arXiv:2509.05841v2 Announce Type: replace-cross Abstract: We give an overview on the emerging applications of GenAI in the financial industry, especially within investment banks. Inherent to these exciting opportunities is a new realm of risks that must be managed properly. By heeding both the Yin and Yang sides of GenAI, we can accelerate its organic growth while safeguarding the entire financial industry during this nascent era of AI.

arXiv:2603.22469v1 Announce Type: cross Abstract: The growing complexity of modern control tasks calls for controllers that can react online as objectives and disturbances change, while preserving closed-loop stability. Recent approaches for improving the performance of nonlinear systems while preserving closed-loop stability rely on time-invariant recurrent neural-network controllers, but offer no principled way to update the controller during operation. Most importantly, switching from one stabilizing policy to another can itself destabilize the closed-loop. We address this problem by introducing a stability-preserving update mechanism for nonlinear, neural-network-based controllers. Each controller is modeled as a causal operator with bounded $\ell_p$-gain, and we derive gain-based conditions under which the controller may be updated online. These conditions yield two practical update schemes, time-scheduled and state-triggered, that guarantee the closed-loop remains $\ell_p$-stable after any number of updates. Our analysis further shows that stability is decoupled from controller optimality, allowing approximate or early-stopped controller synthesis. We demonstrate the approach on nonlinear systems with time-varying objectives and disturbances, and show consistent performance improvements over static and naive online baselines while guaranteeing stability.

arXiv:2603.22479v1 Announce Type: cross Abstract: Defining a constructive process to build general capabilities for language models in an automatic manner is considered an open problem in artificial intelligence. Towards this, we consider the problem of building a curriculum of tasks that grows a model via relevant skill discovery. We provide a concrete framework for this task, using a family of tasks called cross-entropy games, which we postulate is universal in a suitable sense. We show that if it is possible to grow the curriculum for relevant skill discovery by iterating a greedy optimization algorithm, then, under natural assumptions, there is essentially only one meta-objective possible (up to a few hyperparameters). We call the resulting process cognitive training. We postulate that, given sufficiently capable language models as players and meta-samplers and sufficient training time, cognitive training provides a principled way to relevant skill discovery; and hence to the extent general capabilities are achievable via greedy curriculum learning, cognitive training would be a solution.

arXiv:2603.23249v1 Announce Type: cross Abstract: Efficient scheduling of directed acyclic graphs (DAGs) in heterogeneous environments is challenging due to resource capacities and dependencies. In practice, the need for adaptability across environments with varying resource pools and task types, alongside rapid schedule generation, complicates these challenges. We propose WeCAN, an end-to-end reinforcement learning framework for heterogeneous DAG scheduling that addresses task--pool compatibility coefficients and generation-induced optimality gaps. It adopts a two-stage single-pass design: a single forward pass produces task--pool scores and global parameters, followed by a generation map that constructs schedules without repeated network calls. Its weighted cross-attention encoder models task--pool interactions gated by compatibility coefficients, and is size-agnostic to environment fluctuations. Moreover, widely used list-scheduling maps can incur generation-induced optimality gaps from restricted reachability. We introduce an order-space analysis that characterizes the reachable set of generation maps via feasible schedule orders, explains the mechanism behind generation-induced gaps, and yields sufficient conditions for gap elimination. Guided by these conditions, we design a skip-extended realization with an analytically parameterized decreasing skip rule, which enlarges the reachable order set while preserving single-pass efficiency. Experiments on computation graphs and real-world TPC-H DAGs demonstrate improved makespan over strong baselines, with inference time comparable to classical heuristics and faster than multi-round neural schedulers.

