomial in the number of players n . We also apply our main results to a multi-dimensional market game. Our results can be viewed as giving, for a rich class of games, a more robust version of the Revelation Principle, in that we work with weaker informational assumptions (no common prior), yet provide a stronger solution concept (ex-post Nash versus Bayes Nash equilibrium). In comparison to previous work, our main conceptual contribution is showing that weak mediators are a game theoretic object that exist in a wide variety of games – previously, they were only known to exist in traffic routing games. We also give the first weak mediator that can implement an equilibrium optimizing a linear objective function, rather than implementing a possibly worst-case Nash equilibrium.

In this paper, we introduce and examine two new models for competitive contagion in networks, a game-theoretic generalization of the viral marketing problem. In our setting, firms compete to maximize their market share in a network of consumers whose adoption decisions are stochastically determined by the choices of their neighbors. Building on the switching-selecting framework introduced by Goyal and Kearns, we first introduce a new model in which the payoff to firms comprises not only the number of vertices who adopt their (competing) technologies, but also the network connectivity among those nodes. For a general class of stochastic dynamics driving the local adoption process, we derive upper bounds on (1) the (pure strategy) Price of Anarchy (PoA), which measures the inefficiency of resource use at equilibrium, and (2) the Budget Multiplier, which captures the extent to which the network amplifies the imbalances in the firms' initial budgets. These bounds depend on the firm budgets and the maximum degree of the network, but no other structural properties. In addition, we give general conditions under which the PoA and the Budget Multiplier can be unbounded. We also introduce a model in which budgeting decisions are endogenous, rather than externally given as is typical in the viral marketing problem. In this setting, the firms are allowed to choose the number of seeds to initially infect (at a fixed cost per seed), as well as which nodes to select as seeds. In sharp contrast to the results of Goyal and Kearns, we show that for almost any local adoption dynamics, there exists a family of graphs for which the PoA and Budget Multiplier are unbounded.

We consider problems of approximate inference in which the query of interest is given by a complex formula (such as a formula in disjunctive formal form (DNF)) over a joint distribution given by a graphical model. We give a general reduction showing that (approximate) marginal inference for a class of distributions yields approximate inference for DNF queries, and extend our techniques to accommodate even more complex queries, and dense graphical models with variational inference, under certain conditions. Our results unify and generalize classical inference techniques (which are generally restricted to simple marginal queries) and approximate counting methods such as those introduced by Karp, Luby and Madras (which are generally restricted to product distributions).

We introduce and study new models for learning from contagion processes in a network. A learning algorithm is allowed to either choose or passively observe an initial set of seed infections. This seed set then induces a final set of infections resulting from the underlying stochastic contagion dynamics. Our models differ from prior work in that detailed vertex-by-vertex timestamps for the spread of the contagion are not observed. The goal of learning is to infer the unknown network structure. Our main theoretical results are efficient and provably correct algorithms for exactly learning trees. We provide empirical evidence that our algorithm performs well more generally on realistic sparse graphs.

We propose a state-based variant of the classical online learning problem of tracking the best expert. In our setting, the actions of the algorithm and experts correspond to local moves through a continuous and bounded state space. At each step, Nature chooses payoffs as a function of each player’s current position and action. Our model therefore integrates the problem of prediction with expert advice with the stateful formalisms of reinforcement learning. Traditional no-regret learning approaches no longer apply, but we propose a simple algorithm that provably achieves no-regret when the state space is any convex Euclidean region. Our algorithm combines techniques from online learning with results from the literature on pursuit-evasion games. We describe a quantitative trading application in which the convex region captures inventory risk constraints, and local moves limit market impact. Using historical market data, we show experimentally that our algorithm has a strong advantage over classic no-regret approaches.

We study the problem of implementing equilibria of complete information games in settings of incomplete information, and address this problem using "recommender mechanisms." A recommender mechanism is one that does not have the power to enforce outcomes or to force participation, rather it only has the power to suggestion outcomes on the basis of voluntary participation. We show that despite these restrictions, recommender mechanisms can implement equilibria of complete information games in settings of incomplete information under the condition that the game is large---i.e. that there are a large number of players, and any player's action affects any other's payoff by at most a small amount.

