Transaction cost optimization (TCO) of online portfolio selection is crucial in computing science, due to the significant impact of transaction costs in practical short-term trading. Moreover, sparsity of portfolio vector is often desired to enhance stability and decrease risk. However, there is a lack of models considering transaction costs and sparsity simultaneously in the literature. In this paper, we first propose a $K$-sparse TCO model that minimizes the negative return and transaction costs while keeping the portfolio vector being $K$-sparse. Noting that the model is NP-hard due to the $K$-sparse constraint, we bypass this difficulty by reformulating the sparse model to a nonsmooth difference of convex (DC) optimization problem. We show that both problems are equivalent by proving that the penalty parameter is large enough. Then, to overcome the difficulty caused by the nonsmoothness and the simplex constraint of the model, we develop a proximal subgradient algorithm (PSGA) to solve the DC problem and apply the alternating direction of multipliers (ADMM) to compute the proximity operator of the corresponding function. Furthermore, we establish the global convergence of the entire sequence generated by PSGA through showing the surrogate function satisfies the Kurdyka-Łojasiewicz (KL) property. In addition, by showing the KL exponent of the surrogate function is $1/2$, we establish the R-linear convergence rate of PSGA for any arbitrary initiaal point. Finally, we compare our proposed algorithm with other state-of-the-art strategies on four benchmark real-market data sets, with the numerical results showing that the proposed algorithm achieves lower risk while keeping higher return than classical TCO models.
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Applications · Health / Medicine
Predicting spatial gene expression from routine H\&E makes high-resolution molecular profiling accessible at scale, especially for large retrospective cohorts. However, current models mostly treat gene expression as a series of pointwise tasks. While effective for numerical fitting, this approach overlooks biological structures: the functional coordination between genes and their organized distribution across tissue. We reframe this task as structured distribution modeling and introduce \textbf{FLAG}, a diffusion-based framework designed to preserve these biological relationships. To capture such structures, a natural strategy would be to jointly model gene expression and their spatial interactions. However, we identify a critical \textbf{Gene Dimension Curse}: such joint modeling fails in high-dimensional gene spaces. This motivates FLAG, which conditions the generative process using a novel spatial graph encoder to ensure gene-spatial topographic coherence and a Gene Foundation Model (GFM) alignment to maintain high gene-gene structural fidelity. To rigorously assess our approach, we propose a structural evaluation metrics, including Gene Structural Correlation (\textbf{GSC}) and Spatial Structural Correlation (\textbf{SSC}). Our experiments demonstrate that FLAG is highly competitive with or superior to state-of-the-art models in traditional accuracy (PCC/MSE), while achieving significantly enhanced structural fidelity in capturing both gene-gene and gene-spatial relationships.
Applications · Computer Vision
Online Reinforcement Learning (RL) offers a promising avenue for complex image editing but is currently constrained by the scarcity of reliable and fine-grained reward signals. Existing evaluators frequently struggle with a critical perception gap we term "Attention Collapse," where models neglect cross-image comparisons and fail to capture fine-grained details, resulting in inaccurate perception and miscalibrated scores. To address these limitations, we propose **SpatialReward**, a reward model that enforces precise verification via explicit spatial reasoning. By anchoring reasoning to predicted edit regions, SpatialReward grounds semantic judgments in pixel-level evidence, significantly enhancing evaluative accuracy. Trained on a curated 260k spatial-aware dataset, our model achieves state-of-the-art performance on MMRB2 and EditReward-Bench, and outperforms proprietary evaluators on our proposed **MultiEditReward-Bench**. Furthermore, SpatialReward serves as a robust signal in online RL, boosting OmniGen2 by +0.90 on GEdit-Bench—surpassing the leading discriminative model and doubling the gain of GPT-4.1 (+0.45). These results demonstrate that spatial reasoning is essential for unlocking effective alignment in image editing.