arXiv:2508.07494v2 Announce Type: replace-cross Abstract: The generalization of the Koopman operator to systems with control input and the derivation of a nonlinear fundamental lemma are two open problems that play a key role in the development of data-driven control methods for nonlinear systems. In this paper we derive a novel solution to these problems based on basis functions expansion in a product Hilbert space constructed as the tensor product between the Hilbert spaces of the state and input observable functions, respectively. We identify relaxed invariance conditions that guarantee existence of a bounded linear operator, i.e., the generalized Koopman operator, from the constructed product Hilbert space to the Hilbert space corresponding to the lifted state propagated forward in time. Compared to classical Koopman invariance conditions, measure preservation is not required. Moreover, we derive a nonlinear fundamental lemma by exploiting the constructed exact infinite-dimensional bilinear Koopman representation and Hankel operators. The effectiveness of the developed generalized Koopman embedding is illustrated on the Van der Pol oscillator and in predictive control of a soft-robotic manipulator model.

arXiv:2603.19465v2 Announce Type: replace-cross Abstract: We analyze a fixed-point iteration $v \leftarrow \phi(v)$ arising in the optimization of a regularized nuclear norm objective involving the Hadamard product structure, posed in DMR+22 in the context of an optimization problem over the space of algorithms in private machine learning. We prove that the iteration $v^{(k+1)} = \text{diag}((D_{v^{(k)}}^{1/2} M D_{v^{(k)}}^{1/2})^{1/2})$ converges monotonically to the unique global optimizer of the potential function $J(v) = 2 \text{Tr}((D_v^{1/2} M D_v^{1/2})^{1/2}) - \sum v_i$, closing a problem left open there. The bulk of this proof was provided by Gemini 3, subject to some corrections and interventions. Gemini 3 also sketched the initial version of this note. Thus, it represents as much a commentary on the practical use of AI in mathematics as it represents the closure of a small gap in the literature. As such, we include a small narrative description of the prompting process, and some resulting principles for working with AI to prove mathematics.

arXiv:2510.09816v2 Announce Type: replace Abstract: Recent experiments in neuroscience reveal that task-relevant variables are often encoded in approximately orthogonal subspaces of neural population activity. These disentangled, or abstract, representations have been observed in multiple brain areas and across different species. These representations have been shown to support out of distribution generalization and rapid learning of novel tasks. The mechanisms by which these representations emerge remain poorly understood, especially in the case of supervised task behavior. Here, we show mathematically that abstract representations of latent variables are guaranteed to appear in the hidden layer of feedforward nonlinear networks when they are trained on tasks that depend directly on these latent variables. These learned abstract representations reflect the semantics of the input stimuli. To show this, we reformulate the usual optimization over the network weights into a mean field optimization problem over the distribution of neural preactivations. We then apply this framework to finite-width ReLU networks and show that the hidden layer of these networks will exhibit an abstract representation at all global minima of the task objective. Finally, we extend our findings to two broad families of activation functions as well as deep feedforward architectures. Together, our results provide an explanation for the widely observed abstract representations in both the brain and artificial neural networks. In addition, the general framework that we develop here provides a mathematically tractable toolkit for understanding the emergence of different kinds of representations in task-optimized, feature-learning network models.

arXiv:2603.07733v1 Announce Type: new Abstract: This work investigated the capabilities of different models, including the Llama-3 series of models and CHATGPT, with different forms of expression in solving discrete optimization problems by testing natural language datasets. In contrast to formal datasets with a limited scope of parameters, our dataset included a variety of problem types in discrete optimization problems and featured a wide range of parameter magnitudes, including instances with large parameter sets, integrated with augmented data. It aimed to (1) provide an overview of LLMs' ability in large-scale problems, (2) offer suggestions to those who want to solve discrete optimization problems automatically, and (3) regard the performance as a benchmark for future research. These datasets included original, expanded and augmented datasets. Among these three datasets, the original and augmented ones aimed for evaluation while the expanded one may help finetune a new model. In the experiment, comparisons were made between strong and week models, CoT methods and No-CoT methods on various datasets. The result showed that stronger model performed better reasonably. Contrary to general agreement, it also showed that CoT technique was not always effective regarding the capability of models and disordered datasets improved performance of models on easy to-understand problems, even though they were sometimes with high variance, a manifestation of instability. Therefore, for those who seek to enhance the automatic resolution of discrete optimization problems, it is recommended to consult the results, including the line charts presented in the Appendix, as well as the conclusions drawn in this study for relevant suggestions.

