We present \textsf{ModularSubsetSelection} (MSS), a new algorithm for locally differentially private (LDP) frequency estimation. Given a universe of size $k$ and $n$ users, our $\varepsilon$-LDP mechanism encodes each input via a Residue Number System (RNS) over $\ell$ pairwise-coprime moduli $m_0, \ldots, m_{\ell-1}$, and reports a randomly chosen index $j \in [\ell]$ along with the perturbed residue using the statistically optimal \textsf{SubsetSelection} (SS) (Wang et al. 2016). This design reduces the user communication cost from $Θ\bigl(ω\log_2(k/ω)\bigr)$ bits required by standard SS (with $ω\approx k/(e^\varepsilon+1)$) down to $\lceil \log_2 \ell \rceil + \lceil \log_2 m_j \rceil$ bits, where $m_j < k$. Server-side decoding runs in $Θ(n + r k \ell)$ time, where $r$ is the number of LSMR (Fong and Saunders 2011) iterations. In practice, with well-conditioned moduli (\textit{i.e.}, constant $r$ and $\ell = Θ(\log k)$), this becomes $Θ(n + k \log k)$. We prove that MSS achieves worst-case MSE within a constant factor of state-of-the-art protocols such as SS and \textsf{ProjectiveGeometryResponse} (PGR) (Feldman et al. 2022) while avoiding the algebraic prerequisites and dynamic-programming decoder required by PGR. Empirically, MSS matches the estimation accuracy of SS, PGR, and \textsf{RAPPOR} (Erlingsson, Pihur, and Korolova 2014) across realistic $(k, \varepsilon)$ settings, while offering faster decoding than PGR and shorter user messages than SS. Lastly, by sampling from multiple moduli and reporting only a single perturbed residue, MSS achieves the lowest reconstruction-attack success rate among all evaluated LDP protocols.

Linear Predictive Clustering (LPC) partitions samples based on shared linear relationships between feature and target variables, with numerous applications including marketing, medicine, and education. Greedy optimization methods, commonly used for LPC, alternate between clustering and linear regression but lack global optimality. While effective for separable clusters, they struggle in non-separable settings where clusters overlap in feature space. In an alternative constrained optimization paradigm, Bertsimas and Shioda (2007) formulated LPC as a Mixed-Integer Program (MIP), ensuring global optimality regardless of separability but suffering from poor scalability. This work builds on the constrained optimization paradigm to introduce two novel approaches that improve the efficiency of global optimization for LPC. By leveraging key theoretical properties of separability, we derive near-optimal approximations with provable error bounds, significantly reducing the MIP formulation's complexity and improving scalability. Additionally, we can further approximate LPC as a Quadratic Pseudo-Boolean Optimization (QPBO) problem, achieving substantial computational improvements in some settings. Comparative analyses on synthetic and real-world datasets demonstrate that our methods consistently achieve near-optimal solutions with substantially lower regression errors than greedy optimization while exhibiting superior scalability over existing MIP formulations.

Learning intersections of halfspaces is a central problem in Computational Learning Theory. Even for just two halfspaces, it remains a major open question whether learning is possible in polynomial time with respect to the margin $γ$ of the data points and their dimensionality $d$. The best-known algorithms run in quasi-polynomial time $d^{O(\log(1/γ))}$, and it has been shown that this complexity is unavoidable for any algorithm relying solely on correlational statistical queries (CSQ). In this work, we introduce a novel algorithm that provably circumvents the CSQ hardness barrier. Our approach applies to a broad class of distributions satisfying a natural, previously studied, factorizability assumption. Factorizable distributions lie between distribution-specific and distribution-free settings, and significantly extend previously known tractable cases. Under these distributions, we show that CSQ-based methods still require quasipolynomial time even for weakly learning, whereas our algorithm achieves $poly(d,1/γ)$ time by leveraging more general statistical queries (SQ), establishing a strong separation between CSQ and SQ for this simple realizable PAC learning problem. Our result is grounded in a rigorous analysis utilizing a novel duality framework that characterizes the moment tensor structure induced by the marginal distributions. Building on these structural insights, we propose new, efficient learning algorithms. These algorithms combine a refined variant of Jennrich's Algorithm with PCA over random projections of the moment tensor, along with a gradient-descent-based non-convex optimization framework.