We introduce and consider the problem of effectively organizing a population of workers of varying abilities. We assume that arriving tasks for the workforce are homogeneous, and that each is characterized by an unknown and one- dimensional difficulty value x ∈ [0, 1]. Each worker i is characterized by their ability wi ∈ [0, 1], and can solve the task if and only if x ≤ wi. If a worker is unable to solve a given task it must be forwarded to a worker of greater ability. For a given set of worker abilities W and a distribution P over task difficulty, we are interested in the problem of designing efficient forwarding structures for W and P. We give efficient algorithms and structures that simultaneously (approximately) minimize both the maximum workload of any worker, and the number of workers that need to attempt a task. We identify broad conditions under which workloads diminish rapidly with the workforce size, yet only a constant number of workers attempt each task.

We show that parametric multi-armed bandit (MAB) problems with large state and action spaces can be algorithmically reduced to the supervised learning model known as Knows What It Knows or KWIK learning. We give matching impossibility results showing that the KWIK learnability requirement cannot be replaced by weaker supervised learning assumptions. We provide such results in both the standard parametric MAB setting, as well as for a new model in which the action space is finite but growing with time.

We consider a number of classical and new computational problems regarding marginal distributions, and inference in models specifying a full joint distribution. We prove general and efficient reductions between a number of these problems, which demonstrate that algorithmic progress in inference automatically yields progress for “pure data” problems. Our main technique involves formulating the problems as linear programs, and proving that the dual separation oracle for the Ellipsoid Method is provided by the target problem. This technique may be of independent interest in probabilistic inference.

For a number of years we have been conducting controlled human-subject experiments in distributed social computation in networks with only limited and local communication. These experiments cast a number of traditional computational, economic and sociological problems (including graph coloring, consensus, independent set, networked bargaining, biased voting and network formation) as games of strategic interaction in which subjects have financial incentives to collectively "compute" global solutions. I will overview and summarize the many behavioral findings from this line of experimentation. I will give particular emphasis to the novel data the experiments have generated, and the analyses this data has permitted, including quantitative studies of subject "personality" traits such as stubbornness, altruism, and patience, and whether those traits seem helpful or harmful to individual and collective performance.

We report on an extensive series of behavioral experiments in which 36 human subjects collectively build a communication network over which they must solve a competitive coordination task for monetary compensation. There is a cost for creating network links, thus creating a tension between link expenditures and collective and individual incentives. Our most striking finding is the poor performance of the subjects, especially compared to our long series of prior experiments. We demonstrate that the subjects built difficult networks for the coordination task, and compare the structural properties of the built networks to standard generative models of social networks. We also provide extensive analysis of the individual and collective behavior of the subjects, including free riding and factors influencing edge purchasing decisions.

We develop a game-theoretic framework for the study of competition between firms who have budgets to "seed" the initial adoption of their products by consumers located in a social network. The payoffs to the firms are the eventual number of adoptions of their product through a competitive stochastic diffusion process in the network. This framework yields a rich class of competitive strategies, which depend in subtle ways on the stochastic dynamics of adoption, the relative budgets of the players, and the underlying structure of the social network.

Market making refers broadly to trading strategies that seek to profit by providing liquidity to other traders, while avoiding accumulating a large net position in a stock. In this paper, we study the profitability of market making strategies in a variety of timeseries models for the evolution of a stock's price. We first provide a precise theoretical characterization of the profitability of a simple and natural market making algorithm in the absence of any stochastic assumptions on price evolution. This characterization exhibits a trade-off between the positive effect of local price fluctuations and the negative effect of net price change. We then use this general characterization to prove that market making is generally profitable on mean reverting time series --- time series with a tendency to revert to a long-term average. Mean reversion has been empirically observed in many markets, especially foreign exchange and commodities. We show that the slightest mean reversion yields positive expected profit, and also obtain stronger profit guarantees for a canonical stochastic mean reverting process, known as the Ornstein-Uhlenbeck (OU) process, as well as other stochastic mean reverting series studied in the finance literature. We also show that market making remains profitable in expectation for the OU process even if some realistic restrictions on trading frequency are placed on the market maker.

We report on a series of highly controlled human subject experiments in networked bargaining. The basic interaction between two players is the decision of how to share a mutual payment; we extend this to situate the players in a network. Various theories predict, to different levels of uniqueness, what the shares will be. We analyze our experimental results from three points of view: social efficiency, nodal differences, and human differences; and contrast our behavioral results with the theories.

We consider models for bargaining in social networks, in which players are represented by vertices and edges represent bilateral opportunities for deals between pairs of players. Each deal yields some fixed wealth if its two players can agree on how to divide it; otherwise it yields no wealth. In such a setting, Chakraborty and Kearns (WINE 2008) introduced a simple axiomatic model that stipulates an equilibrium concept in which all players are rationally satisfied with their shares. We further explore that equilibrium concept here. In particular, we give an FPTAS to compute approximate equilibrium in bipartite graphs. We also show that equilibrium is not unique, and give conditions that ensure uniqueness on regular graphs. Finally, we explore the effect of network structure on solutions given by our model, using simulation methods and statistical analysis.

