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Researcher profile

Steve Hanneke

Steve Hanneke contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine Learningmath.STData Structures and AlgorithmsStatistics TheoryComputer Science and Game TheoryArtificial IntelligenceComputationCryptography and SecurityInformation Theory

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Published work

15 published item(s)

preprint2026arXiv

On Characterizing Learnability for Adversarial Noisy Bandits

We study adversarial noisy bandits given a known function class $\mathcal{F}$. In each round, the adversary selects a function $f \in \mathcal{F}$, the learner chooses an arm, and then observes a noisy reward determined by the chosen arm and the function $f$. The goal is to minimize the cumulative regret $R(T)$, defined as the difference between the learner's performance and that of the best fixed arm in hindsight over $T$ rounds. We say that a function class $\mathcal{F}$ is learnable if there exists an algorithm achieving sublinear regret. Our main results concern characterizing learnability. The main quantity appearing in our characterization is a convexified variant of the generalized maximin volume introduced by Hanneke and Wang (2025). For oblivious adversaries, we characterize learnability in terms of this convexified generalized maximin volume. For adaptive adversaries, we show that the same quantity characterizes learnability when the arm space is countable. Our analysis builds on a connection between convexified generalized maximin volume and the existence of simple hitting sets. We further conjecture that the same quantity also characterizes learnability when the arm space is uncountable, via its relation to a new complexity measure, which we call the distribution covering number. This notion can be viewed as a strengthened form of the hitting set that still admits efficient learning via the multiplicative weights algorithm. We also pose a number of relevant open questions regarding this problem.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

On the Learning Curves of Revenue Maximization

Learning curves are a fundamental primitive in supervised learning, describing how an algorithm's performance improves with more data and providing a quantitative measure of its generalization ability. Formally, a learning curve plots the decay of an algorithm's error for a fixed underlying distribution as a function of the number of training samples. Prior work on revenue-maximizing learning algorithms, starting with the seminal work of Cole and Roughgarden [STOC, 2014], adopts a distribution-free perspective, which parallels the PAC learning framework in learning theory. This approach evaluates performance against the hardest possible sequence of valuation distributions, one for each sample size, effectively defining the upper envelope of learning curves over all possible distributions, thus leading to error bounds that do not capture the shape of the learning curves. In this work we initiate the study of learning curves for revenue maximization and provide a near-complete characterization of their rate of decay in the basic setting of a single item and a single buyer. In the absence of any restriction on the valuation distribution, we show that there exists a Bayes-consistent algorithm, meaning that its learning curve converges to zero for any arbitrary valuation distribution as the number of samples $n \to \infty$. However, this convergence must be arbitrarily slow, even if the optimal revenue is finite. In contrast, if the optimal revenue is achieved by a finite price, then the optimal rate of decay is roughly $1/\sqrt{n}$. Finally, for distributions supported on discrete sets of values, we show that learning curves decay almost exponentially fast, a rate unattainable under the PAC framework.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Realizable Bayes-Consistency for General Metric Losses

We study strong universal Bayes-consistency in the realizable setting for learning with general metric losses, extending classical characterizations beyond $0$-$1$ classification (Bousquet et al., 2020; Hanneke et al., 2021) and real-valued regression (Attias et al., 2024). Given an instance space $(X,ρ)$, a label space $(Y,\ell)$ with possibly unbounded loss, and a hypothesis class $H \subseteq Y^{X}$, we resolve the realizable case of an open problem presented in Tsir Cohen and Kontorovich (2022). Specifically, we find the necessary and sufficient conditions on the hypothesis class $H$ under which there exists a distribution-free learning rule whose risk converges almost surely to the best-in-class risk (which is zero) for every realizable data-generating distribution. Our main contribution is this sharp characterization in terms of a combinatorial obstruction: Similarly to Attias et al. (2024), we introduce the notion of an infinite non-decreasing $(γ_k)$-Littlestone tree, where $γ_k \to \infty$. This extends the Littlestone tree structure used in Bousquet et al. (2020) to the metric loss setting.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Regret-Oracle Complexity Tradeoffs in Agnostic Online Learning

