Multimodal Domain Generalization (MMDG) leverages the complementary strengths of multiple modalities to enhance model generalization on unseen domains. A central challenge in multimodal learning is optimization imbalance, where modalities converge at different speeds during training. This imbalance leads to unequal gradient contributions, allowing some modalities to dominate the learning process while others lag behind. Existing balancing strategies typically regulate each modality’s gradient contribution based on its classification performance on the source domain to alleviate this issue. However, relying solely on source-domain accuracy neglects a key insight in MMDG: modalities that excel on the source domain may generalize poorly to unseen domains, limiting cross-domain gains. To overcome this limitation, we propose Gradient Modulation Projection (GMP), a unified strategy that promotes balanced optimization in MMDG. GMP first decouples gradients associated with classification and domain-invariance objectives. It then modulates each modality’s gradient based on semantic and domain confidence. Moreover, GMP dynamically adjusts gradient projections by tracking the relative strength of each task, mitigating conflicts between classification and domain-invariant learning within modality-specific encoders. Extensive experiments demonstrate that GMP achieves state-of-the-art performance and integrates flexibly with diverse MMDG methods, significantly improving generalization across multiple benchmarks.

In this work, we study the learning theory of reward modeling with pairwise comparison data using deep neural networks. We establish a novel non-asymptotic regret bound for deep reward estimators in a non-parametric setting, which depends explicitly on the network architecture. Furthermore, to underscore the critical importance of clear human beliefs, we introduce a margin-type condition that assumes the conditional winning probability of the optimal action in pairwise comparisons is significantly distanced from 1/2. This condition enables a sharper regret bound, which substantiates the empirical efficiency of Reinforcement Learning from Human Feedback and highlights clear human beliefs in its success. Notably, this improvement stems from high-quality pairwise comparison data implied by the margin-type condition, is independent of the specific estimators used, and thus applies to various learning algorithms and models.

We consider the problem of density-ratio estimation using Bregman Divergence with Deep ReLU feedforward neural networks (BDD). We establish non-asymptotic error bounds for BDD density-ratio estimators, which are minimax optimal up to a logarithmic factor when the data distribution has finite support. As an application of our theoretical findings, we propose an estimator for the KL-divergence that is asymptotically normal, leveraging our convergence results for the deep density-ratio estimator and a data-splitting method. We also extend our results to cases with unbounded support and unbounded density ratios. Furthermore, we show that the BDD density-ratio estimator can mitigate the curse of dimensionality when data distributions are supported on an approximately low-dimensional manifold. Our results are applied to investigate the convergence properties of the telescoping density-ratio estimator proposed by Rhodes (2020). We provide sufficient conditions under which it achieves a lower error bound than a single-ratio estimator. Moreover, we conduct simulation studies to validate our main theoretical results and assess the performance of the BDD density-ratio estimator.

Multimodal semantic segmentation integrates information from multiple data sources to perform pixel-wise semantic classification of images. By fusing complementary information from modalities such as thermal infrared and depth with traditional RGB, robust and reliable predictions can be achieved. However, existing methods still face challenges such as insufficient utilization of cross-modal information, limitations imposed by local receptive fields, and cross-category semantic confusion. To address these issues, this paper proposes the AS-Mamba framework, which introduces a Mamba structure with linear complexity into the field of image segmentation. Firstly, an Asymmetric Cross Mamba module is constructed to dynamically integrate complementary information from RGB and X modalities through an attention weight allocation strategy and enhanced cross-modal semantic alignment, enabling crossmodal dynamic interaction and global feature fusion. Secondly, a Spatial Perception Mamba module is designed to enhance fine-grained feature representation through a spatial-channel attention mechanism. Experiments on RGB-Thermal and RGBDepth tasks demonstrate the superiority of AS-Mamba.

We investigate the complexity of deep neural networks through the lens of functional equivalence, which posits that different parameterizations can yield the same network function. Leveraging the equivalence property, we present a novel bound on the covering number for deep neural networks, which reveals that the complexity of neural networks can be reduced. Additionally, we demonstrate that functional equivalence benefits optimization, as overparameterized networks tend to be easier to train since increasing network width leads to a diminishing volume of the effective parameter space. These findings can offer valuable insights into the phenomenon of overparameterization and have implications for understanding generalization and optimization in deep learning.

We propose a penalized nonparametric approach to estimating the quantile regression process (QRP) in a nonseparable model using rectifier quadratic unit (ReQU) activated deep neural networks and introduce a novel penalty function to enforce non-crossing of quantile regression curves. We establish the non-asymptotic excess risk bounds for the estimated QRP and derive the mean integrated squared error for the estimated QRP under mild smoothness and regularity conditions. To establish these non-asymptotic risk and estimation error bounds, we also develop a new error bound for approximating $C^s$ smooth functions with $s >1$ and their derivatives using ReQU activated neural networks. This is a new approximation result for ReQU networks and is of independent interest and may be useful in other problems. Our numerical experiments demonstrate that the proposed method is competitive with or outperforms two existing methods, including methods using reproducing kernels and random forests for nonparametric quantile regression.

We derive a novel approximation error bound with explicit prefactor for Sobolev-regular functions using deep convolutional neural networks (CNNs). The bound is non-asymptotic in terms of the network depth and filter lengths, in a rather flexible way. For Sobolev-regular functions which can be embedded into the H\"older space, the prefactor of our error bound depends on the ambient dimension polynomially instead of exponentially as in most existing results, which is of independent interest. We also establish a new approximation result when the target function is supported on an approximate lower-dimensional manifold. We apply our results to establish non-asymptotic excess risk bounds for classification using CNNs with convex surrogate losses, including the cross-entropy loss, the hinge loss (SVM), the logistic loss, the exponential loss and the least squares loss. We show that the classification methods with CNNs can circumvent the curse of dimensionality if input data is supported on a neighborhood of a low-dimensional manifold.