Applications · Computer Vision
Open-set supervised anomaly detection (OSAD) aims to identify unseen anomalies using limited anomalous supervision. However, existing prototype-based methods typically model normal data via a unimodal Gaussian prior, failing to capture inherent multi-modality and resulting in blurred decision boundaries. To address this, we propose Mixture Prototype Flow Matching (MPFM), a framework that learns a continuous transformation from normal feature distributions to a structured Gaussian mixture prototype space. Departing from traditional flow-based approaches that rely on a single velocity vector, MPFM explicitly models the velocity field as a Gaussian mixture prior where each component corresponds to a distinct normal class. This design facilitates mode-aware and semantically coherent distribution transport. Furthermore, we introduce a Mutual Information Maximization Regularizer (MIMR) to prevent prototype collapse and maximize normal-anomaly separability. Extensive experiments demonstrate that MPFM achieves state-of-the-art performance across diverse benchmarks under both single- and multi-anomaly settings.
Deep Learning · Large Language Models
While LLMs have seen substantial improvement in reasoning capabilities, they also sometimes overthink, generating unnecessary reasoning steps, particularly under uncertainty, given ill-posed or ambiguous queries. We introduce statistically principled early stopping methods that monitor uncertainty signals during generation to mitigate this issue. Our first approach is nonparametric and provides finite-sample guarantees on the probability of halting too early on well-posed queries. Our second approach is parametric: it models inter-arrival times of uncertainty keywords as a renewal process and applies sequential testing for stopping. We conduct empirical evaluations on reasoning tasks across several domains and models. Our results indicate that uncertainty-aware early stopping can improve both efficiency and reliability in LLM reasoning. The performance varies across domains, and we observe especially significant gains for math reasoning.
Theory · Reinforcement Learning and Planning
We analytically solve the Mountain Car problem, a canonical benchmark in RL, and derive an optimal control solution, closing a gap after 36 years. This enables us to reveal two surprising insights: The optimal control is quite simple, yet modern RL agents display a large gap to optimality. Motivated by the analysis of the optimal control, we introduce Chebyshev policies as a universal (i.e. dense) class of RL policies from first principles. They can be trained as drop-in replacements of neural nets, reducing the regret by a factor of 4.18, while requiring 268 times fewer parameters, fostering sample efficiency, explainability and real-time capability. Chebyshev policies are evaluated on further RL environments, including a real-world non-linear motion control testbed. They consistently improve performance over neural nets with PPO, ARS and REINFORCE. Our results demonstrate how Chebyshev policies offer a compelling and lightweight alternative or addition to neural nets for low-dimensional control tasks.
Maximum likelihood is fundamental to supervised learning but it cannot be directly applied in correctness-based problems with non-differentiable sampling. In these settings, reinforcement learning (RL) is typically used to maximize expected reward. We show that for binary correctness tasks, expected-reward RL is a first-order approximation of the maximum likelihood objective, yielding vanishing learning signal on low-success inputs. We introduce **Maximum Likelihood Reinforcement Learning (MaxRL)**, a compute-indexed family of sampling-based objectives derived from a pass@k expansion of the likelihood, which interpolates between standard RL and exact maximum likelihood as compute increases. MaxRL admits a simple unbiased policy-gradient estimator whose optimized objective improves with additional compute. Across multiple domains, MaxRL consistently outperforms standard RL and GRPO, achieving higher $pass@1$ and substantially improved $pass@k$.
We study reinforcement learning with delayed state observation, where the agent observes the current state after some random number of time steps. We propose an algorithm that combines the augmentation method and the upper confidence bound approach. For tabular Markov decision processes (MDPs), we derive a regret bound of $\tilde{\mathcal{O}}(H \sqrt{D_{\max} SAK})$, where $S$ and $A$ are the cardinalities of the state and action spaces, $H$ is the time horizon, $K$ is the number of episodes, and $D_{\max}$ is the maximum length of the delay. We also provide a matching lower bound up to logarithmic factors, showing the optimality of our approach. Our analytical framework formulates this problem as a special case of a broader class of MDPs, where their transition dynamics decompose into a known component and an unknown but structured component. We establish general results for this abstract setting, which may be of independent interest.
