We study the bandit-feedback version of online principal component analysis (Bandit PCA): in each round t = 1,…,T, the adversary selects a d × d symmetric gain matrix G_t with spectrum in [0,1] and rank at most r; the learner simultaneously selects a unit vector w_t ∈ S^d-1 and receives the reward w_t^⊤ G_t w_t. The learner receives no other feedback, and aims to minimize the regret against the best unit vector in hindsight. This problem was introduced by Kotlowski and Neu (2019), who gave an algorithm with regret O(d√(rT log T)) and showed the lower bound of Ω(r√(T/log T)). We improve upon both of these bounds and essentially bridge the gap between them, establishing the minimax regret of order r√(dT) up to polylogarithmic factors in d and T. The upper bound is attained by a novel algorithm, which combines online mirror descent on the spectrahedron of (real) density matrices with a multiscale exploration scheme in which the eigenspaces with different spectral magnitudes are updated at different rates. For the lower bound, we construct an adaptive adversary that refines a hidden large-reward subspace based on the learner's actions, in such a way that low regret is impossible without estimating the subspace; as a result, lower-bounding the regret reduces to studying the arising subspace estimation problem. Finally, we discuss connections of Bandit PCA with adaptive-measurement quantum tomography.

A celebrated result of Pollard proves asymptotic consistency for $k$-means clustering when the population distribution has finite variance. In this work, we point out that the population-level $k$-means clustering problem is, in fact, well-posed under the weaker assumption of a finite expectation, and we investigate whether some form of asymptotic consistency holds in this setting. As we illustrate in a variety of negative results, the complete story is quite subtle; for example, the empirical $k$-means cluster centers may fail to converge even if there exists a unique set of population $k$-means cluster centers. A detailed analysis of our negative results reveals that inconsistency arises because of an extreme form of cluster imbalance, whereby the presence of outlying samples leads to some empirical $k$-means clusters possessing very few points. We then give a collection of positive results which show that some forms of asymptotic consistency, under only the assumption of finite expectation, may be recovered by imposing some a priori degree of balance among the empirical $k$-means clusters.

There has been much recent interest in understanding the continuum from adversarial to stochastic settings in online learning, with various frameworks including smoothed settings proposed to bridge this gap. We consider the more general and flexible framework of distributionally constrained adversaries in which instances are drawn from distributions chosen by an adversary within some constrained distribution class [RST11]. Compared to smoothed analysis, we consider general distributional classes which allows for a fine-grained understanding of learning settings between fully stochastic and fully adversarial for which a learner can achieve non-trivial regret. We give a characterization for which distribution classes are learnable in this context against both oblivious and adaptive adversaries, providing insights into the types of interplay between the function class and distributional constraints on adversaries that enable learnability. In particular, our results recover and generalize learnability for known smoothed settings. Further, we show that for several natural function classes including linear classifiers, learning can be achieved without any prior knowledge of the distribution class – in other words, a learner can simultaneously compete against any constrained adversary within learnable distribution classes.

We study the impact of sharing exploration in multi-armed bandits in a grouped setting where a set of groups have overlapping feasible action sets [Baek and Farias '24]. In this grouped bandit setting, groups share reward observations, and the objective is to minimize the collaborative regret, defined as the maximum regret across groups. This naturally captures applications in which one aims to balance the exploration burden between groups or populations – it is known that standard algorithms can lead to significantly imbalanced exploration cost between groups. We address this problem by introducing an algorithm Col-UCB that dynamically coordinates exploration across groups. We show that Col-UCB achieves both optimal minimax and instance-dependent collaborative regret up to logarithmic factors. These bounds are adaptive to the structure of shared action sets between groups, providing insights into when collaboration yields significant benefits over each group learning their best action independently.

We study repeated resource allocation with strategic agents, where monetary transfers are disallowed and the planner has no prior information on agents' utility distributions. Inspired by the costly state verification literature, we assume the planner can request costly audits on the winning agent after allocation, revealing their true utility but without the ability to revoke the allocation. We design a mechanism achieving T-independent 𝒪(K^2) regret in social welfare while requesting 𝒪(K^3 log T) audits in expectation, where K is the number of agents and T is the number of rounds. We further show an Ω(K) lower bound on the regret and an Ω(1) lower bound on the number of audits required for low regret. We also generalize our mechanism and analysis to imperfect audit models. Algorithmically, we show that incentivizing truthful behavior relies on accurately estimating agents' truthful winning probability online. To achieve this, we impose future punishments via adaptive audits; we also introduce an incentive-aligned flagging component allowing agents to flag biased estimates, which we prove is in their best interest. Analytically, without distributional information, the revelation principle cannot dictate a truth-telling equilibrium. Instead, we characterize a Perfect Bayesian Equilibrium via a reduction to an auxiliary game with only benign strategies. The technical tools developed herein can be of independent interest for other robust mechanism design problems where the revelation principle is inapplicable.

