Evaluating chatbot fairness is crucial given their rapid proliferation, yet typical chatbot tasks (e.g., resume writing, entertainment) diverge from the institutional decision-making tasks (e.g., resume screening) which have traditionally been central to discussion of algorithmic fairness. The open-ended nature and diverse use-cases of chatbots necessitate novel methods for bias assessment. This paper addresses these challenges by introducing a scalable counterfactual approach to evaluate "first-person fairness," meaning fairness toward chatbot users based on demographic characteristics. Our method employs a Language Model as a Research Assistant (LMRA) to yield quantitative measures of harmful stereotypes and qualitative analyses of demographic differences in chatbot responses. We apply this approach to assess biases in six of our language models across millions of interactions, covering sixty-six tasks in nine domains and spanning two genders and four races. Independent human annotations corroborate the LMRA-generated bias evaluations. This study represents the first large-scale fairness evaluation based on real-world chat data. We highlight that post-training reinforcement learning techniques significantly mitigate these biases. This evaluation provides a practical methodology for ongoing bias monitoring and mitigation.

Multicalibration is a notion of fairness for predictors that requires them to provide calibrated predictions across a large set of protected groups. Multicalibration is known to be a distinct goal than loss minimization, even for simple predictors such as linear functions. In this work, we consider the setting where the protected groups can be represented by neural networks of size $k$, and the predictors are neural networks of size $n > k$. We show that minimizing the squared loss over all neural nets of size $n$ implies multicalibration for all but a bounded number of unlucky values of $n$. We also give evidence that our bound on the number of unlucky values is tight, given our proof technique. Previously, results of the flavor that loss minimization yields multicalibration were known only for predictors that were near the ground truth, hence were rather limited in applicability. Unlike these, our results rely on the expressivity of neural nets and utilize the representation of the predictor.

Recent language models generate false but plausible-sounding text with surprising frequency. Such “hallucinations” are an obstacle to the usability of language-based AI systems and can harm people who rely upon their outputs. This work shows that there is an inherent statistical lower-bound on the rate that pretrained language models hallucinate certain types of facts, having nothing to do with the transformer LM architecture or data quality. For “arbitrary” facts whose veracity cannot be determined from the training data, we show that hallucinations must occur at a certain rate for language models that satisfy a statistical calibration condition appropriate for generative language models. Specifically, if the maximum probability of any fact is bounded, we show that the probability of generating a hallucination is close to the fraction of facts that occur exactly once in the training data (a “Good-Turing” estimate), even assuming ideal training data without errors. One conclusion is that models pretrained to be sufficiently good predictors (i.e., calibrated) may require post-training to mitigate hallucinations on the type of arbitrary facts that tend to appear once in the training set. However, our analysis also suggests that there is no statistical reason that pretraining will lead to hallucination on facts that tend to appear more than once in the training data (like references to publications such as articles and books, whose hallucinations have been particularly notable and problematic) or on systematic facts (like arithmetic calculations). Therefore, different architectures and learning algorithms may mitigate these latter types of hallucinations.

We introduce a new type of test, called a Turing Experiment (TE), for evaluating to what extent a given language model, such as GPT models, can simulate different aspects of human behavior. A TE can also reveal consistent distortions in a language model’s simulation of a specific human behavior. Unlike the Turing Test, which involves simulating a single arbitrary individual, a TE requires simulating a representative sample of participants in human subject research. We carry out TEs that attempt to replicate well-established findings from prior studies. We design a methodology for simulating TEs and illustrate its use to compare how well different language models are able to reproduce classic economic, psycholinguistic, and social psychology experiments: Ultimatum Game, Garden Path Sentences, Milgram Shock Experiment, and Wisdom of Crowds. In the first three TEs, the existing findings were replicated using recent models, while the last TE reveals a “hyper-accuracy distortion” present in some language models (including ChatGPT and GPT-4), which could affect downstream applications in education and the arts.

Neural networks are capable of translating between languages—in some cases even between two languages where there is little or no access to parallel translations, in what is known as Unsupervised Machine Translation (UMT). Given this progress, it is intriguing to ask whether machine learning tools can ultimately enable understanding animal communication, particularly that of highly intelligent animals. We propose a theoretical framework for analyzing UMT when no parallel translations are available and when it cannot be assumed that the source and target corpora address related subject domains or posses similar linguistic structure. We exemplify this theory with two stylized models of language, for which our framework provides bounds on necessary sample complexity; the bounds are formally proven and experimentally verified on synthetic data. These bounds show that the error rates are inversely related to the language complexity and amount of common ground. This suggests that unsupervised translation of animal communication may be feasible if the communication system is sufficiently complex.

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.

This work offers a novel theoretical perspective on why, despite numerous attempts, adversarial approaches to generative modeling (e.g., GANs) have not been as successful for certain generation tasks, particularly sequential tasks such as Natural Language Generation, as they have in others, such as Computer Vision. In particular, on sequential data such as text, maximum-likelihood approaches are significantly more utilized than GANs. We show that, while it may seem that maximizing likelihood is inherently different than minimizing distinguishability, this distinction is largely an artifact of the limited representational capacity of the model family, for a wide class of adversarial objectives. We give a theoretical model in which minimizing KL-divergence (i.e., maximizing likelihood) is a more efficient approach to effectively minimizing the same distinguishability criteria that adversarial models seek to optimize. Reductions show that minimizing distinguishability can be seen as simply boosting likelihood for certain families of models including n-gram models and neural networks with a softmax output layer. To achieve a full polynomial-time reduction, a novel next-token distinguishability model is considered. Some preliminary empirical evidence is also provided to substantiate our theoretical analyses.

Neural networks (NNs) struggle to efficiently solve certain problems, such as learning parities, even when there are simple learning algorithms for those problems. Can NNs discover learning algorithms on their own? We exhibit a NN architecture that, in polynomial time, learns as well as any efficient learning algorithm describable by a constant-sized program. For example, on parity problems, the NN learns as well as Gaussian elimination, an efficient algorithm that can be succinctly described. Our architecture combines both recurrent weight sharing between layers and convolutional weight sharing to reduce the number of parameters down to a constant, even though the network itself may have trillions of nodes. While in practice the constants in our analysis are too large to be directly meaningful, our work suggests that the synergy of Recurrent and Convolutional NNs (RCNNs) may be more natural and powerful than either alone, particularly for concisely parameterizing discrete algorithms.

