This paper investigates convex-concave minimax optimization problems where only the function value access is allowed. We introduce a class of Hessianaware quantum zeroth-order methods that can find the ǫ-saddle point within O(d2/3ǫ 2/3) function value oracle calls. This represents an improvement of d1/3ǫ 1/3 over the O(dǫ 1) upper bound of classical zeroth-order methods, where d denotes the problem dimension. We extend these results to µ-stronglyconvex µ-strongly-concave minimax problems using a restart strategy, and show a speedup of d1/3µ 1/3 compared to classical zeroth-order methods. The acceleration achieved by our methods stems from the construction of efficient quantum estimators for the Hessian and the subsequent design of efficient Hessian-aware algorithms. In addition, we apply such ideas to non-convex optimization, leading to a reduction in the query complexity compared to classical methods.

We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Increments (MBI) oracle and the Riemannian Heat-kernel (RHK) oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with ε-accuracy requires O(log(1/ε)) iterations in Kullback-Leibler divergence assuming access to exact oracles and O(log2(1/ε))iterations in the total variation metric assuming access to sufficiently accurate inexact oracles.

Machine learning can greatly benefit from providing learning algorithms with pairs of contrastive training examples--typically pairs of instances that differ only slightly, yet have different class labels. Intuitively, the difference in the instances helps explain the difference in the class labels. This paper proposes a theoretical framework in which the effect of various types of contrastive examples on active learners is studied formally. The focus is on the sample complexity of learning concept classes and how it is influenced by the choice of contrastive examples. We illustrate our results with geometric concept classes and classes of Boolean functions. Interestingly, we reveal a connection between learning from contrastive examples and the classical model of self-directed learning.

We study matching markets with ties, where workers on one side of the market may have tied preferences over jobs, determined by their matching utilities. Unlike classical two-sided markets with strict preferences, no single stable matching exists that is utility-maximizing for all workers. To address this challenge, we introduce the Optimal Stable Share (OSS)-ratio, which measures the ratio of a worker's maximum achievable utility in any stable matching to their utility in a given matching. We prove that distributions over only stable matchings can incur linear utility losses, i.e., an Ω(N) OSS-ratio, where N is the number of workers. To overcome this, we design an algorithm that efficiently computes a distribution over (possibly non-stable) matchings, achieving an asymptotically tight O(logN) OSS-ratio. When exact utilities are unknown, our second algorithm guarantees workers a logarithmic approximation of their optimal utility under bounded instability. Finally, we extend our offline approximation results to a bandit learning setting where utilities are only observed for matched pairs. In this setting, we consider worker-optimal stable regret, design an adaptive algorithm that smoothly interpolates between markets with strict preferences and those with statistical ties, and establish a lower bound revealing the fundamental trade-off between strict and tied preference regimes.

We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles, and demonstrate the first dimension-independent query complexities (up to poly-logarithmic factors) for zeroth-order convex optimization in both smooth and nonsmooth settings. Interestingly, only using noisy function evaluation oracles, we match the first-order query complexities of classical gradient descent, thereby exhibiting exponential separation between quantum and classical zeroth-order optimization. We then generalize these algorithms to work in non-Euclidean settings by using quantum (sub)gradient estimation to instantiate mirror descent and its variants, including dual averaging and mirror prox. By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games. We identify a parameter regime in which our zero-sum games algorithm is faster than any existing classical or quantum approach.

Online portfolio selection seeks to determine a sequence of allocations to maximize capital growth. Classical universal strategies asymptotically match the best constant-rebalanced portfolio but ignore potential forecasts, whereas heuristic methods often collapse when belief fails. We formalize this tension in a learningaugmented setting in which an investor observes (possibly erroneous) predictions prior to each decision moment, and we introduce the Rebalanced Arithmetic Mean portfolio with predictions (RAM). Under arbitrary return sequences, we prove that RAM captures at least a constant fraction of the hindsight-optimal wealth when forecasts are perfect while still exceeding the geometric mean of the sequence even when the predictions are adversarial. Comprehensive experiments on largescale equity data strengthen our theory, spanning both synthetic prediction streams and production-grade machine-learning models. RAM advantages over universalportfolio variants equipped with side information across various regimes. These results demonstrate that modest predictive power can be reliably converted into tangible gains without sacrificing worst-case guarantees.

Projection-free first-order methods, e.g., the celebrated Frank-Wolfe (FW) algorithms, have emerged as powerful tools for optimization over simple convex sets such as polyhedra, because of their scalability, fast convergence, and iteration-wise feasibility without costly projections. However, extending these methods effectively to general compact convex sets remains challenging and largely open, as FW methods rely on expensive linear optimization oracles (LOO), while penalty-based methods often struggle with poor feasibility. We tackle this open challenge by presenting Hom-PGD, a novel projection-free method without expensive (optimization) oracles. Our method constructs a homeomorphism between the convex constraint set and a unit ball, transforming the original problem into an equivalent ball-constrained formulation, thus enabling efficient gradient-based optimization while preserving the original problem structure. We prove that Hom-PGD attains optimal convergence rates matching gradient descent with constant step-size to find an ϵ-approximate (stationary) solution: O(log(1/ϵ))for strongly convex objectives, O(ϵ 1) for convex objectives, and O(ϵ 2) for non-convex objectives. Meanwhile, Hom-PGD enjoys a low per-iteration complexity of O(n2), without expensive oracles like LOO or projection, where nis the input size. Our framework further extends to certain non-convex sets, broadening its applicability in practical optimization scenarios with complex constraints. Extensive numerical experiments demonstrate that Hom-PGD achieves comparable convergence rates to state-of-theart projection-free methods, while significantly reducing per-iteration runtime (up to 5 orders of magnitude faster) and thus the total problem-solving time.

In this paper, we study differentially private mechanisms for functions whose outputs lie in a Euclidean Jordan algebra. Euclidean Jordan algebras capture many important mathematical structures and form the foundation of linear programming, second-order cone programming, and semidefinite programming. Our main contribution is a generic Gaussian mechanism for such functions, with sensitivity measured in ℓ2, ℓ1, and ℓ norms. Notably, this framework includes the important case where the function outputs are symmetric matrices, and sensitivity is measured in the Frobenius, nuclear, or spectral norm. We further derive private algorithms for solving symmetric cone programs under various settings, using a combination of the multiplicative weights update method and our generic Gaussian mechanism. As an application, we present differentially private algorithms for semidefinite programming, resolving a major open question posed by [Hsu, Roth, Roughgarden, and Ullman, ICALP 2014].

We develop optimization methods which offer new trade-offs between the number of gradient and Hessian computations needed to compute the critical point of a nonconvex function. We provide a method that for a twice-differentiable f: Rd R with L2-Lipschitz Hessian, an input initial point with -bounded sub-optimality, and a sufficiently small ϵ > 0, outputs an ϵ-critical point, i.e., a point xsuch that

We study the problem of computing the exact median by leveraging side information to minimize costly, exact comparisons. We analyze this problem in two key settings: (1) using predictions from unreliable "weak" oracles, and (2) exploiting known structural information in the form of a partial order. In the classical setting, we introduce a modified LazySelect algorithm that combines weak comparisons with occasional strong comparisons through majority voting. We show that this hybrid strategy has near-linear running time and can achieve high-probability correctness using only sublinear strong comparisons, even when the weak oracle is only slightly better than random guessing. Our theoretical results hold under the persistent comparison model, where resampling will not amplify the probability of correctness. In the partially ordered setting, we generalize the notion of median to directed acyclic graphs (DAGs) and show that the complexity of median selection depends heavily on the DAG's width. We complement our analysis with extensive experiments on synthetic data.