Digital labor markets are increasingly used to procure diverse forms of human input, from data annotation to food delivery. A central concern in such markets is the ability of platforms to suppress wages by exploiting the abundance of low-cost labor. To better understand labor outcomes, we introduce a novel posted-price procurement model with coverage objectives. In our model a platform seeks to acquire a set of M tasks while minimizing wait time, as well as total spending. Workers are sampled from a population and complete a task if the posted price is higher than their estimated cost of labor. First, we show that with a simple pricing strategy the platform can cover all categories in time O(M), while paying only a O(log(M)/M) fraction of the total cost of labor by exploiting high worker uncertainty about their costs. Then, we study the impact of collective action to prevent the exploitation of workers. We show how a small, strategically chosen coalition of workers that commits to a price floor forces the platform's total spending from logarithmic to linear in M, substantially reducing the power of the platform to suppress wages. In contrast, a randomly sampled coalition of equal size remains largely ineffective. We complement our theory with synthetic experiments that demonstrate the benefit of targeted recruitment for collective action across different market regimes. More broadly, our results provide a theoretical foundation for understanding how to organize collective bargaining for promoting welfare in digital labor markets.

We study a fundamental problem in the evaluation of large language models that we call training on the test task. Unlike wrongful practices like training on the test data, leakage, or data contamination, training on the test task is not a malpractice. Rather, the term describes a growing set of techniques to include task-relevant data in the pretraining stage of a language model. We demonstrate that training on the test task confounds both relative model evaluations and claims about emergent capabilities. We argue that the seeming superiority of one model family over another may be explained by a different degree of training on the test task. To this end, we propose an effective method to adjust for the effect of training on the test task on benchmark evaluations. Put simply, to fine-tune each model under comparison on the same task-relevant data before evaluation. Lastly, we show that instances of emergent behavior disappear gradually as models train on the test task. Our work promotes a new perspective on the evaluation of large language models with broad implications for benchmarking and the study of emergent capabilities.

Annotation and classification of legal text are central components of empirical legal research. Traditionally, these tasks are often delegated to trained research assistants. Motivated by the advances in language modeling, empirical legal scholars are increasingly turning to commercial models, hoping that it will alleviate the significant cost of human annotation. In this work, we present a comprehensive analysis of large language models’ current abilities to perform legal annotation tasks. To do so, we construct CaselawQA, a benchmark comprising 260 legal text classification tasks, nearly all new to the machine learning community. We demonstrate that commercial models, such as GPT-4.5 and Claude 3.7 Sonnet, achieve non-trivial accuracy but generally fall short of the performance required for legal work. We then demonstrate that small, lightly fine-tuned models vastly outperform commercial models. A few dozen to a few hundred labeled examples are usually enough to achieve higher accuracy. Our work points to a viable alternative to the predominant practice of prompting commercial models. For concrete legal annotation tasks with some available labeled data, researchers are likely better off using a fine-tuned open-source model. Code, datasets, and fine-tuned models are available at https://github.com/socialfoundations/lawma.

High quality annotations are increasingly a bottleneck in the explosively growing machine learning ecosystem. Scalable evaluation methods that avoid costly annotation have therefore become an important research ambition. Many hope to use strong existing models in lieu of costly labels to provide cheap model evaluations. Unfortunately, this method of using models as judges introduces biases, such as self-preferencing, that can distort model comparisons. An emerging family of debiasing tools promises to fix these issues by using a few high quality labels to debias a large number of model judgments. In this paper, we study how far such debiasing methods, in principle, can go. Our main result shows that when the judge is no more accurate than the evaluated model, no debiasing method can decrease the required amount of ground truth labels by more than half. Our result speaks to the severe limitations of the LLM-as-a-judge paradigm at the evaluation frontier where the goal is to assess newly released models that are possibly better than the judge. Through an empirical evaluation, we demonstrate that the sample size savings achievable in practice are even more modest than what our theoretical limit suggests. Along the way, our work provides new observations about debiasing methods for model evaluation, and points out promising avenues for future work.

Improving social welfare is a complex challenge requiring policymakers to optimize objectives across multiple time horizons. Evaluating the impact of such policies presents a fundamental challenge, as those that appear suboptimal in the short run may yield significant long-term benefits. We tackle this challenge by analyzing the long-term dynamics of two prominent policy frameworks: Rawlsian policies, which prioritize those with the greatest need, and utilitarian policies, which maximize immediate welfare gains. Conventional wisdom suggests these policies are at odds, as Rawlsian policies are assumed to come at the cost of reducing the average social welfare, which their utilitarian counterparts directly optimize. We challenge this assumption by analyzing these policies in a sequential decision-making framework where individuals' welfare levels stochastically decay over time, and policymakers can intervene to prevent this decay. Under reasonable assumptions, we prove that interventions following Rawlsian policies can outperform utilitarian policies in the long run, even when the latter dominate in the short run. We characterize the exact conditions under which Rawlsian policies can outperform utilitarian policies. We further illustrate our theoretical findings using simulations, which highlight the risks of evaluating policies based solely on their short-term effects. Our results underscore the necessity of considering long-term horizons in designing and evaluating welfare policies; the true efficacy of even well-established policies may only emerge over time.

Two narratives about machine learning ecosystems grew out of recent algorithmic fairness discourse. In one, dubbed \emph{monoculture}, algorithmic ecosystems tend toward homogeneity akin to a single model making all decisions. Individuals then face the risk of systematic exclusion with no recourse. In the other, \emph{model multiplicity}, many models solve the same task with similar accuracy, causing excessive variation in outcomes. Both narratives are compelling, yet, seemingly at odds: model multiplicity can’t exist in a strict monoculture. In this work, we conduct a comprehensive empirical evaluation to test both claims. We work from the premise that increasingly decision makers will use large language models for consequential prediction tasks. We therefore examine 50 language models, open source models ranging in size from 1B to 141B parameters and state-of-the-art commercial models, under 4 different prompt variations, and across 6 different prediction tasks. Evaluating both new and old quantitative measures of monoculture and multiplicity, we find the empirical landscape sits between the two extremes. Each narrative finds some empirical support, but neither is dominant. Systematic exclusion with no recourse is rare, but model similarity is real. Even when starting from a single model, prompt variation induces some diversity in predictions. Our results contribute critical empirical grounding to ongoing debates and point toward a middle ground between monoculture and multiplicity as the most realistic outcome.

Large language model (LLM) evaluation is increasingly costly, prompting interest in methods that speed up evaluation by shrinking benchmark datasets. Benchmark prediction (also called efficient LLM evaluation) aims to select a small subset of evaluation points and predict overall benchmark performance from that subset. In this paper, we systematically assess the strengths and limitations of 11 benchmark prediction methods across 19 diverse benchmarks. First, we identify a highly competitive baseline: Take a random sample and fit a regression model on the sample to predict missing entries. Outperforming most existing methods, this baseline challenges the assumption that careful subset selection is necessary for benchmark prediction. Second, we discover that all existing methods crucially depend on model similarity. They work best when interpolating scores among similar models. The effectiveness of benchmark prediction sharply declines when new models have higher accuracy than previously seen models. In this setting of extrapolation, none of the previous methods consistently beat a simple average over random samples. To improve over the sample average, we introduce a new method inspired by augmented inverse propensity weighting. This method consistently outperforms the random sample average even for extrapolation. However, its performance still relies on model similarity and the gains are modest in general. This shows that benchmark prediction fails just when it is most needed: at the evaluation frontier, where the goal is to evaluate new models of unknown capabilities.