Domain generalization is the problem of machine learning when the training data and the test data come from different “domains” (data distributions). We propose an elementary theoretical model of the domain generalization problem, introducing the concept of a meta-distribution over domains. In our model, the training data available to a learning algorithm consist of multiple datasets, each from a single domain, drawn in turn from the meta-distribution. We show that our model can capture a rich range of learning phenomena specific to domain generalization for three different settings: learning with Massart noise, learning decision trees, and feature selection. We demonstrate approaches that leverage domain generalization to reduce computational or data requirements in each of these settings. Experiments demonstrate that our feature selection algorithm indeed ignores spurious correlations and improves generalization.

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.

We present a transductive learning algorithm that takes as input training examples from a distribution P and arbitrary (unlabeled) test examples, possibly chosen by an adversary. This is unlike prior work that assumes that test examples are small perturbations of P. Our algorithm outputs a selective classifier, which abstains from predicting on some examples. By considering selective transductive learning, we give the first nontrivial guarantees for learning classes of bounded VC dimension with arbitrary train and test distributions—no prior guarantees were known even for simple classes of functions such as intervals on the line. In particular, for any function in a class C of bounded VC dimension, we guarantee a low test error rate and a low rejection rate with respect to P. Our algorithm is efficient given an Empirical Risk Minimizer (ERM) for C. Our guarantees hold even for test examples chosen by an unbounded white-box adversary. We also give guarantees for generalization, agnostic, and unsupervised settings.

In contexts such as college admissions, hiring, and image search, decision-makers often aspire to formulate selection criteria that yield both high-quality and diverse results. However, simultaneously optimizing for quality and diversity can be challenging, especially when the decision-maker does not know the true quality of any criterion and instead must rely on heuristics and intuition. We introduce an algorithmic framework that takes as input a user's selection criterion, which may yield high-quality but homogeneous results. Using an application-specific notion of substitutability, our algorithms suggest similar criteria with more diverse results, in the spirit of statistical or demographic parity. For instance, given the image search query "chairman", it suggests alternative queries which are similar but more gender-diverse, such as "chairperson". In the context of college admissions, we apply our algorithm to a dataset of students' applications and rediscover Texas's "top 10% rule": the input criterion is an ACT score cutoff, and the output is a class rank cutoff, automatically accepting the students in the top decile of their graduating class. Historically, this policy has been effective in admitting students who perform well in college and come from diverse backgrounds. We complement our empirical analysis with learning-theoretic guarantees for estimating the true diversity of any criterion based on historical data.

This paper presents an algorithm for enumerating biases in word embeddings. The algorithm exposes a large number of offensive associations related to sensitive features such as race and gender on publicly available embeddings, including a supposedly "debiased" embedding. These biases are concerning in light of the widespread use of word embeddings. The associations are identified by geometric patterns in word embeddings that run parallel between people's names and common lower-case tokens. The algorithm is highly unsupervised: it does not even require the sensitive features to be pre-specified. This is desirable because: (a) many forms of discrimination?such as racial discrimination-are linked to social constructs that may vary depending on the context, rather than to categories with fixed definitions; and (b) it makes it easier to identify biases against intersectional groups, which depend on combinations of sensitive features. The inputs to our algorithm are a list of target tokens, e.g. names, and a word embedding. It outputs a number of Word Embedding Association Tests (WEATs) that capture various biases present in the data. We illustrate the utility of our approach on publicly available word embeddings and lists of names, and evaluate its output using crowdsourcing. We also show how removing names may not remove potential proxy bias.

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

Algorithms often must solve sequences of closely related problems. If the algorithm runs a standard procedure with worst-case runtime guarantees on each instance, it will fail to take advantage of valuable structure shared across the problem instances. When a commuter drives from work to home, for example, there are typically only a handful of routes that will ever be the shortest path. A naïve algorithm that does not exploit this common structure may spend most of its time checking roads that will never be in the shortest path. More generally, we can often ignore large swaths of the search space that will likely never contain an optimal solution. We present an algorithm that learns to maximally prune the search space on repeated computations, thereby reducing runtime while provably outputting the correct solution each period with high probability. Our algorithm employs a simple explore-exploit technique resembling those used in online algorithms, though our setting is quite different. We prove that, with respect to our model of pruning search spaces, our approach is optimal up to constant factors. Finally, we illustrate the applicability of our model and algorithm to three classic problems: shortest-path routing, string search, and linear programming. We present experiments confirming that our simple algorithm is effective at significantly reducing the runtime of solving repeated computations.

While humor is often thought to be beyond the reach of Natural Language Processing, we show that several aspects of single-word humor correlate with simple linear directions in Word Embeddings. In particular: (a) the word vectors capture multiple aspects discussed in humor theories from various disciplines; (b) each individual’s sense of humor can be represented by a vector, which can predict differences in people’s senses of humor on new, unrated, words; and (c) upon clustering humor ratings of multiple demographic groups, different humor preferences emerge across the different groups. Humor ratings are taken from the work of Engelthaler and Hills (2017) as well as from an original crowdsourcing study of 120,000 words. Our dataset further includes annotations for the theoretically-motivated humor features we identify.

Recently proposed models which learn to write computer programs from data use either input/output examples or rich execution traces. Instead, we argue that a novel alternative is to use a glass-box scoring function, given as a program itself that can be directly inspected. Glass-box optimization covers a wide range of problems, from computing the greatest common divisor of two integers, to learning-to-learn problems. In this paper, we present an intelligent search system which learns, given the partial program and the glass-box problem, the probabilities over the space of programs. We empirically demonstrate that our informed search procedure leads to significant improvements compared to brute-force program search, both in terms of accuracy and time. For our experiments we use rich context free grammars inspired by number theory, text processing, and algebra. Our results show that (i) running our framework iteratively can considerably increase the number of problems solved, (ii) our framework can improve itself even in domain agnostic scenarios, and (iii) it can solve problems that would be otherwise too slow to solve with brute-force search.

Most systems and learning algorithms optimize average performance or average loss – one reason being computational complexity. However, many objectives of practical interest are more complex than simply average loss. This arises, for example, when balancing performance or loss with fairness across people. We prove that, from a computational perspective, optimizing arbitrary objectives that take into account performance over a small number of groups is not significantly harder to optimize than average performance. Our main result is a polynomial-time reduction that uses a linear optimizer to optimize an arbitrary (Lipschitz continuous) function of performance over a (constant) number of possibly-overlapping groups. This includes fairness objectives over small numbers of groups, and we further point out that other existing notions of fairness such as individual fairness can be cast as convex optimization and hence more standard convex techniques can be used. Beyond learning, our approach applies to multi-objective optimization, more generally.