arXiv:2603.06597v1 Announce Type: cross Abstract: This paper proposes a two-time scale neurodynamic duplex approach to solve distributionally robust geometric joint chance-constrained optimization problems. The probability distributions of the row vectors are not known in advance and belong to a certain distributional uncertainty set. In our paper, we study three uncertainty sets for the unknown distributions. The neurodynamic duplex is designed based on three projection equations. The main contribution of our work is to propose a neural network-based method to solve distributionally robust joint chance-constrained optimization problems that converges in probability to the global optimum without the use of standard state-of-the-art solving methods. We show that neural networks can be used to solve multiple instances of a problem. In the numerical experiments, we apply the proposed approach to solve a problem of shape optimisation and a telecommunication problem.

arXiv:2603.07821v1 Announce Type: cross Abstract: Along with the rapid development of new urban mobility options like ride-sharing over the past decade, on-demand micro-transit services stand out as a middle ground, bridging the gap between fixed-line mass transit and single-request ride-hailing, balancing ridership maximization and travel time minimization. Micro-transit adoption can have significant social impact. It improves urban sustainability, through lower energy consumption and reduced emissions, while enhancing equitable mobility access for disadvantaged communities, thanks to its lower vehicle miles per passenger, flexible schedules, and affordable pricing. However, effective operation of micro-transit services requires planning geo-fenced zones in advance, which involves solving a challenging combinatorial optimization problem. Existing approaches enumerate candidate zones first and selects a fixed number of optimal zones in the second step. In this paper, we generalize the Micro-Transit Zoning Problem (MZP) to allow a global budget rather than imposing a size limit for candidate zones. We also design a Column Generation (CG) framework to solve the problem and several pricing heuristics to accelerate computation. Extensive numerical experiments across major U.S. cities demonstrate that our approach produces higher-quality solutions more efficiently and scales better in the generalized setting.

arXiv:2603.07998v1 Announce Type: cross Abstract: This work introduces the Drag-Aware Aerodynamic Manipulability (DAAM), a geometric framework for control allocation in redundant multirotors. By equipping the propeller spin-rate space with a Riemannian metric based on the remaining symmetric acceleration capacity of each motor, the formulation explicitly accounts for motor torque limits and aerodynamic drag. Mapping this metric through the nonlinear thrust law to the generalized force space yields a state-dependent manipulability volume. The log-determinant of this volume acts as a natural barrier function, strictly penalizing drag-induced saturation and low-spin thrust loss. Optimizing this volume along the allocation fibers provides a redundancy resolution strategy inherently invariant to arbitrary coordinate scaling in the generalized-force space. Analytically, we prove that the resulting optimal allocations locally form smooth embedded manifolds, and we geometrically characterize the global jump discontinuities that inevitably arise from physical actuator limits and spin-rate sign transitions.

arXiv:2603.08494v1 Announce Type: cross Abstract: Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces. In this setting, the optimal first-order improvement direction emerges as a pseudoinverse-weighted gradient, revealing how constraints induce a distorted ascent geometry. We further demonstrate that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression, and establish a compatibility principle that characterizes the existence of common admissible directions across multiple objectives. The resulting framework unifies gradient projection, spectral truncation, and multi-objective feasibility within a single geometric structure.

