Gradient equilibrium (GEQ) is a recently introduced online optimization framework that generalizes first-order stationarity from offline optimization, and abstracts problems like online conformal prediction. While GEQ has curious similarities with known online learning frameworks, such as regret minimization, prior work has shown that GEQ error and regret are incomparable as objectives, leaving open a precise understanding of how GEQ fits into the broader online learning landscape. In this work, we show that GEQ is equivalent to Blackwell approachability in the algorithmic sense. That is, a Blackwell approachability problem can always be solved using queries to a black-box GEQ oracle, with no asymptotic loss in the oracle’s error rate, and vice versa. Taken together with known equivalences between approachability, regret minimization, and calibration, these results imply an equivalence between GEQ and these frameworks, as well. Hence, while GEQ guarantees are semantically different from known online learning guarantees, GEQ algorithms are equally powerful primitives as classical regret minimization and calibration algorithms. Our reductions are efficient and can be used to transfer refined guarantees, such as optimism and strong adaptivity, from regret minimization to GEQ. Our techniques can also be used to identify necessary and sufficient conditions for GEQ, and to establish reductions between different notions of GEQ with unconstrained and constrained decision sets.

A canonical desideratum for prediction problems is that performance guarantees should hold not just on average over the population, but also for meaningful subpopulations within the overall population. But what constitutes a meaningful subpopulation? In this work, we take the perspective that relevant subpopulations should be defined with respect to the clusters that naturally emerge from the distribution of individuals for which predictions are being made. In this view, a population refers to a mixture model whose components constitute the relevant subpopulations. We suggest two formalisms for capturing per-subgroup guarantees: first, by attributing each individual to the component from which they were most likely drawn, given their features; and second, by attributing each individual to all components in proportion to their relative likelihood of having been drawn from each component. Using online calibration as a case study, we study a multi-objective algorithm that provides guarantees for each of these formalisms by handling all plausible underlying subpopulation structures simultaneously, and achieve an $O(T^{1/2})$ rate even when the subpopulations are not well-separated. In comparison, the more natural cluster-then-predict approach that first recovers the structure of the subpopulations and then makes predictions suffers from a $O(T^{2/3})$ rate and requires the subpopulations to be separable. Along the way, we prove that providing per-subgroup calibration guarantees for underlying clusters can be easier than learning the clusters: separation between median subgroup features is required for the latter but not the former.

After pre-training, large language models are aligned with human preferences based on pairwise comparisons. State-of-the-art alignment methods (such as PPO-based RLHF and DPO) are built on the assumption of aligning with a single preference model, despite being deployed in settings where users have diverse preferences. As a result, it is not even clear that these alignment methods produce models that satisfy users \emph{on average} --- a minimal requirement for pluralistic alignment. Drawing on social choice theory and modeling users' comparisons through individual Bradley-Terry (BT) models, we introduce an alignment method's \emph{distortion}: the worst-case ratio between the optimal achievable average utility, and the average utility of the learned policy. The notion of distortion helps draw sharp distinctions between alignment methods: \emph{Nash Learning from Human Feedback} achieves the minimax optimal distortion of $(\frac{1}{2} + o(1)) \cdot \beta$ (for the BT temperature $\beta$), robustly across utility distributions, distributions of comparison pairs, and permissible KL divergences from the reference policy. RLHF and DPO, by contrast, suffer $\geq (1 - o(1)) \cdot \beta$ distortion already without a KL constraint, and $e^{\Omega(\beta)}$ or even unbounded distortion in the full setting, depending on how comparison pairs are sampled.

We study the tradeoff between sample complexity and round complexity in *on-demand sampling*, where the learning algorithm adaptively samples from $k$ distributions over a limited number of rounds. In the realizable setting of Multi-Distribution Learning (MDL), we show that the optimal sample complexity of an $r$-round algorithm scales approximately as $dk^{\Theta(1/r)} / \epsilon$. For the general agnostic case, we present an algorithm that achieves near-optimal sample complexity of $\widetilde O((d + k) / \epsilon^2)$ within $\widetilde O(\sqrt{k})$ rounds. Of independent interest, we introduce a new framework, Optimization via On-Demand Sampling (OODS), which abstracts the sample-adaptivity tradeoff and captures most existing MDL algorithms. We establish nearly tight bounds on the round complexity in the OODS setting. The upper bounds directly yield the $\widetilde O(\sqrt{k})$-round algorithm for agnostic MDL, while the lower bounds imply that achieving sub-polynomial round complexity would require fundamentally new techniques that bypass the inherent hardness of OODS.

Finetuning provides a scalable and cost-effective means of customizing language models for specific tasks or response styles, with greater reliability than prompting or in-context learning. In contrast, the conventional wisdom is that injecting knowledge via finetuning results in brittle performance and poor generalization. We argue that the dichotomy of "task customization" (e.g., instruction tuning) and "knowledge injection" (e.g., teaching new facts) is a distinction without a difference. We instead identify concrete factors that explain the heterogeneous effectiveness observed with finetuning. To this end, we conduct a large-scale experimental study of finetuning the frontier Gemini v1.5 model family on a spectrum of datasets that are artificially engineered to interpolate between the strengths and failure modes of finetuning. Our findings indicate that question-answer training data formats provide much stronger knowledge generalization than document/article-style training data, numerical information can be harder for finetuning to retain than categorical information, and models struggle to apply finetuned knowledge during multi-step reasoning even when trained on similar examples---all factors that render ``knowledge injection'' to be especially difficult, even after controlling for considerations like data augmentation and information volume. On the other hand, our findings also indicate that it is not fundamentally more difficult to finetune information about a real-world event than information about writing style.