Prediction sets can wrap around any ML model to cover unknown test outcomes with a guaranteed probability. Yet, it remains unclear how to use them optimally for downstream decision-making. Here, we propose a decision-theoretic framework that seeks to minimize the expected loss (risk) against a worst-case distribution consistent with the prediction set's coverage guarantee. We first characterize the minimax optimal policy for a fixed prediction set, showing that it balances the worst-case loss inside the set with a penalty for potential losses outside the set. Building on this, we derive the optimal prediction set construction that minimizes the resulting robust risk subject to a coverage constraint. Finally, we introduce Risk-Optimal Conformal Prediction (ROCP), a practical algorithm that targets these risk-minimizing sets while maintaining finite-sample distribution-free marginal coverage. Empirical evaluations on medical diagnosis and safety-critical decision-making tasks demonstrate that ROCP reduces critical mistakes compared to baselines, particularly when out-of-set errors are costly.
Applications · Language, Speech and Dialog
SQL is central to enterprise data engineering, yet generating fully correct SQL code in a single attempt remains difficult—even for experienced developers and advanced \ttsql LLMs—often requiring multiple debugging iterations. We introduce \textbf{\ourbench}, the first benchmark for enterprise-level SQL reasoning and debugging. Our benchmark is built upon two key innovations: (1) an \textbf{automated construction workflow} that employs reverse engineering to systematically inject realistic bugs into large-scale SQL code, enabling scalable and diverse benchmark generation; and (2) an \textbf{execution-free evaluation framework} tailored for enterprise settings, providing fast, accurate, and resource-efficient assessment. \ourbench comprises $469$ \ourbenchsyn queries featuring syntax errors with explicit error messages, and $516$ \ourbenchsem queries targeting semantic errors where codes fails to meet user intent. The queries are highly complex, averaging over $140$ lines, and featuring deep and wide abstract syntax trees (average width $>11$, depth $>8.7$). Evaluation of nearly $30$ LLMs reveals a substantial performance gap: the best-performing model, Claude-4-Sonnet, achieves only $36.46\%$ accuracy on \ourbenchsyn and $32.17\%$ on \ourbenchsem, while most models score below $20\%$. We further explore four solution strategies, identify key challenges, and outline promising directions for enterprise SQL debugging with LLMs.
We consider anonymous multi-agent path finding (MAPF) where a set of robots is tasked to travel to a set of targets on a finite, connected graph. We show that MAPF can be cast as a special class of multi-marginal optimal transport (MMOT) problems with an underlying Markovian structure, under which the exponentially large MMOT collapses to a linear program (LP) polynomial in size. Focusing on the anonymous setting, we establish conditions under which the corresponding LP is feasible, totally unimodular, and yields min-cost, integral~$(\{0,1\})$ transports that do not overlap in both space and time. To adapt the approach to large-scale problems, we cast the MAPF-MMOT in a probabilistic framework via Schrödinger bridges. Under standard assumptions, we show that the Schrödinger bridge formulation reduces to an entropic regularization of the corresponding MMOT that admits an iterative Sinkhorn-type solution. The Schrödinger bridge, being a probabilistic framework, provides a shadow (fractional) transport that we use as a template to solve a reduced LP and demonstrate that it results in near-optimal, integral transports at a significant reduction in complexity. Extensive experiments highlight the optimality and scalability of the proposed approaches.
Applications · Chemistry, Physics, and Earth Sciences
Deep learning-based methods have shown remarkable effectiveness in solving PDEs, largely due to their ability to enable fast simulations once trained. However, despite the availability of high-performance computing infrastructure, many critical applications remain constrained by the substantial computational costs associated with generating large-scale, high-quality datasets and training models. In this work, inspired by studies on the structure of Green's functions for elliptic PDEs, we introduce Neural-HSS, a parameter-efficient architecture built upon the Hierarchical Semi-Separable (HSS) matrix structure that is provably data-efficient for a broad class of PDEs. We theoretically analyze the proposed architecture, proving that it satisfies exactness properties even in very low-data regimes. We also investigate its connections with other architectural primitives, such as the Fourier neural operator layer and convolutional layers. We experimentally validate the data efficiency of Neural-HSS on the three-dimensional Poisson equation over a grid of two million points, demonstrating its superior ability to learn from data generated by elliptic PDEs in the low-data regime while outperforming baseline methods. Finally, we demonstrate its capability to learn from data arising from a broad class of PDEs in diverse domains, including electromagnetism, fluid dynamics, and biology.