A generative model may generate utter nonsense when it is fit to maximize the likelihood of observed data. This happens due to “model error,” i.e., when the true data generating distribution does not fit within the class of generative models being learned. To address this, we propose a model of active distribution learning using a binary invalidity oracle that identifies some examples as clearly invalid, together with random positive examples sampled from the true distribution. The goal is to maximize the likelihood of the positive examples subject to the constraint of (almost) never generating examples labeled invalid by the oracle. Guarantees are agnostic compared to a class of probability distributions. We first show that proper learning may require exponentially many queries to the invalidity oracle. We then give an improper distribution learning algorithm that uses only polynomially many queries.

Reusing passwords across multiple websites is a common practice that compromises security. Recently, Blum and Vempala have proposed password strategies to help people calculate, in their heads, passwords for different sites without dependence on third-party tools or external devices. Thus far, the security and efficiency of these "mental algorithms" has been analyzed only theoretically. But are such methods usable? We present the first usability study of humanly computable password strategies, involving a learning phase (to learn a password strategy), then a rehearsal phase (to login to a few websites), and multiple follow-up tests. In our user study, with training, participants were able to calculate a deterministic eight-character password for an arbitrary new website in under 20 seconds.

The emergence of personal computing devices offers both a challenge and opportunity for displaying text: small screens can be hard to read, but also support higher resolution. To fit content on a small screen, text must be small. This small text size can make computing devices unusable, in particular to low-vision users, whose vision is not correctable with glasses. Usability is also decreased for sighted users straining to read the small letters, especially without glasses at hand. We propose animated scripts called livefonts for displaying English with improved legibility for all users. Because paper does not support animation, traditional text is static. However, modern screens support animation, and livefonts capitalize on this capability. We evaluate our livefont variations' legibility through a controlled lab study with low-vision and sighted participants, and find our animated scripts to be legible across vision types at approximately half the size (area) of traditional letters, while previous smartfonts (static alternate scripts) did not show a significant legibility advantage for low-vision users. We evaluate the learnability of our livefont with low-vision and sighted participants, and find it to be comparably learnable to static smartfonts after two thousand practice sentences.

The blind application of machine learning runs the risk of amplifying biases present in data. Such a danger is facing us with word embedding, a popular framework to represent text data as vectors which has been used in many machine learning and natural language processing tasks. We show that even word embeddings trained on Google News articles exhibit female/male gender stereotypes to a disturbing extent. This raises concerns because their widespread use, as we describe, often tends to amplify these biases. Geometrically, gender bias is first shown to be captured by a direction in the word embedding. Second, gender neutral words are shown to be linearly separable from gender definition words in the word embedding. Using these properties, we provide a methodology for modifying an embedding to remove gender stereotypes, such as the association between the words receptionist and female, while maintaining desired associations such as between the words queen and female. Using crowd-worker evaluation as well as standard benchmarks, we empirically demonstrate that our algorithms significantly reduce gender bias in embeddings while preserving the its useful properties such as the ability to cluster related concepts and to solve analogy tasks. The resulting embeddings can be used in applications without amplifying gender bias.

A system capable of suggesting multi-word phrases while someone is writing could supply ideas about content and phrasing and allow those ideas to be inserted efficiently. Meanwhile, statistical language modeling has provided various approaches to predicting phrases that users type. We introduce a simple extension to the familiar mobile keyboard suggestion interface that presents phrase suggestions that can be accepted by a repeated-tap gesture. In an extended composition task, we found that phrases were interpreted as suggestions that affected the content of what participants wrote more than conventional single-word suggestions, which were interpreted as predictions. We highlight a design challenge: how can a phrase suggestion system make valuable suggestions rather than just accurate predictions'

As small displays on devices like smartwatches become increasingly common, many people have difficulty reading the text on these displays. Vision conditions like presbyopia that result in blurry near vision make reading small text particularly hard. We design multiple different scripts for displaying English text, legible at small sizes even when blurry, for small screens such as smartphones and smartwatches. These "smartfonts" redesign visual character presentations to improve the reading experience. Like cursive, Grade 1 Braille, and ordinary fonts, they preserve orthography and spelling. They have the potential to enable people to read more text comfortably on small screens, e.g., without reading glasses. To simulate presbyopia, we blur images and evaluate their legibility using paid crowdsourcing. We also evaluate the difficulty of learning to read smartfonts and observe a learnability/legibility trade-off. Our most learnable smartfont can be read at roughly half the speed of Latin after two thousand practice sentences. It is also legible smaller than half the size of traditional Latin (i.e. "English") when blurry.

We introduce an unsupervised approach to efficiently discover the underlying features in a data set via crowdsourcing. Our queries ask crowd members to articulate a feature common to two out of three displayed examples. In addition, we ask the crowd to provide binary labels for these discovered features on the remaining examples. The triples are chosen adaptively based on the labels of the previously discovered features on the data set. This approach is motivated by a formal framework of feature elicitation that we introduce and analyze in this paper. In two natural models of features, hierarchical and independent, we show that a simple adaptive algorithm recovers all features with less labor than any nonadaptive algorithm. The savings are as a result of automatically avoiding the elicitation of redundant features or synonyms. Experimental results validate the theoretical findings and the usefulness of this approach.

Personalization is a way for computers to support people’s diverse interests and needs by providing content tailored to the individual. While strides have been made in algorithmic approaches to personalization, most require access to a significant amount of data. However, even when data is limited online crowds can be used to infer an individual’s personal preferences. Aided by the diversity of tastes among online crowds and their ability to understand others, we show that crowdsourcing is an effective on-demand tool for personalization. Unlike typical crowdsourcing approaches that seek a ground truth, we present and evaluate two crowdsourcing approaches designed to capture personal preferences. The first, taste-matching, identifies workers with similar taste to the requester and uses their taste to infer the requester’s taste. The second, taste-grokking, asks workers to explicitly predict the requester’s taste based on training examples. These techniques are evaluated on two subjective tasks, personalized image recommendation and tailored textual summaries. Taste-matching and taste-grokking both show improvement over the use of generic workers, and have different benefits and drawbacks depending on the complexity of the task and the variability of the taste space.

Learning programs is a timely and interesting challenge. In Programming by Example (PBE), a system attempts to infer a program from input and output examples alone, by searching for a composition of some set of base functions. We show how machine learning can be used to speed up this seemingly hopeless search problem, by learning weights that relate textual features describing the provided input-output examples to plausible sub-components of a program. This generic learning framework lets us address problems beyond the scope of earlier PBE systems. Experiments on a prototype implementation show that learning improves search and ranking on a variety of text processing tasks found on help forums.

When dealing with subjective, noisy, or otherwise nebulous features, the “wisdom of crowds” suggests that one may benefit from multiple judgments of the same feature on the same object. We give theoretically-motivated ""feature multi-selection"" algorithms that choose, among a large set of candidate features, not only which features to judge but how many times to judge each one. We demonstrate the effectiveness of this approach for linear regression on a crowdsourced learning task of predicting people’s height and weight from photos, using features such as ""gender"" and ""estimated weight"" as well as culturally fraught ones such as ""attractive"".