Recent advancements in large language models (LLMs) have demonstrated remarkable text generation capabilities. However, controlling specific attributes of generated text remains challenging without architectural modifications or extensive fine-tuning. Current methods typically toggle a single, basic attribute but struggle with precise multi-attribute control. In scenarios where attribute requirements conflict, existing methods lack coordination mechanisms, causing interference between desired attributes. Furthermore, these methods fail to incorporate iterative optimization processes in the controlled generation pipeline. To address these limitations, we propose Conflict-aware, Composite, and Collaborative Controlled Text Generation (C$^3$TG), a two-phase framework for fine-grained, multi-dimensional text attribute control. During generation, C$^3$TG selectively pairs the LLM with the required attribute classifiers from the 17 available dimensions and employs weighted KL-divergence to adjust token probabilities. The optimization phase then leverages an energy function combining classifier scores and penalty terms to resolve attribute conflicts through iterative feedback, enabling precise control over multiple dimensions simultaneously while preserving natural text flow. Experiments show that C$^3$TG significantly outperforms baselines across multiple metrics including attribute accuracy, linguistic fluency, and output diversity, while simultaneously reducing toxicity. These results establish C$^3$TG as an effective and flexible solution for multi-dimensional text attribute control that requires no costly model modifications.

The curse of dimensionality presents a pervasive challenge in optimization problems, with exponential expansion of the search space rapidly causing traditional algorithms to become inefficient or infeasible. An adaptive sampling strategy is presented to accelerate optimization in this domain as an alternative to uniform quasi-Monte Carlo (QMC) methods. This method, referred to as Hyperellipsoid Density Sampling (HDS), generates its sequences by defining multiple hyperellipsoids throughout the search space. HDS uses three types of unsupervised learning algorithms to circumvent high-dimensional geometric calculations, producing an intelligent, non-uniform sample sequence that exploits statistically promising regions of the parameter space and improves final solution quality in high-dimensional optimization problems. A key feature of the method is optional Gaussian weights, which may be provided to influence the sample distribution towards known locations of interest. This capability makes HDS versatile for applications beyond optimization, providing a focused, denser sample distribution where models need to concentrate their efforts on specific, non-uniform regions of the parameter space. The method was evaluated against Sobol, a standard QMC method, using differential evolution (DE) on the 29 CEC2017 benchmark test functions. The results show statistically significant improvements in solution geometric mean error (p < 0.05), with average performance gains ranging from 3% in 30D to 37% in 10D. This paper demonstrates the efficacy of HDS as a robust alternative to QMC sampling for high-dimensional optimization.

AI inference scaling is often tuned through 1D heuristics (a fixed reasoning pass) or 2D bivariate trade-offs (e.g., accuracy vs. compute), which fail to consider cost and latency constraints. We introduce a 3D optimization framework that jointly calibrates accuracy, cost, and latency within a unified decision space, enabling constraints-aware inference scaling. Using Monte Carlo simulations across three representative scenarios and nine simulated large language models, we evaluate four optimization methods to address the 3D multi-objective optimization (MOO) problem. Framing inference scaling in MOO shapes a feasible space that 1D and 2D optimizations fail to capture, enabling environment-adaptive selection of the inference scaling~$k$. Results show that knee-point optimization based on Pareto frontiers achieves the best balance, while accuracy-maximization remains favorable when accuracy is prioritized. Our results further show that smaller models, when combined with optimal inference scaling, can match or exceed the performance of larger models at a fraction of the cost. The framework establishes a theoretical foundation for deployment-aware inference scaling across diverse operational conditions.

