In explainable AI, DNN gradients are used to interpret the prediction; in safetycritical control systems, gradients could encode safety constraints; in scientificcomputing applications, gradients could encode physical invariants. While recent work on provable editing of DNNs has focused on input-output constraints, the problem of enforcing hard constraints on DNN gradients remains unaddressed. We present ProGrad, the first efficient approach for editing the parameters of a DNN to provably enforce hard constraints on the DNN gradients.

We study bilevel optimization problems where the lower-level problems are strongly convex and have coupled linear constraints. To overcome the potential nonsmoothness of the hyper-objective and the computational challenges associated with the Hessian matrix, we utilize penalty and augmented Lagrangian methods to reformulate the original problem as a single-level one. Especially, we establish a strong theoretical connection between the reformulated function and the original hyper-objective by characterizing the closeness of their values and derivatives. Based on this reformulation, we propose a single-loop, first-order algorithm for linearly constrained bilevel optimization (SFLCB). We provide rigorous analyses of its non-asymptotic convergence rates, showing an improvement over prior double-loop algorithms - form O(ϵ 3 log(ϵ 1))to O(ϵ 3). The experiments corroborate our theoretical findings and demonstrate the practical efficiency of the proposed SFLCB algorithm.

Transformer models have become the dominant backbone for sequence modeling, leveraging self-attention to produce contextualized token representations. These are typically aggregated into fixed-size vectors via pooling operations for downstream tasks. While much of the literature has focused on attention mechanisms, the role of pooling remains underexplored despite its critical impact on model behavior. In this paper, we introduce a theoretical framework that rigorously characterizes the expressivity of Transformer-based models equipped with widely used pooling methods by deriving closed-form bounds on their representational capacity and the ability to distinguish similar inputs. Our analysis extends to different variations of attention formulations, demonstrating that these bounds hold across diverse architectural variants. We empirically evaluate pooling strategies across tasks requiring both global and local contextual understanding, spanning three major modalities: computer vision, natural language processing, and time-series analysis. Results reveal consistent trends in how pooling choices affect accuracy, sensitivity, and optimization behavior. Our findings unify theoretical and empirical perspectives, providing practical guidance for selecting or designing pooling mechanisms suited to specific tasks. This work positions pooling as a key architectural component in Transformer models and lays the foundation for more principled model design beyond attention alone.

Visual anomaly detection is significant in safety-critical and reliability-sensitive scenarios. Prior studies mainly emphasize the design and training of scoring functions, while little effort has been devoted to constructing decision rules based on these score functions. A recent work Ma et al. (2025b) highlights this issue and proposes the SAC-BL algorithm to address it. This method consists of a strong anomaly constraint (SAC) network and a betting-like (BL) algorithm serving as the decision rule. The SAC-BL algorithm can control the false discovery rate (FDR).

In this paper, we study the Karush-Kuhn-Tucker (KKT) points of the associated maximum-margin problem in homogeneous neural networks, including fullyconnected and convolutional neural networks. In particular, We investigates the relationship between such KKT points across networks of different widths generated. We introduce and formalize the KKT point embedding principle, establishing that KKT points of a homogeneous network's max-margin problem (PΦ) can be embedded into the KKT points of a larger network's problem (P Φ) via specific linear isometric transformations. We rigorously prove this principle holds for neuron splitting in fully-connected networks and channel splitting in convolutional neural networks. Furthermore, we connect this static embedding to the dynamics of gradient flow training with smooth losses. We demonstrate that trajectories initiated from appropriately mapped points remain mapped throughout training and that the resulting ω-limit sets of directions are correspondingly mapped, thereby preserving the alignment with KKT directions dynamically when directional convergence occurs. We conduct several experiments to justify that trajectories are preserved. Our findings offer insights into the effects of network width, parameter redundancy, and the structural connections between solutions found via optimization in homogeneous networks of varying sizes.

Learning-to-Defer (L2D) methods route each query either to a predictive model or to external experts. While existing work studies this problem in batch settings, real-world deployments require handling streaming data, changing expert availability, and shifting expert distribution. We introduce the first online L2D algorithm for multiclass classification with bandit feedback and a dynamically varying pool of experts. Our method achieves regret guarantees of $O((n+n_e)T^{2/3})$ in general and $O((n+n_e)\sqrt{T})$ under a low-noise condition, where $T$ is the time horizon, $n$ is the number of labels, and $n_e$ is the number of distinct experts observed across rounds. The analysis builds on novel $\mathcal{H}$-consistency bounds for the online framework, combined with first-order methods for online convex optimization. Experiments on synthetic and real-world datasets demonstrate that our approach effectively extends standard Learning-to-Defer to settings with varying expert availability and reliability.

Wide neural networks in the feature-learning regime drive modern deep learning, and yet they remain far less studied than their kernel-regime counterparts. We consider a critical yet under-explored difference between these two regimes: the regulariser and prior implied by gradient flow training. This canonical regularisation property is well-studied in kernel regime networks -- of all the infinite global minima, gradient flow selects exactly the vanishing ridge solution -- and underpins the celebrated NN-GP correspondence, precisely allowing the modelling of noise during training. However, we prove ridge regularisation biases gradient flow in feature-learning regime networks, even in the infinitesimal limit of vanishing regularisation. Over training, ridge distorts the inductive bias of the network, with a particular damage done to pretrained networks where the implicit prior is informative. We resolve this by axiomatising the canonical regulariser as a regime-agnostic function-space energy and lift, which uniquely identifies ridge in the kernel regime, and crucially generalises to the feature-learning regime. By studying the Riemannian geometry of feature-learning networks, we derive geodesic ridge from our framework, generalising ridge to the feature-learning regime. Correspondingly, we prove the canonical function-space prior is a Riemannian Gibbs Process, generalising the more familiar Gaussian Process. As a practical contribution, we propose arc ridge as a minimax-robust, scalable surrogate to geodesic ridge, revealing a deep relationship between early stopping and canonical regularisation across learning regimes. Finally, we demonstrate the consequences of our theory empirically on both image processing and NLP transfer-learning problems.

Length-dependent logit rescaling is widely used to stabilize long-context self-attention, but existing analyses and methods suggest conflicting inverse-temperature laws for the context length $n$, ranging from $(\log n)^{1/2}$ to $\log n$ and $(\log n)^2$. We provide a general theory showing that the desirable scale is determined by the gap-counting function $N_n$ of each attention row. Counting how many competitors lie within each gap from the maximum, we define an upper-tail accumulation scale and prove that it gives the critical inverse-temperature scale for softmax concentration: below this scale, the top competitors remain unseparated, whereas above it, the attention entropy collapses. This framework unifies prior scaling laws as different $N_n$ and yields a direct diagnostic for attention-score families, from idealized theoretical models to more practical transformers.

Graph neural networks (GNNs) are the de facto standard deep learning architectures for machine learning on graphs. This has led to a large body of work analyzing the capabilities and limitations of these models, particularly pertaining to their representation and extrapolation capacity. We offer a novel theoretical perspective on the representation and extrapolation capacity of GNNs, by answering the question: how do GNNs behave as the number of graph nodes become very large? Under mild assumptions, we show that when we draw graphs of increasing size from the Erd os-Rényi model, the probability that such graphs are mapped to a particular output by a class of GNN classifiers tends to either zero or to one. This class includes the popular graph convolutional network architecture. The result establishes'zero-one laws' for these GNNs, and analogously to other convergence laws, entails theoretical limitations on their capacity. We empirically verify our results, observing that the theoretical asymptotic limits are evident already on relatively small graphs.