Agnostic online learning is classically solved via a reduction to the realizable setting, utilizing Littlestone's Standard Optimal Algorithm (SOA) as a base learner. However, the SOA is computationally intractable to execute even for a single round. To overcome this barrier, recent work in oracle-efficient online learning replaces the SOA with a realizable base learner that accesses the concept class exclusively through an offline empirical risk minimization (ERM) oracle. While such agnostic learners achieve near-optimal expected regret, they suffer from a doubly-exponential oracle complexity of $O\big(T^{2^{O(d_\mathrm{LD})}}\big)$, where $d_\mathrm{LD}$ is the Littlestone dimension and $T$ is the number of rounds. In this work, we significantly improve this oracle complexity while relying on an even weaker primitive: a weak-consistency oracle, which merely decides whether a given labeled dataset is realizable. At the core of our approach is an adaptive and dynamic agnostic-to-realizable reduction that actively prunes non-realizable label sequences on the fly. By using the VC dimension ($d_\mathrm{VC}$) to bound the number of dynamically maintained active paths, our algorithm reduces the total query complexity down to $O(T^{d_\mathrm{VC}+1})$ while perfectly preserving near-optimal expected regret. Crucially, this dynamic pruning also yields a memory reduction over the standard reduction. Furthermore, we formally quantify the regret--oracle complexity tradeoff, providing upper bounds that smoothly interpolate between restricted query budgets and attainable expected regret. We complement these with lower bounds proving that any learner restricted to $Q = o(\sqrt{T})$ queries must suffer an expected regret of $Ω(T/Q)$.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Uniform Convergence Beyond Glivenko-Cantelli

We characterize conditions under which collections of distributions on $\{0,1\}^\mathbb{N}$ admit uniform estimation of their mean. Prior work from Vapnik and Chervonenkis (1971) has focused on uniform convergence using the empirical mean estimator, leading to the principle known as $P-$ Glivenko-Cantelli. We extend this framework by moving beyond the empirical mean estimator and introducing Uniform Mean Estimability, also called UME-learnability, which captures when a collection permits uniform mean estimation by any arbitrary estimator. We work on the space created by the mean vectors of the collection of distributions. For each distribution, the mean vector records the expected value in each coordinate. We show that separability of the mean vectors is a sufficient condition for UME-learnability. However, we show that separability of the mean vectors is not necessary for UME-learnability by constructing a collection of distributions whose mean vectors are non-separable yet UME-learnable using techniques fundamentally different from those used in our separability-based analysis. Finally, we establish that countable unions of UME-learnable collections are also UME-learnable, solving the conjecture posed in Cohen et al. (2025).

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

What is Learnable in Valiant's Theory of the Learnable?

Valiant's 1984 paper is widely credited with introducing the PAC learning model, but it, in fact, introduced a different model: unlike PAC learning, the learner receives only positives, may issue membership queries, and must output a hypothesis with no false positives. Prior work characterized variants, including the case without queries. We revisit Valiant's original model and ask: *Which classes are learnable in it?* For every finite domain, including Valiant's Boolean-hypercube setting, we show that a class is learnable if and only if every realizable positive sample can be certified by a poly-size adaptive query-compression scheme. This is a new variant of sample compression where the learner certifies samples via a short interaction with the membership oracle. Our characterization shows that learnability in Valiant's model is strictly sandwiched between learnability in the PAC model and the variant of Valiant's model without membership queries. This is one of the rare cases where introducing membership queries changes the set of learnable classes, and not just the sample or computational complexity. Next, we study the natural extension of the model to arbitrary domains. While we do not obtain an exact characterization, our techniques readily generalize and show that the same strict sandwiching persists. Finally, we show that $d$-dimensional halfspaces, which are not learnable without queries, are learnable with queries: we give a $\mathrm{poly}(d) \tilde{O}(1/ε)$ sample and $\mathrm{poly}(d) \mathrm{polylog}(1/ε)$ query algorithm, and prove that at least $Ω(d)$ samples or queries are necessary. To our knowledge, this is the first algorithm for halfspaces in Valiant's model. Together, these results uncover a surprisingly rich theory behind Valiant's original notion of learnability and introduce ideas that may be of independent interest in learning theory.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Contextual Bandits and Optimistically Universal Learning