This work resolves the following question in non-Euclidean statistics: Is it possible to consistently estimate the Fr\'echet mean set of an unknown population distribution, with respect to the Hausdorff metric, when given access to independent identically-distributed samples? Our affirmative answer is based on a careful analysis of the ``relaxed empirical Fr\'echet mean set estimators'' which identify the set of near-minimizers of the empirical Fr\'echet functional and where the amount of ``relaxation'' vanishes as the number of data tends to infinity. Our main theoretical results include exact descriptions of which relaxation rates give weak consistency and which give strong consistency, as well as the construction of a ``two-step estimator'' which (assuming only the finiteness of certain moments and a mild condition on the metric entropy of the underlying metric space) adaptively finds the fastest possible relaxation rate for strongly consistent estimation. Our main practical result is simply that researchers working with non-Euclidean data in the real world can be better off computing relaxed empirical Fr\'echet mean sets rather than unrelaxed empirical Fr\'echet mean sets.

Classical results in statistical learning typically consider two extreme data-generating models: i.i.d. instances from an unknown distribution, or fully adversarial instances, often much more challenging statistically. To bridge the gap between these models, recent work introduced the smoothed framework, in which at each iteration an adversary generates an instance from a distribution constrained to have density bounded by.. -1 compared to some fixed base measure... This framework interpolates between the i.i.d. and adversarial cases, depending on the value of... For the classical online prediction problem, most prior results in smoothed online learning rely on the arguably strong assumption that the base measure.. is known to the learner, contrasting with standard settings in the PAC learning or consistency literature. We consider the general agnostic problem in which the base measure is unknown and values are arbitrary. In this direction, [12] showed that empirical risk minimization has sublinear regret under the well-specified assumption. We propose an algorithm R-Cover based on recursive coverings which is the first to guarantee sublinear regret for agnostic smoothed online learning without prior knowledge of.. and without the well-specified assumption. For classification, we prove that R-Cover has adaptive regret O (v..... /..) for function classes with VC dimension.., which is optimal up to logarithmic factors. For regression, we establish that R-Cover has sublinear oblivious regret for function classes with polynomial fat-shattering dimension growth.

The [Formula: see text]traveling salesman problem (k-TSP) seeks a tour of minimal length that visits a subset of [Formula: see text] points. The traveling repairman problem (TRP) seeks a complete tour with minimal latency. This paper provides constant-factor probabilistic approximations of both problems. We first show that the optimal length of the k-TSP path grows at a rate of [Formula: see text]. The proof provides a constant-factor approximation scheme, which solves a TSP in a high-concentration zone, leveraging large deviations of local concentrations. Then, we show that the optimal TRP latency grows at a rate of [Formula: see text]. This result extends the classic Beardwood–Halton–Hammersley theorem to the TRP. Again, the proof provides a constant-factor approximation scheme, which visits zones by decreasing order of probability density. We discuss practical implications of this result in the design of transportation and logistics systems. Finally, we propose dedicated notions of fairness—randomized population-based fairness for the k-TSP and geographic fairness for the TRP—and give algorithms to balance efficiency and fairness. Funding: This work was partly supported by [Grant ONR N00014-18-1-2122] and the Singapore National Research Foundation through the Singapore–Massachusetts Institute of Technology Alliance for Research and Technology Centre for Future Urban Mobility. Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2021.0286 .

We study the problem in which a central planner sequentially allocates a single resource to multiple strategic agents using their utility reports at each round, but without using any monetary transfers. We consider general agent utility distributions and two standard settings: a finite horizon T and an infinite horizon with γ discounts. We provide general tools to characterize the convergence rate between the optimal mechanism for the central planner and the first-best allocation if true agent utilities were available. This heavily depends on the utility distributions, yielding rates anywhere between 1/√(T) and 1/T for the finite-horizon setting, and rates faster than √(1-γ), including exponential rates for the infinite-horizon setting as agents are more patient γ→ 1. On the algorithmic side, we design mechanisms based on the promised-utility framework to achieve these rates and leverage structure on the utility distributions. Intuitively, the more flexibility the central planner has to reward or penalize any agent while incurring little social welfare cost, the faster the convergence rate. In particular, discrete utility distributions typically yield the slower rates 1/√(T) and √(1-γ), while smooth distributions with density typically yield faster rates 1/T (up to logarithmic factors) and 1-γ.