Text processing, tedious and error-prone even for programmers, remains one of the most alluring targets of Programming by Example. An examination of real-world text processing tasks found on help forums reveals that many such tasks, beyond simple string manipulation, involve latent hierarchical structures.

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 revisit classic algorithmic search and optimization problems from the perspective of competition. Rather than a single optimizer minimizing expected cost, we consider a zero-sum game in which a search problem is presented to two players, whose only goal is to outperform the opponent. Such games are typically exponentially large zero-sum games, but they often have a rich structure. We provide general techniques by which such structure can be leveraged to find minmax-optimal and approximate minmax-optimal strategies. We give examples of ranking, hiring, compression, and binary search duels, among others. We give bounds on how often one can beat the classic optimization algorithms in such duels.

In this paper, we consider the capacity C of the binary deletion channel for the limiting case where the deletion probability p goes to 0. It is known that for any p <; 1/2, the capacity satisfies C ≥ 1-H(p), where H is the standard binary entropy. We show that this lower bound is essentially tight in the limit, by providing an upper bound C ≤ 1-(1-o(1))H(p), where the o(1) term is understood to be vanishing as p goes to 0. Our proof utilizes a natural counting argument that should prove helpful in analyzing related channels.

We study strategic games in the tradition of cooperative game theory, where players may make binding agreements that involve side payments. We use a decomposition of strategic games into competitive and cooperative components to define and axiomatize a solution that unifies earlier solutions and generalizes to the case of incomplete information. In the latter case, we give incentive compatible mechanisms that achieve first-best efficiency.

We consider computational aspects of alternating move games, repeated games in which players take actions at alternating time steps rather than playing simultaneously. We show that alternating move games are more tractable than simultaneous move games: we give an FPTAS for computing an ε-approximate equilibrium of an alternating move game with any number of players. In contrast, it is known that for k ≥ 3 players, there is no FPTAS for computing Nash equilibria of simultaneous move repeated games unless P = PPAD. We also consider equilibria in memoryless strategies, which are guaranteed to exist in two player games. We show that for the special case of k = 2 players, all but a negligible fraction of games admit an equilibrium in pure memoryless strategies that can be found in polynomial time. Moreover, we give a PTAS to compute an ε-approximate equilibrium in pure memoryless strategies in any 2 player game that admits an exact equilibrium in pure memoryless strategies.

Given data drawn from a mixture of multivariate Gaussians, a basic problem is to accurately estimate the mixture parameters. We provide a polynomial-time algorithm for this problem for the case of two Gaussians in $n$ dimensions (even if they overlap), with provably minimal assumptions on the Gaussians, and polynomial data requirements. In statistical terms, our estimator converges at an inverse polynomial rate, and no such estimator (even exponential time) was known for this problem (even in one dimension). Our algorithm reduces the n-dimensional problem to the one-dimensional problem, where the method of moments is applied. One technical challenge is proving that noisy estimates of the first six moments of a univariate mixture suffice to recover accurate estimates of the mixture parameters, as conjectured by Pearson (1894), and in fact these estimates converge at an inverse polynomial rate.

We give a new model of learning motivated by smoothed analysis (Spielman and Teng, 2001). In this model, we analyze two new algorithms, for PAC-learning DNFs and agnostically learning decision trees, from random examples drawn from a constant-bounded product distributions. These two problems had previously been solved using membership queries (Jackson, 1995; Gopalan et al, 2005). Our analysis demonstrates that the "heavy" Fourier coefficients of a DNF suffice to recover the DNF. We also show that a structural property of the Fourier spectrum of any boolean function over "typical" product distributions. In a second model, we consider a simple new distribution over the boolean hypercube, one which is symmetric but is not the uniform distribution, from which we can learn O(log n)-depth decision trees in polynomial time.

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.

In an online linear optimization problem, on each period t, an online algorithm chooses $s_t\in\mathcal{S}$ from a fixed (possibly infinite) set $\mathcal{S}$ of feasible decisions. Nature (who may be adversarial) chooses a weight vector $w_t\in\mathbb{R}^n$, and the algorithm incurs cost $c(s_t,w_t)$, where c is a fixed cost function that is linear in the weight vector. In the full-information setting, the vector $w_t$ is then revealed to the algorithm, and in the bandit setting, only the cost experienced, $c(s_t,w_t)$, is revealed. The goal of the online algorithm is to perform nearly as well as the best fixed $s\in\mathcal{S}$ in hindsight. Many repeated decision-making problems with weights fit naturally into this framework, such as online shortest-path, online traveling salesman problem (TSP), online clustering, and online weighted set cover. Previously, it was shown how to convert any efficient exact offline optimization algorithm for such a problem into an efficient online algorithm in both the full-information and the bandit settings, with average cost nearly as good as that of the best fixed $s\in\mathcal{S}$ in hindsight. However, in the case where the offline algorithm is an approximation algorithm with ratio $\alpha >1$, the previous approach worked only for special types of approximation algorithms. We show how to convert any offline approximation algorithm for a linear optimization problem into a corresponding online approximation algorithm, with a polynomial blowup in runtime. If the offline algorithm has an $\alpha$-approximation guarantee, then the expected cost of the online algorithm on any sequence is not much larger than $\alpha$ times that of the best $s\in\mathcal{S}$, where the best is chosen with the benefit of hindsight. Our main innovation is combining Zinkevich's algorithm for convex optimization with a geometric transformation that can be applied to any approximation algorithm. Standard techniques generalize the above result to the bandit setting, except that a “barycentric spanner” for the problem is also (provably) necessary as input. Our algorithm can also be viewed as a method for playing large repeated games, where one can compute only approximate best responses, rather than best responses.

We give a query algorithm for agnostically learning decision trees with respect to the uniform distribution on inputs. Given black-box access to an *arbitrary* binary function f on the n-dimensional hypercube, our algorithm finds a function that agrees with f on almost (within an epsilon fraction) as many inputs as the best size-t decision tree, in time poly(n,t,1ε).

The motivating problem is agnostically learning parity functions, i.e., parity with arbitrary or adversarial noise. Specifically, given random labeled examples from an *arbitrary* distribution, we would like to produce an hypothesis whose accuracy nearly matches the accuracy of the best parity function. Our algorithm runs in time 2O(n/log n), which matches the best known for the easier cases of learning parities with random classification noise (Blum et al, 2003) and for agnostically learning parities over the uniform distribution on inputs (Feldman et al, 2006).