Seven years ago, researchers proposed a postprocessing method to equalize the error rates of a model across different demographic groups. The work launched hundreds of papers purporting to improve over the postprocessing baseline. We empirically evaluate these claims through thousands of model evaluations on several tabular datasets. We find that the fairness-accuracy Pareto frontier achieved by postprocessing contains all other methods we were feasibly able to evaluate. In doing so, we address two common methodological errors that have confounded previous observations. One relates to the comparison of methods with different unconstrained base models. The other concerns methods achieving different levels of constraint relaxation. At the heart of our study is a simple idea we call unprocessing that roughly corresponds to the inverse of postprocessing. Unprocessing allows for a direct comparison of methods using different underlying models and levels of relaxation.

Many recent efforts augment language models with retrieval, by adding retrieved data to the input context. For this approach to succeed, the retrieved data must be added at both training and test time. Moreover, as input length grows linearly with the size of retrieved data, cost in computation and memory grows quadratically for modern Transformers. To avoid these complications, we simply fine-tune the model on retrieved data at test time, using its standard training setup. We build a large-scale distributed index based on text embeddings of the Pile dataset. For each test input, our system retrieves its neighbors and fine-tunes the model on their text. Surprisingly, retrieving and training on as few as 20 neighbors, each for only one gradient iteration, drastically improves performance across more than 20 language modeling tasks in the Pile. For example, test-time training with nearest neighbors significantly narrows the performance gap between a small GPT-2 and a GPT-Neo model more than 10 times larger. Sufficient index quality and size, however, are necessary. Our work establishes a first baseline of test-time training for language modeling.

Regulators and academics are increasingly interested in the causal effect that algorithmic actions of a digital platform have on user consumption. In pursuit of estimating this effect from observational data, we identify a set of assumptions that permit causal identifiability without assuming randomized platform actions. Our results are applicable to platforms that rely on machine-learning-powered predictions and leverage knowledge from historical data. The key novelty of our approach is to explicitly model the dynamics of consumption over time, exploiting the repeated interaction of digital platforms with their participants to prove our identifiability results. By viewing the platform as a controller acting on a dynamical system, we can show that exogenous variation in consumption and appropriately responsive algorithmic control actions are sufficient for identifying the causal effect of interest. We complement our claims with an analysis of ready-to-use finite sample estimators and empirical investigations. More broadly, our results deriving identifiability conditions tailored to digital platform settings illustrate a fruitful interplay of control theory and causal inference.

We study how to best spend a budget of noisy labels to compare the accuracy of two binary classifiers. It’s common practice to collect and aggregate multiple noisy labels for a given data point into a less noisy label via a majority vote. We prove a theorem that runs counter to conventional wisdom. If the goal is to identify the better of two classifiers, we show it’s best to spend the budget on collecting a single label for more samples. Our result follows from a non-trivial application of Cramér’s theorem, a staple in the theory of large deviations. We discuss the implications of our work for the design of machine learning benchmarks, where they overturn some time-honored recommendations. In addition, our results provide sample size bounds superior to what follows from Hoeffding’s bound.

Algorithmic predictions are emerging as a promising solution concept for efficiently allocating societal resources. Fueling their use is an underlying assumption that such systems are necessary to identify individuals for interventions. We propose a principled framework for assessing this assumption: Using a simple mathematical model, we evaluate the efficacy of prediction-based allocations in settings where individuals belong to larger units such as hospitals, neighborhoods, or schools. We find that prediction-based allocations outperform baseline methods using aggregate unit-level statistics only when between-unit inequality is low and the intervention budget is high. Our results hold for a wide range of settings for the price of prediction, treatment effect heterogeneity, and unit-level statistics’ learnability. Combined, we highlight the potential limits to improving the efficacy of interventions through prediction.

Many applications of RCTs involve the presence of multiple treatment administrators—from field experiments to online advertising—that compete for the subjects’ attention. In the face of competition, estimating a causal effect becomes difficult, as the position at which a subject sees a treatment influences their response, and thus the treatment effect. In this paper, we build a game-theoretic model of agents who wish to estimate causal effects in the presence of competition, through a bidding system and a utility function that minimizes estimation error. Our main technical result establishes an approximation with a tractable objective that maximizes the sample value obtained through strategically allocating budget on subjects. This allows us to find an equilibrium in our model: we show that the tractable objective has a pure Nash equilibrium, and that any Nash equilibrium is an approximate equilibrium for our general objective that minimizes estimation error under broad conditions. Conceptually, our work successfully combines elements from causal inference and game theory to shed light on the equilibrium behavior of experimentation under competition.

We examine multi-task benchmarks in machine learning through the lens of social choice theory. We draw an analogy between benchmarks and electoral systems, where models are candidates and tasks are voters. This suggests a distinction between cardinal and ordinal benchmark systems. The former aggregate numerical scores into one model ranking; the latter aggregate rankings for each task. We apply Arrow’s impossibility theorem to ordinal benchmarks to highlight the inherent limitations of ordinal systems, particularly their sensitivity to the inclusion of irrelevant models. Inspired by Arrow’s theorem, we empirically demonstrate a strong trade-off between diversity and sensitivity to irrelevant changes in existing multi-task benchmarks. Our result is based on new quantitative measures of diversity and sensitivity that we introduce. Sensitivity quantifies the impact that irrelevant changes to tasks have on a benchmark. Diversity captures the degree of disagreement in model rankings across tasks. We develop efficient approximation algorithms for both measures, as exact computation is computationally challenging. Through extensive experiments on seven cardinal benchmarks and eleven ordinal benchmarks, we demonstrate a clear trade-off between diversity and stability: The more diverse a multi-task benchmark, the more sensitive to trivial changes it is. Additionally, we show that the aggregated rankings of existing benchmarks are highly unstable under irrelevant changes. The codes and data are available at https://socialfoundations.github.io/benchbench/.

Current question-answering benchmarks predominantly focus on accuracy in realizable prediction tasks.Conditioned on a question and answer-key, does the most likely token match the ground truth?Such benchmarks necessarily fail to evaluate LLMs' ability to quantify ground-truth outcome uncertainty.In this work, we focus on the use of LLMs as risk scores for unrealizable prediction tasks.We introduce folktexts, a software package to systematically generate risk scores using LLMs, and evaluate them against US Census data products.A flexible API enables the use of different prompting schemes, local or web-hosted models, and diverse census columns that can be used to compose custom prediction tasks.We evaluate 17 recent LLMs across five proposed benchmark tasks.We find that zero-shot risk scores produced by multiple-choice question-answering have high predictive signal but are widely miscalibrated.Base models consistently overestimate outcome uncertainty, while instruction-tuned models underestimate uncertainty and produce over-confident risk scores.In fact, instruction-tuning polarizes answer distribution regardless of true underlying data uncertainty.This reveals a general inability of instruction-tuned models to express data uncertainty using multiple-choice answers.A separate experiment using verbalized chat-style risk queries yields substantially improved calibration across instruction-tuned models.These differences in ability to quantify data uncertainty cannot be revealed in realizable settings, and highlight a blind-spot in the current evaluation ecosystem that folktexts covers.