This paper introduces the concept of leakage-robust Bayesian persuasion. Situated between public Bayesian persuasion and private Bayesian persuasion, leakage-robust persuasion considers a setting where one or more signals privately communicated by a sender to the receivers may be leaked. We study the design of leakage-robust Bayesian persuasion schemes and quantify the price of robustness using two formalisms:

Collaboration is crucial for reaching collective goals. However, its potential for effectiveness is often undermined by the strategic behavior of individual agents — a fact that is captured by a high Price of Stability (PoS) in recent literature [BHPS21]. Implicit in the traditional PoS analysis is the assumption that agents have full knowledge of how their tasks relate to one another. We offer a new perspective on bringing about efficient collaboration across strategic agents using information design. Inspired by the increasingly important role collaboration plays in machine learning (such as platforms for collaborative federated learning and data cooperatives), we propose a framework in which the platform possesses more information about how the agents’ tasks relate to each other than the agents themselves. Our results characterize how and to what degree such platforms can leverage their information advantage and steer strategic agents towards efficient collaboration.

Motivated by the emergence of decentralized machine learning (ML) ecosystems, we study the delegation of data collection. Taking the field of contract theory as our starting point, we design optimal and near-optimal contracts that deal with two fundamental information asymmetries that arise in decentralized ML: uncertainty in the assessment of model quality and uncertainty regarding the optimal performance of any model. We show that a principal can cope with such asymmetry via simple linear contracts that achieve $1-1/\epsilon$ fraction of the optimal utility. To address the lack of a priori knowledge regarding the optimal performance, we give a convex program that can adaptively and efficiently compute the optimal contract. We also analyze the optimal utility and linear contracts for the more complex setting of multiple interactions.

A common explanation for negative user impacts of content recommender systems is misalignment between the platform’s objective and user welfare. In this work, we show that misalignment in the platform’s objective is not the only potential cause of unintended impacts on users: even when the platform’s objective is fully aligned with user welfare, the platform’s learning algorithm can induce negative downstream impacts on users. The source of these user impacts is that different pieces of content may generate observable user reactions (feedback information) at different rates; these feedback rates may correlate with content properties, such as controversiality or demographic similarity of the creator, that affect the user experience. Since differences in feedback rates can impact how often the learning algorithm engages with different content, the learning algorithm may inadvertently promote content with certain such properties. Using the multi-armed bandit framework with probabilistic feedback, we examine the relationship between feedback rates and a learning algorithm’s engagement with individual arms for different no-regret algorithms. We prove that no-regret algorithms can exhibit a wide range of dependencies: if the feedback rate of an arm increases, some no-regret algorithms engage with the arm more, some no-regret algorithms engage with the arm less, and other no-regret algorithms engage with the arm approximately the same number of times. From a platform design perspective, our results highlight the importance of looking beyond regret when measuring an algorithm’s performance, and assessing the nature of a learning algorithm’s engagement with different types of content as well as their resulting downstream impacts.

Black-box finetuning is an emerging interface for adapting state-of-the-art language models to user needs. However, such access may also let malicious actors undermine model safety. To demonstrate the challenge of defending finetuning interfaces, we introduce covert malicious finetuning, a method to compromise model safety via finetuning while evading detection. Our method constructs a malicious dataset where every individual datapoint appears innocuous, but finetuning on the dataset teaches the model to respond to encoded harmful requests with encoded harmful responses. Applied to GPT-4, our method produces a finetuned model that acts on harmful instructions 99% of the time and avoids detection by defense mechanisms such as dataset inspection, safety evaluations, and input/output classifiers. Our findings question whether black-box finetuning access can be secured against sophisticated adversaries.

A fundamental shortcoming of the concept of Nash equilibrium is its computational intractability: approximating Nash equilibria in normal-form games is PPAD-hard. In this paper, inspired by the ideas of smoothed analysis, we introduce a relaxed variant of Nash equilibrium called σ-smooth Nash equilibrium, for a {smoothness parameter} σ. In a σ-smooth Nash equilibrium, players only need to achieve utility at least as high as their best deviation to a σ-smooth strategy, which is a distribution that does not put too much mass (as parametrized by σ) on any fixed action. We distinguish two variants of σ-smooth Nash equilibria: strong σ-smooth Nash equilibria, in which players are required to play σ-smooth strategies under equilibrium play, and weak σ-smooth Nash equilibria, where there is no such requirement. We show that both weak and strong σ-smooth Nash equilibria have superior computational properties to Nash equilibria: when σ as well as an approximation parameter ϵ and the number of players are all constants, there is a {constant-time} randomized algorithm to find a weak ϵ-approximate σ-smooth Nash equilibrium in normal-form games. In the same parameter regime, there is a polynomial-time deterministic algorithm to find a strong ϵ-approximate σ-smooth Nash equilibrium in a normal-form game. These results stand in contrast to the optimal algorithm for computing ϵ-approximate Nash equilibria, which cannot run in faster than quasipolynomial-time, subject to complexity-theoretic assumptions. We complement our upper bounds by showing that when either σ or ϵ is an inverse polynomial, finding a weak ϵ-approximate σ-smooth Nash equilibria becomes computationally intractable. Our results are the first to propose a variant of Nash equilibrium which is computationally tractable, allows players to act independently, and which, as we discuss, is justified by an extensive line of work on individual choice behavior in the economics literature.

We study a communication game between a sender and receiver. The sender chooses one of her signals about the state of the world (i.e., an anecdote) and communicates it to the receiver who takes an action affecting both players. The sender and receiver both care about the state of the world but are also influenced by personal preferences, so their ideal actions can differ. We characterize perfect Bayesian equilibria. The sender faces a temptation to persuade: she wants to select a biased anecdote to influence the receiver’s action. Anecdotes are still informative to the receiver (who will debias at equilibrium) but the attempt to persuade comes at the cost of precision. This gives rise to informational homophily where the receiver prefers to listen to like-minded senders because they provide higher-precision signals. Communication becomes polarized when the sender is an expert with access to many signals, with the sender choosing extreme outlier anecdotes at equilibrium (unless preferences are perfectly aligned). This polarization dissipates all the gains from communication with an increasingly well-informed sender when the anecdote distribution is heavy-tailed. Experts therefore face a curse of informedness: receivers will prefer to listen to less-informed senders who cannot pick biased signals as easily.