Projection methods aim to reduce the dimensionality of the optimization instance, thereby improving the scalability of high-dimensional problems. Recently, Sakaue and Oki proposed a data-driven approach for linear programs (LPs), where the projection matrix is learned from observed problem instances drawn from an application-specific distribution of problems. We analyze the generalization guarantee for the data-driven projection matrix learning for convex quadratic programs (QPs). Unlike in LPs, the optimal solutions of convex QPs are not confined to the vertices of the feasible polyhedron, and this complicates the analysis of the optimal value function. To overcome this challenge, we demonstrate that the solutions of convex QPs can be localized within a feasible region corresponding to a special active set, utilizing Caratheodory's theorem. Building on such observation, we propose the unrolled active set method, which models the computation of the optimal value as a Goldberg-Jerrum (GJ) algorithm with bounded complexities, thereby establishing learning guarantees. We then further extend our analysis to other settings, including learning to match the optimal solution and input-aware setting, where we learn a mapping from QP problem instances to projection matrices.

The implicit hitting set (IHS) approach offers a general framework for solving computationally hard combinatorial optimization problems declaratively. IHS iterates between a decision oracle used for extracting sources of inconsistency and an optimizer for computing so-called hitting sets (HSs) over the accumulated sources of inconsistency. While the decision oracle is language-specific, the optimizers is usually instantiated through integer programming. We explore alternative algorithmic techniques for hitting set optimization based on different ways of employing pseudo-Boolean (PB) reasoning as well as stochastic local search. We extensively evaluate the practical feasibility of the alternatives in particular in the context of pseudo-Boolean (0-1 IP) optimization as one of the most recent instantiations of IHS. Highlighting a trade-off between efficiency and reliability, while a commercial IP solver turns out to remain the most effective way to instantiate HS computations, it can cause correctness issues due to numerical instability; in fact, we show that exact HS computations instantiated via PB reasoning can be made competitive with a numerically exact IP solver. Furthermore, the use of PB reasoning as a basis for HS computations allows for obtaining certificates for the correctness of IHS computations, generally applicable to any IHS instantiation in which reasoning in the declarative language at hand can be captured in the PB-based proof format we employ.

Abstract-- Differential drive mobile manipulators combine the mobility of wheeled bases with the manipulation capability of multi-joint arms, enabling versatile applications but posing considerable challenges for trajectory planning due to their high-dimensional state space and nonholonomic constraints. This paper introduces T opA Y, an optimization-based planning framework designed for efficient and safe trajectory generation for differential drive mobile manipulators. The framework employs a hierarchical initial value acquisition strategy, including topological paths search for the base and parallel sampling for the manipulator . A polynomial trajectory representation with arc length-yaw parameterization is also proposed to reduce optimization complexity while preserving dynamic feasibility. Extensive simulation and real-world experiments validate that T opA Y achieves higher planning efficiency and success rates than state-of-the-art method in dense and complex scenarios. The source code is released at https://github.com/T Differential drive mobile manipulator (DDMoMa), comprising multi-joint manipulator(s) mounted on a differential drive base (DDB), integrates rich manipulation ability of manipulators and mobility of wheeled robots.

The ability to design molecules while preserving similarity to a target molecule and/or property is crucial for various applications in drug discovery, chemical design, and biology. We introduce in this paper an efficient training-free method for navigating and sampling from the molecular space with a generative Chemical Language Model (CLM), while using the molecular similarity to the target as a guide. Our method leverages the contextual representations learned from the CLM itself to estimate the molecular similarity, which is then used to adjust the autoregressive sampling strategy of the CLM. At each step of the decoding process, the method tracks the distance of the current generations from the target and updates the logits to encourage the preservation of similarity in generations. We implement the method using a recently proposed $\sim$47M parameter SMILES-based CLM, GP-MoLFormer, and therefore refer to the method as GP-MoLFormer-Sim, which enables a test-time update of the deep generative policy to reflect the contextual similarity to a set of guide molecules. The method is further integrated into a genetic algorithm (GA) and tested on a set of standard molecular optimization benchmarks involving property optimization, molecular rediscovery, and structure-based drug design. Results show that, GP-MoLFormer-Sim, combined with GA (GP-MoLFormer-Sim+GA) outperforms existing training-free baseline methods, when the oracle remains black-box. The findings in this work are a step forward in understanding and guiding the generative mechanisms of CLMs.