Surveys have recently gained popularity as a tool to study large language models. By comparing models’ survey responses to those of different human reference populations, researchers aim to infer the demographics, political opinions, or values best represented by current language models. In this work, we critically examine language models' survey responses on the basis of the well-established American Community Survey by the U.S. Census Bureau. Evaluating 43 different language models using de-facto standard prompting methodologies, we establish two dominant patterns. First, models' responses are governed by ordering and labeling biases, for example, towards survey responses labeled with the letter “A”. Second, when adjusting for these systematic biases through randomized answer ordering, models across the board trend towards uniformly random survey responses, irrespective of model size or training data. As a result, models consistently appear to better represent subgroups whose aggregate statistics are closest to uniform for the survey under consideration, leading to potentially misguided conclusions about model alignment.

The power of digital platforms is at the center of major ongoing policy and regulatory efforts. To advance existing debates, we designed and executed an experiment to measure the performative power of online search providers. Instantiated in our setting, performative power quantifies the ability of a search engine to steer web traffic by rearranging results. To operationalize this definition we developed a browser extension that performs unassuming randomized experiments in the background. These randomized experiments emulate updates to the search algorithm and identify the causal effect of different content arrangements on clicks. Analyzing tens of thousands of clicks, we discuss what our robust quantitative findings say about the power of online search engines, using the Google Shopping antitrust investigation as a case study. More broadly, we envision our work to serve as a blueprint for how the recent definition of performative power can help integrate quantitative insights from online experiments with future investigations into the economic power of digital platforms.

We study how well machine learning models trained on causal features generalize across domains. We consider 16 prediction tasks on tabular datasets covering applications in health, employment, education, social benefits, and politics. Each dataset comes with multiple domains, allowing us to test how well a model trained in one domain performs in another. For each prediction task, we select features that have a causal influence on the target of prediction. Our goal is to test the hypothesis that models trained on causal features generalize better across domains. Without exception, we find that predictors using all available features, regardless of causality, have better in-domain and out-of-domain accuracy than predictors using causal features. Moreover, even the absolute drop in accuracy from one domain to the other is no better for causal predictors than for models that use all features. In addition, we show that recent causal machine learning methods for domain generalization do not perform better in our evaluation than standard predictors trained on the set of causal features. Likewise, causal discovery algorithms either fail to run or select causal variables that perform no better than our selection. Extensive robustness checks confirm that our findings are stable under variable misclassification.

We initiate a principled study of algorithmic collective action on digital platforms that deploy machine learning algorithms. We propose a simple theoretical model of a collective interacting with a firm’s learning algorithm. The collective pools the data of participating individuals and executes an algorithmic strategy by instructing participants how to modify their own data to achieve a collective goal. We investigate the consequences of this model in three fundamental learning-theoretic settings: nonparametric optimal learning, parametric risk minimization, and gradient-based optimization. In each setting, we come up with coordinated algorithmic strategies and characterize natural success criteria as a function of the collective’s size. Complementing our theory, we conduct systematic experiments on a skill classification task involving tens of thousands of resumes from a gig platform for freelancers. Through more than two thousand model training runs of a BERT-like language model, we see a striking correspondence emerge between our empirical observations and the predictions made by our theory. Taken together, our theory and experiments broadly support the conclusion that algorithmic collectives of exceedingly small fractional size can exert significant control over a platform’s learning algorithm.

We introduce the notion of performative power, which measures the ability of a firm operating an algorithmic system, such as a digital content recommendation platform, to cause change in a population of participants. We relate performative power to the economic study of competition in digital economies. Traditional economic concepts struggle with identifying anti-competitive patterns in digital platforms not least due to the complexity of market definition. In contrast, performative power is a causal notion that is identifiable with minimal knowledge of the market, its internals, participants, products, or prices.Low performative power implies that a firm can do no better than to optimize their objective on current data. In contrast, firms of high performative power stand to benefit from steering the population towards more profitable behavior. We confirm in a simple theoretical model that monopolies maximize performative power. A firm's ability to personalize increases performative power, while competition and outside options decrease performative power. On the empirical side, we propose an observational causal design to identify performative power from discontinuities in how digital platforms display content. This allows to repurpose causal effects from various studies about digital platforms as lower bounds on performative power. Finally, we speculate about the role that performative power might play in competition policy and antitrust enforcement in digital marketplaces.

When reasoning about strategic behavior in a machine learning context it is tempting to combine standard microfoundations of rational agents with the statistical decision theory underlying classification. In this work, we argue that a direct combination of these ingredients leads to brittle solution concepts of limited descriptive and prescriptive value. First, we show that rational agents with perfect information produce discontinuities in the aggregate response to a decision rule that we often do not observe empirically. Second, when any positive fraction of agents is not perfectly strategic, desirable stable points—where the classifier is optimal for the data it entails—no longer exist. Third, optimal decision rules under standard microfoundations maximize a measure of negative externality known as social burden within a broad class of assumptions about agent behavior. Recognizing these limitations we explore alternatives to standard microfoundations for binary classification. We describe desiderata that help navigate the space of possible assumptions about agent responses, and we then propose the noisy response model. Inspired by smoothed analysis and empirical observations, noisy response incorporates imperfection in the agent responses, which we show mitigates the limitations of standard microfoundations. Our model retains analytical tractability, leads to more robust insights about stable points, and imposes a lower social burden at optimality.

Although the fairness community has recognized the importance of data, researchers in the area primarily rely on UCI Adult when it comes to tabular data. Derived from a 1994 US Census survey, this dataset has appeared in hundreds of research papers where it served as the basis for the development and comparison of many algorithmic fairness interventions. We reconstruct a superset of the UCI Adult data from available US Census sources and reveal idiosyncrasies of the UCI Adult dataset that limit its external validity. Our primary contribution is a suite of new datasets derived from US Census surveys that extend the existing data ecosystem for research on fair machine learning. We create prediction tasks relating to income, employment, health, transportation, and housing. The data span multiple years and all states of the United States, allowing researchers to study temporal shift and geographic variation. We highlight a broad initial sweep of new empirical insights relating to trade-offs between fairness criteria, performance of algorithmic interventions, and the role of distribution shift based on our new datasets. Our findings inform ongoing debates, challenge some existing narratives, and point to future research directions.

Linear dynamical systems are a fundamental and powerful parametric model class. However, identifying the parameters of a linear dynamical system is a venerable task, permitting provably efficient solutions only in special cases. This work shows that the eigenspectrum of unknown linear dynamics can be identified without full system identification. We analyze a computationally efficient and provably convergent algorithm to estimate the eigenvalues of the state-transition matrix in a linear dynamical system.When applied to time series clustering, our algorithm can efficiently cluster multi-dimensional time series with temporal offsets and varying lengths, under the assumption that the time series are generated from linear dynamical systems. Evaluating our algorithm on both synthetic data and real electrocardiogram (ECG) signals, we see improvements in clustering quality over existing baselines.