High-quality, personalized recommendations are a key feature in many online systems. Since these systems often have explicit knowledge of social network structures, the recommendations may incorporate this information. This paper focuses on networks that represent trust and recommendation systems that incorporate these trust relationships. The goal of a trust-based recommendation system is to generate personalized recommendations by aggregating the opinions of other users in the trust network.In analogy to prior work on voting and ranking systems, we use the axiomatic approach from the theory of social choice. We develop a set of five natural axioms that a trust-based recommendation system might be expected to satisfy. Then, we show that no system can simultaneously satisfy all the axioms. However, for any subset of four of the five axioms we exhibit a recommendation system that satisfies those axioms. Next we consider various ways of weakening the axioms, one of which leads to a unique recommendation system based on random walks. We consider other recommendation systems, including systems based on personalized PageRank, majority of majorities, and minimum cuts, and search for alternative axiomatizations that uniquely characterize these systems.Finally, we determine which of these systems are incentive compatible, meaning that groups of agents interested in manipulating recommendations can not induce others to share their opinion by lying about their votes or modifying their trust links. This is an important property for systems deployed in a monetized environment.

We give a computationally efficient algorithm that learns (under distributional assumptions) a halfspace in the difficult agnostic framework of Kearns, Schapire, and Sellie [Mach. Learn., 17 (1994), pp. 115–141], where a learner is given access to a distribution on labelled examples but where the labelling may be arbitrary (similar to malicious noise). It constructs a hypothesis whose error rate on future examples is within an additive $\epsilon$ of the optimal halfspace, in time poly$(n)$ for any constant $\epsilon>0$, for the uniform distribution over $\{-1,1\}^n$ or unit sphere in $\mathbb R^n,$ as well as any log-concave distribution in $\mathbb R^n$. It also agnostically learns Boolean disjunctions in time $2^{\tilde{O}(\sqrt{n})}$ with respect to any distribution. Our algorithm, which performs $L_1$ polynomial regression, is a natural noise-tolerant arbitrary-distribution generalization of the well-known “low-degree” Fourier algorithm of Linial, Mansour, and Nisan. We observe that significant improvements on the running time of our algorithm would yield the fastest known algorithm for learning parity with noise, a challenging open problem in computational learning theory.

In an online linear optimization problem, on each period t, an online algorithm chooses st ∈ S from a fixed (possibly infinite) set S of feasible decisions. Nature (who may be adversarial) chooses a weight vector wt ∈ R, and the algorithm incurs cost c(st,wt), where c is a fixed cost function that is linear in the weight vector. In the full-information setting, the vector wt is then revealed to the algorithm, and in the bandit setting, only the cost experienced, c(st,wt), is revealed. The goal of the online algorithm is to perform nearly as well as the best fixed s ∈ S in hindsight. Many repeated decision-making problems with weights fit naturally into this framework, such as online shortest-path, online TSP, online clustering, and online weighted set cover.

In recent years, there has been a proliferation of theoretical graph models, e.g., preferential attachment and small-world models, motivated by real-world graphs such as the Internet topology. To address the natural question of which model is best for a particular data set, we propose a model selection criterion for graph models. Since each model is in fact a probability distribution over graphs, we suggest using Maximum Likelihood to compare graph models and select their parameters. Interestingly, for the case of graph models, computing likelihoods is a difficult algorithmic task. However, we design and implement MCMC algorithms for computing the maximum likelihood for four popular models: a power-law random graph model, a preferential attachment model, a small-world model, and a uniform random graph model. We hope that this novel use of ML will objectify comparisons between graph models.