When learning in strategic environments, a key question is whether agents can overcome uncertainty about their preferences to achieve outcomes they could have achieved absent any uncertainty. Can they do this solely through interactions with each other? We focus this question on the ability of agents to attain the value of their Stackelberg optimal strategy and study the impact of information asymmetry. We study repeated interactions in fully strategic environments where players' actions are decided based on learning algorithms that take into account their observed histories and knowledge of the game. We study the pure Nash equilibria (PNE) of a meta-game where players choose these algorithms as their actions. We demonstrate that if one player has perfect knowledge about the game, then any initial informational gap persists. That is, while there is always a PNE in which the informed agent achieves her Stackelberg value, there is a game where no PNE of the meta-game allows the partially informed player to achieve her Stackelberg value. On the other hand, if both players start with some uncertainty about the game, the quality of information alone does not determine which agent can achieve her Stackelberg value. In this case, the concept of information asymmetry becomes nuanced and depends on the game's structure. Overall, our findings suggest that repeated strategic interactions alone cannot facilitate learning effectively enough to earn an uninformed player her Stackelberg value.

We study calibration measures in a sequential prediction setup. In addition to rewarding accurate predictions (completeness) and penalizing incorrect ones (soundness), an important desideratum of calibration measures is *truthfulness*, a minimal condition for the forecaster not to be incentivized to exploit the system. Formally, a calibration measure is truthful if the forecaster (approximately) minimizes the expected penalty by predicting the conditional expectation of the next outcome, given the prior distribution of outcomes. We conduct a taxonomy of existing calibration measures. Perhaps surprisingly, all of them are far from being truthful. We introduce a new calibration measure termed the *Subsampled Smooth Calibration Error (SSCE)*, which is complete and sound, and under which truthful prediction is optimal up to a constant multiplicative factor. In contrast, under existing calibration measures, there are simple distributions on which a polylogarithmic (or even zero) penalty is achievable, while truthful prediction leads to a polynomial penalty.

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by \(\tfrac{1}{\sigma }\) times that of the uniform distribution; nature then samples an input from this distribution. Here, σ is a parameter that interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold for adaptive adversaries that can base their choice of input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm’s current state; this appears to rule out the standard proof approaches in smoothed analysis. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems: (1) Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of σ-smooth distributions and the hypothesis class has VC dimension d . We bound the regret by \(\tilde{O}(\sqrt {T d\ln (1/\sigma)} + d\ln (T/\sigma))\) and provide a near-matching lower bound. Our result shows that under smoothed analysis, learnability against adaptive adversaries is characterized by the finiteness of the VC dimension. This is as opposed to the worst-case analysis, where online learnability is characterized by Littlestone dimension (which is infinite even in the extremely restricted case of one-dimensional threshold functions). Our results fully answer an open question of Rakhlin et al. [ 64 ]. (2) Online discrepancy minimization: We consider the setting of the online Komlós problem, where the input is generated from an adaptive sequence of σ-smooth and isotropic distributions on the ℓ 2 unit ball. We bound the ℓ ∞ norm of the discrepancy vector by \(\tilde{O}(\ln ^2(\frac{nT}{\sigma }))\) . This is as opposed to the worst-case analysis, where the tight discrepancy bound is \(\Theta (\sqrt {T/n})\) . We show such \(\mathrm{polylog}(nT/\sigma)\) discrepancy guarantees are not achievable for non-isotropic σ-smooth distributions. (3) Dispersion in online optimization: We consider online optimization with piecewise Lipschitz functions where functions with ℓ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is \(({\sigma }/{\sqrt {T\ell }}, \tilde{O}(\sqrt {T\ell }))\) -dispersed. That is, every ball of radius \({\sigma }/{\sqrt {T\ell }}\) is split by \(\tilde{O}(\sqrt {T\ell })\) of the partitions made by these functions. This result matches the dispersion parameters of Balcan et al. [ 13 ] for oblivious smooth adversaries, up to logarithmic factors. On the other hand, worst-case sequences are trivially (0, T )-dispersed. 1

Competition between traditional platforms is known to improve user utility by aligning the platform's actions with user preferences. But to what extent is alignment exhibited in data-driven marketplaces? To study this question from a theoretical perspective, we introduce a duopoly market where platform actions are bandit algorithms and the two platforms compete for user participation. A salient feature of this market is that the quality of recommendations depends on both the bandit algorithm and the amount of data provided by interactions from users. This interdependency between the algorithm performance and the actions of users complicates the structure of market equilibria and their quality in terms of user utility. Our main finding is that competition in this market does not perfectly align market outcomes with user utility. Interestingly, market outcomes exhibit misalignment not only when the platforms have separate data repositories, but also when the platforms have a shared data repository. Nonetheless, the data sharing assumptions impact what mechanism drives misalignment and also affect the specific form of misalignment (e.g. the quality of the best-case and worst-case market outcomes). More broadly, our work illustrates that competition in digital marketplaces has subtle consequences for user utility that merit further investigation.

Multi-distribution learning is a natural generalization of PAC learning to settings with multiple data distributions. There remains a significant gap between the known upper and lower bounds for PAC-learnable classes. In particular, though we understand the sample complexity of learning a VC dimension $d$ class on $k$ distributions to be $O(\epsilon^{-2} \ln(k) (d + k) + \min \{\epsilon^{-1} d k, \epsilon^{-4} \ln(k) d\})$, the best lower bound is $\Omega(\epsilon^{-2}(d + k \ln(k)))$. We discuss recent progress on this problem and some hurdles that are fundamental to the use of game dynamics in statistical learning.