Consequential decision-making incentivizes individuals to strategically adapt their behavior to the specifics of the decision rule. While a long line of work has viewed strategic adaptation as gaming and attempted to mitigate its effects, recent work has instead sought to design classifiers that incentivize individuals to improve a desired quality. Key to both accounts is a cost function that dictates which adaptations are rational to undertake. In this work, we develop a causal framework for strategic adaptation. Our causal perspective clearly distinguishes between gaming and improvement and reveals an important obstacle to incentive design. We prove any procedure for designing classifiers that incentivize improvement must inevitably solve a non-trivial causal inference problem. We show a similar result holds for designing cost functions that satisfy the requirements of previous work. With the benefit of hindsight, our results show much of the prior work on strategic classification is causal modeling in disguise.

When predictions support decisions they may influence the outcome they aim to predict. We call such predictions performative; the prediction influences the target. Performativity is a well-studied phenomenon in policy-making that has so far been neglected in supervised learning. When ignored, performativity surfaces as undesirable distribution shift, routinely addressed with retraining. We develop a risk minimization framework for performative prediction bringing together concepts from statistics, game theory, and causality. A conceptual novelty is an equilibrium notion we call performative stability. Performative stability implies that the predictions are calibrated not against past outcomes, but against the future outcomes that manifest from acting on the prediction. Our main results are necessary and sufficient conditions for the convergence of retraining to a performatively stable point of nearly minimal loss. In full generality, performative prediction strictly subsumes the setting known as strategic classification. We thus also give the first sufficient conditions for retraining to overcome strategic feedback effects.

While real-world decisions involve many competing objectives, algorithmic decisions are often evaluated with a single objective function. In this paper, we study algorithmic policies which explicitly trade off between a private objective (such as profit) and a public objective (such as social welfare). We analyze a natural class of policies which trace an empirical Pareto frontier based on learned scores, and focus on how such decisions can be made in noisy or data-limited regimes. Our theoretical results characterize the optimal strategies in this class, bound the Pareto errors due to inaccuracies in the scores, and show an equivalence between optimal strategies and a rich class of fairness-constrained profit-maximizing policies. We then present empirical results in two different contexts — online content recommendation and sustainable abalone fisheries — to underscore the generality of our approach to a wide range of practical decisions. Taken together, these results shed light on inherent trade-offs in using machine learning for decisions that impact social welfare.

In this paper, we propose Test-Time Training, a general approach for improving the performance of predictive models when training and test data come from different distributions. We turn a single unlabeled test sample into a self-supervised learning problem, on which we update the model parameters before making a prediction. This also extends naturally to data in an online stream. Our simple approach leads to improvements on diverse image classification benchmarks aimed at evaluating robustness to distribution shifts.

In performative prediction, the choice of a model influences the distribution of future data, typically through actions taken based on the model's predictions. We initiate the study of stochastic optimization for performative prediction. What sets this setting apart from traditional stochastic optimization is the difference between merely updating model parameters and deploying the new model. The latter triggers a shift in the distribution that affects future data, while the former keeps the distribution as is. Assuming smoothness and strong convexity, we prove rates of convergence for both greedily deploying models after each stochastic update (greedy deploy) as well as for taking several updates before redeploying (lazy deploy). In both cases, our bounds smoothly recover the optimal $O(1/k)$ rate as the strength of performativity decreases. Furthermore, they illustrate how depending on the strength of performative effects, there exists a regime where either approach outperforms the other. We experimentally explore the trade-off on both synthetic data and a strategic classification simulator.

We ask how many measurements of the accuracy on a multiclass benchmark are needed to achieve a given amount of overfitting.

Excessive reuse of holdout data can lead to overfitting. However, there is little concrete evidence of significant overfitting due to holdout reuse in popular multiclass benchmarks today. Known results show that, in the worst-case, revealing the accuracy of $k$ adaptively chosen classifiers on a data set of size $n$ allows to create a classifier with bias of $\Theta(\sqrt{k/n})$ for any binary prediction problem. We show a new upper bound of $\tilde O(\max\{\sqrt{k\log(n)/(mn)}, k/n\})$ on the worst-case bias that any attack can achieve in a prediction problem with $m$ classes. Moreover, we present an efficient attack that achieve a bias of $\Omega(\sqrt{k/(m^2 n)})$ and improves on previous work for the binary setting ($m=2$). We also present an inefficient attack that achieves a bias of $\tilde\Omega(k/n)$. Complementing our theoretical work, we give new practical attacks to stress-test multiclass benchmarks by aiming to create as large a bias as possible with a given number of queries. Our experiments show that the additional uncertainty of prediction with a large number of classes indeed mitigates the effect of our best attacks.

We clarify what fairness guarantees we can and cannot expect to follow from unconstrained machine learning. Specifically, we show that in many settings, unconstrained learning on its own implies group calibration, that is, the outcome variable is conditionally independent of group membership given the score. A lower bound confirms the optimality of our upper bound. Moreover, we prove that as the excess risk of the learned score decreases, the more strongly it violates separation and independence, two other standard fairness criteria. Our results challenge the view that group calibration necessitates an active intervention, suggesting that often we ought to think of it as a byproduct of unconstrained machine learning.

Adaptive data analysis is frequently criticized for its pessimistic generalization guarantees. The source of these pessimistic bounds is a model that permits arbitrary, possibly adversarial analysts that optimally use information to bias results. While being a central issue in the field, still lacking are notions of natural analysts that allow for more optimistic bounds faithful to the reality that typical analysts aren’t adversarial. In this work, we propose notions of natural analysts that smoothly interpolate between the optimal non-adaptive bounds and the best-known adaptive generalization bounds. To accomplish this, we model the analyst’s knowledge as evolving according to the rules of an unknown dynamical system that takes in revealed information and outputs new statistical queries to the data. This allows us to restrict the analyst through different natural control-theoretic notions. One such notion corresponds to a recency bias, formalizing an inability to arbitrarily use distant information. Another complementary notion formalizes an anchoring bias, a tendency to weight initial information more strongly. Both notions come with quantitative parameters that smoothly interpolate between the non-adaptive case and the fully adaptive case, allowing for a rich spectrum of intermediate analysts that are neither non-adaptive nor adversarial. Natural not only from a cognitive perspective, we show that our notions also capture standard optimization methods, like gradient descent in various settings. This gives a new interpretation to the fact that gradient descent tends to overfit much less than its adaptive nature might suggest.

Static classification has been the predominant focus of the study of fairness in machine learning. While most models do not consider how decisions change populations over time, it is conventional wisdom that fairness criteria promote the long-term well-being of groups they aim to protect. This work studies the interaction of static fairness criteria with temporal indicators of well-being. We show a simple one-step feedback model in which common criteria do not generally promote improvement over time, and may in fact cause harm. Our results highlight the importance of temporal modeling in the evaluation of fairness criteria, suggesting a range of new challenges and trade-offs.

Excessive reuse of test data has become commonplace in today's machine learning workflows. Popular benchmarks, competitions, industrial scale tuning, among other applications, all involve test data reuse beyond guidance by statistical confidence bounds. Nonetheless, recent replication studies give evidence that popular benchmarks continue to support progress despite years of extensive reuse. We proffer a new explanation for the apparent longevity of test data: Many proposed models are similar in their predictions and we prove that this similarity mitigates overfitting. Specifically, we show empirically that models proposed for the ImageNet ILSVRC benchmark agree in their predictions well beyond what we can conclude from their accuracy levels alone. Likewise, models created by large scale hyperparameter search enjoy high levels of similarity. Motivated by these empirical observations, we give a non-asymptotic generalization bound that takes similarity into account, leading to meaningful confidence bounds in practical settings.