We report on an extensive series of highly controlled human subject experiments in networked trade. Our point of departure is a simple and well-studied bipartite network exchange model, for which previous work has established a detailed equilibrium theory relating wealth to network topology. A notable feature of this theory is its prediction that there may be significant local variation in equilibrium wealths and prices purely as a result of structural asymmetries in the network.

We provide provably privacy-preserving versions of belief propagation, Gibbs sampling, and other local algorithms — distributed multiparty protocols in which each party or vertex learns only its final local value, and absolutely nothing else.

We present the first large-scale empirical application of reinforcement learning to the important problem of optimized trade execution in modern financial markets. Our experiments are based on 1.5 years of millisecond time-scale limit order data from NASDAQ, and demonstrate the promise of reinforcement learning methods to market microstructure problems. Our learning algorithm introduces and exploits a natural "low-impact " factorization of the state space. 1.

We consider the problem of learning accurate models from multiple sources of "nearby" data. Given distinct samples from multiple data sources and estimates of the dissimilarities between these sources, we provide a general theory of which samples should be used to learn models for each source. This theory is applicable in a broad decision-theoretic learning framework, and yields results for classification and regression generally, and for density estimation within the exponential family. A key component of our approach is the development of approximate triangle inequalities for expected loss, which may be of independent interest.

We introduce a game-theoretic model for network formation inspired by earlier stochastic models that mix localized and long-distance connectivity. In this model, players may purchase edges at distance d at a cost of d , and wish to minimize the sum of their edge purchases and their average distance to other players. In this model, we show there is a striking "small world" threshold phenomenon: in two dimensions, if < 2 then every Nash equilibrium results in a network of constant diameter (independent of network size), and if > 2 then every Nash equilibrium results in a network whose diameter grows as a root of the network size, and thus is unbounded. We contrast our results with those of Kleinberg [8] in a stochastic model, and empirically investigate the "navigability" of equilibrium networks. Our theoretical results all generalize to higher dimensions.

We study the stability properties of the dynamics of the standard continuous limit-order mechanism that is used in modern equity markets. We ask whether such mechanisms are susceptible to "buttery effects" --- the iniction of large changes on common measures of market activity by only small perturbations of the order sequence. We show that the answer depends strongly on whether the market consists of "absolute" traders (who determine their prices independent of the current order book state) or "relative" traders (who determine their prices relative to the current bid and ask). We prove that while the absolute trader model enjoys provably strong stability properties, the relative trader model is vulnerable to great instability. Our theoretical results are supported by large-scale experiments using limit order data from INET, a large electronic exchange for NASDAQ stocks.

We study a natural extension of classical evolutionary game theory to a setting in which pairwise interactions are restricted to the edges of an undirected graph or network. We generalize the definition of an evolutionary stable strategy (ESS), and show a pair of complementary results that exhibit the power of randomization in our setting: subject to degree or edge density conditions, the classical ESS of any game are preserved when the graph is chosen randomly and the mutation set is chosen adversarially, or when the graph is chosen adversarially and the mutation set is chosen randomly. We examine natural strengthenings of our generalized ESS definition, and show that similarly strong resultsnare not possible for them.

We initiate the study of learning from multiple sources of limited data, each of which may be corrupted at a different rate. We develop a com- plete theory of which data sources should be used for two fundamental problems: estimating the bias of a coin, and learning a classifier in the presence of label noise. In both cases, efficient algorithms are provided for computing the optimal subset of data.

We examine the marriage of recent probabilistic generative models for social networks with classical frameworks from mathematical eco- nomics. We are particularly interested in how the statistical structure of such networks influences global economic quantities such as price vari- ation. Our findings are a mixture of formal analysis, simulation, and experiments on an international trade data set from the United Nations.

We introduce new online models for two important aspectsof modern financial markets: Volume Weighted Average Pricetrading and limit order books. We provide an extensivestudy of competitive algorithms in these models and relatethem to earlier online algorithms for stock trading.

nspired by events ranging from 9/11 to the collapse of the accounting firm Arthur Ander- sen, economists Kunreuther and Heal [5] recently introduced an interesting game-theoretic model for problems of interdependent security (IDS), in which a large number of players must make individual investment decisions related to security — whether physical, finan- cial, medical, or some other type — but in which the ultimate safety of each participant may depend in a complex way on the actions of the entire population. A simple example is the choice of whether to install a fire sprinkler system in an individual condominium in a large building. While such a system might greatly reduce the chances of the owner’s prop- erty being destroyed by a fire originating within their own unit, it might do little or nothing to reduce the chances of damage caused by fires originating in other units (since sprinklers can usually only douse small fires early). If “enough” other unit owners have not made the investment in sprinklers, it may be not cost-effective for any individual to do so.