Large language models trained for safety and harmlessness remain susceptible to adversarial misuse, as evidenced by the prevalence of “jailbreak” attacks on early releases of ChatGPT that elicit undesired behavior. Going beyond recognition of the issue, we investigate why such attacks succeed and how they can be created. We hypothesize two failure modes of safety training: competing objectives and mismatched generalization. Competing objectives arise when a model’s capabilities and safety goals conflict, while mismatched generalization occurs when safety training fails to generalize to a domain for which capabilities exist. We use these failure modes to guide jailbreak design and then evaluate state-of-the-art models, including OpenAI’s GPT-4 and Anthropic’s Claude v1.3, against both existing and newly designed attacks. We find that vulnerabilities persist despite the extensive red-teaming and safety-training efforts behind these models. Notably, new attacks utilizing our failure modes succeed on every prompt in a collection of unsafe requests from the models’ red-teaming evaluation sets and outperform existing ad hoc jailbreaks. Our analysis emphasizes the need for safety-capability parity—that safety mechanisms should be as sophisticated as the underlying model—and argues against the idea that scaling alone can resolve these safety failure modes.

We initiate the study of smoothed analysis for the sequential probability assignment problem with contexts. We study information-theoretically optimal minmax rates as well as a framework for algorithmic reduction involving the maximum likelihood estimator oracle. Our approach establishes a general-purpose reduction from minimax rates for sequential probability assignment for smoothed adversaries to minimax rates for transductive learning. This leads to optimal (logarithmic) fast rates for parametric classes and classes with finite VC dimension. On the algorithmic front, we develop an algorithm that efficiently taps into the MLE oracle, for general classes of functions. We show that under general conditions this algorithmic approach yields sublinear regret.

We provide a unifying framework for the design and analysis of multi-calibrated predictors. By placing the multi-calibration problem in the general setting of multi-objective learning---where learning guarantees must hold simultaneously over a set of distributions and loss functions---we exploit connections to game dynamics to achieve state-of-the-art guarantees for a diverse set of multi-calibration learning problems. In addition to shedding light on existing multi-calibration guarantees and greatly simplifying their analysis, our approach also yields improved guarantees, such as error tolerances that scale with the square-root of group size versus the constant tolerances guaranteed by prior works, and improving the complexity of $k$-class multi-calibration by an exponential factor of $k$ versus Gopalan et al.. Beyond multi-calibration, we use these game dynamics to address emerging considerations in the study of group fairness and multi-distribution learning.

In this paper, we introduce a generalization of the standard Stackelberg Games (SGs) framework: _Calibrated Stackelberg Games_. In CSGs, a principal repeatedly interacts with an agent who (contrary to standard SGs) does not have direct access to the principal's action but instead best responds to _calibrated forecasts_ about it. CSG is a powerful modeling tool that goes beyond assuming that agents use ad hoc and highly specified algorithms for interacting in strategic settings to infer the principal's actions and thus more robustly addresses real-life applications that SGs were originally intended to capture. Along with CSGs, we also introduce a stronger notion of calibration, termed _adaptive calibration_, that provides fine-grained any-time calibration guarantees against adversarial sequences. We give a general approach for obtaining adaptive calibration algorithms and specialize them for finite CSGs. In our main technical result, we show that in CSGs, the principal can achieve utility that converges to the optimum Stackelberg value of the game both in _finite_ and _continuous_ settings and that no higher utility is achievable. Two prominent and immediate applications of our results are the settings of learning in Stackelberg Security Games and strategic classification, both against _calibrated_ agents.

As the scale of machine learning models increases, trends such as scaling laws anticipate consistent downstream improvements in predictive accuracy. However, these trends take the perspective of a single model-provider in isolation, while in reality providers often compete with each other for users. In this work, we demonstrate that competition can fundamentally alter the behavior of these scaling trends, even causing overall predictive accuracy across users to be non-monotonic or decreasing with scale. We define a model of competition for classification tasks, and use data representations as a lens for studying the impact of increases in scale. We find many settings where improving data representation quality (as measured by Bayes risk) decreases the overall predictive accuracy across users (i.e., social welfare) for a marketplace of competing model-providers. Our examples range from closed-form formulas in simple settings to simulations with pretrained representations on CIFAR-10. At a conceptual level, our work suggests that favorable scaling trends for individual model-providers need not translate to downstream improvements in social welfare in marketplaces with multiple model providers.

Online learning of revenue-optimal auctions is a fundamental problem in mechanism design without priors. Nevertheless, all the existing positive results assume that the auctioneer optimizes over a parameterized class of auctions, such as pricings and auctions with reserves. This is perhaps not surprising given that natural correlations that occur in online sequences pose a challenge to characterizing a succinct class of revenue-optimal auctions. This has left behind a significant gap in our understanding of online-learnability of general classes of non-parametric auctions.

In online marketplaces, customers have access to hundreds of reviews for a single product. Buyers often use reviews from other customers that share their type---such as height for clothing, skin type for skincare products, and location for outdoor furniture---to estimate their values, which they may not know a priori. Customers with few relevant reviews may hesitate to make a purchase except at a low price, so for the seller, there is a tension between setting high prices and ensuring that there are enough reviews so that buyers can confidently estimate their values. Simultaneously, sellers may use reviews to gauge the demand for items they wish to sell.

We study the stochastic vertex cover problem. In this problem, G = (V, E) is an arbitrary known graph, and G⋆ is an unknown random subgraph of G where each edge e is realized independently with probability p. Edges of G⋆ can only be verified using edge queries. The goal in this problem is to find a minimum vertex cover of G⋆ using a small number of queries.

We study the design of computationally efficient online learning algorithms under smoothed analysis. In this setting, at every step, an adversary generates a sample from an adaptively chosen distribution whose density is upper bounded by $1/\sigma$ times the uniform density. Given access to an offline optimization (ERM) oracle, we give the first computationally efficient online algorithms whose sublinear regret depends only on the pseudo/VC dimension $d$ of the class and the smoothness parameter $\sigma$. In particular, we achieve \emph{oracle-efficient} regret bounds of $ O ( \sqrt{T d\sigma^{-1}} ) $ for learning real-valued functions and $ O ( \sqrt{T d\sigma^{-\frac{1}{2}} } )$ for learning binary-valued functions. Our results establish that online learning is computationally as easy as offline learning, under the smoothed analysis framework. This contrasts the computational separation between online learning with worst-case adversaries and offline learning established by [HK16].Our algorithms also achieve improved bounds for some settings with binary-valued functions and worst-case adversaries. These include an oracle-efficient algorithm with $O ( \sqrt{T(d |\mathcal{X}|)^{1/2} })$ regret that refines the earlier $O ( \sqrt{T|\mathcal{X}|})$ bound of [DS16] for finite domains, and an oracle-efficient algorithm with $O(T^{3/4} d^{1/2})$ regret for the transductive setting.