arXiv:2408.00329v2 Announce Type: replace-cross Abstract: Deep neural networks (DNNs) are vulnerable to small adversarial perturbations of the inputs, posing a significant challenge to their reliability and robustness. Empirical methods such as adversarial training can defend against particular attacks but remain vulnerable to more powerful attacks. Alternatively, Lipschitz networks provide certified robustness to unseen perturbations but lack sufficient expressive power. To harness the advantages of both approaches, we design a novel two-step Optimal Transport induced Adversarial Defense (OTAD) model that can fit the training data accurately while preserving the local Lipschitz continuity. First, we train a DNN with a regularizer derived from optimal transport theory, yielding a discrete optimal transport map linking data to its features. By leveraging the map's inherent regularity, we interpolate the map by solving the convex integration problem (CIP) to guarantee the local Lipschitz property. OTAD is extensible to diverse architectures of ResNet and Transformer, making it suitable for complex data. For efficient computation, the CIP can be solved through training neural networks. OTAD opens a novel avenue for developing reliable and secure deep learning systems through the regularity of optimal transport maps. Empirical results demonstrate that OTAD can outperform other robust models on diverse datasets.

arXiv:2505.18362v3 Announce Type: replace-cross Abstract: We develop a general theoretical framework for optimal probability density control on standard measure spaces, aimed at addressing large-scale multi-agent control problems. In particular, we establish a maximum principle (MP) for control problems posed on infinite-dimensional spaces of probability distributions and control vector fields. We further derive the Hamilton--Jacobi--Bellman equation for the associated value functional defined on the space of probability distributions. Both results are presented in a concise form and supported by rigorous mathematical analysis, enabling efficient numerical treatment of these problems. Building on the proposed MP, we introduce a scalable numerical algorithm that leverages deep neural networks to handle high-dimensional settings. The effectiveness of the approach is demonstrated through several multi-agent control examples involving domain obstacles and inter-agent interactions.

arXiv:2603.03759v1 Announce Type: cross Abstract: Many large-scale platforms and networked control systems have a centralized decision maker interacting with a massive population of agents under strict observability constraints. Motivated by such applications, we study a cooperative Markov game with a global agent and $n$ homogeneous local agents in a communication-constrained regime, where the global agent only observes a subset of $k$ local agent states per time step. We propose an alternating learning framework $(\texttt{ALTERNATING-MARL})$, where the global agent performs subsampled mean-field $Q$-learning against a fixed local policy, and local agents update by optimizing in an induced MDP. We prove that these approximate best-response dynamics converge to an $\widetilde{O}(1/\sqrt{k})$-approximate Nash Equilibrium, while yielding a separation in the sample complexities between the joint state space and action space. Finally, we validate our results in numerical simulations for multi-robot control and federated optimization.

arXiv:2510.26303v3 Announce Type: replace-cross Abstract: Adam [Kingma & Ba, 2015] is the de facto optimizer in deep learning, yet its theoretical understanding remains limited. Prior analyses show that Adam favors solutions aligned with $\ell_\infty$-geometry, but these results are restricted to the full-batch regime. In this work, we study the implicit bias of incremental Adam (using one sample per step) for logistic regression on linearly separable data, and show that its bias can deviate from the full-batch behavior. As an extreme example, we construct datasets on which incremental Adam provably converges to the $\ell_2$-max-margin classifier, in contrast to the $\ell_\infty$-max-margin bias of full-batch Adam. For general datasets, we characterize its bias using a proxy algorithm for the $\beta_2 \to 1$ limit. This proxy maximizes a data-adaptive Mahalanobis-norm margin, whose associated covariance matrix is determined by a data-dependent dual fixed-point formulation. We further present concrete datasets where this bias reduces to the standard $\ell_2$- and $\ell_\infty$-max-margin classifiers. As a counterpoint, we prove that Signum [Bernstein et al., 2018] converges to the $\ell_\infty$-max-margin classifier for any batch size. Overall, our results highlight that the implicit bias of Adam crucially depends on both the batching scheme and the dataset, while Signum remains invariant.

