Extracting a flat clustering solution from a hierarchy is a common task in practical cluster analysis and can be formulated as an optimisation problem. Existing approaches focus on finding a single optimal solution. We introduce FOSC-X, a framework for extracting the top-M globally optimal flat clusterings from local, non-horizontal cuts of a hierarchical cluster tree, while optionally enforcing constraints on the number of clusters. This enables automatic identification of multiple high-quality alternative clusterings that capture different aspects of the hierarchical structure. Without constraints, the top-M problem can be solved in polynomial time using dynamic programming, exploiting the property that locally optimal partial candidates within subtrees can be combined to form globally optimal solutions while automatically determining the number of clusters. However, this can lead to solutions with numbers of clusters that are ultimately undesirable -- e.g., too large to be meaningful or practically analysed within a particular application domain. Imposing cluster-count constraints breaks the optimality property underlying the unconstrained dynamic programming approach, since locally optimal partial candidates may no longer combine into feasible globally optimal solutions. FOSC-X addresses this challenge through a dynamic programming strategy that maintains compact sets of feasible candidates using lower and upper feasibility bounds while pruning infeasible or dominated combinations. The resulting method guarantees optimal rankings of the top-M solutions with linear-time complexity in the number of cluster nodes and dataset size, both with and without cluster-count constraints. Experiments show that FOSC-X efficiently reveals alternative clustering structures overlooked by single-solution extraction methods.

Clustering is a fundamental task in both machine learning and data mining. Among various methods, edge-colored clustering (ECC) has emerged as a useful approach for handling categorical data. Given a hypergraph with (hyper)edges labeled by colors, ECC aims to assign vertex colors to minimize the number of edges where the vertex color differs from the edge's color. However, traditional ECC has inherent limitations, as it enforces a nonoverlapping and exhaustive clustering. To tackle these limitations, three versions of ECC have been studied: LOCALECC and GLOBALECC, which allow overlapping clusters, and ROBUSTECC, which accounts for vertex outliers.

Although receiving notable improvements, current multi-view clustering (MVC) techniques generally rely on feature library mechanisms to propagate accumulated knowledge from historical views to newly-arrived data, which overlooks the information pertaining to basis embedding within each view. Moreover, the mapping paradigm inevitably alters the values of learned landmarks and built affinities due to the uninterruption nature, accordingly disarraying the hierarchical cluster structures. To mitigate these two issues, we in the paper provide a named BSTM algorithm. Concretely, we firstly synchronize with the distinct dimensions by introducing a group of specialized projectors, and then establish unified anchors for all views collected so far to capture intrinsic patterns. Afterwards, departing from per-view architectures, we devise a shared bipartite graph construction via indicators to quantify similarity, which not only avoids redundant data-recalculations but alleviates the representation distortion caused by fusion.

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.

Training LLMs relies on distributed implementations using multiple GPUs to compute gradients in parallel with sharded optimizers. However, synchronizing gradients in data parallel setups introduces communication overhead that grows with the number of workers, limiting parallelization efficiency. Local optimization algorithms reduce communications but incur high memory costs as they prevent optimizer state sharding, hindering scalability. To address this, we propose ACcumulate while COmmunicate (ACCO), a memory-efficient optimization algorithm for distributed LLM training. By synchronizing delayed gradients while computing new ones, ACCO reduces GPU idle time and supports heterogeneous hardware. To mitigate the convergence issues caused by delayed updates, we introduce a novel technique ensuring training dynamics align with standard distributed optimization. Compared to ZeRO-1, our approach is significantly faster and scales effectively across heterogeneous hardware.