Societal and real-world considerations such as robustness, fairness, social welfare and multi-agent tradeoffs have given rise to multi-distribution learning paradigms, such as collaborative [Blum et al. 2017], group distributionally robust [Sagawa et al. 2019], and fair federated learning [Mohri et al. 2019]. In each of these settings, a learner seeks to minimize its worstcase loss over a set of $n$ predefined distributions, while using as few samples as possible. In this paper, we establish the optimal sample complexity of these learning paradigms and give algorithms that meet this sample complexity. Importantly, our sample complexity bounds exceed that of the sample complexity of learning a single distribution only by an additive factor of $\frac{n\log(n)}{\epsilon^2}$. These improve upon the best known sample complexity of agnostic federated learning by Mohri et al. 2019 by a multiplicative factor of $n$, the sample complexity of collaborative learning by Nguyen and Zakynthinou 2018 by a multiplicative factor $\frac{\log(n)}{\epsilon^3}$, and give the first sample complexity bounds for the group DRO objective of Sagawa et al. 2019. To achieve optimal sample complexity, our algorithms learn to sample and learn from distributions on demand. Our algorithm design and analysis extends stochastic optimization techniques to solve zero-sum games in a new stochastic setting.

New Framework for Learning Against Long-Lived, Forward-Looking Agents Repeated Stackelberg games are a canonical model for strategic principal-agent interactions. Learning in these games is well studied against myopic agents who greedily maximize their per-round payoff. However, complications arise with nonmyopic agents because they may strategically deviate from best responding to mislead the principal. In “Learning in Stackelberg Games with Nonmyopic Agents,” Haghtalab, Lykouris, Nietert, and Wei provide a general framework that reduces learning in the presence of nonmyopic agents to robust bandit optimization against myopic agents. This leads to a challenge of designing minimally reactive bandit algorithms, which balance the statistical efficiency of the principal’s learning algorithm against its effectiveness at inducing near-best responses. The authors tackle this challenge across problem domains, including security games, dynamic pricing, and strategic classification. Along the way, they uncover a structural property for learning in security games, enabling them to improve the state-of-the-art query complexity with n targets from [Formula: see text] to a near-optimal [Formula: see text].

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by a multiplicative factor from the uniform distribution; nature then samples an input from this distribution. This interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold for adaptive adversaries that can base their choice of an input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm's current state; this appears to rule out the standard proof approaches in smoothed analysis. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems: Online learning, Online discrepancy and Dispersion in online optimization. We show that in these setting, we can get bounds that match bounds we can get for non-adaptive adversaries.

In recent years, federated learning has been embraced as an approach for bringing about collaboration across large populations of learning agents. However, little is known about how collaboration protocols should take agents’ incentives into account when allocating individual resources for communal learning in order to maintain such collaborations. Inspired by game theoretic notions, this paper introduces a framework for incentive-aware learning and data sharing in federated learning. Our stable and envy-free equilibria capture notions of collaboration in the presence of agents interested in meeting their learning objectives while keeping their own sample collection burden low. For example, in an envy-free equilibrium, no agent would wish to swap their sampling burden with any other agent and in a stable equilibrium, no agent would wish to unilaterally reduce their sampling burden. In addition to formalizing this framework, our contributions include characterizing the structural properties of such equilibria, proving when they exist, and showing how they can be computed. Furthermore, we compare the sample complexity of incentive-aware collaboration with that of optimal collaboration when one ignores agents’ incentives.

Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. Although these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this discrepancy as well as increase the modeling scope of submodular optimization. In this work, we consider robust submodular maximization with structured combinatorial constraints and give efficient algorithms with provable guarantees. Our approach is applicable to constraints defined by single or multiple matroids and knapsack as well as distributionally robust criteria. We consider both the offline setting where the data defining the problem are known in advance and the online setting where the input data are revealed over time. For the offline setting, we give a general (nearly) optimal bicriteria approximation algorithm that relies on new extensions of classical algorithms for submodular maximization. For the online version of the problem, we give an algorithm that returns a bicriteria solution with sublinear regret. Summary of Contribution: Constrained submodular maximization is one of the core areas in combinatorial optimization with a wide variety of applications in operations research and computer science. Over the last decades, both communities have been interested on the design and analysis of new algorithms with provable guarantees. Sensor location, influence maximization and data summarization are some of the applications of submodular optimization that lie at the intersection of the aforementioned communities. Particularly, our work focuses on optimizing several submodular functions simultaneously. We provide new insights and algorithms to the offline and online variants of the problem which significantly expand the related literature. At the same time, we provide a computational study that supports our theoretical results.

Motivated by applications such as college admission and insurance rate determination, we study a classification problem where the inputs are controlled by strategic individuals who can modify their features at a cost. A learner can only partially observe the features, and aims to classify individuals with respect to a quality score. The goal is to design a classification mechanism that maximizes the overall quality score in the population, taking any strategic updating into account. When scores are linear and mechanisms can assign their own scores to agents, we show that the optimal classifier is an appropriate projection of the quality score. For the more restrictive task of binary classification via linear thresholds, we construct a (1/4)-approximation to the optimal classifier when the underlying feature distribution is sufficiently smooth and admits an oracle for finding dense regions. We extend our results to settings where the prior distribution is unknown and must be learned from samples.