arXiv:2506.03230v2 Announce Type: replace-cross Abstract: Fine-tuning is a critical step for adapting large language models (LLMs) to domain-specific downstream tasks. To mitigate the substantial computational and memory costs of full-model fine-tuning, Parameter-Efficient Fine-Tuning (PEFT) methods have been proposed to update only a small subset of model parameters. However, performance gaps between PEFT approaches and full-model fine-tuning still exist. In this work, we present DiaBlo, a simple yet effective PEFT approach that updates only the diagonal blocks of selected model weight matrices. Unlike Low-Rank Adaptation (LoRA) and its variants, DiaBlo eliminates the need for low-rank matrix products, thereby avoiding the reliance on auxiliary initialization schemes or customized optimization strategies to improve convergence. This design leads to stable and robust convergence while maintaining comparable memory efficiency and training speed to LoRA. Moreover, we provide theoretical guarantees showing that, under mild low-rank conditions, DiaBlo is more expressive than LoRA in the linear problem and converges to a stationary point of the general nonlinear full fine-tuning. Through extensive experiments across a range of tasks, including commonsense reasoning, arithmetic reasoning, code generation, and safety alignment, we show that fine-tuning only diagonal blocks is sufficient for strong and consistent performance. DiaBlo not only achieves competitive accuracy but also preserves high memory efficiency and fast fine-tuning speed. Codes are available at https://github.com/ziyangjoy/DiaBlo.

arXiv:2603.00047v2 Announce Type: replace-cross Abstract: The alignment tax is widely discussed but has not been formally characterized. We provide a geometric theory of the alignment tax in representation space. Under linear representation assumptions, we define the alignment tax rate as the squared projection of the safety direction onto the capability subspace and derive the Pareto frontier governing safety-capability tradeoffs, parameterized by a single quantity of the principal angle between the safety and capability subspaces. We prove this frontier is tight and show it has a recursive structure. safety-safety tradeoffs under capability constraints are governed by the same equation, with the angle replaced by the partial correlation between safety objectives given capability directions. We derive a scaling law decomposing the alignment tax into an irreducible component determined by data structure and a packing residual that vanishes as $O(m'/d)$ with model dimension $d$, and establish conditions under which capability preservation mediates or resolves conflicts between safety objectives.

arXiv:2603.01731v2 Announce Type: replace-cross Abstract: Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential equations, contrasting well-established numerical methods with new AI-based techniques, specifically Physics-Informed Neural Networks (PINNs). We first analyze the logistic differential equation, using its closed-form solution to verify numerical schemes and validate PINN performance. We then address the Porous Medium Equation (PME), a nonlinear partial differential equation with no general closed-form solution, building strong solvers of the direct problem and testing techniques for parameter estimation in the inverse problem. Our results suggest that PINNs can closely estimate solutions at competitive computational cost, and thus propose an effective tool for solving both direct and inverse problems for complex systems.

arXiv:2602.19439v1 Announce Type: new Abstract: Problem Definition. Supply chain optimization models frequently become infeasible because of modeling errors. Diagnosis and repair require scarce OR expertise: analysts must interpret solver diagnostics, trace root causes across echelons, and fix formulations without sacrificing operational soundness. Whether AI agents can perform this task remains untested. Methodology/Results. OptiRepair splits this task into a domain-agnostic feasibility phase (iterative IIS-guided repair of any LP) and a domain-specific validation phase (five rationality checks grounded in inventory theory). We test 22 API models from 7 families on 976 multi-echelon supply chain problems and train two 8B-parameter models using self-taught reasoning with solver-verified rewards. The trained models reach 81.7% Rational Recovery Rate (RRR) -- the fraction of problems resolved to both feasibility and operational rationality -- versus 42.2% for the best API model and 21.3% on average. The gap concentrates in Phase 1 repair: API models average 27.6% recovery rate versus 97.2% for trained models. Managerial Implications. Two gaps separate current AI from reliable model repair: solver interaction (API models restore only 27.6% of infeasible formulations) and operational rationale (roughly one in four feasible repairs violate supply chain theory). Each requires a different intervention: solver interaction responds to targeted training; operational rationale requires explicit specification as solver-verifiable checks. For organizations adopting AI in operational planning, formalizing what "rational" means in their context is the higher-return investment.