We consider the contextual bandit problem on general action and context spaces, where the learner's rewards depend on their selected actions and an observable context. This generalizes the standard multi-armed bandit to the case where side information is available, e.g., patients' records or customers' history, which allows for personalized treatment. We focus on consistency -- vanishing regret compared to the optimal policy -- and show that for large classes of non-i.i.d. contexts, consistency can be achieved regardless of the time-invariant reward mechanism, a property known as universal consistency. Precisely, we first give necessary and sufficient conditions on the context-generating process for universal consistency to be possible. Second, we show that there always exists an algorithm that guarantees universal consistency whenever this is achievable, called an optimistically universal learning rule. Interestingly, for finite action spaces, learnable processes for universal learning are exactly the same as in the full-feedback setting of supervised learning, previously studied in the literature. In other words, learning can be performed with partial feedback without any generalization cost. The algorithms balance a trade-off between generalization (similar to structural risk minimization) and personalization (tailoring actions to specific contexts). Lastly, we consider the case of added continuity assumptions on rewards and show that these lead to universal consistency for significantly larger classes of data-generating processes.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Robustly-reliable learners under poisoning attacks

Data poisoning attacks, in which an adversary corrupts a training set with the goal of inducing specific desired mistakes, have raised substantial concern: even just the possibility of such an attack can make a user no longer trust the results of a learning system. In this work, we show how to achieve strong robustness guarantees in the face of such attacks across multiple axes. We provide robustly-reliable predictions, in which the predicted label is guaranteed to be correct so long as the adversary has not exceeded a given corruption budget, even in the presence of instance targeted attacks, where the adversary knows the test example in advance and aims to cause a specific failure on that example. Our guarantees are substantially stronger than those in prior approaches, which were only able to provide certificates that the prediction of the learning algorithm does not change, as opposed to certifying that the prediction is correct, as we are able to achieve in our work. Remarkably, we provide a complete characterization of learnability in this setting, in particular, nearly-tight matching upper and lower bounds on the region that can be certified, as well as efficient algorithms for computing this region given an ERM oracle. Moreover, for the case of linear separators over logconcave distributions, we provide efficient truly polynomial time algorithms (i.e., non-oracle algorithms) for such robustly-reliable predictions. We also extend these results to the active setting where the algorithm adaptively asks for labels of specific informative examples, and the difficulty is that the adversary might even be adaptive to this interaction, as well as to the agnostic learning setting where there is no perfect classifier even over the uncorrupted data.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Universal Online Learning with Unbounded Losses: Memory Is All You Need

We resolve an open problem of Hanneke on the subject of universally consistent online learning with non-i.i.d. processes and unbounded losses. The notion of an optimistically universal learning rule was defined by Hanneke in an effort to study learning theory under minimal assumptions. A given learning rule is said to be optimistically universal if it achieves a low long-run average loss whenever the data generating process makes this goal achievable by some learning rule. Hanneke posed as an open problem whether, for every unbounded loss, the family of processes admitting universal learning are precisely those having a finite number of distinct values almost surely. In this paper, we completely resolve this problem, showing that this is indeed the case. As a consequence, this also offers a dramatically simpler formulation of an optimistically universal learning rule for any unbounded loss: namely, the simple memorization rule already suffices. Our proof relies on constructing random measurable partitions of the instance space and could be of independent interest for solving other open questions. We extend the results to the non-realizable setting thereby providing an optimistically universal Bayes consistent learning rule.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Universally Consistent Online Learning with Arbitrarily Dependent Responses

This work provides an online learning rule that is universally consistent under processes on (X,Y) pairs, under conditions only on the X process. As a special case, the conditions admit all processes on (X,Y) such that the process on X is stationary. This generalizes past results which required stationarity for the joint process on (X,Y), and additionally required this process to be ergodic. In particular, this means that ergodicity is superfluous for the purpose of universally consistent online learning.

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preprint2021arXiv

Adversarially Robust Learning with Unknown Perturbation Sets

We study the problem of learning predictors that are robust to adversarial examples with respect to an unknown perturbation set, relying instead on interaction with an adversarial attacker or access to attack oracles, examining different models for such interactions. We obtain upper bounds on the sample complexity and upper and lower bounds on the number of required interactions, or number of successful attacks, in different interaction models, in terms of the VC and Littlestone dimensions of the hypothesis class of predictors, and without any assumptions on the perturbation set.