We conduct the first large meta-analysis of overfitting due to test set reuse in the machine learning community. Our analysis is based on over one hundred machine learning competitions hosted on the Kaggle platform over the course of several years. In each competition, numerous practitioners repeatedly evaluated their progress against a holdout set that forms the basis of a public ranking available throughout the competition. Performance on a separate test set used only once determined the final ranking. By systematically comparing the public ranking with the final ranking, we assess how much participants adapted to the holdout set over the course of a competition. Our study shows, somewhat surprisingly, little evidence of substantial overfitting. These findings speak to the robustness of the holdout method across different data domains, loss functions, model classes, and human analysts.

Fairness in machine learning has predominantly been studied in static classification settings without concern for how decisions change the underlying population over time. Conventional wisdom suggests that fairness criteria promote the long-term well-being of those groups they aim to protect. We study how static fairness criteria interact with temporal indicators of well-being, such as long-term improvement, stagnation, and decline in a variable of interest. We demonstrate that even in a one-step feedback model, common fairness criteria in general do not promote improvement over time, and may in fact cause harm in cases where an unconstrained objective would not. We completely characterize the delayed impact of three standard criteria, contrasting the regimes in which these exhibit qualitatively different behavior. In addition, we find that a natural form of measurement error broadens the regime in which fairness criteria perform favorably. Our results highlight the importance of measurement and temporal modeling in the evaluation of fairness criteria, suggesting a range of new challenges and trade-offs.

Saliency methods have emerged as a popular tool to highlight features in an input deemed relevant for the prediction of a learned model. Several saliency methods have been proposed, often guided by visual appeal on image data. In this work, we propose an actionable methodology to evaluate what kinds of explanations a given method can and cannot provide. We find that reliance, solely, on visual assessment can be misleading. Through extensive experiments we show that some existing saliency methods are independent both of the model and of the data generating process. Consequently, methods that fail the proposed tests are inadequate for tasks that are sensitive to either data or model, such as, finding outliers in the data, explaining the relationship between inputs and outputs that the model learned, and debugging the model. We interpret our findings through an analogy with edge detection in images, a technique that requires neither training data nor model. Theory in the case of a linear model and a single-layer convolutional neural network supports our experimental findings.

We prove that stochastic gradient descent efficiently converges to the global optimizer of the maximum likelihood objective of an unknown linear time-invariant dynamical system from a sequence of noisy observations generated by the system. Even though the objective function is non-convex, we provide polynomial running time and sample complexity bounds under strong but natural assumptions. Linear systems identification has been studied for many decades, yet, to the best of our knowledge, these are the first polynomial guarantees for the problem we consider.

Recent work on fairness in machine learning has focused on various statistical discrimination criteria and how they trade off. Most of these criteria are observational: They depend only on the joint distribution of predictor, protected attribute, features, and outcome. While convenient to work with, observational criteria have severe inherent limitations that prevent them from resolving matters of fairness conclusively. Going beyond observational criteria, we frame the problem of discrimination based on protected attributes in the language of causal reasoning. This viewpoint shifts attention from "What is the right fairness criterion?" to "What do we want to assume about our model of the causal data generating process?" Through the lens of causality, we make several contributions. First, we crisply articulate why and when observational criteria fail, thus formalizing what was before a matter of opinion. Second, our approach exposes previously ignored subtleties and why they are fundamental to the problem. Finally, we put forward natural causal non-discrimination criteria and develop algorithms that satisfy them.

We show that parametric models trained by a stochastic gradient method (SGM) with few iterations have vanishing generalization error. We prove our results by arguing that SGM is algorithmically stable in the sense of Bousquet and Elisseeff. Our analysis only employs elementary tools from convex and continuous optimization. We derive stability bounds for both convex and non-convex optimization under standard Lipschitz and smoothness assumptions. Applying our results to the convex case, we provide new insights for why multiple epochs of stochastic gradient methods generalize well in practice. In the non-convex case, we give a new interpretation of common practices in neural networks, and formally show that popular techniques for training large deep models are indeed stability-promoting. Our findings conceptually underscore the importance of reducing training time beyond its obvious benefit.

Machine learning relies on the assumption that unseen test instances of a classification problem follow the same distribution as observed training data. However, this principle can break down when machine learning is used to make important decisions about the welfare (employment, education, health) of strategic individuals. Knowing information about the classifier, such individuals may manipulate their attributes in order to obtain a better classification outcome. As a result of this behavior -- often referred to as gaming -- the performance of the classifier may deteriorate sharply. Indeed, gaming is a well-known obstacle for using machine learning methods in practice; in financial policy-making, the problem is widely known as Goodhart's law. In this paper, we formalize the problem, and pursue algorithms for learning classifiers that are robust to gaming.

This special section comprises seven fully refereed papers whose extended abstracts were presented at the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013) in Berkeley, California, October 27--29, 2013. The unrefereed conference versions of these papers were published by IEEE in the FOCS 2013 proceedings. The regular conference program consisted of 79 papers chosen from among 281 submissions. These were selected by a program committee consisting of Alexandr Andoni, Maria-Florina Balcan, Kenneth L. Clarkson, Moritz Hardt, Johan Hast\aad, Nicole Immorlica, Yuval Ishai, Yael Tauman Kalai, Tali Kaufman, Jonathan Kelner, Adam Klivans, Amit Kumar, Yury Makarychev, Raghu Meka, Peter Bro Miltersen, Seffi Naor, Harald Räcke, Omer Reingold, Mohit Singh, Madhu Sudan, Paul Valiant, Virginia Vassilevska Williams, and John Watrous. The program committee was chaired by Omer Reingold. The papers invited to this special section were also selected with the input of the program committee. The seven papers in this section span a broad range of topics, including cryptography, approximation algorithms, dynamic algorithms, hardness of approximation, arithmetic complexity theory, quantum complexity, and learning theory. Each paper underwent an extensive refereeing process; we thank both the authors and the anonymous referees for their efforts. In addition, we would like to thank SICOMP Editor-in-Chief Leonard Schulman and SIAM Senior Publications Coordinator Heather Blythe for their help in preparing this special section.

The organizer of a machine learning competition faces the problem of maintaining an accurate leaderboard that faithfully represents the quality of the best submission of each competing team. What makes this estimation problem particularly challenging is its sequential and adaptive nature. As participants are allowed to repeatedly evaluate their submissions on the leaderboard, they may begin to overfit to the holdout data that supports the leaderboard. Few theoretical results give actionable advice on how to design a reliable leaderboard. Existing approaches therefore often resort to poorly understood heuristics such as limiting the bit precision of answers and the rate of re-submission. In this work, we introduce a notion of leaderboard accuracy tailored to the format of a competition. We introduce a natural algorithm called the Ladder and demonstrate that it simultaneously supports strong theoretical guarantees in a fully adaptive model of estimation, withstands practical adversarial attacks, and achieves high utility on real submission files from a Kaggle competition. Notably, we are able to sidestep a powerful recent hardness result for adaptive risk estimation that rules out algorithms such as ours under a seemingly very similar notion of accuracy. On a practical note, we provide a completely parameter-free variant of our algorithm that can be deployed in a real competition with no tuning required whatsoever.