Practical and pervasive needs for robustness and privacy in algorithms have inspired the design of online adversarial and differentially private learning algorithms. The primary quantity that characterizes learnability in these settings is the Littlestone dimension of the class of hypotheses [Ben-David et al., 2009, Alon et al., 2019]. This characterization is often interpreted as an impossibility result because classes such as linear thresholds and neural networks have infinite Littlestone dimension. In this paper, we apply the framework of smoothed analysis [Spielman and Teng, 2004], in which adversarially chosen inputs are perturbed slightly by nature. We show that fundamentally stronger regret and error guarantees are possible with smoothed adversaries than with worst-case adversaries. In particular, we obtain regret and privacy error bounds that depend only on the VC dimension and the bracketing number of a hypothesis class, and on the magnitudes of the perturbations.

The k -center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances: a 2-approximation for symmetric k -center and an O (log * ( k ))-approximation for the asymmetric version. Therefore, to improve on these ratios, one must go beyond the worst case. In this work, we take this approach and provide strong positive results both for the asymmetric and symmetric k -center problems under a natural input stability (promise) condition called α-perturbation resilience [15], which states that the optimal solution does not change under any α-factor perturbation to the input distances. We provide algorithms that give strong guarantees simultaneously for stable and non-stable instances: Our algorithms always inherit the worst-case guarantees of clustering approximation algorithms and output the optimal solution if the input is 2-perturbation resilient. In particular, we show that if the input is only perturbation resilient on part of the data, our algorithm will return the optimal clusters from the region of the data that is perturbation resilient while achieving the best worst-case approximation guarantee on the remainder of the data. Furthermore, we prove that our result is tight by showing symmetric k -center under (2 − ϵ)-perturbation resilience is hard unless NP = RP . The impact of our results is multifaceted. First, to our knowledge, asymmetric k -center is the first problem that is hard to approximate to any constant factor in the worst case, yet can be optimally solved in polynomial time under perturbation resilience for a constant value of α. This is also the first tight result for any problem under perturbation resilience, i.e., this is the first time the exact value of α for which the problem switches from being NP-hard to efficiently computable has been found. Furthermore, our results illustrate a surprising relationship between symmetric and asymmetric k -center instances under perturbation resilience. Unlike approximation ratio, for which symmetric k -center is easily solved to a factor of 2 but asymmetric k -center cannot be approximated to any constant factor, both symmetric and asymmetric k -center can be solved optimally under resilience to 2-perturbations. Finally, our guarantees in the setting where only part of the data satisfies perturbation resilience make these algorithms more applicable to real-life instances.

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

Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this discrepancy as well as increase the modeling scope of submodular optimization. In this work, we consider robust submodular maximization with structured combinatorial constraints and give efficient algorithms with provable guarantees. Our approach is applicable to constraints defined by single or multiple matroids, knapsack as well as distributionally robust criteria. We consider both the offline setting where the data defining the problem is known in advance as well as the online setting where the input data is revealed over time. For the offline setting, we give a nearly optimal bi-criteria approximation algorithm that relies on new extensions of the classical greedy algorithm. For the online version of the problem, we give an algorithm that returns a bi-criteria solution with sub-linear regret.

The Lazy Shortest Path (LazySP) class consists of motion-planning algorithms that only evaluate edges along candidate shortest paths between the source and target. These algorithms were designed to minimize the number of edge evaluations in settings where edge evaluation dominates the running time of the algorithm such as manipulation in cluttered environments and planning for robots in surgical settings; but how close to optimal are LazySP algorithms in terms of this objective? Our main result is an analytical upper bound, in a probabilistic model, on the number of edge evaluations required by LazySP algorithms; a matching lower bound shows that these algorithms are asymptotically optimal in the worst case.

In this work we study loss functions for learning and evaluating probability distributions over large discrete domains. Unlike classification or regression where a wide variety of loss functions are used, in the distribution learning and density estimation literature, very few losses outside the dominant \emph{log loss} are applied. We aim to understand this fact, taking an axiomatic approach to the design of loss functions for distributions. We start by proposing a set of desirable criteria that any good loss function should satisfy. Intuitively, these criteria require that the loss function faithfully evaluates a candidate distribution, both in expectation and when estimated on a few samples. Interestingly, we observe that \emph{no loss function} possesses all of these criteria. However, one can circumvent this issue by introducing a natural restriction on the set of candidate distributions. Specifically, we require that candidates are \emph{calibrated} with respect to the target distribution, i.e., they may contain less information than the target but otherwise do not significantly distort the truth. We show that, after restricting to this set of distributions, the log loss and a large variety of other losses satisfy the desired criteria. These results pave the way for future investigations of distribution learning that look beyond the log loss, choosing a loss function based on application or domain need.

Voting systems typically treat all voters equally. We argue that perhaps they should not: Voters who have supported good choices in the past should be given higher weight than voters who have supported bad ones. To develop a formal framework for desirable weighting schemes, we draw on no-regret learning. Specifically, given a voting rule, we wish to design a weighting scheme such that applying the voting rule, with voters weighted by the scheme, leads to choices that are almost as good as those endorsed by the best voter in hindsight. We derive possibility and impossibility results for the existence of such weighting schemes, depending on whether the voting rule and the weighting scheme are deterministic or randomized, as well as on the social choice axioms satisfied by the voting rule.

In recent years crowdsourcing has become the method of choice for gathering labeled training data for learning algorithms. Standard approaches to crowdsourcing view the process of acquiring labeled data separately from the process of learning a classifier from the gathered data. This can give rise to computational and statistical challenges. For example, in most cases there are no known computationally efficient learning algorithms that are robust to the high level of noise that exists in crowdsourced data, and efforts to eliminate noise through voting often require a large number of queries per example. In this paper, we show how by interleaving the process of labeling and learning, we can attain computational efficiency with much less overhead in the labeling cost. In particular, we consider the \em realizable setting where there exists a true target function in $\mathcal{F}$ and consider a pool of labelers. When a noticeable fraction of the labelers are \emphperfect, and the rest behave arbitrarily, we show that any $\mathcal{F}$ that can be efficiently learned in the traditional \em realizable PAC model can be learned in a computationally efficient manner by querying the crowd, despite high amounts of noise in the responses. Moreover, we show that this can be done while each labeler only labels a constant number of examples and the number of labels requested per example, on average, is a constant. When no perfect labelers exist, a related task is to find a set of the labelers which are \emphgood but not perfect. We show that we can identify all good labelers, when at least the majority of labelers are good.

