Optimality

Quantum kernel methods are among the leading candidates for achieving quantum advantage in supervised learning. A key bottleneck is the cost of inference: evaluating a trained model on new data requires estimating a weighted sum $\sum_{i=1}^N α_i k(x,x_i)$ of $N$ kernel values to additive precision $\varepsilon$, where $α$ is the vector of trained coefficients. The standard approach estimates each term independently via sampling, yielding a query complexity of $O(N\lVertα\rVert_2^2/\varepsilon^2)$. In this work, we identify two independent axes for improvement: (1) How individual kernel values are estimated (sampling versus quantum amplitude estimation), and (2) how the sum is approximated (term-by-term versus via a single observable), and systematically analyze all combinations thereof. The query-optimal combination, encoding the full inference sum as the expectation value of a single observable and applying quantum amplitude estimation, achieves a query complexity of $O(\lVertα\rVert_1/\varepsilon)$, removing the dependence on $N$ from the query count and yielding a quadratic improvement in both $\lVertα\rVert_1$ and $\varepsilon$. We prove a matching lower bound of $Ω(\lVertα\rVert_1/\varepsilon)$, establishing query-optimality of our approach up to logarithmic factors. Beyond query complexity, we also analyze how these improvements translate into gate costs and show that the query-optimal strategy is not always optimal in practice from the perspective of gate complexity. Our results provide both a query-optimal algorithm and a practically optimal choice of strategy depending on hardware capabilities, along with a complete landscape of intermediate methods to guide practitioners. All algorithms require only amplitude estimation as a subroutine and are thus natural candidates for early-fault-tolerant implementations.

Robotic path planning problems are often NP-hard, and practical solutions typically rely on approximation algorithms with provable performance guarantees for general cases. While designing such algorithms is challenging, formally proving their approximation optimality is even more demanding, which requires domain-specific geometric insights and multi-step mathematical reasoning over complex operational constraints. Recent Large Language Models (LLMs) have demonstrated strong performance on mathematical reasoning benchmarks, yet their ability to assist with research-level optimality proofs in robotic path planning remains under-explored. In this work, we introduce the first benchmark for evaluating LLMs on approximation-ratio proofs of robotic path planning algorithms. The benchmark consists of 34 research-grade proof tasks spanning diverse planning problem types and complexity levels, each requiring structured reasoning over algorithm descriptions, problem constraints, and theoretical guarantees. Our evaluation of state-of-the-art proprietary and open-source LLMs reveals that even the strongest models struggle to produce fully valid proofs without external domain knowledge. However, providing LLMs with task-specific in-context lemmas substantially improves reasoning quality, a factor that is more effective than generic chain-of-thought prompting or supplying the ground-truth approximation ratio as posterior knowledge. We further provide fine-grained error analysis to characterize common logical failures and hallucinations, and demonstrate how each error type can be mitigated through targeted context augmentation.

In general reinforcement learning, all established optimal agents, including AIXI, are model-based, explicitly maintaining and using environment models. This paper introduces Universal AI with Q-Induction (AIQI), the first model-free agent proven to be asymptotically $\varepsilon$-optimal in general RL. AIQI performs universal induction over distributional action-value functions, instead of policies or environments like previous works. Under a grain of truth condition, we prove that AIQI is strong asymptotically $\varepsilon$-optimal and asymptotically $\varepsilon$-Bayes-optimal. We also apply our novel proof techniques to show asymptotic $\varepsilon$-optimality of Self-AIXI without any ad-hoc assumptions. Our results significantly expand the diversity of known universal agents.

