We resolve a 30-year-old open problem concerning the power of unlabeled data in online learning by tightly quantifying the gap between transductive and standard online learning. We prove that for every concept class $\mathcal{H}$ with Littlestone dimension $d$, the transductive mistake bound is at least $\Omega(\sqrt{d})$. This establishes an exponential improvement over previous lower bounds of $\Omega(\log \log d)$, $\Omega(\sqrt{\log d})$, and $\Omega(\log d)$, respectively due to Ben-David, Kushilevitz, and Mansour (1995, 1997) and Hanneke, Moran, and Shafer (2023). We also show that our bound is tight: for every $d$, there exists a class of Littlestone dimension $d$ with transductive mistake bound $O(\sqrt{d})$. Our upper bound also improves the previous best known upper bound of $(2/3) \cdot d$ from Ben-David et al. (1997). These results demonstrate a quadratic gap between transductive and standard online learning, thereby highlighting the benefit of advanced access to the unlabeled instance sequence. This stands in stark contrast to the PAC setting, where transductive and standard learning exhibit similar sample complexities.

We study a sequential decision-making problem on a $n$-node graph $\mathcal{G}$ where each node has an unknown label from a finite set $\mathbf{\Omega}$, drawn from a joint distribution $\mathcal{P}$ that is Markov with respect to $\mathcal{G}$. At each step, selecting a node reveals its label and yields a label-dependent reward. The goal is to adaptively choose nodes to maximize expected accumulated discounted rewards. We impose a frontier exploration constraint, where actions are limited to neighbors of previously selected nodes, reflecting practical constraints in settings such as contact tracing and robotic exploration. We design a Gittins index-based policy that applies to general graphs and is provably optimal when $\mathcal{G}$ is a forest.

We study the task of noiseless linear regression under Gaussian covariates in the presence of additive oblivious contamination. Specifically, we are given i.i.d.\ samples from a distribution $(x, y)$ on $\mathbb R^d \times \mathbb R$ with $x \sim \mathcal N(0,I_d)$ and $y = x^\top \beta + z$, where $z$ is drawn from an unknown distribution that is independent of $x$.

We study online and transductive online learning in settings where the learner can interact with the concept class only via Empirical Risk Minimization (ERM) or weak consistency oracles on arbitrary subsets of the instance domain. This contrasts with standard online models, where the learner has full knowledge of the concept class. The ERM oracle returns a hypothesis that minimizes the loss on a given subset, while the weak consistency oracle returns only a binary signal indicating whether the subset is realizable by a concept in the class. The learner's performance is measured by the number of mistakes and oracle calls. In the standard online setting with ERM access, we establish tight lower bounds in both the realizable and agnostic cases: $\Omega(2^{d_\mathrm{LD}})$ mistakes and $\Omega(\sqrt{T 2^{d_\mathrm{LD}}})$ regret, respectively, where $T$ is the number of timesteps and $d_\mathrm{LD}$ is the Littlestone dimension of the class. We further show how existing results for online learning with ERM access translate to the setting with a weak consistency oracle, at the cost of increasing the number of oracle calls by $O(T)$. We then consider the transductive online model, where the instance sequence is known in advance but labels are revealed sequentially. For general Littlestone classes, we show that the optimal mistake bound in the realizable case and in the agnostic case can be achieved using $O(T^{d_\mathrm{VC}+1})$ weak consistency oracle calls, where $d_\mathrm{VC}$ is the VC dimension of the class. On the negative side, we show that $\Omega(T)$ weak consistency queries are necessary for transductive online learnability, and that $\Omega(T)$ ERM queries are necessary to avoid exponential dependence on the Littlestone dimension.

The gradient descent (GD) has been one of the most common optimizer in machine learning. In particular, the loss landscape of a neural network is typically sharpened during the initial phase of training, making the training dynamics hover on the edge of stability. This is beyond our standard understanding of GD convergence in the stable regime where arbitrarily chosen stepsize is sufficiently smaller than the edge of stability. Recently, Wu et al. (COLT2024) have showed that GD converges with arbitrary stepsize under linearly separable logistic regression. Although their analysis hinges on the self-bounding property of the logistic loss, which seems to be a cornerstone to establish a modified descent lemma, our pilot study shows that other loss functions without the self-bounding property can make GD converge with arbitrary stepsize. To further understand what property of a loss function matters in GD, we aim to show arbitrary-stepsize GD convergence for a general loss function based on the framework of \emph{Fenchel--Young losses}. We essentially leverage the classical perceptron argument to derive the convergence rate for achieving $\epsilon$-optimal loss, which is possible for a majority of Fenchel--Young losses. Among typical loss functions, the Tsallis entropy achieves the GD convergence rate $T=\Omega(\epsilon^{-1/2})$, and the R{\'e}nyi entropy achieves the far better rate $T=\Omega(\epsilon^{-1/3})$. We argue that these better rate is possible because of \emph{separation margin} of loss functions, instead of the self-bounding property.