We give query complexity lower bounds for convex optimization and the related feasibility problem. We show that quadratic memory is necessary to achieve the optimal oracle complexity for first-order convex optimization. In particular, this shows that center-of-mass cutting-planes algorithms in dimension $d$ which use $\tilde O(d^2)$ memory and $\tilde O(d)$ queries are Pareto-optimal for both convex optimization and the feasibility problem, up to logarithmic factors. Precisely, we prove that to minimize $1$-Lipschitz convex functions over the unit ball to $1/d^4$ accuracy, any deterministic first-order algorithms using at most $d^{2-\delta}$ bits of memory must make $\tilde\Omega(d^{1+\delta/3})$ queries, for any $\delta\in[0,1]$. For the feasibility problem, in which an algorithm only has access to a separation oracle, we show a stronger trade-off: for at most $d^{2-\delta}$ memory, the number of queries required is $\tilde\Omega(d^{1+\delta})$. This resolves a COLT 2019 open problem of Woodworth and Srebro.

In this paper we provide oracle complexity lower bounds for finding a point in a given set using a memory-constrained algorithm that has access to a separation oracle. We assume that the set is contained within the unit d-dimensional ball and contains a ball of known radius epsilon > 0. This setup is commonly referred to as the feasibility problem. We show that to solve feasibility problems with accuracy c >= e(-do(1)), any deterministic algorithm either uses d(1+delta) bits of memory or must make at least 1/(d(0.01 delta)epsilon(2) 1-delta/1+1.01 delta -o(1)) oracle queries, for any delta is an element of [0, 1]. Additionally, we show that randomized algorithms either use d1+d memory or make at least 1/(d(2 delta)c(2(1-4 delta)-o(1))) queries for any delta is an element of [0, 1/4]. Because gradient descent only uses linear memory O(d ln 1/epsilon) but makes O(1/epsilon(2)) queries, our results imply that it is Pareto-optimal in the oracle complexity/memory tradeoff. Further, our results show that the oracle complexity for deterministic algorithms is always polynomial in 1/epsilon if the algorithm has less than quadratic memory in d. This reveals a sharp phase transition since with quadratic O(d(2) ln 1/epsilon) memory, cutting plane methods only require O(d ln 1/epsilon) queries.

This paper addresses the statistical problem of estimating the infinite-norm deviation from the empirical mean to the distribution mean for high-dimensional distributions on $\{0,1\}^d$, potentially with $d=\infty$. Unlike traditional bounds as in the classical Glivenko-Cantelli theorem, we explore the instance-dependent convergence behavior. For product distributions, we provide the exact non-asymptotic behavior of the expected maximum deviation, revealing various regimes of decay. In particular, these tight bounds demonstrate the necessity of a previously proposed factor for an upper bound, answering a corresponding COLT 2023 open problem. We also consider general distributions on $\{0,1\}^d$ and provide the tightest possible bounds for the maximum deviation of the empirical mean given only the mean statistic. Along the way, we prove a localized version of the Dvoretzky-Kiefer-Wolfowitz inequality. Additionally, we present some results for two other cases, one where the deviation is measured in some $q$-norm, and the other where the distribution is supported on a continuous domain $[0,1]^d$, and also provide some high-probability bounds for the maximum deviation in the independent Bernoulli case.

Cohen and Kontorovich (COLT 2023) initiated the study of what we call here the Binomial Empirical Process: the maximal absolute value of a sequence of inhomogeneous normalized and centered binomials. They almost fully analyzed the case where the binomials are independent, and the remaining gap was closed by Blanchard and Voráček (ALT 2024). In this work, we study the much more general and challenging case with correlations. In contradistinction to Gaussian processes, whose behavior is characterized by the covariance structure, we discover that, at least somewhat surprisingly, for binomial processes covariance does not even characterize convergence. Although a full characterization remains out of reach, we take the first steps with nontrivial upper and lower bounds in terms of covering numbers.