The Winner Determination Problem (WDP) in combinatorial auctions is NP-hard, and no existing method reliably predicts which instances will defeat fast greedy heuristics. The ML-for-combinatorial-optimization community has focused on learning to \emph{replace} solvers, yet recent evidence shows that graph neural networks (GNNs) rarely outperform well-tuned classical methods on standard benchmarks. We pursue a different objective: learning to predict \emph{when} a given instance is hard for greedy allocation, enabling instance-dependent algorithm selection. We design a 20-dimensional structural feature vector and train a lightweight MLP hardness classifier that predicts the greedy optimality gap with mean absolute error 0.033, Pearson correlation 0.937, and binary classification accuracy 94.7% across three random seeds. For instances identified as hard -- those exhibiting ``whale-fish'' trap structure where greedy provably fails -- we deploy a heterogeneous GNN specialist that achieves ${\approx}0\%$ optimality gap on all six adversarial configurations tested (vs.\ 3.75--59.24% for greedy). A hybrid allocator combining the hardness classifier with GNN and greedy solvers achieves 0.51% overall gap on mixed distributions. Our honest evaluation on CATS benchmarks confirms that GNNs do not outperform Gurobi (0.45--0.71 vs.\ 0.20 gap), motivating the algorithm selection framing. Learning \emph{when} to deploy expensive solvers is more tractable than learning to replace them.

Sampling-based motion planners (SBMPs) are widely used for robot motion planning with complex kinodynamic constraints in high-dimensional spaces, yet their serial computation design results in planning speeds that scale poorly with problem complexity. Recent efforts to parallelize SBMPs have achieved significant speedups in finding feasible solutions; however, they provide no guarantees of optimizing an objective function. We introduce Kino-PAX$^{+}$, a massively parallel kinodynamic SBMP with asymptotic near-optimal guarantees. Kino-PAX$^{+}$ builds a sparse tree of dynamically feasible trajectories by decomposing traditionally serial operations into three massively parallel subroutines. The algorithm focuses computation on the most promising nodes within local neighborhoods for propagation and refinement, enabling rapid improvement of solution cost. We prove that, while maintaining probabilistic $δ$-robust completeness, this focus on promising nodes ensures asymptotic $δ$-robust near-optimality. Our results show that Kino-PAX$^{+}$ finds solutions up to three orders of magnitude faster than existing serial methods and achieves lower solution costs

To promote cooperation in Multi-Agent Reinforcement Learning, the reward signals of all agents can be aggregated together, forming global rewards that are commonly known as the fully cooperative setting. However, global rewards are usually noisy because they contain the contributions of all agents, which have to be resolved in the credit assignment process. On the other hand, using local reward benefits from faster learning due to the separation of agents' contributions, but can be suboptimal as agents myopically optimize their own reward while disregarding the global optimality. In this work, we propose a method that combines the merits of both approaches. By using a graph of interaction between agents, our method discerns the individual agent contribution in a more fine-grained manner than a global reward, while alleviating the cooperation problem with agents' local reward. We also introduce a practical approach for approximating such a graph. Our experiments demonstrate the flexibility of the approach, enabling improvements over the traditional local and global reward settings.

In this paper, we study the private and robust alignment of language models from a theoretical perspective by establishing upper bounds on the suboptimality gap in both offline and online settings. We consider preference labels subject to privacy constraints and/or adversarial corruption, and analyze two distinct interplays between them: privacy-first and corruption-first. For the privacy-only setting, we show that log loss with an MLE-style algorithm achieves near-optimal rates, in contrast to conventional wisdom. For the joint privacy-and-corruption setting, we first demonstrate that existing offline algorithms in fact provide stronger guarantees -- simultaneously in terms of corruption level and privacy parameters -- than previously known, which further yields improved bounds in the corruption-only regime. In addition, we also present the first set of results for private and robust online alignment. Our results are enabled by new uniform convergence guarantees for log loss and square loss under privacy and corruption, which we believe have broad applicability across learning theory and statistics.

We consider the design of smoothings of the (coordinate-wise) max function in $\mathbb{R}^d$ in the infinity norm. The LogSumExp function $f(x)=\ln(\sum^d_i\exp(x_i))$ provides a classical smoothing, differing from the max function in value by at most $\ln(d)$. We provide an elementary construction of a lower bound, establishing that every overestimating smoothing of the max function must differ by at least $\sim 0.8145\ln(d)$. Hence, LogSumExp is optimal up to small constant factors. However, we provide strictly stronger smoothings showing the entropy-based LogSumExp approach is not exactly optimal. In small dimensions, we propose exactly optimal smoothings, attaining our lower bound.