We examine correlated equilibria in the recently introduced formalism of graphical games, a succinct representation for multiplayer games. We establish a natural and powerful relationship between the graphical structure of a multiplayer game and a certain Markov network representing distributions over joint actions. Our first main result establishes that this Markov network succinctly represents all correlated equilibria of the graphical game up to expected payoff equivalence. Our second main result provides a general algorithm for computing correlated equilibria in a graphical game based on its associated Markov network. For a special class of graphical games that includes trees, this algorithm runs in time polynomial in the graphical game representation (which is polynomial in the number of players and exponential in the graph degree).

Over the last several years, a number of authors have developed graphtheoretic or network models for large-population game theory. In such models, each player or organization is represented by a vertex in a graph, and payoffs are determined by the actions of only those in the neighborhood of a player. This allows the detailed specification of social, organizational, biological and other types of structure in the strategic interaction of the population.In this talk, I will survey these models and the attendant algorithms for certain basic computations, including Nash and correlated equilibria. Generalizations to macroeconomic models will be discussed, as well as potential connections to social network theory and the emerging field of neuroeconomics.

We establish a new hardness result that shows that the difficulty of plan- ning in factored Markov decision processes is representational rather than just computational. More precisely, we give a fixed family of fac- tored MDPs with linear rewards whose optimal policies and value func- tions simply cannot be represented succinctly in any standard parametric form. Previous hardness results indicated that computing good policies from the MDP parameters was difficult, but left open the possibility of succinct function approximation for any fixed factored MDP. Our result applies even to policies which yield a polynomially poor approximation to the optimal value, and highlights interesting connectionswith the com- plexity class of Arthur-Merlin games.

We introduce NashProp, an iterative and local message-passing algo- rithm for computing Nash equilibria in multi-player games represented by arbitrary undirected graphs. We provide a formal analysis and exper- imental evidence demonstrating that NashProp performs well on large graphical games with many loops, often converging in just a dozen itera- tions on graphs with hundreds of nodes. NashProp generalizes the tree algorithm of (Kearns et al. 2001), and can be viewed as similar in spirit to belief propagation in probabilis- tic inference, and thus complements the recent work of (Vickrey and Koller 2002), who explored a junction tree approach. Thus, as for prob- abilistic inference, we have at least two promising general-purpose ap- proaches to equilibria computation in graphs.

Designing the dialogue policy of a spoken dialogue system involves many nontrivial choices. This paper presents a reinforcement learning approach for automatically optimizing a dialogue policy, which addresses the technical challenges in applying reinforcement learning to a working dialogue system with human users. We report on the design, construction and empirical evaluation of NJFun, an experimental spoken dialogue system that provides users with access to information about fun things to do in New Jersey. Our results show that by optimizing its performance via reinforcement learning, NJFun measurably improves system performance.

We report on the use of reinforcement learning with Cobot, a software agent residing in the well-known online community LambdaMOO. Our initial work on Cobot (Isbell et al.2000) provided him with the ability to collect social statistics and report them to users. Here we describe an application of RL allowing Cobot to take proactive actions in this complex social environment, and adapt behavior from multiple sources of human reward. After 5 months of training, and 3171 reward and punishment events from 254 different LambdaMOO users, Cobot learned nontrivial preferences for a number of users, modifing his behavior based on his current state. Here we describe LambdaMOO and the state and action spaces of Cobot, and report the statistical results of the learning experiment.

We describe a new algorithm for computing a Nash equilibrium in graphical games, a compact representation for multi-agent systems that we introduced in previous work. The algorithm is the first to compute equilibria both efficiently and exactly for a non-trivial class of graphical games.

The First Trading Agent Competition (TAC) was held from June 22nd to July 8th, 2000. TAC was designed to create a benchmark problem in the complex domain of e-marketplaces and to motivate researchers to apply unique approaches to a common task. This article describes ATTac-2000, the first-place finisher in TAC. ATTac-2000 uses a principled bidding strategy that includes several elements of adaptivity. In addition to the success at the competition, isolated empirical results are presented indicating the robustness and effectiveness of ATTac-2000's adaptive strategy.

