Synthetically-generated data plays an increasingly larger role in training large language models. However, while synthetic data has been found to be useful, studies have also shown that without proper curation it can cause LLM performance to plateau, or even "collapse", after many training iterations. In this paper, we formalize this question and develop a theoretical framework to investigate how much curation is needed in order to ensure that LLM performance continually improves. Our analysis is inspired by boosting, a classic machine learning technique that leverages a very weak learning algorithm to produce an arbitrarily good classifier. The approach we analyze subsumes many recently proposed methods for training LLMs on synthetic data, and thus our analysis sheds light on why they are successful, and also suggests opportunities for future improvement. We present experiments that validate our theory, and show that dynamically focusing labeling resources on the most challenging examples --- in much the same way that boosting focuses the efforts of the weak learner --- leads to improved performance.

When applying differential privacy to sensitive data, we can often improve performance using external information such as other sensitive data, public data, or human priors. We propose to use the learning-augmented algorithms (or algorithms with predictions) framework---previously applied largely to improve time complexity or competitive ratios---as a powerful way of designing and analyzing privacy-preserving methods that can take advantage of such external information to improve utility. This idea is instantiated on the important task of multiple quantile release, for which we derive error guarantees that scale with a natural measure of prediction quality while (almost) recovering state-of-the-art prediction-independent guarantees. Our analysis enjoys several advantages, including minimal assumptions about the data, a natural way of adding robustness, and the provision of useful surrogate losses for two novel ''meta'' algorithms that learn predictions from other (potentially sensitive) data. We conclude with experiments on challenging tasks demonstrating that learning predictions across one or more instances can lead to large error reductions while preserving privacy.

Consider a setting where we wish to automate an expensive task with a machine learning algorithm using a limited labeling resource. In such settings, examples routed for labeling are often out of scope for the machine learning algorithm. For example, in a spam detection setting, human reviewers not only provide labeled data but are such high-quality detectors of spam that examples routed to them no longer require machine evaluation. As a consequence, the distribution of examples routed to the machine is intimately tied to the process generating labels. We introduce a formalization of this setting, and give an algorithm that simultaneously learns a model and decides when to request a label by leveraging ideas from both the abstention and active learning literatures. We prove an upper bound on the algorithm's label complexity and a matching lower bound for any algorithm in this setting. We conduct a thorough set of experiments including an ablation study to test different components of our algorithm. We demonstrate the effectiveness of an efficient version of our algorithm over margin sampling on a variety of datasets.

We propose and analyze algorithms to solve a range of learning tasks under user-level differential privacy constraints. Rather than guaranteeing only the privacy of individual samples, user-level DP protects a user's entire contribution ($m \ge 1$ samples), providing more stringent but more realistic protection against information leaks. We show that for high-dimensional meanestimation, empirical risk minimization with smooth losses, stochastic convex optimization, and learning hypothesis classes with finite metric entropy, the privacy cost decreases as $O(1/\sqrt{m})$ as users provide more samples. In contrast, when increasing the number of users $n$, the privacy cost decreases at a faster $O(1/n)$ rate. We complement these results with lower bounds showing the minimax optimality of our algorithms for mean estimation and stochastic convex optimization. Our algorithms rely on novel techniques for private mean estimation in arbitrary dimension with error scaling as the concentration radius $\tau$ of the distribution rather than the entire range.

Online active learning (AL) algorithms often assume immediate access to a label once a query has been made. However, due to practical constraints, the labels of these queried examples are generally only available in “batches”. In this work, we present an analysis for a generic class of batch online AL algorithms, which reveals that the effects of batching are in fact mild and only result in an additional label complexity term that is quasilinear in the batch size. To our knowledge, this provides the first theoretical justification for such algorithms and we show how they can be applied to batch variants of three canonical online AL algorithms: IWAL, ORIWAL, and DHM. Finally, we also present empirical results across several benchmark datasets that corroborate these theoretical insights.

A centrally differentially private algorithm maps raw data to differentially private outputs. In contrast, a locally differentially private algorithm may only access data through public interaction with data holders, and this interaction must be a differentially private function of the data. We study the intermediate model of \emph{pan-privacy}. Unlike a locally private algorithm, a pan-private algorithm receives data in the clear. Unlike a centrally private algorithm, the algorithm receives data one element at a time and must maintain a differentially private internal state while processing this stream. First, we show that pan-privacy against multiple intrusions on the internal state is equivalent to sequentially interactive local privacy. Next, we contextualize pan-privacy against a single intrusion by analyzing the sample complexity of uniformity testing over domain $[k]$. Focusing on the dependence on $k$, centrally private uniformity testing has sample complexity $\Theta(\sqrt{k})$, while noninteractive locally private uniformity testing has sample complexity $\Theta(k)$. We show that the sample complexity of pan-private uniformity testing is $\Theta(k^{2/3})$. By a new $\Omega(k)$ lower bound for the sequentially interactive setting, we also separate pan-private from sequentially interactive locally private and multi-intrusion pan-private uniformity testing.