We consider the design of computationally efficient online learning algorithms in an adversarial setting in which the learner has access to an offline optimization oracle. We present an algorithm called Generalized Follow-the-Perturbed-Leader and provide conditions under which it is oracle-efficient while achieving vanishing regret. Our results make significant progress on an open problem raised by Hazan and Koren [31], who showed that oracle-efficient algorithms do not exist in general [30] and asked whether one can identify properties under which oracle-efficient online learning may be possible. Our auction-design framework considers an auctioneer learning an optimal auction for a sequence of adversarially selected valuations with the goal of achieving revenue that is almost as good as the optimal auction in hindsight, among a class of auctions. We give oracle-efficient learning results for: (1) VCG auctions with bidder-specific reserves in single-parameter settings, (2) envy-free item pricing in multi-item auctions, and (3) s-level auctions of Morgenstern and Roughgarden [43] for single-item settings. The last result leads to an approximation of the overall optimal Myerson auction when bidders’ valuations are drawn according to a fast-mixing Markov process, extending prior work that only gave such guarantees for the i.i.d. setting. Finally, we derive various extensions, including: (1) oracle-efficient algorithms for the contextual learning setting in which the learner has access to side information (such as bidder demographics), (2) learning with approximate oracles such as those based on Maximal-in-Range algorithms, and (3) no-regret bidding in simultaneous auctions, resolving an open problem of Daskalakis and Syrgkanis [14].

We introduce a collaborative PAC learning model, in which k players attempt to learn the same underlying concept. We ask how much more information is required to learn an accurate classifier for all players simultaneously. We refer to the ratio between the sample complexity of collaborative PAC learning and its non-collaborative (single-player) counterpart as the overhead. We design learning algorithms with O(ln(k)) and O(ln^2(k)) overhead in the personalized and centralized variants our model. This gives an exponential improvement upon the naive algorithm that does not share information among players. We complement our upper bounds with an Omega(ln(k)) overhead lower bound, showing that our results are tight up to a logarithmic factor.

We study a variant of online linear optimization where the player receives a hint about the loss function at the beginning of each round. The hint is given in the form of a vector that is weakly correlated with the loss vector on that round. We show that the player can benefit from such a hint if the set of feasible actions is sufficiently round. Specifically, if the set is strongly convex, the hint can be used to guarantee a regret of O(log(T)), and if the set is q-uniformly convex for q\in(2,3), the hint can be used to guarantee a regret of o(sqrt{T}). In contrast, we establish Omega(sqrt{T}) lower bounds on regret when the set of feasible actions is a polyhedron.

We revisit the problem of designing optimal, individually rational matching mechanisms (in a general sense, allowing for cycles in directed graphs), where each player—who is associated with a subset of vertices—matches as many of his own vertices when he opts into the matching mechanism as when he opts out. We offer a new perspective on this problem by considering an arbitrary graph, but assuming that vertices are associated with players at random. Our main result asserts that, under certain conditions, any fixed optimal matching is likely to be individually rational up to lower-order terms. We also show that a simple and practical mechanism is (fully) individually rational, and likely to be optimal up to lower-order terms. We discuss the implications of our results for market design in general, and kidney exchange in particular.

We study the \emphapproximate recovery problem: Given corrupted 1-bit measurements of the form sign(w^* ⋅x_i), recover a vector w that is a good approximation to w^* ∈\Re^d. This problem has been studied by both the learning theory and signal processing communities. In learning theory, this is known as the problem of \emphlearning halfspaces with noise, and in signal processing, as \emph1-bit compressed sensing, in which there is an additional assumption that w^* is t-sparse. The challenge in both cases is to design computationally efficient algorithms that are tolerant to large amounts of noise under realistic noise models. Furthermore, in the case of 1-bit compressed sensing, we require the number of measurements x_i to scale polynomially in t and only polylogarithmically in d, the ambient dimension. In this work, we introduce algorithms with nearly optimal guarantees for both problems under two realistic noise models, \emphbounded (Massart) noise and \emphadversarial (agnostic) noise, when the measurements x_i’s are drawn from any isotropic log-concave distribution. In bounded (Massart) noise, an adversary can flip the measurement of each point x with probability η(x)≤η< 1/2. For this problem, we present an efficient algorithm that returns w such that \|w- w^*\|_2 ≤εin time poly(d, \frac 1 ε) for \emphany constant η< 1/2. This improves significantly over the best known result of Awasthi et al. 2015, in this space that required the noise to be as small as η≈10^-6. We then introduce an attribute-efficient variant of this algorithm for 1-bit compressed sensing that achieves the same guarantee with poly(t, \log(d), \frac 1 ε) measurements when \|w^*\|_0≤t. For adversarial (agnostic) noise, where any νfraction of measurements can be corrupted, we provide an algorithm that returns w such that \|w-w^*\|_2 ≤O(ν) + ε, with \tildeΩ( \frac t ε^3 \polylog(d)) measurements. Our results improve on the best known approximation results in this space and under some regimes improve on the sample complexity of the existing results. Furthermore, this is the first result of its kind in 1-bit compressed sensing that goes beyond the Gaussian marginal distribution and works for any isotrpic log-concave distribution.