A central question in game theory and artificial intelligence is how a rational agent should behave in a complex environment, given that it cannot perform unbounded computations. We study strategic aspects of this question by formulating a simple model of a game with additional costs (computational or otherwise) for each strategy. First we connect this to zero-sum games, proving a counter-intuitive generalization of the classic min-max theorem to zero-sum games with the addition of strategy costs. We then show that potential games with strategy costs remain potential games. Both zero-sum and potential games with strategy costs maintain a very appealing property: simple learning dynamics converge to equilibrium. 1 The Approach and Basic Model How should an intelligent agent play a complicated game like chess, given that it does not have unlimited time to think? This question reflects one fundamental aspect of "bounded rationality," a term coined by Herbert Simon [1]. However, bounded rationality has proven to be a slippery concept to formalize (prior work has focused largely on finite automata playing simple repeated games such as prisoner's dilemma, e.g. [2, 3, 4, 5]). This paper focuses on the strategic aspects of decisionmaking in complex multi-agent environments, i.e., on how a player should choose among strategies of varying complexity, given that its opponents are making similar decisions. Our model applies to general strategic games and allows for a variety of complexities that arise in real-world applications. For this reason, it is applicable to one-shot games, to extensive games, and to repeated games, and it generalizes existing models such as repeated games played by finite automata. To easily see that bounded rationality can drastically affect the outcome of a game, consider the following factoring game. Player 1 chooses an n-bit number and sends it to Player 2, who attempts to find its prime factorization. If Player 2 is correct, he is paid 1 by Player 1, otherwise he pays 1 to Player 1. Ignoring complexity costs, the game is a trivial win for Player 2. However, for large n, the game should is essentially a win for Player 1, who can easily output a large random number that Player 2 cannot factor (under appropriate complexity assumptions). In general, the outcome of a game (even a zero-sum game like chess) with bounded rationality is not so clear. To concretely model such games, we consider a set of available strategies along with strategy costs. Consider an example of two players preparing to play a computerized chess game for $100K prize. Suppose the players simultaneously choose among two available options: to use a $10K program A or an advanced program B, which costs $50K. We refer to the row chooser as white and to the column chooser as black, with the corresponding advantages reflected by the win probabilities of white described in Table 1a. For example, when both players use program A, white wins 55% of the time and black wins 45% of the time (we ignore draws). The players naturally want to choose strategies to maximize their expected net payoffs, i.e., their expected payoff minus their cost. Each cell in Table 1b contains a pair of payoffs in units of thousands of dollars; the first is white's net expected payoff and the second is black's. a) A B A 55% 93% B 13% 51% b) A (-10) B (-50) A (-10) 45, 35 43,-3 B (-50) 3, 37 1,-1 Figure 1: a) Table of first-player winning probabilities based on program choices. b) Table of expected net earnings in thousands of dollars. The unique equilibrium is (A,B) which strongly favors the second player. A surprising property is evident in the above game. Everything about the game seems to favor white. Yet due to the (symmetric) costs, at the unique Nash equilibrium (A,B) of Table 1b, black wins 87% of the time and nets $34K more than white. In fact, it is a dominant strategy for white to play A and for black to play B. To see this, note that playing B increases white's probability of winning by 38%, independent of what black chooses. Since the pot is $100K, this is worth $38K in expectation, but B costs $40K more than A. On the other hand, black enjoys a 42% increase in probability of winning due to B, independent of what white does, and hence is willing to pay the extra $40K. Before formulating the general model, we comment on some important aspects of the chess example. First, traditional game theory states that chess can be solved in "only" two rounds of elimination of dominated strategies [10], and the outcome with optimal play should always be the same: either a win for white or a win for black. This theoretical prediction fails in practice: in top play, the outcome is very nondeterministic with white winning roughly twice as often as black. The game is too large and complex to be solved by brute force. Second, we have been able to analyze the above chess program selection example exactly because we formulated as a game with a small number of available strategies per player. Another formulation that would fit into our model would be to include all strategies of chess, with some reasonable computational costs. However, it is beyond our means to analyze such a large game. Third, in the example above we used monetary software cost to illustrate a type of strategy cost. But the same analysis could accommodate many other types of costs that can be measured numerically and subtracted from the payoffs, such as time or effort involved in the development or execution of a strategy, and other resource costs. Additional examples in this paper include the number of states in a finite automaton, the number of gates in a circuit, and the number of turns on a commuter's route. Our analysis is limited, however, to cost functions that depend only on the strategy of the player and not the strategy chosen by its opponent. For example, if our players above were renting computers A or B and paying for the time of actual usage, then the cost of using A would depend on the choice of computer made by the opponent. Generalizing the example above, we consider a normal form game with the addition of strategy costs, a player-dependent cost for playing each available strategy. Our main results regard two important classes of games: constant-sum and potential games. Potential games with strategy costs remain potential games. While two-person constant-sum games are no longer constant, we give a basic structural description of optimal play in these games. Lastly, we show that known learning dynamics converge in both classes of games. 2 Definition of strategy costs We first define an N -person normal-form game G = (N , S, p) consisting of finite sets of (available) pure strategies S = (S1 , . . . , SN ) for the N players, and a payoff function p : S1 . . . SN RN . Players simultaneously choose strategies si Si after which player i is rewarded with pi (s1 , . . . , sN ). A randomized or mixed strategy i for player i is a probability distribution over its pure strategies Si , x . x i i = R|Si | : = 1, xj 0 j We extend p to 1 . . . N in the natural way, i.e., pi (1 , . . . , N ) = E[pi (s1 , . . . , sN )] where each si is drawn from i , independently. Denote by s-i = (s1 , s2 , . . . , si-1 , si+1 , . . . , sN ) and similarly for -i . A best response by player i to -i is i i such that pi (i , -i ) = maxi i pi (i , -i ). A (mixed strategy) Nash equilibrium of G is a vector of strategies (1 , . . . , N ) 1 . . . N such that each i is a best response to -i . We now define G-c , the game G with strategy costs c = (c1 , . . . , cN ), where ci : Si R. It is simply an N -person normal-form game G-c = (N , S, p-c ) with the same sets of pure strategies as G, but with a new payoff function p-c : S1 . . . SN RN where, p-c (s1 , . . . , sN ) = pi (s1 , . . . , sN ) - ci (si ), for i = 1, . . . , N . i We similarly extend ci to i in the natural way. 3 Two-person constant-sum games with strategy costs Recall that a game is constant-sum (k -sum for short) if at every combination of individual strategies, the players' payoffs sum to some constant k. Two-person k -sum games have some important properties, not shared by general sum games, which result in more effective game-theoretic analysis. In particular, every k -sum game has a unique value v R. A mixed strategy for player 1 is called optimal if it guarantees payoff v against any strategy of player 2. A mixed strategy for player 2 is optimal if it guarantees k - v against any strategy of player 1. The term optimal is used because optimal strategies guarantee as much as possible (v + k - v = k ) and playing anything that is not optimal can result in a lesser payoff, if the opponent responds appropriately. (This fact is easily illustrated in the game rock-paper-scissors randomizing uniformly among the strategies guarantees each player 50% of the pot, while playing anything other than uniformly random enables the opponent to win strictly more often.) The existence of optimal strategies for both players follows from the min-max theorem. An easy corollary is that the Nash equilibria of a k -sum game are exchangeable: they are simply the cross-product of the sets of optimal mixed strategies for both players. Lastly, it is well-known that equilibria in two-person k -sum games can be learned in repeated play by simple dynamics that are guaranteed to converge [17]. With the addition of strategy costs, a k -sum game is no longer k -sum and hence it is not clear, at first, what optimal strategies there are, if any. (Many examples of general-sum games do not have optimal strategies.) We show the following generalization of the above properties for zero-sum games with strategies costs. Theorem 1. Let G be a finite two-person k -sum game and G-c be the game with strategy costs c = (c1 , c2 ). 1. There is a value v R for G-c and nonempty sets OPT1 and OPT2 of optimal mixed strategies for the two players. OPT1 is the set of strategies that guarantee player 1 payoff v - c2 (2 ), against any strategy 2 chosen by player 2. Similarly, OPT2 is the set of strategies that guarantee player 2 payoff k - v - c1 (1 ) against any 1 . 2. The Nash equilibria of G-c are exchangeable: the set of Nash equilibria is OPT1 OPT2 . 3. The set of net payoffs possible at equilibrium is an axis-parallel rectangle in R2 . For zero-sum games, the term optimal strategy was natural: the players could guarantee v and k - v , respectively, and this is all that there was to share. Moreover, it is easy to see that only pairs of optimal strategies can have the Nash equilibria property, being best responses to each other. In the case of zero-sum games with strategy costs, the optimal structure is somewhat counterintuitive. First, it is strange that the amount guaranteed by either player depends on the cost of the other player's action, when in reality each player pays the cost of its own action. Second, it is not even clear why we call these optimal strategies. To get a feel for this latter issue, notice that the sum of the net payoffs to the two players is always k - c1 (1 ) - c2 (2 ), which is exactly the total of what optimal strategies guarantee, v - c2 (2 ) + k - v - c1 (1 ). Hence, if both players play what we call optimal strategies, then neither player can improve and they are at Nash equilibrium. On the other hand, suppose player 1 selects a strategy 1 that does not guarantee him payoff at least v - c2 (2 ). This means that there is some response 2 by player 2 for which player 1's payoff is < v - c2 (2 ) and hence player 2's payoff is > k - v - c1 (1 ). Thus player 2's best response to 1 must give player 2 payoff > k - v - c1 (1 ) and leave player 1 with < v - c2 (2 ). The proof of the theorem (the above reasoning only implies part 2 from part 1) is based on the following simple observation. Consider the k -sum game H = (N , S, q ) with the following payoffs: q1 (s1 , s2 ) = p1 (s1 , s2 ) - c1 (s1 ) + c2 (s2 ) = p-c (s1 , s2 ) + c2 (s2 ) 1 q2 (s1 , s2 ) = p2 (s1 , s2 ) - c2 (s1 ) + c1 (s1 ) = p-c (s1 , s2 ) + c1 (s1 ) 2 That is to say, Player 1 pays its strategy cost to Player 2 and vice versa. It is easy to verify that, 1 , 1 1 , 2 2 q1 (1 , 2 ) - q1 (1 , 2 ) = p-c (1 , 2 ) - p-c (1 , 2 ) 1 1 (1) This means that the relative advantage in switching strategies in games G-c and H are the same. In particular, 1 is a best response to 2 in G-c if and only if it is in H . A similar equality holds for player 2's payoffs. Note that these conditions imply that the games G-c and H are strategically equivalent in the sense defined by Moulin and Vial [16]. Proof of Theorem 1. Let v be the value of the game H . For any strategy 1 that guarantees player 1 payoff v in H , 1 guarantees player 1 v - c2 (2 ) in G-c . This follows from the definition of H . Similarly, any strategy 2 that guarantees player 2 payoff k - v in H will guarantee k - v - c1 (1 ) in G-c . Thus the sets OPT1 and OPT2 are non-empty. Since v - c2 (2 ) + k - v - c1 (1 ) = k - c1 (1 ) - c2 (2 ) is the sum of the payoffs in G-c , nothing greater can be guaranteed by either player. Since the best responses of G-c and H are the same, the Nash equilibria of the two games are the same. Since H is a k -sum game, its Nash equilibria are exchangeable, and thus we have part 2. (This holds for any game that is strategically equivalent to k -sum.) Finally, the optimal mixed strategies OPT1 , OPT2 of any k -sum game are convex sets. If we look at the achievable costs of the mixed strategies in OPTi , by the definition of the cost of a mixed strategy, this will be a convex subset of R, i.e., an interval. By parts 1 and 2, the set of achievable net payoffs at equilibria of G-c are therefore the cross-product of intervals. To illustrate Theorem 1 graphically, Figure 2 gives a 4 4 example with costs of 1, 2, 3, and 4, respectively. It illustrates a situation with multiple optimal strategies. Notice that player 1 is completely indifferent between its optimal choices A and B, and player 2 is completely indifferent between C and D. Thus the only question is how kind they would like to be to their opponent. The (A,C) equilibrium is perhaps most natural as it is yields the highest payoffs for both parties. Note that the proof of the above theorem actually shows that zero-sum games with costs share additional appealing properties of zero-sum games. For example, computing optimal strategies is a polynomial time-computation in an n n game, as it amounts to computing the equilibria of H . We next show that they also have appealing learning properties, though they do not share all properties of zero-sum games.1 3.1 Learning in repeated two-person k -sum games with strategy costs Another desirable property of k -sum games is that, in repeated play, natural learning dynamics converge to the set of Nash equilibria. Before we state the analogous conditions for k -sum games with costs, we briefly give a few definitions. A repeated game is one in which players chooses a sequence of strategies vectors s1 , s2 , . . ., where each st = (st , . . . , st ) is a strategy vector of some 1 N fixed stage game G = (N , S, p). Under perfect monitoring, when selecting an action in any period the players know all the previous selected actions.As we shall discuss, it is possible to learn to play without perfect monitoring as well. 1 One property that is violated by the chess example is the "advantage of an advantage" property. Say Player 1 has the advantage over Player 2 in a square game if p1 (s1 , s2 ) p2 (s2 , s1 ) for all strategies s1 , s2 . At equilibrium of a k-sum game, a player with the advantage must have a payoff at least as large as its opponent. This is no longer the case after incorporating strategy costs, as seen in the chess example, where Player 1 has the advantage (even including strategy costs), yet his equilibrium payoff is smaller than 2's. a) A B C D b) A (-1) B (-2) C (-3) D (-4) A 6, 4 7, 3 7.5, 2.5 8.5, 1.5 A (-1) 5, 3 5, 2 4.5, 1.5 4.5, 0.5 B 5, 5 6, 4 6.5, 3.5 7, 3 B (-2) 4, 3 4, 2 3.5, 1.5 3, 1 C 3, 7 4, 6 4.5, 5.5 5.5, 4.5 C (-3) 2, 4 2, 3 1.5, 2.5 1.5, 1.5 D 2, 8 3, 7 3.5, 6.5 4.5, 5.5 D (-4) 1, 4 1, 3 0.5, 2.5 0.5, 1.5 PLAYER 2 NET PAYOFF

We give the first algorithm that (under distributional assumptions) efficiently learns halfspaces in the notoriously difficult agnostic framework of Kearns, Schapire, & Sellie, where a learner is given access to labeled examples drawn from a distribution, without restriction on the labels (e.g. adversarial noise). The algorithm constructs a hypothesis whose error rate on future examples is within an additive /spl epsi/ of the optimal halfspace, in time poly(n) for any constant /spl epsi/ > 0, under the uniform distribution over {-1, 1}/sup n/ or the unit sphere in /spl Ropf//sup n/ , as well as under any log-concave distribution over /spl Ropf/ /sup n/. It also agnostically learns Boolean disjunctions in time 2/sup O~(/spl radic/n)/ with respect to any distribution. The new algorithm, essentially L/sub 1/ polynomial regression, is a noise-tolerant arbitrary distribution generalization of the "low degree" Fourier algorithm of Linial, Mansour, & Nisan. We also give a new algorithm for PAC learning halfspaces under the uniform distribution on the unit sphere with the current best bounds on tolerable rate of "malicious noise".

It is well-known that everything that is learnable in the difficult online setting, where an arbitrary sequences of examples must be labeled one at a time, is also learnable in the batch setting, where examples are drawn independently from a distribution. We show a result in the opposite di- rection. We give an efficient conversion algorithm from batch to online that is transductive: it uses future unlabeled data. This demonstrates the equivalence between what is properly and efficiently learnable in a batch model and a transductive online model.

Boosting algorithms are procedures that "boost" low-accuracy weak learning algorithms to achieve arbitrarily high accuracy. Over the past decade boosting has been widely used in practice and has become a major research topic in computational learning theory. In this paper we study boosting in the presence of random classification noise, giving both positive and negative results.

We describe a slightly subexponential time algorithm for learning parity functions in the presence of random classification noise, a problem closely related to several cryptographic and coding problems. Our algorithm runs in polynomial time for the case of parity functions that depend on only the first O (log n log log n ) bits of input, which provides the first known instance of an efficient noise-tolerant algorithm for a concept class that is not learnable in the Statistical Query model of Kearns [1998]. Thus, we demonstrate that the set of problems learnable in the statistical query model is a strict subset of those problems learnable in the presence of noise in the PAC model.In coding-theory terms, what we give is a poly( n )-time algorithm for decoding linear k × n codes in the presence of random noise for the case of k = c log n log log n for some c > 0. (The case of k = O (log n ) is trivial since one can just individually check each of the 2 k possible messages and choose the one that yields the closest codeword.)A natural extension of the statistical query model is to allow queries about statistical properties that involve t -tuples of examples, as opposed to just single examples. The second result of this article is to show that any class of functions learnable (strongly or weakly) with t -wise queries for t = O (log n ) is also weakly learnable with standard unary queries. Hence, this natural extension to the statistical query model does not increase the set of weakly learnable functions.

A constant rebalanced portfolio is an investment strategy which keeps the same distribution of wealth among a set of stocks from day to day. There has been much work on Cover's Universal algorithm, which is competitive with the best constant rebalanced portfolio determined in hindsight (D. Helmbold et al., 1995; A. Blum and A. Kalai, 1999; T.M. Cover and E. Ordentlich, 1996). While this algorithm has good performance guarantees, all known implementations are exponential in the number of stocks, restricting the number of stocks used in experiments. We present an efficient implementation of the Universal algorithm that is based on non-uniform random walks that are rapidly mixing (D. Applegate and R. Kannanm, 1991). This same implementation also works for non-financial applications of the Universal algorithm, such as data compression (T.M. Cover, 1886) and language modeling (A. Kalai et al., 1999).

We describe a slightly sub-exponential time algorithm for learning parity functions in the presence of random classification noise. This results in a polynomial-time algorithm for the case of parity functions that depend on only the first O(log n log log n) bits of input. This is the first known instance of an efficient noise-tolerant algorithm for a concept class that is provably not learnable in the Statistical Query model of Kearns [7]. Thus, we demonstrate that the set of problems learnable in the statistical query model is a strict subset of those problems learnable in the presence of noise in the PAC model. In coding-theory terms, what we give is a poly(n)-time algorithm for decoding linear k × n codes in the presence of random noise for the case of k = clog n log log n for some c > 0. (The case of k --- O(log n) is trivial since one can just individually check each of the 2 k possible messages and choose the one that yields the closest codeword.) A natural extension of the statistical query model is to allow queries about statistical properties that involve t-tuples of examples (as opposed to single examples). The second result of this paper is to show that any class of functions learnable (strongly or weakly) with t-wise queries for t = O(log n) is also weakly learnable with standard unary queries. Hence this natural extension to the statistical query model does not increase the set of weakly learnable functions.

The empirical error on a test set, the hold-out estimate, often is a more reliable estimate of generalization error than the observed error on the training set, the training estimate.K-fold cross validation is used in practice with the hope of being more accurate than the hold-out estimate without reducing the number of training examples.We argue that the k-fold estimate does in fact achieve this goal.Specifically, we show that for any nontrivial learning problem and learning algorithm that is insensitive to example ordering, the k-fold estimate is strictly more accurate than a single hold-out estimate on l/k of the data, for 2 < k < n (It = 12 is leave-one-out), based on its variance and all higher moments.Previous bounds were termed sanitycheck because they compared the k-fold estimate to the training estimate and, further, restricted the VC dimension and required a notion of hypothesis stability [2].In order to avoid these dependencies, we consider a k-fold hypothesis that is a randomized combination or average of the LT individual hypotheses.We introduceprogressive validation as another possible improvement on the hold-out estimate.This estimate of the generalization error is, in many ways, as good as that of a single hold-out, but it uses an average of half as many examples for testing.The procedure also involves a hold-out set, but after an example has been tested, it is added to the training set and the learning algorithm is rerun.

We construct an online algorithm for paging that achieves an O(r+log k) competitive ratio when compared to an offline strategy that is allowed the additional ability to "rent" pages at a cost of 1/r. In contrast, the competitive ratio of the Marking algorithm for this scenario is O(r log k). Our algorithm can be thought of in the standard setting as having a "fine-grained" competitive ratio, achieving an O(1) ratio when the request sequence consists of a small number of working sets, gracefully decaying to O(log k) as this number increases. Our result is a generalization of the result by Y. Bartal et al. (1997) that one can achieve an O(r+log n) ratio for the unfair n-state uniform-space Metrical Task System problem. That result was a key component of the polylog(n) competitive randomized algorithm given in that paper for the general Metrical Task System problem. One motivation of this work is that it may be a first step toward achieving a polylog(k) randomized competitive ratio for the much more difficult k-server problem.

Multiple language models are combined for many tasks in language modeling, such as domain and topic adaptation. In this work, we compare on-line algorithms from machine learning to existing algorithms for combining language models. On-line algorithms developed for this problem have parameters that are updated dynamically to adapt to a data set during evaluation. On-line analysis provides guarantees that these algorithms will perform nearly as well as the best model chosen in hindsight from a large class of models, e.g., the set of all static mixtures. We describe several on-line algorithms and present results comparing these techniques with existing language modeling combination methods on the task of domain adaptation. We demonstrate that, in some situations, on-line techniques can significantly outperform static mixtures (by over 10% in terms of perplexity) and are especially effective when the nature of the test data is unknown or changes over time.

The notion of a virtual sensor for optimal 3D reconstruction is introduced. Instead of planar perspective images that collect many rays at a fixed viewpoint, omnivergent cameras collect a small number of rays at many different viewpoints. The resulting 2D manifold of rays are arranged into two multiple-perspective images for stereo reconstruction. We call such images omnivergent images, and the process of reconstructing the scene from such images, omnivergent stereo. This procedure is shown to produce 3D scene models with minimal reconstruction error due to the fact that for any point in the 3D scene, two rays with maximum vergence angle can be found in the omnivergent images. Furthermore, omnivergent images are shown to have horizontal epipolar lines, enabling the application of traditional stereo matching algorithms, without modification. Three types of omnivergent virtual sensors are presented: spherical omnivergent cameras, center-strip cameras and dual-strip cameras.

A constant rebalanced portfolio is an investment strategy which keeps the same distribution of wealth among a set of stocks from period to period.Recently there has been work on on-line investment strategies that are competitive with the best constant rebalanced portfolio determined in hindsight [2, 10, 3, 4, 13, 51.For the universal algorithm of Cover [2], we provide a simple analysis which naturally extends to the case of a fixed percentage transaction cost (commission), answering a question raised in [2, 10, 3, 4, 13, 51.In addition, we present a simple randomized implementation that is significantly faster in practice.We conclude by explaining how these algorithms can be applied to other problems, such as combining the predictions of statistical language models, where the resulting guarantees are more striking.