The structure of naming systems in natural languages hinges on a trade-off between high informativeness and low complexity. Focusing on the domain of kinship naming, we analyze such trade-off while addressing simplifying assumptions of prior work, namely: (i) universal communicative need across languages, and (ii) optimal listeners. To that aim, we collect data from four different languages, and analyze how different communicative needs and variations in the listener model influence the informativeness--complexity trade-off. Adopting a referential game setup from emergent communication, we further show that trade-off optimality is not only theoretically achievable but also emerges empirically in learned communication systems.

Large reasoning models (LRMs) often consume excessive tokens, inflating computational cost and latency. More broadly, in goal reaching sequential decision problems we often want to reach the goal quickly, and LRM reasoning can be viewed through this lens. We challenge the assumption that longer responses improve accuracy. By penalizing reasoning tokens using a discounted reinforcement learning setup (interpretable as a small token cost) and analyzing Blackwell optimality in restricted policy classes, we encourage concise yet accurate reasoning, analogous to preferring shorter successful trajectories in a stochastic shortest path problem. Experiments confirm our theoretical results that this approach shortens chains of thought while preserving accuracy.

We study the dynamic allocation of indivisible resources to strategic agents under long-term constraints, where the planner aims to maximize social welfare, satisfy multiple constraints, and elicit near-truthful reports. We find standard primal-dual methods fragile in this setting: agents easily manipulate their reports to distort dual variables, sacrificing social efficiency for individual utility. To address this, we propose the Incentive-Aware Primal-Dual (IAPD) framework. On the primal side, we integrate three components to suppress manipulation: a VCG-based payment neutralizes immediate misreporting benefits, while epoch-based lazy updates and random exploration together ensure potential future gains are outweighed by immediate penalties. On the dual side, to overcome a learning barrier due to lazy updates -- which we call the "price of incentives" -- we design a novel optimistic online learning algorithm, O-FTRL-FP. It utilizes a fixed-point oracle to resolve the circular dependency between optimistic dual variables and the resulting allocations. Ultimately, our mechanism attains $\tilde{\mathcal O}(\sqrt T)$ social welfare regret, satisfies all long-term constraints, and induces a near-truthful equilibrium. It also smoothly generalizes to multi-unit multi-demand allocation problems. Notably, this $\tilde{\mathcal O}(\sqrt T)$ regret near-matches the non-strategic $Ω(\sqrt T)$ lower bound, demonstrating that incentive-awareness can be accommodated at nearly no cost.

Merely pursuing performance may adversely affect safety, while a conservative policy for safe exploration will degrade the performance. How to guarantee both safety and performance in learning-based control problems is an interesting yet challenging issue. This paper aims to enhance system performance with a safety guarantee by solving reinforcement learning (RL)-based optimal control problems for nonlinear systems subject to high-relative-degree state constraints and unknown time-varying disturbance/actuator faults. A new type of control barrier functions (CBFs), termed high-order reciprocal-based control barrier function, is proposed to handle high-relative-degree constraints, which extends the design of CBFs to enforce robust safety without knowing the disturbance bound. The concept of gradient similarity is proposed to quantify the relationship between safety and performance. Finally, gradient manipulation and adaptive mechanisms are introduced in the model-based safe RL framework to enhance the performance with a safety guarantee. Two simulation examples illustrate the efficacy of the proposed algorithms.

Local SGD is a popular optimization method in distributed learning, often outperforming other algorithms in practice, including mini-batch SGD. Despite this success, theoretically proving the dominance of local SGD in settings with reasonable data heterogeneity has been difficult, creating a significant gap between theory and practice. In this paper, we provide new lower bounds for local SGD under existing first-order data heterogeneity assumptions, showing that these assumptions are insufficient to prove the effectiveness of local update steps. Furthermore, under these same assumptions, we demonstrate the min-max optimality of accelerated mini-batch SGD, which fully resolves our understanding of distributed optimization for several problem classes. Our results emphasize the need for better models of data heterogeneity to understand the effectiveness of local SGD in practice. Towards this end, we consider higher-order smoothness and heterogeneity assumptions, providing new upper bounds that imply the dominance of local SGD over mini-batch SGD when data heterogeneity is low.

Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.