The dialogue strategies used by a spoken dialogue system strongly influence performance and user satisfaction. An ideal system would not use a single fixed strategy, but would adapt to the circumstances at hand. To do so, a system must be able to identify dialogue properties that suggest adaptation. This paper focuses on identifying situations where the speech recognizer is performing poorly. We adopt a machine learning approach to learn rules from a dialogue corpus for identifying these situations. Our results show a significant improvement over the baseline and illustrate that both lower-level acoustic features and higher-level dialogue features can affect the performance of the learning algorithm.

In this article we prove sanity-check bounds for the error of the leave-one-out cross-validation estimate of the generalization error: that is, bounds showing that the worst-case error of this estimate is not much worse than that of the training error estimate. The name sanity check refers to the fact that although we often expect the leave-one-out estimate to perform considerably better than the training error estimate, we are here only seeking assurance that its performance will not be considerably worse. Perhaps surprisingly, such assurance has been given only for limited cases in the prior literature on cross-validation. Any nontrivial bound on the error of leave-one-out must rely on some notion of algorithmic stability. Previous bounds relied on the rather strong notion of hypothesis stability, whose application was primarily limited to nearest-neighbor and other local algorithms. Here we introduce the new and weaker notion of error stability and apply it to obtain sanity-check bounds for leave-one-out for other classes of learning algorithms, including training error minimization procedures and Bayesian algorithms. We also provide lower bounds demonstrating the necessity of some form of error stability for proving bounds on the error of the leave-one-out estimate, and the fact that for training error minimization algorithms, in the worst case such bounds must still depend on the Vapnik-Chervonenkis dimension of the hypothesis class.

We study the problem of determining, for a class of functions H, whether an unknown target function f is contained in H or is "far" from any function in H. Thus, in contrast to problems of learning, where we must construct a good approximation to f in H on the basis of sample data, in problems of testing we are only required to determine the existence of a good approximation.Our main results demonstrate that, over the domain [0, lld for constant d, the number of examples required for testing grows only as O(S~/~+' ) (where 6 is any small constant), for both decision trees of size s and a special class of neural networks with s hidden units.This is in contrast to the Q(s) examples required for learning these same classes.Our tests are based on combinatorial constructions demonstrating that these classes can be approximated by small classes of coarse partitions of space, and rely on repeated application of the well-known Birthday Paradox.Permission to make digital or hard copies of all or part ofthis work for pt~sonal or ch..sroot~ USC is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies hear this notice and the full citation on the first page.To copy

In the last decade or so, many of the central problems of “classical” artificial intelligence — such as learning, planning and logical inference — have been reformulated in statistical or probabilistic frameworks. The benefits of this trend include the adoption of a common set of mathematical tools for the various AI subdisciplines, increased attention on algorithmic issues, and an emphasis on approximate methods for some notoriously hard exact AI problems. The trend also provides excellent opportunities for researchers from theoretical computer science to contribute to and influence AI. In this tutorial, I will survey these probabilistic frameworks and the basic computational problems posed in several well-developed areas of AI. I will describe some of the algorithms developed for these problems, overview what is formally known about them (and also what is suspected but not proven), and try to give a flavor of the mathematical techniques involved. The tutorial will be self-contained, designed to be accessible to anyone in the theory community, with an emphasis on the interesting open problems. Likely topics include:

In this paper, we study the problem of learning in the presence of classification noise in the probabilistic learning model of Valiant and its variants. In order to identify the class of “robust” learning algorithms in the most general way, we formalize a new but related model of learning from statistical queries . Intuitively, in this model a learning algorithm is forbidden to examine individual examples of the unknown target function, but is given acess to an oracle providing estimates of probabilities over the sample space of random examples. One of our main results shows that any class of functions learnable from statistical queries is in fact learnable with classification noise in Valiant's model, with a noise rate approaching the information-theoretic barrier of 1/2. We then demonstrate the generality of the statistical query model, showing that practically every class learnable in Valiant's model and its variants can also be learned in the new model (and thus can be learned in the presence of noise). A notable exception to this statement is the class of parity functions, which we prove is not learnable from statistical queries, and for which no noise-tolerant algorithm is known.

: In this paper we prove sanity-check bounds for the error of the leave-one-out crossvalidation estimate of the generalization error: that is, bounds showing that the worst-case error of this estimate is not much worse than that of the training error estimate. The name sanity-check refers to the fact that although we often expect the leave-one-out estimate to perform considerably better than the training error estimate, we are here only seeking assurance that its performance will not be considerably worse. Perhaps surprisingly, such assurance has been given only for rather limited cases in the prior literature on cross-validation. Any nontrivial bound on the error of leave-one-out must rely on some notion of algorithmic stability. Previous bounds relied on the rather strong notion of hypothesis stability, whose application was primarily limited to nearest-neighbor and other local algorithms. Here we introduce the new and weaker notion of error stability, and apply it to obtain sanity-c...