We propose a family of recursive cutting-plane algorithms to solve feasibility problems with constrained memory, which can also be used for first-order convex optimization. Precisely, in order to find a point within a ball of radius $\epsilon$ with a separation oracle in dimension $d$ -- or to minimize $1$-Lipschitz convex functions to accuracy $\epsilon$ over the unit ball -- our algorithms use $\mathcal O(\frac{d^2}{p}\ln \frac{1}{\epsilon})$ bits of memory, and make $\mathcal O((C\frac{d}{p}\ln \frac{1}{\epsilon})^p)$ oracle calls, for some universal constant $C \geq 1$. The family is parametrized by $p\in[d]$ and provides an oracle-complexity/memory trade-off in the sub-polynomial regime $\ln\frac{1}{\epsilon}\gg\ln d$. While several works gave lower-bound trade-offs (impossibility results) -- we explicit here their dependence with $\ln\frac{1}{\epsilon}$, showing that these also hold in any sub-polynomial regime -- to the best of our knowledge this is the first class of algorithms that provides a positive trade-off between gradient descent and cutting-plane methods in any regime with $\epsilon\leq 1/\sqrt d$. The algorithms divide the $d$ variables into $p$ blocks and optimize over blocks sequentially, with approximate separation vectors constructed using a variant of Vaidya's method. In the regime $\epsilon \leq d^{-\Omega(d)}$, our algorithm with $p=d$ achieves the information-theoretic optimal memory usage and improves the oracle-complexity of gradient descent.

We study the fundamental limits of learning in contextual bandits, where a learner's rewards depend on their actions and a known context, which extends the canonical multi-armed bandit to the case where side-information is available. We are interested in universally consistent algorithms, which achieve sublinear regret compared to any measurable fixed policy, without any function class restriction. For stationary contextual bandits, when the underlying reward mechanism is time-invariant, Blanchard et. al (2022) characterized learnable context processes for which universal consistency is achievable; and further gave algorithms ensuring universal consistency whenever this is achievable, a property known as optimistic universal consistency. It is well understood, however, that reward mechanisms can evolve over time, possibly adversarially, and depending on the learner's actions. We show that optimistic universal learning for contextual bandits with adversarial rewards is impossible in general, contrary to all previously studied settings in online learning -- including standard supervised learning. We also give necessary and sufficient conditions for universal learning under various adversarial reward models, and an exact characterization for online rewards. In particular, the set of learnable processes for these reward models is still extremely general -- larger than i.i.d., stationary or ergodic -- but in general strictly smaller than that for supervised learning or stationary contextual bandits, shedding light on new adversarial phenomena.

We provide algorithms for regression with adversarial responses under large classes of non-i.i.d. instance sequences, on general separable metric spaces, with provably minimal assumptions. We also give characterizations of learnability in this regression context. We consider universal consistency which asks for strong consistency of a learner without restrictions on the value responses. Our analysis shows that such an objective is achievable for a significantly larger class of instance sequences than stationary processes, and unveils a fundamental dichotomy between value spaces: whether finite-horizon mean estimation is achievable or not. We further provide optimistically universal learning rules, i.e., such that if they fail to achieve universal consistency, any other algorithms will fail as well. For unbounded losses, we propose a mild integrability condition under which there exist algorithms for adversarial regression under large classes of non-i.i.d. instance sequences. In addition, our analysis also provides a learning rule for mean estimation in general metric spaces that is consistent under adversarial responses without any moment conditions on the sequence, a result of independent interest.

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.

This technical report provides additional results for the main paper “Probabilistic bounds on the k-Traveling Salesman Problem (k-TSP) and the Traveling Repairman Problem (TRP)”. For the k-TSP, we extend the probabilistic bounds derived in the main paper to the case of distributions with general densities. For the TRP, we propose a utility-based notion of fairness and derive constant-factor probabilistic bounds for this objective, thus extending the TRP bounds from the main paper to non-linear utilities.

In this paper, we analyze the number of neurons and training parameters that a neural network needs to approximate multivariate functions of bounded second mixed derivatives --- Korobov functions. We prove upper bounds on these quantities for shallow and deep neural networks, drastically lessening the curse of dimensionality. Our bounds hold for general activation functions, including ReLU. We further prove that these bounds nearly match the minimal number of parameters any continuous function approximator needs to approximate Korobov functions, showing that neural networks are near-optimal function approximators.