We investigate the problem of learning an unknown probability distribution over a discrete population from random samples. Our goal is to design efficient algorithms that simultaneously achieve low error in total variation norm while guaranteeing Differential Privacy to the individuals of the population.We describe a general approach that yields near sample-optimal and computationally efficient differentially private estimators for a wide range of well-studied and natural distribution families. Our theoretical results show that for a wide variety of structured distributions there exist private estimation algorithms that are nearly as efficient - both in terms of sample size and running time - as their non-private counterparts. We complement our theoretical guarantees with an experimental evaluation. Our experiments illustrate the speed and accuracy of our private estimators on both synthetic mixture models and a large public data set.

Overfitting is the bane of data analysts, even when data are plentiful. Formal approaches to understanding this problem focus on statistical inference and generalization of individual analysis procedures. Yet the practice of data analysis is an inherently interactive and adaptive process: new analyses and hypotheses are proposed after seeing the results of previous ones, parameters are tuned on the basis of obtained results, and datasets are shared and reused. An investigation of this gap has recently been initiated by the authors in (Dwork et al., 2014), where we focused on the problem of estimating expectations of adaptively chosen functions.In this paper, we give a simple and practical method for reusing a holdout (or testing) set to validate the accuracy of hypotheses produced by a learning algorithm operating on a training set. Reusing a holdout set adaptively multiple times can easily lead to overfitting to the holdout set itself. We give an algorithm that enables the validation of a large number of adaptively chosen hypotheses, while provably avoiding overfitting. We illustrate the advantages of our algorithm over the standard use of the holdout set via a simple synthetic experiment.We also formalize and address the general problem of data reuse in adaptive data analysis. We show how the differential-privacy based approach in (Dwork et al., 2014) is applicable much more broadly to adaptive data analysis. We then show that a simple approach based on description length can also be used to give guarantees of statistical validity in adaptive settings. Finally, we demonstrate that these incomparable approaches can be unified via the notion of approximate max-information that we introduce. This, in particular, allows the preservation of statistical validity guarantees even when an analyst adaptively composes algorithms which have guarantees based on either of the two approaches.

A great deal of effort has been devoted to reducing the risk of spurious scientific discoveries, from the use of sophisticated validation techniques, to deep statistical methods for controlling the false discovery rate in multiple hypothesis testing. However, there is a fundamental disconnect between the theoretical results and the practice of data analysis: the theory of statistical inference assumes a fixed collection of hypotheses to be tested, or learning algorithms to be applied, selected non-adaptively before the data are gathered, whereas in practice data is shared and reused with hypotheses and new analyses being generated on the basis of data exploration and the outcomes of previous analyses.

We consider the problem of identifying the parameters of an unknown mixture of two arbitrary d-dimensional gaussians from a sequence of independent random samples. Our main results are upper and lower bounds giving a computationally efficient moment-based estimator with an optimal convergence rate, thus resolving a problem introduced by Pearson (1894). Denoting by σ2 the variance of the unknown mixture, we prove that Θ(σ12) samples are necessary and sufficient to estimate each parameter up to constant additive error when d=1. Our upper bound extends to arbitrary dimension d>1 up to a (provably necessary) logarithmic loss in d using a novel---yet simple---dimensionality reduction technique. We further identify several interesting special cases where the sample complexity is notably smaller than our optimal worst-case bound. For instance, if the means of the two components are separated by Ω(σ) the sample complexity reduces to O(σ2) and this is again optimal.

Matrix Completion is the problem of recovering an unknown real-valued low-rank matrix from a subsample of its entries. Important recent results show that the problem can be solved efficiently under the assumption that the unknown matrix is incoherent and the subsample is drawn uniformly at random. Are these assumptions necessary? It is well known that Matrix Completion in its full generality is NP-hard. However, little is known if we make additional assumptions such as incoherence and permit the algorithm to output a matrix of slightly higher rank. In this paper we prove that Matrix Completion remains computationally intractable even if the unknown matrix has rank 4 but we are allowed to output any constant rank matrix, and even if additionally we assume that the unknown matrix is incoherent and are shown 90% of the entries. This result relies on the conjectured hardness of the 4-Coloring problem. We also consider the positive semidefinite Matrix Completion problem. Here we show a similar hardness result under the standard assumption that \mathrmP\ne \mathrmNP. Our results greatly narrow the gap between existing feasibility results and computational lower bounds. In particular, we believe that our results give the first complexity-theoretic justification for why distributional assumptions are needed beyond the incoherence assumption in order to obtain positive results. On the technical side, we contribute several new ideas on how to encode hard combinatorial problems in low-rank optimization problems. We hope that these techniques will be helpful in further understanding the computational limits of Matrix Completion and related problems.

We give the first algorithm for Matrix Completion that achieves running time and sample complexity that is polynomial in the rank of the unknown target matrix, \emphlinear in the dimension of the matrix, and \emphlogarithmic in the condition number of the matrix. To the best of our knowledge, all previous algorithms either incurred a quadratic dependence on the condition number of the unknown matrix or a quadratic dependence on the dimension of the matrix. Our algorithm is based on a novel extension of Alternating Minimization which we show has theoretical guarantees under standard assumptions even in the presence of noise.

Alternating minimization is a widely used and empirically successful heuristic for matrix completion and related low-rank optimization problems. Theoretical guarantees for alternating minimization have been hard to come by and are still poorly understood. This is in part because the heuristic is iterative and non-convex in nature. We give a new algorithm based on alternating minimization that provably recovers an unknown low-rank matrix from a random subsample of its entries under a standard incoherence assumption. Our results reduce the sample size requirements of the alternating minimization approach by at least a quartic factor in the rank and the condition number of the unknown matrix. These improvements apply even if the matrix is only close to low-rank in the Frobenius norm. Our algorithm runs in nearly linear time in the dimension of the matrix and, in a broad range of parameters, gives the strongest sample bounds among all subquadratic time algorithms that we are aware of. Underlying our work is a new robust convergence analysis of the well-known Power Method for computing the dominant singular vectors of a matrix. This viewpoint leads to a conceptually simple understanding of alternating minimization. In addition, we contribute a new technique for controlling the coherence of intermediate solutions arising in iterative algorithms based on a smoothed analysis of the QR factorization. These techniques may be of interest beyond their application here.

We show that, under a standard hardness assumption, there is no computationally efficient algorithm that given n samples from an unknown distribution can give valid answers to n3+o(1) adaptively chosen statistical queries. A statistical query asks for the expectation of a predicate over the underlying distribution, and an answer to a statistical query is valid if it is "close" to the correct expectation over the distribution. Our result stands in stark contrast to the well known fact that exponentially many statistical queries can be answered validly and efficiently if the queries are chosen non-adaptively (no query may depend on the answers to previous queries). Moreover, Dwork et al. [1], showed how to accurately answer exponentially many adaptively chosen statistical queries via a computationally inefficient algorithm. They also gave efficient algorithm that can answer nearly n2 adaptively chosen queries, which shows our result is almost quantitatively tight. Conceptually, our result demonstrates that achieving statistical validity alone can be a source of computational intractability in adaptive settings. For example, in the modern large collaborative research environment, data analysts typically choose a particular approach based on previous findings. False discovery occurs if a research finding is supported by the data but not by the underlying distribution. While the study of preventing false discovery in Statistics is decades old, to the best of our knowledge our result is the first to demonstrate a computational barrier. In particular, our result suggests that the perceived difficulty of preventing false discovery in today's collaborative research environment may be inherent.