This paper describes new and efficient algorithms for learning deterministic finite automata. Our approach is primarily distinguished by two features: (1) the adoption of an average-case setting to model the “typical” labeling of a finite automaton, while retaining a worst-case model for the underlying graph of the automaton, along with (2) a learning model in which the learner is not provided with the means to experiment with the machine, but rather must learn solely by observing the automaton's output behavior on a random input sequence. The main contribution of this paper is in presenting the first efficient algorithms for learning non-trivial classes of automata in an entirely passive learning model. We adopt an on-line learning model in which the learner is asked to predict the output of the next state, given the next symbol of the random input sequence; the goal of the learner is to make as few prediction mistakes as possible. Assuming the learner has a means of resetting the target machine to a fixed start state, we first present an efficient algorithm that makes an expected polynomial number of mistakes in this model. Next, we show how this first algorithm can be used as a subroutine by a second algorithm that also makes a polynomial number of mistakes even in the absence of a reset. Along the way, we prove a number of combinatorial results for randomly labeled automata. We also show that the labeling of the states and the bits of the input sequence need not be truly random, but merely semi - random . Finally, we discuss an extension of our results to a model in which automata are used to represent distributions over binary strings.

We give a theoretical and experimental analysis of the generalization error of cross validation using two natural measures of the problem under consideration. The approximation rate measures the accuracy to which the target function can be ideally approximated as a function of the number of parameters, and thus captures the complexity of the target function with respect to the hypothesis model. The estimation rate measures the deviation between the training and generalization errors as a function of the number of parameters, and thus captures the extent to which the hypothesis model suffers from overfitting. Using these two measures, we give a rigorous and general bound on the error of the simplest form of cross validation. The bound clearly shows the dangers of making γ —the fraction of data saved for testing—too large or too small. By optimizing the bound with respect to γ, we then argue that the following qualitative properties of cross-validation behavior should be quite robust to significant changes in the underlying model selection problem: When the target function complexity is small compared to the sample size, the performance of cross validation is relatively insensitive to the choice of γ. The importance of choosing γ optimally increases, and the optimal value for γ decreases, as the target function becomes more complex relative to the sample size. There is nevertheless a single fixed value for γ that works nearly optimally for a wide range of target function complexity.

We analyze the performance of top-down algorithms for decision tree learning, such as those employed by the widely used C4.5 and CART software packages.Our main result is a proof that such algorithms are boosling algorithms.By this we mean that if the functions that label the internal nodes of the decision tree can weakly approximate the unknown target function, then the top-down algorithms we study will amplify this weak advantage to build a tree achieving any desired level of accuracy.The bounds we obtain for this amplification show an interesting dependence on the splitting criterion used by the top-down algorithm.More precisely, if the functions used to label the internal nodes have error 1/2 -v as approximations to the target function, then for the splitting criteria used by CART and C4.5, trees of size (1/e) o(U7'~) and (1/e) '(1%( li')172) (respectively) suffice to drive the error below e.Thus (for example), small constant advantage over random guessing is amplified to constant error with trees of constant size.For a new splitting criterion suggested by our analysis, the much stronger bound of(1/6)0(1172) (which is polynomial in 1/c) is obtained.The differing bounds have a natural explanation in terms of concavity properties of the splitting criterion.The primary contribution of this work is in proving that some popular and empirically successful heuristics that are based on first principles meet the criteria of an independently motivated theoretical model.

In the model selection problem... The goal of this paper is to provide such a comparison, and more importantly, to describe the general conclusions to which it has led. Relying on evidence that is approximately equally divided between controlled experimental results and related formal analysis, we compare three well-known model selection algorithms and attempt to identify their relative and absolute strengths and weaknesses, and we provide some general methods for analyzing the behavior and performance of model selection algorithms. Our hope is that these results will help the informed practitioner make an educated choice of model selection algorithm (perhaps based in part on some known properties of the model selection problem confronting them). The summary of the paper follows. In Section 2, we provide a formalization of the model selection problem. In this formalization, we isolate the problem of choosing the appropriate complexity...

