Generalization is a key challenge in machine learning, specifically in reasoning tasks, where models are expected to solve problems more complex than those encountered during training. Existing approaches typically train reasoning models in an end-to-end fashion, directly mapping input instances to solutions. While this allows models to learn useful heuristics from data, it often results in limited generalization beyond the training distribution. In this work, we propose a novel approach to reasoning generalization by learning energy landscapes over the solution spaces of smaller, more tractable subproblems. At test time, we construct a global energy landscape for a given problem by combining the energy functions of multiple subproblems. This compositional approach enables the incorporation of additional constraints during inference, allowing the construction of energy landscapes for problems of increasing difficulty. To improve the sample quality from this newly constructed energy landscape, we introduce Parallel Energy Minimization (PEM). We evaluate our approach on a wide set of reasoning problems. Our method outperforms existing state-of-the-art methods, demonstrating its ability to generalize to larger and more complex problems.

In online inverse linear optimization, a learner observes time-varying sets of feasible actions and an agent's optimal actions, selected by solving linear optimization over the feasible actions. The learner sequentially makes predictions of the agent's true linear objective function, and their quality is measured by the regret, the cumulative gap between optimal objective values and those achieved by following the learner's predictions. A seminal work by Bärmann et al. (2017) obtained a regret bound of O( T), where T is the time horizon. Subsequently, the regret bound has been improved to O(n4 lnT) by Besbes et al. (2021, 2025) and to O(nlnT) by Gollapudi et al. (2021), where nis the dimension of the ambient space of objective vectors. However, these logarithmic-regret methods are highly inefficient when T is large, as they need to maintain regions specified by O(T) constraints, which represent possible locations of the true objective vector.

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.

The rise of smart manufacturing under Industry 4.0 introduces mass customization and dynamic production, demanding more advanced and flexible scheduling techniques. The flexible job-shop scheduling problem (FJSP) has attracted significant attention due to its complex constraints and strong alignment with real-world production scenarios. Current deep reinforcement learning (DRL)-based approaches to FJSP predominantly employ constructive methods. While effective, they often fall short of reaching (near-)optimal solutions. In contrast, improvement-based methods iteratively explore the neighborhood of initial solutions and are more effective in approaching optimality.

We consider binary classification restricted to a class of continuous piecewise linear functions whose decision boundaries are (possibly nonconvex) starshaped polyhedral sets, supported on a fixed polyhedral simplicial fan. We investigate the expressivity of these function classes and describe the combinatorial and geometric structure of the loss landscape, most prominently the sublevel sets, for two loss-functions: the 0/1-loss (discrete loss) and a log-likelihood loss function. In particular, we give explicit bounds on the VC dimension of this model, and concretely describe the sublevel sets of the discrete loss as chambers in a hyperplane arrangement. For the log-likelihood loss, we give sufficient conditions for the optimum to be unique, and describe the geometry of the optimum when varying the rate parameter of the underlying exponential probability distribution.

Visual encoders have become fundamental components in modern computer vision pipelines. However, ensuring robustness against adversarial perturbations remains a critical challenge. Recent efforts have explored both supervised and unsupervised adversarial fine-tuning strategies. We identify two key limitations in these approaches: (i) they often suffer from instability, especially during the early stages of fine-tuning, resulting in suboptimal convergence and degraded performance on clean data, and (ii) they exhibit a suboptimal trade-off between robustness and clean data accuracy, hindering the simultaneous optimization of both objectives. To overcome these challenges, we propose Lagrangian-Optimized Robust Embeddings (LORE), a novel unsupervised adversarial fine-tuning framework.

We investigate optimal transport (OT) for measures on graph metric spaces with different total masses. To mitigate the limitations of traditional Lp geometry, Orlicz-Wasserstein (OW) and generalized Sobolev transport (GST) employ Orlicz geometric structure, leveraging convex functions to capture nuanced geometric relationships and remarkably contribute to advance certain machine learning approaches. However, both OW and GST are restricted to measures with equal total mass, limiting their applicability to real-world scenarios where mass variation is common, and input measures may have noisy supports, or outliers. To address unbalanced measures, OW can either incorporate mass constraints or marginal discrepancy penalization, but this leads to a more complex two-level optimization problem. Additionally, GST provides a scalable yet rigid framework, which poses significant challenges to extend GST to accommodate nonnegative measures.

Nowadays deep models are required to be versatile due to the increasing realistic needs. Multi-task learning (MTL) offers an efficient way for this purpose to learn multiple tasks simultaneously with a single model. However, prior MTL solutions often focus on resolving conflicts and imbalances during optimization, which may not outperform simple linear scalarization strategies [Xin et al., 2022]. Instead of altering the optimization trajectory, this paper leverages mode connectivity to efficiently approach the Pareto front and identify the desired trade-off point. Unlike Pareto Front Learning (PFL), which aims to align with the entire Pareto front, we focus on effectively and efficiently exploring optimal trade-offs. However, three challenges persist: (1) the low-loss path can neither fully traverse trade-offs nor align with user preference due to its randomness, (2) commonly adopted Bézier curves in mode connectivity are ill-suited to navigating the complex loss landscapes of deep models, and (3) poor scalability to large-scale task scenarios. To address these challenges, we adopt non-uniform rational B-Splines (NURBS) to model mode connectivity, allowing for more flexible and precise curve optimization. Additionally, we introduce an order-aware objective to explore task loss tradeoffs and employ a task grouping strategy to enhance scalability under massive task scenarios. Extensive experiments on key MTL datasets demonstrate that our proposed method, EXTRA(EXplore TRAde-offs), effectively identifies the desired point on the Pareto front and achieves state-of-the-art performance.

The substantial memory demands of pre-training and fine-tuning large language models (LLMs) require memory-efficient optimization algorithms. One promising approach is layer-wise optimization, which treats each transformer block as a single layer and optimizes it sequentially, while freezing the other layers to save optimizer states and activations. Although effective, these methods ignore the varying importance of the modules within each layer, leading to suboptimal performance. Moreover, layer-wise sampling provides only limited memory savings, as at least one full layer must remain active during optimization. To overcome these limitations, we propose Module-wise Importance SAmpling (MISA), a novel method that divides each layer into smaller modules and assigns importance scores to each module. MISA uses a weighted random sampling mechanism to activate modules, provably reducing gradient variance compared to layer-wise sampling. Additionally, we establish an O(1/ K)convergence rate under non-convex and stochastic conditions, where K is the total number of block updates, and provide a detailed memory analysis showcasing MISA's superiority over existing baseline methods.

Verifiers can improve language model (LM) capabilities by providing feedback or selecting the best response from a pool of generated candidates. Currently, high-quality verifiers are either unscalable (e.g., humans) or limited in utility (e.g., tools like Lean for formal proofs). While LM judges and reward models have become broadly useful as general-purpose verifiers, a significant performance gap remains between them and oracle verifiers. To help close this gap, we introduce WEAVER, a framework for designing a strong verifier by combining multiple weak, imperfect verifiers. First we find that weighted ensembles of verifiers, which typically require learning from labeled data, significantly outperform unweighted combinations due to differences in the verifiers. To reduce the dependency on labeled data, WEAVER leverages weak supervision to estimate each verifier's accuracy and combines their outputs into a unified score that better reflects true response quality.