The ''trace reconstruction'' problem asks, given an unknown binary string $x$ and a channel that repeatedly returns ''traces'' of $x$ with each bit randomly deleted with some probability $p$, how many traces are needed to recover $x$? There is an exponential gap between the best known upper and lower bounds for this problem. Many variants of the model have been introduced in hopes of motivating or revealing new approaches to narrow this gap. We study the variant of circular trace reconstruction introduced by Narayanan and Ren (ITCS 2021), in which traces undergo a random cyclic shift in addition to random deletions. We show an improved lower bound of $\tildeΩ(n^5)$ for circular trace reconstruction. This contrasts with the (previously) best known lower bounds of $\tildeΩ(n^3)$ in the circular case and $\tildeΩ(n^{3/2})$ in the linear case. Our bound shows the indistinguishability of traces from two sparse strings $x,y$ that each have a constant number of nonzeros. Can this technique be extended significantly? How hard is it to reconstruct a sparse string $x$ under a cyclic deletion channel? We resolve these questions by showing, using Fourier techniques, that $\tilde{O}(n^6)$ traces suffice for reconstructing any constant-sparse string in a circular deletion channel, in contrast to the upper bound of $\exp(\tilde{O}(n^{1/3}))$ for general strings in the circular deletion channel. This shows that new algorithms or new lower bounds must focus on non-constant-sparse strings.

We study the problem of multiclass online learning under \emph{bandit feedback} within the framework of \emph{universal learning} [Bousquet, Hanneke, Moran, van Handel, and Yehudayoff; STOC ’21]. \par In multiclass online learning under bandit feedback, it is well known that no concept class $\mathcal{C}$ is \emph{uniformly} learnable when the effective label space is unbounded, or in other words, no online learner guarantees a finite bound on the expected number of mistakes holding uniformly over all realizable data sequences. In contrast, surprisingly, we show that in the case of \emph{universal} learnability of concept classes $\mathcal{C}$, there is an exact equivalence between multiclass online learnability under bandit feedback and full supervision, in both the realizable and agnostic settings. \par More specifically, our first main contribution is a theory that establishes an inherent dichotomy in multiclass online learning under bandit feedback within the realizable setting. In particular, for any concept class $\mathcal{C}$ even when the effective label space is unbounded, we have: (1) If $\mathcal{C}$ does not have an infinite multiclass Littlestone tree, then there is a deterministic online learner that makes only finitely many mistakes against any realizable adversary, crucially without placing a uniform bound on the number of mistakes. (2) If $\mathcal{C}$ has an infinite multiclass Littlestone tree, then there is a strategy for the realizable adversary that forces any learner, including randomized, to make linear expected number of mistakes. Furthermore, our second main contribution reveals a similar trend in the agnostic setting.

We investigate prophet inequalities with competitive ratios approaching 1, seeking to generalize k-uniform matroids. We first show that large girth does not suffice: for all k, there exists a matroid of girth ≥ k and a prophet inequality instance on that matroid whose optimal competitive ratio is 1/2. Next, we show k-fold matroid unions do suffice: we provide a prophet inequality with competitive ratio 1-O(√{(log k)/k}) for any k-fold matroid union. Our prophet inequality follows from an online contention resolution scheme. The key technical ingredient in our online contention resolution scheme is a novel bicriterion concentration inequality for arbitrary monotone 1-Lipschitz functions over independent items which may be of independent interest. Applied to our particular setting, our bicriterion concentration inequality yields "Chernoff-strength" concentration for a 1-Lipschitz function that is not (approximately) self-bounding.

We study the problem of multiclass learning with bandit feedback in both the i.i.d. batch and adversarial online models. In the *uniform* learning framework, it is well known that no hypothesis class $\mathcal{H}$ is learnable in either model when the effective number of labels is unbounded. In contrast, within the *universal* learning framework, recent works by (Hanneke et al., 2025b) and (Hanneke et al., 2025a) have established surprising exact equivalences between learnability under bandit feedback and full supervision in both the i.i.d. batch and adversarial online models, respectively. This raises a natural question: What happens in the *non-uniform* learning framework, which lies between the uniform and universal learning frameworks? Our contributions are twofold: (1) We provide a combinatorial characterization of learnable hypothesis classes in both models, in the realizable and agnostic settings, within the non-uniform learning framework. Notably, this includes elementary and natural hypothesis classes, such as a countably infinite collection of constant functions over some domain that is learnable in both models. (2) We construct a hypothesis class that is non-uniformly learnable under full supervision in the adversarial online model (and thus also in the i.i.d. batch model), but not non-uniformly learnable under bandit feedback in the i.i.d. batch model (and thus also not in the adversarial online model). This serves as our main novel technical contribution that reveals a fundamental distinction between the non-uniform and universal learning frameworks.

We provide a full characterization of the concept classes that are optimistically universally online learnable with {0, 1} labels. The notion of optimistically universal online learning was defined in [Hanneke, 2021] in order to understand learnability under minimal assumptions. In this paper, following the philosophy behind that work, we investigate two questions, namely, for every concept class: (1) What are the minimal assumptions on the data process admitting online learnability? (2) Is there a learning algorithm which succeeds under every data process satisfying the minimal assumptions? Such an algorithm is said to be optimistically universal for the given concept class. We resolve both of these questions for all concept classes, and moreover, as part of our solution we design general learning algorithms for each case. Finally, we extend these algorithms and results to the agnostic case, showing an equivalence between the minimal assumptions on the data process for learnability in the agnostic and realizable cases, for every concept class, as well as the equivalence of optimistically universal learnability.

We study fair resource allocation when the resources contain a mixture of divisible and indivisible goods, focusing on the well-studied fairness notion of maximin share fairness (MMS). With only indivisible goods, a full MMS allocation may not exist, but a constant multiplicative approximate allocation always does. We analyze how the MMS approximation guarantee would be affected when the resources to be allocated also contain divisible goods. In particular, we show that the worst-case MMS approximation guarantee with mixed goods is no worse than that with only indivisible goods. However, there exist problem instances to which adding some divisible resources would strictly decrease the MMS approximation ratios of the instances. On the algorithmic front, we propose a constructive algorithm that will always produce an \alpha-MMS allocation for any number of agents, where \alpha takes values between 1/2 and 1 and is a monotonically increasing function determined by how agents value the divisible goods relative to their MMS values.