Differentially private learning algorithms protect individual participants in the training dataset by guaranteeing that their presence does not significantly change the resulting model. In order to make this promise, such algorithms need to know the maximum contribution that can be made by a single user: the more data an individual can contribute, the more noise will need to be added to protect them. While most existing analyses assume that the maximum contribution is known and fixed in advance{—}indeed, it is often assumed that each user contributes only a single example{—}we argue that in practice there is a meaningful choice to be made. On the one hand, if we allow users to contribute large amounts of data, we may end up adding excessive noise to protect a few outliers, even when the majority contribute only modestly. On the other hand, limiting users to small contributions keeps noise levels low at the cost of potentially discarding significant amounts of excess data, thus introducing bias. Here, we characterize this trade-off for an empirical risk minimization setting, showing that in general there is a “sweet spot” that depends on measurable properties of the dataset, but that there is also a concrete cost to privacy that cannot be avoided simply by collecting more data.

The covariance matrix of a dataset is a fundamental statistic that can be used for calculating optimum regression weights as well as in many other learning and data analysis settings. For datasets containing private user information, we often want to estimate the covariance matrix in a way that preserves differential privacy. While there are known methods for privately computing the covariance matrix, they all have one of two major shortcomings. Some, like the Gaussian mechanism, only guarantee (epsilon, delta)-differential privacy, leaving a non-trivial probability of privacy failure. Others give strong epsilon-differential privacy guarantees, but are impractical, requiring complicated sampling schemes, and tend to perform poorly on real data. In this work we propose a new epsilon-differentially private algorithm for computing the covariance matrix of a dataset that addresses both of these limitations. We show that it has lower error than existing state-of-the-art approaches, both analytically and empirically. In addition, the algorithm is significantly less complicated than other methods and can be efficiently implemented with rejection sampling.

We introduce a novel repeated Inverse Reinforcement Learning problem: the agent has to act on behalf of a human in a sequence of tasks and wishes to minimize the number of tasks that it surprises the human by acting suboptimally with respect to how the human would have acted. Each time the human is surprised, the agent is provided a demonstration of the desired behavior by the human. We formalize this problem, including how the sequence of tasks is chosen, in a few different ways and provide some foundational results.

We consider the \emph{Threshold Bandit} setting, a variant of the classical multi-armed bandit problem in which the reward on each round depends on a piece of side information known as a \emph{threshold value}. The learner selects one of $K$ actions (arms), this action generates a random sample from a fixed distribution, and the action then receives a unit payoff in the event that this sample exceeds the threshold value. We consider two versions of this problem, the \emph{uncensored} and \emph{censored} case, that determine whether the sample is always observed or only when the threshold is not met. Using new tools to understand the popular UCB algorithm, we show that the uncensored case is essentially no more difficult than the classical multi-armed bandit setting. Finally we show that the censored case exhibits more challenges, but we give guarantees in the event that the sequence of threshold values is generated optimistically.

Credit networks provide a flexible model of distributed trust, which supports transactions between untrusted counterparties through paths of intermediaries. We extend this model by introducing interest rates (prices on lines of credit), both as a means to incentivize credit issuance and to provide a framework for modeling networks of financial relationships. Including interest rates poses a new constraint on transactions, as intermediaries will route payments only if the interest received covers any interest paid. We account for these constraints in an efficient algorithm for finding the maximum transaction flow between two agents in a financial network. There are generally many feasible payment paths serving a given transaction, and we show that the policy for selecting among such paths can have a substantial effect on liquidity, as measured by steady-state probability of transaction success. Finally, we consider the situation where the transaction source can choose among heuristic path selection mechanisms, in order to maximize their payoff. Through empirical game-theoretic analysis, we find that routing is inefficient due to the positive externality of choices promoting network liquidity. However, agent choices do reflect some consideration of overall network liquidity, in addition to their own interest payments.

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

We consider a budgeted variant of the problem of learning from expert advice with N experts. Each queried expert incurs a cost and there is a given budget B on the total cost of experts that can be queried in any prediction round. We provide an online learning algorithm for this setting with regret after T prediction rounds bounded by O(sqrt(C log(N)T/B)), where C is the total cost of all experts. We complement this upper bound with a nearly matching lower bound Omega(sqrt(CT/B)) on the regret of any algorithm for this problem. We also provide experimental validation of our algorithm.

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

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

Inspired by real-time ad exchanges for online display advertising, we consider the problem of inferring a buyer's value distribution for a good when the buyer is repeatedly interacting with a seller through a posted-price mechanism. We model the buyer as a strategic agent, whose goal is to maximize her long-term surplus, and we are interested in mechanisms that maximize the seller's long-term revenue. We present seller algorithms that are no-regret when the buyer discounts her future surplus --- i.e. the buyer prefers showing advertisements to users sooner rather than later. We also give a lower bound on regret that increases as the buyer's discounting weakens and shows, in particular, that any seller algorithm will suffer linear regret if there is no discounting.