We develop order-equivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the face-poset viewpoint. We (i) characterize all linear order-equivariant maps, (ii) build OENN layers, and (iii) prove a universal approximation theorem (UAT) for continuous order-equivariant maps, which is a new result even when restricted to sheaf neural networks (for which no UAT was known before). We illustrate the framework on graph and sheaf models. Our results can also be seen as extending the UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well.
Applications · Chemistry, Physics, and Earth Sciences
Surrogate models for topology optimization (TO) exhibit highly variable out-of-distribution (OOD) generalization under distribution shifts such as changing loads or boundary conditions, yet the source of this variability remains unclear. We hypothesize that OOD performance is governed by how much information the conditioning signal preserves about the adjoint sensitivity (reduced gradient) that drives classical TO. Modeling the TO pipeline as a causal Markov chain, the Data Processing Inequality establishes that, under this abstraction, the sensitivity field is an information-theoretically optimal conditioning signal for topology prediction. However, computing exact adjoint sensitivities can be expensive or unavailable in practice; we observe that certain physical fields can approximate sensitivities through monotone transformations. To formalize this, we introduce \textbf{pseudo-sensitivities} to characterize which fields enable generalization versus those that are information-poor. We then show that a sensitivity-conditioned Bernoulli flow-matching generator empirically confirms these predictions: conditioning on sensitivities yields state-of-the-art OOD performance, while increasingly distant physical fields degrade toward raw parameter conditioning. We further benchmark against competitive baselines, and find the same ordering of conditioning signals and the same OOD trends. Results hold across structural TO benchmarks under load shifts and our new CFD-TO dataset under boundary-condition shifts such as multi-outlet configurations.
Deep Learning · Large Language Models
Large reasoning models (LRMs) are often evaluated using metrics such as final-answer accuracy or token count. However, identical scores on these metrics can hide fundamentally different reasoning structures. To address this limitation, we introduce a scalable LRM benchmark of logic puzzles and a pipeline that converts unstructured traces into verifiable reasoning graphs of claims and dependencies. This turns reasoning into a structured, measurable object whose topology can be quantitatively analyzed. Building on this, we define a reasoning efficiency metric that quantifies how concentrated the model's logical flow is. Our analysis on open-source reasoning models shows that structural measurements separate behaviors that token count and accuracy conflate, providing a practical tool for diagnosing failure modes and comparing how reasoning scales with puzzle difficulty.
Research on the expressive power of neural networks has identified the minimum depth and width of neural networks that enable universal approximation and memorization. However, existing results are derived under exact arithmetic and cannot be directly applied to real implementations on computers, which can only use a finite set of numbers and inexact machine operations with round-off errors. In this work, we study floating-point ReLU networks that have floating-point parameters and use floating-point operations. Specifically, we investigate their minimum depth and width to represent all functions from the set of floating-point vectors $\mathbb F^d$ to the set of floating-point numbers $\mathbb F$. We first show that the minimum depth for representing all functions from $\mathbb F^d$ to $\mathbb F$ is exactly three, where two layers can be sufficient if we consider a smaller domain and/or codomain. We further show that the minimum width for representing all functions from $\mathbb F^d$ to $\mathbb F$ lies between $2d$ and $2d+4$. In addition, if we restrict the domain to non-negative floats, it lies between $d$ and $d+4$, where it can be smaller for a smaller domain, even beyond $d$. Our results show that the existing results analyzed under exact arithmetic do not extend to the floating-point setup.