Bayesian Optimization is a widely used method for optimizing expensive black-box functions, relying on probabilistic surrogate models such as Gaussian Processes. The quality of the surrogate model is crucial for good optimization performance, especially in the few-shot setting where only a small number of batches of points can be evaluated. In this setting, the initialization plays a critical role in shaping the surrogate's predictive quality and guiding subsequent optimization. Despite this, practitioners typically rely on (quasi-)random designs to cover the input space. However, such approaches neglect two key factors: (a) space-filling designs may not be desirable to reduce predictive uncertainty, and (b) efficient hyperparameter learning during initialization is essential for high-quality prediction, which may conflict with space-filling designs. To address these limitations, we propose Hyperparameter-Informed Predictive Exploration (HIPE), a novel acquisition strategy that balances predictive uncertainty reduction with hyperparameter learning using information-theoretic principles. We derive a closed-form expression for HIPE in the Gaussian Process setting and demonstrate its effectiveness through extensive experiments in active learning and few-shot BO. Our results show that HIPE outperforms standard initialization strategies in terms of predictive accuracy, hyperparameter identification, and subsequent optimization performance, particularly in large-batch, few-shot settings relevant to many real-world Bayesian Optimization applications.

Online bidding is a classic optimization problem, with several applications in online decision-making, the design of interruptible systems, and the analysis of approximation algorithms. In this work, we study online bidding under learningaugmented settings that incorporate stochasticity, in either the prediction oracle or the algorithm itself. In the first part, we study bidding under distributional predictions, and find Pareto-optimal algorithms that offer the best-possible tradeoff between the consistency and the robustness of the algorithm. In the second part, we study the power and limitations of randomized bidding algorithms, by presenting upper and lower bounds on the consistency/robustness tradeoffs. Previous works focused predominantly on oracles that do not leverage stochastic information on the quality of the prediction, and deterministic algorithms.

We consider the basic problem of learning Single-Index Models with respect to the square loss under the Gaussian distribution in the presence of adversarial label noise. Our main contribution is the first computationally efficient algorithm for this learning task, achieving a constant factor approximation, that succeeds for the class of all monotone activations with bounded moment of order 2 + ζ, for ζ > 0. This class in particular includes all monotone Lipschitz functions and even discontinuous functions like (possibly biased) halfspaces. Prior work for the case of unknown activation either does not attain constant factor approximation or succeeds for a substantially smaller family of activations. The main conceptual novelty of our approach lies in developing an optimization framework that steps outside the boundaries of usual gradient methods and instead identifies a useful vector field to guide the algorithm updates by directly leveraging the problem structure, properties of Gaussian spaces, and regularity of monotone functions.

Large reasoning models (LRMs) like OpenAI-o1 have shown impressive capabilities in natural language reasoning. However, these models frequently demonstrate inefficiencies or inaccuracies when tackling complex mathematical operations. While integrating computational tools such as Code Interpreters (CIs) offers a promising solution, it introduces a critical challenge: a conflict between the model's internal, probabilistic reasoning and the external, deterministic knowledge provided by the CI, which often leads models to unproductive deliberation. To overcome this, we introduce CoRT (Code-Optimized Reasoning Training), a post-training framework designed to teach LRMs to effectively utilize CIs. We propose Hint-Engineering, a new data synthesis strategy that strategically injects diverse hints at optimal points within reasoning paths. This approach generates high-quality, code-integrated reasoning data specifically tailored to optimize LRMCI interaction. Using this method, we have synthesized 30 high-quality samples to post-train models ranging from 1.5B to 32B parameters through supervised fine-tuning.

Many emerging applications--such as adversarial training, AI alignment, and robust optimization--can be framed as zero-sum games between neural nets, with von Neumann-Nash equilibria (NE) capturing the desirable system behavior. While such games often involve non-convex non-concave objectives, empirical evidence shows that simple gradient methods frequently converge, suggesting a hidden geometric structure. In this paper, we provide a theoretical framework that explains this phenomenon through the lens of hidden convexity and overparameterization. We identify sufficient conditions--spanning initialization, training dynamics, and network width--that guarantee global convergence to a NE in a broad class of non-convex min-max games. To our knowledge, this is the first such result for games that involve two-layer neural networks. Technically, our approach is twofold: (a) we derive a novel path-length bound for the alternating gradient descent-ascent scheme in min-max games; and (b) we show that the reduction from a hidden convex-concave geometry to two-sided Polyak-Łojasiewicz (PL) min-max condition hold with high probability under overparameterization, using tools from random matrix theory.