The $k$-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances. Therefore to improve on these ratios, one must go beyond the worst case. In this work, we take this approach and provide strong positive results both for the asymmetric and symmetric $k$-center problems under a natural input stability (promise) condition called $α$-perturbation resilience [Bilu and Linia 2012], which states that the optimal solution does not change under any alpha-factor perturbation to the input distances. We provide algorithms that give strong guarantees simultaneously for stable and non-stable instances: our algorithms always inherit the worst-case guarantees of clustering approximation algorithms, and output the optimal solution if the input is $2$-perturbation resilient. Furthermore, we prove our result is tight by showing symmetric $k$-center under $(2-ε)$-perturbation resilience is hard unless $NP=RP$. The impact of our results are multifaceted. This is the first tight result for any problem under perturbation resilience. Furthermore, our results illustrate a surprising relationship between symmetric and asymmetric $k$-center instances under perturbation resilience. Unlike approximation ratio, for which symmetric $k$-center is easily solved to a factor of 2 but asymmetric $k$-center cannot be approximated to any constant factor, both symmetric and asymmetric $k$-center can be solved optimally under resilience to 2-perturbations. Finally, our guarantees in the setting where only part of the data satisfies perturbation resilience makes these algorithms more applicable to real-life instances.

We study the learnability of linear separators in \Re^d in the presence of bounded (a.k.a Massart) noise. This is a realistic generalization of the random classification noise model, where the adversary can flip each example x with probability η(x) ≤η. We provide the first polynomial time algorithm that can learn linear separators to arbitrarily small excess error in this noise model under the uniform distribution over the unit sphere in \Re^d, for some constant value of η. While widely studied in the statistical learning theory community in the context of getting faster convergence rates, computationally efficient algorithms in this model had remained elusive. Our work provides the first evidence that one can indeed design algorithms achieving arbitrarily small excess error in polynomial time under this realistic noise model and thus opens up a new and exciting line of research. We additionally provide lower bounds showing that popular algorithms such as hinge loss minimization and averaging cannot lead to arbitrarily small excess error under Massart noise, even under the uniform distribution. Our work, instead, makes use of a margin based technique developed in the context of active learning. As a result, our algorithm is also an active learning algorithm with label complexity that is only logarithmic in the desired excess error ε.

Starting with the seminal work by Kempe et al., a broad variety of problems, such as targeted marketing and the spread of viruses and malware, have been modeled as selecting a subset of nodes to maximize diffusion through a network. In cyber-security applications, however, a key consideration largely ignored in this literature is stealth. In particular, an attacker often has a specific target in mind, but succeeds only if the target is reached (e.g., by malware) before the malicious payload is detected and corresponding countermeasures deployed. The dual side of this problem is deployment of a limited number of monitoring units, such as cyber-forensics specialists, so as to limit the likelihood of such targeted and stealthy diffusion processes reaching their intended targets. We investigate the problem of optimal monitoring of targeted stealthy diffusion processes, and show that a number of natural variants of this problem are NP-hard to approximate. On the positive side, we show that if stealthy diffusion starts from randomly selected nodes, the defender's objective is submodular, and a fast greedy algorithm has provable approximation guarantees. In addition, we present approximation algorithms for the setting in which an attacker optimally responds to the placement of monitoring nodes by adaptively selecting the starting nodes for the diffusion process. Our experimental results show that the proposed algorithms are highly effective and scalable.

In a Stackelberg Security Game, a defender commits to a randomized deployment of security resources, and an attacker best-responds by attacking a target that maximizes his utility. While algorithms for computing an optimal strategy for the defender to commit to have had a striking real-world impact, deployed applications require significant information about potential attackers, leading to inefficiencies. We address this problem via an online learning approach. We are interested in algorithms that prescribe a randomized strategy for the defender at each step against an adversarially chosen sequence of attackers, and obtain feedback on their choices (observing either the current attacker type or merely which target was attacked). We design no-regret algorithms whose regret (when compared to the best fixed strategy in hindsight) is polynomial in the parameters of the game, and sublinear in the number of times steps.

The stochastic matching problem deals with finding a maximum matching in a graph whose edges are unknown but can be accessed via queries. This is a special case of stochastic k-set packing, where the problem is to find a maximum packing of sets, each of which exists with some probability. In this paper, we provide edge and set query algorithms for these two problems, respectively, that provably achieve some fraction of the omniscient optimal solution. Our main theoretical result for the stochastic matching (i.e., 2-set packing) problem is the design of an adaptive algorithm that queries only a constant number of edges per vertex and achieves a (1-ε) fraction of the omniscient optimal solution, for an arbitrarily small ε > 0. Moreover, this adaptive algorithm performs the queries in only a constant number of rounds. We complement this result with a non-adaptive (i.e., one round of queries) algorithm that achieves a (0.5 - ε) fraction of the omniscient optimum. We also extend both our results to stochastic k-set packing by designing an adaptive algorithm that achieves a (2/k - ε) fraction of the omniscient optimal solution, again with only O(1) queries per element. This guarantee is close to the best known polynomial-time approximation ratio of 3/k+1 -ε for the deterministic k-set packing problem [Furer 2013].

Most work building on the Stackelberg security games model assumes that the attacker can perfectly observe the defender's randomized assignment of resources to targets. This assumption has been challenged by recent papers, which designed tailor-made algorithms that compute optimal defender strategies for security games with limited surveillance. We analytically demonstrate that in zero-sum security games, lazy defenders, who simply keep optimizing against perfectly informed attackers, are almost optimal against diligent attackers, who go to the effort of gathering a reasonable number of observations. This result implies that, in some realistic situations, limited surveillance may not need to be explicitly addressed.

We address the problem of noise management in clustering algorithms. Namely, issues that arise when on top of some cluster structure the data also contains an unstructured set of points. We consider how clustering algorithms can be “robustified" so that they recover the cluster structure in spite of the unstructured part of the input. We introduce some quantitative measures of such robustness that take into account the strength of the embedded cluster structure as well was the mildness of the noise subset. We propose a simple and efficient method to turn any centroid-based clustering algorithm into a noise-robust one, and prove robustness guarantees for our method with respect to these measures. We also prove that more straightforward ways of “robustifying” clustering algorithms fail to achieve similar guarantees.