Social Aspects · Privacy
We design a class of additive noise mechanisms that satisfy $(\varepsilon, \delta)$-differential privacy (DP) for scalar, real-valued query functions with known sensitivities, with a particular focus on moderate and low-privacy regimes. These mechanisms, which we call *mixture mechanisms*, are constructed by mixing multiple Gaussian distributions that share the same variance but differ in their means and mixture weights. The resulting distributions can be interpreted as convex combinations of a zero-mean Gaussian (as used in the analytic Gaussian mechanism) and additional Gaussians whose means depend on the sensitivity of the query function. We derive tight conditions on the variances required for $(\varepsilon, \delta)$-DP and provide efficient algorithms to compute them. Compared to the analytic Gaussian mechanism, our mechanisms yield substantially lower expected noise amplitudes ($l_1$-loss) and variances ($l_2$-loss for zero-mean distributions). In the low-privacy regime that motivates our design, our mechanisms approach optimality, mitigating nearly all of the optimality gap of the analytic Gaussian mechanism.
Low-rank approximation has emerged as a cornerstone technique for model compression and parameter-efficient fine-tuning, enabling substantial reductions in computation and memory without altering model architectures. However, existing approaches often overlook the shifts in feature distributions induced by the approximation process, which can lead to error amplification and unstable inference. We propose a distribution-aware whitening framework that dynamically whitens layer inputs based on the evolving feature distributions, ensuring second-order isotropy of input features. This allows that the discarded components in the low-rank approximation are those with minimal impact on model outputs, thereby minimizing cumulative approximation errors across layers. We theoretically analyze how distribution misalignment leads to error propagation and demonstrate that our approach achieves tighter control over layerwise distortion. Extensive experiments across various large language models demonstrate the superiority of our method in post-training compression. Moreover, our method can also serve as an effective initialization for LoRA-style parameter-efficient fine-tuning. Our findings highlight the importance of considering feature distributions in low-rank approximations, paving the way for reliable and effective model compression strategies.
Social Aspects · Accountability, Transparency, and Interpretability
Large language models can express values in two main ways: (1) $\textit{intrinsic}$ expression, reflecting the model's inherent values learned during training, and (2) $\textit{prompted}$ expression, elicited by explicit prompts. Given their widespread use in value alignment, it is paramount to clearly understand their underlying mechanisms, particularly whether they mostly overlap (as one might expect) or rely on distinct mechanisms, but this remains largely understudied. We analyze this at the mechanistic level using two approaches: (1) $\textit{value vectors}$, feature directions representing value mechanisms extracted from the residual stream, and (2) $\textit{value neurons}$, MLP neurons that contribute to value vectors. We demonstrate that intrinsic and prompted value mechanisms partly share common components crucial for inducing value expression, generalizing across languages and reconstructing theoretical inter-value correlations in the model's internal representations. Yet, as these mechanisms also possess unique elements that fulfill distinct roles, they lead to different degrees of response diversity ($\textit{intrinsic}$ $>$ $\textit{prompted}$) and value steerability ($\textit{prompted}$ $>$ $\textit{intrinsic}$). In particular, components unique to the intrinsic mechanism promote lexical diversity in responses, whereas those specific to the prompted mechanism strengthen instruction following, taking effect even in distant tasks like jailbreaking.
Theory · Deep Learning
We propose a Likelihood Matching approach for training diffusion models by first establishing an equivalence between the likelihood of the target data distribution and a likelihood along the sample path of the reverse diffusion. To efficiently compute the reverse sample likelihood, the equivalence, a quasi-likelihood is considered to approximate each reverse transition density by a Gaussian distribution with matched conditional mean and covariance, respectively. The score and Hessian functions for the diffusion generation are estimated by maximizing the quasi-likelihood, ensuring a consistent matching of both the first two transition moments between every two time points. A stochastic sampler is introduced to facilitate the computation that leverages both the estimated score and Hessian information. We establish consistency of the quasi-maximum likelihood estimation, and provide non-asymptotic convergence guarantees for the proposed sampler, quantifying the rates of the approximation errors due to score and Hessian estimation, dimensionality, and the number of diffusion steps. Empirical and simulation evaluations demonstrate the effectiveness of the proposed Likelihood Matching and validate the theoretical results.