We consider the classic cake-cutting problem of producing fair allocations for $n$ agents, in the Robertson-Webb query model. In this model, it is known that: (i) proportional allocations can be computed using $O(n \log n)$ queries, and this is optimal for deterministic protocols; (ii) envy-free allocations (a subset of proportional allocations) can be computed using $O\left( n^{n^{n^{n^{n^{n}}}}} \right)$ queries, and the best known lower bound is $\Omega(n^2)$; (iii) perfect allocations (a subset of envy-free allocations) cannot be computed using a bounded (in $n$) number of queries. In this work, we introduce two hierarchies of new fairness notions: \newnotioninverse \,(\newnotioninverseabbrev) and \newnotionlinear \,(\newnotionlinearabbrev). An allocation is \newnotioninverseabbrev-$k$ if the allocation is complete and, for any subset of agents $S$ of size at most $k$, every agent $i \in S$ believes the value of all pieces allocated to agents in $S$ to be at least $\frac{1}{n-|S|+1}$, making the union of all pieces allocated to agents not in $S$ at most $\frac{n-|S|}{n-|S|+1}$; for \newnotionlinearabbrev-$k$ allocations, these bounds become $\frac{|S|}{n}$ and $\frac{n-|S|}{n}$, respectively. Intuitively, these notions of fairness ask that, for every agent $i$, the collective value (from the perspective of agent $i$) that a group of agents receives is limited. If the group includes $i$, its value is lower-bounded, and if the group excludes $i$, it is upper-bounded, thus providing the agent some protection against the formation of coalitions.

We consider the graph alignment problem, wherein the objective is to find a vertex correspondence between two graphs that maximizes the edge overlap. The graph alignment problem is an instance of the quadratic assignment problem (QAP), known to be NP-hard in the worst case even to approximately solve. In this paper, we analyze Birkhoff relaxation, a tight convex relaxation of QAP, and present theoretical guarantees on its performance when the inputs follow the Gaussian Wigner Model. More specifically, the weighted adjacency matrices are correlated Gaussian Orthogonal Ensemble with correlation $1/\sqrt{1+\sigma^2}$.

Recent large language models (LLMs) with long-chain-of-thought reasoning--such as DeepSeek-R1--have achieved impressive results on Olympiad-level mathematics benchmarks. However, they often rely on a narrow set of strategies and struggle with problems that require a novel way of thinking. To systematically investigate these limitations, we introduce OMEGA--Out-of-distribution Math Problems Evaluation with 3 Generalization Axes--a controlled yet diverse benchmark designed to evaluate three axes of out-of-distribution generalization, inspired by Boden's typology of creativity: (1) Exploratory--applying known problem-solving skills to more complex instances within the same problem domain; (2) Compositional--combining distinct reasoning skills, previously learned in isolation, to solve novel problems that require integrating these skills in new and coherent ways; and (3) Transformative--adopting novel, often unconventional strategies by moving beyond familiar approaches to solve problems more effectively. OMEGA consists of programmatically generated training-test pairs derived from templated problem generators across geometry, number theory, algebra, combinatorics, logic, and puzzles, with solutions verified using symbolic, numerical, or graphical methods. We evaluate frontier (or top-tier) LLMs and observe sharp performance degradation as problem complexity increases. Moreover, we fine-tune the Qwen-series models across all generalization settings and observe notable improvements in exploratory generalization, while compositional generalization remains limited, and transformative reasoning shows little to no improvement. By isolating and quantifying these fine-grained failures, OMEGA lays the groundwork for advancing LLMs toward genuine mathematical creativity beyond mechanical proficiency.

Algorithms with predictions is a recent framework for decision-making under uncertainty that leverages the power of machine-learned predictions without making any assumption about their quality. The goal in this framework is for algorithms to achieve an improved performance when the predictions are accurate while maintaining acceptable guarantees when the predictions are erroneous. A serious concern with algorithms that use predictions is that these predictions can be biased and, as a result, cause the algorithm to make decisions that are deemed unfair. We show that this concern manifests itself in the classical secretary problem in the learning-augmented setting---the state-of-the-art algorithm can have zero probability of accepting the best candidate, which we deem unfair, despite promising to accept a candidate whose expected value is at least $\max\{\Omega (1), 1 - O(\varepsilon)\}$ times the optimal value, where $\varepsilon$ is the prediction error.We show how to preserve this promise while also guaranteeing to accept the best candidate with probability $\Omega(1)$. Our algorithm and analysis are based on a new ``pegging'' idea that diverges from existing works and simplifies/unifies some of their results. Finally, we extend to the $k$-secretary problem and complement our theoretical analysis with experiments.

Linear temporal logic (LTL) and, more generally, $\omega$-regular objectives are alternatives to the traditional discount sum and average reward objectives in reinforcement learning (RL), offering the advantage of greater comprehensibility and hence explainability. In this work, we study the relationship between these objectives. Our main result is that each RL problem for $\omega$-regular objectives can be reduced to a limit-average reward problem in an optimality-preserving fashion, via (finite-memory) reward machines. Furthermore, we demonstrate the efficacy of this approach by showing that optimal policies for limit-average problems can be found asymptotically by solving a sequence of discount-sum problems approximately. Consequently, we resolve an open problem: optimal policies for LTL and $\omega$-regular objectives can be learned asymptotically.