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preprint2021arXiv

Online Learning with Simple Predictors and a Combinatorial Characterization of Minimax in 0/1 Games

Which classes can be learned properly in the online model? -- that is, by an algorithm that at each round uses a predictor from the concept class. While there are simple and natural cases where improper learning is necessary, it is natural to ask how complex must the improper predictors be in such cases. Can one always achieve nearly optimal mistake/regret bounds using "simple" predictors? In this work, we give a complete characterization of when this is possible, thus settling an open problem which has been studied since the pioneering works of Angluin (1987) and Littlestone (1988). More precisely, given any concept class C and any hypothesis class H, we provide nearly tight bounds (up to a log factor) on the optimal mistake bounds for online learning C using predictors from H. Our bound yields an exponential improvement over the previously best known bound by Chase and Freitag (2020). As applications, we give constructive proofs showing that (i) in the realizable setting, a near-optimal mistake bound (up to a constant factor) can be attained by a sparse majority-vote of proper predictors, and (ii) in the agnostic setting, a near-optimal regret bound (up to a log factor) can be attained by a randomized proper algorithm. A technical ingredient of our proof which may be of independent interest is a generalization of the celebrated Minimax Theorem (von Neumann, 1928) for binary zero-sum games. A simple game which fails to satisfy Minimax is "Guess the Larger Number", where each player picks a number and the larger number wins. The payoff matrix is infinite triangular. We show this is the only obstruction: if a game does not contain triangular submatrices of unbounded sizes then the Minimax Theorem holds. This generalizes von Neumann's Minimax Theorem by removing requirements of finiteness (or compactness), and captures precisely the games of interest in online learning.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A No-Free-Lunch Theorem for MultiTask Learning

Multitask learning and related areas such as multi-source domain adaptation address modern settings where datasets from $N$ related distributions $\{P_t\}$ are to be combined towards improving performance on any single such distribution ${\cal D}$. A perplexing fact remains in the evolving theory on the subject: while we would hope for performance bounds that account for the contribution from multiple tasks, the vast majority of analyses result in bounds that improve at best in the number $n$ of samples per task, but most often do not improve in $N$. As such, it might seem at first that the distributional settings or aggregation procedures considered in such analyses might be somehow unfavorable; however, as we show, the picture happens to be more nuanced, with interestingly hard regimes that might appear otherwise favorable. In particular, we consider a seemingly favorable classification scenario where all tasks $P_t$ share a common optimal classifier $h^*,$ and which can be shown to admit a broad range of regimes with improved oracle rates in terms of $N$ and $n$. Some of our main results are as follows: $\bullet$ We show that, even though such regimes admit minimax rates accounting for both $n$ and $N$, no adaptive algorithm exists; that is, without access to distributional information, no algorithm can guarantee rates that improve with large $N$ for $n$ fixed. $\bullet$ With a bit of additional information, namely, a ranking of tasks $\{P_t\}$ according to their distance to a target ${\cal D}$, a simple rank-based procedure can achieve near optimal aggregations of tasks' datasets, despite a search space exponential in $N$. Interestingly, the optimal aggregation might exclude certain tasks, even though they all share the same $h^*$.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

On the Value of Target Data in Transfer Learning

We aim to understand the value of additional labeled or unlabeled target data in transfer learning, for any given amount of source data; this is motivated by practical questions around minimizing sampling costs, whereby, target data is usually harder or costlier to acquire than source data, but can yield better accuracy. To this aim, we establish the first minimax-rates in terms of both source and target sample sizes, and show that performance limits are captured by new notions of discrepancy between source and target, which we refer to as transfer exponents.

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preprint2020arXiv

Proper Learning, Helly Number, and an Optimal SVM Bound

The classical PAC sample complexity bounds are stated for any Empirical Risk Minimizer (ERM) and contain an extra logarithmic factor $\log(1/ε)$ which is known to be necessary for ERM in general. It has been recently shown by Hanneke (2016) that the optimal sample complexity of PAC learning for any VC class C is achieved by a particular improper learning algorithm, which outputs a specific majority-vote of hypotheses in C. This leaves the question of when this bound can be achieved by proper learning algorithms, which are restricted to always output a hypothesis from C. In this paper we aim to characterize the classes for which the optimal sample complexity can be achieved by a proper learning algorithm. We identify that these classes can be characterized by the dual Helly number, which is a combinatorial parameter that arises in discrete geometry and abstract convexity. In particular, under general conditions on C, we show that the dual Helly number is bounded if and only if there is a proper learner that obtains the optimal joint dependence on $ε$ and $δ$. As further implications of our techniques we resolve a long-standing open problem posed by Vapnik and Chervonenkis (1974) on the performance of the Support Vector Machine by proving that the sample complexity of SVM in the realizable case is $Θ((n/ε)+(1/ε)\log(1/δ))$, where $n$ is the dimension. This gives the first optimal PAC bound for Halfspaces achieved by a proper learning algorithm, and moreover is computationally efficient.

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