We examine the problem of learning to play various games optimally against resource-bounded adversaries, with an explicit emphasis on the computational efficiency of the learning algorithm. We are especially interested in providing efficient algorithms for games other than penny-matching (in which payoff is received for matching the adversary's action in the current round), and for adversaries other than the classically studied finite automata. In particular, we examine games and adversaries for which the learning algorithm's past actions may strongly affect the adversary's future willingness to "cooperate" (that is, permit high payoff), and therefore require carefully planned actions on the part of the learning algorithm. For example, in the game we call contract, both sides play O or 1 on each round, but our side receives payoff only if we play 1 in synchrony with the adversary; unlike penny-matching, playing O in synchrony with the adversary pays nothing. The name of the game is derived from the example of signing a contract, which becomes valid only if both parties sign (play 1).

In this paper we introduce and investigate a mathematically rigorous theory of learning curves that is based on ideas from statistical mechanics. The advantage of our theory over the well-established Vapnik-Chervonenkis theory is that our bounds can be considerably tighter in many cases, and are also more reflective of the true behavior (functional form) of learning curves. This behavior can often exhibit dramatic properties such as phase transitions, as well as power law asymptotics not explained by the VC theory. The disadvantages of our theory are that its application requires knowledge of the input distribution, and it is limited so far to finite cardinality function classes. We illustrate our results with many concrete examples of learning curve bounds derived from our theory.

We present new results, both positive and negative, on the well-studied problem of learning disjunctive normal form (DNF) expressions.We first prove that an algorithm due to Kushilevitz and Mansour ysis of a finite class of boolean functions 011 the hypercube.1

We introduce and investigate a new model of learning probability distributions from independent draws. Our model is inspired by the popular Probably Approximately Correct (PAC) model for learning boolean functions from labeled

Efficient distribution-free learning of Boolean formulas from positive and negative examples is considered. It is shown that classes of formulas that are efficiently learnable from only positive examples or only negative examples have certain closure properties. A new substitution technique is used to show that in the distribution-free case learning DNF (disjunctive normal form formulas) is no harder than learning monotone DNF. We prove that monomials cannot be efficiently learned from negative examples alone, even if the negative examples are uniformly distributed. It is also shown that, if the examples are drawn from uniform distributions, then the class of DNF in which each variable occurs at most once is efficiently weakly learnable (i.e., individual examples are correctly classified with a probability larger than 1/2 + 1/ p , where p is a polynomial in the relevant parameters of the learning problem). We then show an equivalence between the notion of weak learning and the notion of group learning , where a group of examples of polynomial size, either all positive or all negative, must be correctly classified with high probability.

In this paper, we prove the intractability of learning several classes of Boolean functions in the distribution-free model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent , in that they hold regardless of the syntactic form in which the learner chooses to represent its hypotheses. Our methods reduce the problems of cracking a number of well-known public-key cryptosystems to the learning problems. We prove that a polynomial-time learning algorithm for Boolean formulae, deterministic finite automata or constant-depth threshold circuits would have dramatic consequences for cryptography and number theory. In particular, such an algorithm could be used to break the RSA cryptosystem, factor Blum integers (composite numbers equivalent to 3 modulo 4), and detect quadratic residues. The results hold even if the learning algorithm is only required to obtain a slight advantage in prediction over random guessing. The techniques used demonstrate an interesting duality between learning and cryptography. We also apply our results to obtain strong intractability results for approximating a generalization of graph coloring.

Abstract. We introduce a new formal model in which a learning algorithm must combine a collection of potentially poor but statistically independent hypothesis functions in order to approximate an unknown target function arbitrarily well. Our motivation includes the question of how tomake optimal use of multiple independent runs of a mediocre learning algorithm, as well as settings in which the many hypotheses are obtained by a distributed population of identical learning agents. Keywords: 1.

Article Efficient learning of typical finite automata from random walks Share on Authors: Yoav Freund View Profile , Michael Kearns View Profile , Dana Ron View Profile , Ronitt Rubinfeld View Profile , Robert E. Schapire View Profile , Linda Sellie View Profile Authors Info & Claims STOC '93: Proceedings of the twenty-fifth annual ACM symposium on Theory of ComputingJune 1993 Pages 315–324https://doi.org/10.1145/167088.167191Published:01 June 1993 28citation470DownloadsMetricsTotal Citations28Total Downloads470Last 12 Months3Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Article Free Access Share on Efficient noise-tolerant learning from statistical queries Author: Michael Kearns View Profile Authors Info & Claims STOC '93: Proceedings of the twenty-fifth annual ACM symposium on Theory of ComputingJune 1993Pages 392–401https://doi.org/10.1145/167088.167200Published:01 June 1993Publication History 139citation502DownloadsMetricsTotal Citations139Total Downloads502Last 12 Months37Last 6 weeks6 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