We consider a fundamental problem in unsupervised learning called subspace recovery: given a collection of m points in R^n, if many but not necessarily all of these points are contained in a d-dimensional subspace T can we find it? The points contained in T are called inliers and the remaining points are outliers. This problem has received considerable attention in computer science and in statistics. Yet efficient algorithms from computer science are not robust to adversarial outliers, and the estimators from robust statistics are hard to compute in high dimensions. This is a serious and persistent issue not just in this application, but for many other problems in unsupervised learning. Are there algorithms for subspace recovery that are both robust to outliers and efficient? We give an algorithm that finds T when it contains more than a d/n fraction of the points. Hence, for say d = n/2 this estimator is both easy to compute and well-behaved when there are a constant fraction of outliers. We prove that it is small set expansion hard to find T when the fraction of errors is any larger and so our estimator is an optimal compromise between efficiency and robustness. In fact, this basic problem has a surprising number of connections to other areas including small set expansion, matroid theory and functional analysis that we make use of here.

We consider differentially private approximate singular vector computation. Known worst-case lower bounds show that the error of any differentially private algorithm must scale polynomially with the dimension of the singular vector. We are able to replace this dependence on the dimension by a natural parameter known as the coherence of the matrix that is often observed to be significantly smaller than the dimension both theoretically and empirically. We also prove a matching lower bound showing that our guarantee is nearly optimal for every setting of the coherence parameter. Notably, we achieve our bounds by giving a robust analysis of the well-known power iteration algorithm, which may be of independent interest. Our algorithm also leads to improvements in worst-case settings and to better low-rank approximations in the spectral norm.

Linear sketches are powerful algorithmic tools that turn an n-dimensional input into a concise lower-dimensional representation via a linear transformation. Such sketches have seen a wide range of applications including norm estimation over data streams, compressed sensing, and distributed computing. In almost any realistic setting, however, a linear sketch faces the possibility that its inputs are correlated with previous evaluations of the sketch. Known techniques no longer guarantee the correctness of the output in the presence of such correlations. We therefore ask: Are linear sketches inherently non-robust to adaptively chosen inputs? We give a strong affirmative answer to this question. Specifically, we show that no linear sketch approximates the Euclidean norm of its input to within an arbitrary multiplicative approximation factor on a polynomial number of adaptively chosen inputs. The result remains true even if the dimension of the sketch is d=n-o(n) and the sketch is given unbounded computation time. Our result is based on an algorithm with running time polynomial in d that adaptively finds a distribution over inputs on which the sketch is incorrect with constant probability. Our result implies several corollaries for related problems including lp-norm estimation and compressed sensing. Notably, we resolve an open problem in compressed sensing regarding the feasibility of l2/l2-recovery guarantees in presence of computationally bounded adversaries.

Suppose we would like to know all answers to a set of statistical queries $C$ on a data set up to small error, but we can access the data itself only by using statistical queries. A trivial solution is to exhaustively ask all queries in $C$. In this paper, we investigate how and when we can do better than this naïve approach. We show that the number of statistical queries necessary and sufficient for this task is---up to polynomial factors---equal to the agnostic learning complexity of $C$ in Kearns' statistical query (SQ) model. This gives a complete answer to the question when run-time is not a concern. We then show that the problem can be solved efficiently (allowing arbitrary error on a small fraction of queries) whenever the answers to $C$ can be described by a submodular function. This includes many natural concept classes, such as graph cuts and Boolean disjunctions and conjunctions. These results are interesting not only from a learning theoretic point of view, but also from the perspective of privacy-preserving data analysis. In this context, our second result leads to an algorithm that efficiently releases differentially private answers to all Boolean conjunctions with 1% average error. This presents significant progress on a key open problem in privacy-preserving data analysis. Our first result, on the other hand, gives unconditional lower bounds on any differentially private algorithm that admits a (potentially non--privacy-preserving) implementation using only statistical queries. Not only our algorithms but also most known private algorithms can be implemented using only statistical queries and hence are constrained by these lower bounds. Our result therefore isolates the complexity of agnostic learning in the SQ model as a new barrier in the design of differentially private algorithms.

We study fairness in classification, where individuals are classified, e.g., admitted to a university, and the goal is to prevent discrimination against individuals based on their membership in some group, while maintaining utility for the classifier (the university). The main conceptual contribution of this paper is a framework for fair classification comprising (1) a (hypothetical) task-specific metric for determining the degree to which individuals are similar with respect to the classification task at hand; (2) an algorithm for maximizing utility subject to the fairness constraint, that similar individuals are treated similarly. We also present an adaptation of our approach to achieve the complementary goal of "fair affirmative action," which guarantees statistical parity (i.e., the demographics of the set of individuals receiving any classification are the same as the demographics of the underlying population), while treating similar individuals as similarly as possible. Finally, we discuss the relationship of fairness to privacy: when fairness implies privacy, and how tools developed in the context of differential privacy may be applied to fairness.

We initiate a principled study of graph densification. Given a graph G the goal of graph densification is to come up with another graph H that has significantly more edges than G but nevertheless approximates G well with respect to some set of test functions. In this paper we focus on the case of cut and spectral approximations.

We present a new algorithm for differentially private data release, based on a simple combination of the Exponential Mechanism with the Multiplicative Weights update rule. Our MWEM algorithm achieves what are the best known and nearly optimal theoretical guarantees, while at the same time being simple to implement and experimentally more accurate on actual data sets than existing techniques.

This work considers computationally efficient privacy-preserving data release. We study the task of analyzing a database containing sensitive information about individual participants. Given a set of statistical queries on the data, we want to release approximate answers to the queries while also guaranteeing differential privacy—protecting each participant's sensitive data. Our focus is on computationally efficient data release algorithms; we seek algorithms whose running time is polynomial, or at least sub-exponential, in the data dimensionality. Our primary contribution is a computationally efficient reduction from differentially private data release for a class of counting queries, to learning thresholded sums of predicates from a related class. We instantiate this general reduction with algorithms for learning thresholds, obtaining new results for differentially private data release. As two examples, taking {0, 1}d to be the data domain (of dimension d), we obtain differentially private algorithms for: 1. Releasing all k-way conjunction counting queries (or k-way contingency tables). For any given k, the resulting data release algorithm has bounded error as long as the database is of size at least (ignoring the dependence on other parameters). The running time is polynomial in the database size. The best sub-exponential time algorithms known prior to our work required a database of size Õ (dk/2) [Dwork McSherry Nissim and Smith 2006]. 2. Releasing any family of counting queries that is specified by a constant depth AC0 predicate. This algorithm releases accurate answers to a (1 − γ)-fraction of the queries in the family. For any γ ≥ quasipoly (1/d), the algorithm has bounded error as long as the database is of size at least quasipoly(d) (again ignoring the dependence on other parameters). The running time is quasipoly(d). The first learning algorithm uses techniques for representing thresholded sums of predicates as lowdegree polynomial threshold functions. The second learning algorithm is based on a result of Jackson Klivans and Servedio [JKS 2002], and utilizes Fourier analysis of the database viewed as a function mapping queries to answers.

Computing accurate low rank approximations of large matrices is a fundamental data mining task. In many applications however the matrix contains sensitive information about individuals. In such case we would like to release a low rank approximation that satisfies a strong privacy guarantee such as differential privacy. Unfortunately, to date the best known algorithm for this task that satisfies differential privacy is based on naive input perturbation or randomized response: Each entry of the matrix is perturbed independently by a sufficiently large random noise variable, a low rank approximation is then computed on the resulting matrix. We give (the first) significant improvements in accuracy over randomized response under the natural and necessary assumption that the matrix has low coherence. Our algorithm is also very efficient and finds a constant rank approximation of an m x n matrix in time O(mn). Note that even generating the noise matrix required for randomized response already requires time O(mn).

We initiate a study of when the value of mathematical relaxations such as linear and semi-definite programs for constraint satisfaction problems (CSPs) is approximately preserved when restricting the instance to a sub-instance induced by a small random subsample of the variables. Let C be a family of CSPs such as 3SAT, Max-Cut, etc., and let П be a mathematical program that is a relaxation for C, in the sense that for every instance P ∊ C, П(P) is a number in [0, 1] upper bounding the maximum fraction of satisfiable constraints of P. Loosely speaking, we say that subsampling holds for C and П if for every sufficiently dense instance P ∊ C and every ε > 0, if we let P′ be the instance obtained by restricting P to a sufficiently large constant number of variables, then П(P′) ∊ (1 ± ε)П(P). We say that weak subsampling holds if the above guarantee is replaced with П(P′) = 1 − Θ(γ) whenever П(P) = 1 − γ, where Θ hides only absolute constants. We obtain both positive and negative results, showing that: 1. Subsampling holds for the BasicLP and BasicSDP programs. BasicSDP is a variant of the semi-definite program considered by Raghavendra (2008), who showed it gives an optimal approximation factor for every constraint-satisfaction problem under the unique games conjecture. BasicLP is the linear programming analog of BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of unique games type. 3. There are non-unique CSPs for which even weak subsampling fails for the above tighter semi-definite programs. Also there are unique CSPs for which (even weak) subsampling fails for the Sherali-Adams linear programming hierarchy. As a corollary of our weak subsampling for strong semi-definite programs, we obtain a polynomial-time algorithm to certify that random geometric graphs (of the type considered by Feige and Schechtman, 2002) of max-cut value 1 − γ have a cut value at most 1 − γ/10. More generally, our results give an approach to obtaining average-case algorithms for CSPs using semi-definite programming hierarchies.

Suppose we would like to know all answers to a set of statistical queries C on a data set up to small error, but we can only access the data itself using statistical queries. A trivial solution is to exhaustively ask all queries in C. Can we do any better? We show that the number of statistical queries necessary and sufficient for this task is---up to polynomial factors---equal to the agnostic learning complexity of C in Kearns' statistical query (SQ)model. This gives a complete answer to the question when running time is not a concern.

We consider statistical data analysis in the interactive setting. In this setting a trusted curator maintains a database of sensitive information about individual participants, and releases privacy-preserving answers to queries as they arrive. Our primary contribution is a new differentially private multiplicative weights mechanism for answering a large number of interactive counting (or linear) queries that arrive online and may be adaptively chosen. This is the first mechanism with worst-case accuracy guarantees that can answer large numbers of interactive queries and is efficient (in terms of the runtime's dependence on the data universe size). The error is asymptotically optimal in its dependence on the number of participants, and depends only logarithmically on the number of queries being answered. The running time is nearly linear in the size of the data universe. As a further contribution, when we relax the utility requirement and require accuracy only for databases drawn from a rich class of databases, we obtain exponential improvements in running time. Even in this relaxed setting we continue to guarantee privacy for any input database. Only the utility requirement is relaxed. Specifically, we show that when the input database is drawn from a smooth distribution - a distribution that does not place too much weight on any single data item - accuracy remains as above, and the running time becomes poly-logarithmic in the data universe size. The main technical contributions are the application of multiplicative weights techniques to the differential privacy setting, a new privacy analysis for the interactive setting, and a technique for reducing data dimensionality for databases drawn from smooth distributions.

We consider the noise complexity of differentially private mechanisms in the setting where the user asks d linear queries f:Rn -> R non-adaptively. Here, the database is represented by a vector in R and proximity between databases is measured in the l1-metric. We show that the noise complexity is determined by two geometric parameters associated with the set of queries. We use this connection to give tight upper and lower bounds on the noise complexity for any d ≤ n. We show that for d random linear queries of sensitivity 1, it is necessary and sufficient to add l2-error Θ(min d√d/ε,d√(log (n/d))/ε) to achieve ε-differential privacy. Assuming the truth of a deep conjecture from convex geometry, known as the Hyperplane conjecture, we can extend our results to arbitrary linear queries giving nearly matching upper and lower bounds.

We give a simple, more efficient and uniform proof of the hard-core lemma, a fundamental result in complexity theory with applications in machine learning and cryptography. Our result follows from the connection between boosting algorithms and hard-core set constructions discovered by Klivans and Servedio [11]. Informally stated, our result is the following: suppose we fix a family of boolean functions. Assume there is an efficient algorithm which for every input length and every smooth distribution (i.e. one that doesn't assign too much weight to any single input) over the inputs produces a circuit such that the circuit computes the boolean function noticeably better than random. Then, there is an efficient algorithm which for every input length produces a circuit that computes the function correctly on almost all inputs. Our algorithm significantly simplifies previous proofs of the uniform and the non-uniform hard-core lemma, while matching or improving the previously best known parameters. The algorithm uses a generalized multiplicative update rule combined with a natural notion of approximate Bregman projection. Bregman projections are widely used in convex optimization and machine learning. We present an algorithm which efficiently approximates the Bregman projection onto the set of high density measures when the Kullback-Leibler divergence is used as a distance function. Our algorithm has a logarithmic runtime over any domain from which we can efficiently sample. High density measures correspond to smooth distributions which arise naturally, for instance, in the context of online learning. Hence, our technique may be of independent interest.

We show a connection between the semidefinite relaxation of unique games and their behavior under parallel repetition. Specifically,denoting by val(G) the value of a two-prover unique game G, andby sdpval(G) the value of a natural semidefinite program to approximate val(G), we prove that for every l epsi N, if sdpval(G) ges 1-delta, then val(Gl) ges 1-radicsldelta. Here, Gldenotes the l-fold parallel repetition of G, and s=O(log(k/delta)), where k denotes the alphabet size of the game. For the special case where G is an XOR game (i.e., k=2), we obtain the same bound but with s as an absolute constant. Our bounds on s are optimal up to a factor of O(log(1/delta)). For games with a significant gap between the quantities val(G) and sdpval(G), our result implies that val(Gl) may be much larger than val(G)l, giving a counterexample to the strong parallel repetition conjecture. In a recent breakthrough, Raz (FOCS'08) has shown such an example using the max-cut game on oddcycles. Our results are based on a generalization of his techniques.