In this paper a new technique is described for exactly identifying certain classes of read-once Boolean formulas. The method is based on sampling the input-output behavior of the target formula on a probability distribution that is determined by the fixed point of the formula’s amplification function (defined as the probability that a one is output by the formula when each input bit is one independently with probability p). By performing various statistical tests on easily sampled variants of the fixed-point distribution, it is possible to efficiently infer all structural information about any logarithmic-depth formula (with high probability). Results are applied to prove the existence of short universal identification sequences for large classes of formulas. Also described are extensions of these algorithms to handle high rates of noise, and to learn formulas of unbounded depth in Valiant’s model with respect to specific distributions.

In this paper an extension of the distribution-free model of learning introduced by Valiant [Comm. ACM, 27(1984), pp. 1134–1142] that allows the presence of malicious errors in the examples given to a learning algorithm is studied. Such errors are generated by an adversary with unbounded computational power and access to the entire history of the learning algorithm’s computation. Thus, a worst-case model of errors is studied. The results of this research include general methods for bounding the rate of error tolerable by any learning algorithm, efficient algorithms tolerating nontrivial rates of malicious errors, and equivalences between problems of learning with errors and standard combinatorial optimization problems.

In this paper we initiate an investigation of generalizations of the Probably Approximately Correct (PAC) learning model that attempt to significantly weaken the target function assumptions. The ultimate goal in this direction is informally termed agnostic learning, in which we make virtually no assumptions on the target function. The name derives from the fact that as designers of learning algorithms, we give up the belief that Nature (as represented by the target function) has a simple or succinct explanation.

A technique for exactly identifying certain classes of read-once Boolean formulas is introduced. The method is based on sampling the input-output behavior of the target formula on a probability distribution which is determined by the fixed point of the formula's amplification function (defined as the probability that a 1 is output by the formula when each input bit is 1 independently with probability p). By performing various statistical tests on easily sampled variants of the fixed-point distribution, it is possible to infer efficiently all structural information about any logarithmic-depth target family (with high probability). The results are used to prove the existence of short universal identification sequences for large classes of formulas. Extensions of the algorithms to handle high rates of noise and to learn formulas of unbounded depth in L.G. Valiant's (1984) model with respect to specific distributions are described.>

A model of machine learning in which the concept to be learned may exhibit uncertain or probabilistic behavior is investigated. Such probabilistic concepts (or p-concepts) may arise in situations such as weather prediction, where the measured variables and their accuracy are insufficient to determine the outcome with certainty. It is required that learning algorithms be both efficient and general in the sense that they perform well for a wide class of p-concepts and for any distribution over the domain. Many efficient algorithms for learning natural classes of p-concepts are given, and an underlying theory of learning p-concepts is developed in detail.>

Article Free Access Share on Crytographic limitations on learning Boolean formulae and finite automata Authors: M. Kearns Harvard University Harvard UniversityView Profile , L. G. Valiant Harvard University Harvard UniversityView Profile Authors Info & Claims STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computingFebruary 1989 Pages 433–444https://doi.org/10.1145/73007.73049Published:01 February 1989Publication History 144citation661DownloadsMetricsTotal Citations144Total Downloads661Last 12 Months75Last 6 weeks6 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Article Free Access Share on Learning in the presence of malicious errors Authors: Michael Kearns Harvard University Harvard UniversityView Profile , Ming Li Harvard University Harvard UniversityView Profile Authors Info & Claims STOC '88: Proceedings of the twentieth annual ACM symposium on Theory of computingJanuary 1988 Pages 267–280https://doi.org/10.1145/62212.62238Online:01 January 1988Publication History 62citation618DownloadsMetricsTotal Citations62Total Downloads618Last 12 Months31Last 6 weeks9 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Article Free Access Share on On the learnability of Boolean formulae Authors: M. Kearns Harvard University Harvard UniversityView Profile , M. Li Harvard University Harvard UniversityView Profile , L. Pitt University of Illinois University of IllinoisView Profile , L. Valiant Harvard University Harvard UniversityView Profile Authors Info & Claims STOC '87: Proceedings of the nineteenth annual ACM symposium on Theory of computingJanuary 1987 Pages 285–295https://doi.org/10.1145/28395.28426Online:01 January 1987Publication History 157citation769DownloadsMetricsTotal Citations157Total Downloads769Last 12 Months62Last 6 weeks5 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my Alerts New Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF