Federated Learning (FL) is an emerging collaborative machine learning framework where multiple clients train the global model without sharing their own datasets. In FL, the model inconsistency caused by the local data heterogeneity across clients results in the near-orthogonality of client updates, which leads to the global update norm reduction and slows down the convergence. Most previous works focus on eliminating the difference of parameters (or gradients) between the local and global models, which may fail to reflect the model inconsistency due to the complex structure of the machine learning model and the Euclidean space's limitation in meaningful geometric representations.In this paper, we propose FedMRUR by adopting the manifold model fusion scheme and a new global optimizer to alleviate the negative impacts.Concretely, FedMRUR adopts a hyperbolic graph manifold regularizer enforcing the representations of the data in the local and global models are close to each other in a low-dimensional subspace. Because the machine learning model has the graph structure, the distance in hyperbolic space can reflect the model bias better than the Euclidean distance.In this way, FedMRUR exploits the manifold structures of the representations to significantly reduce the model inconsistency.FedMRUR also aggregates the client updates norms as the global update norm, which can appropriately enlarge each client's contribution to the global update, thereby mitigating the norm reduction introduced by the near-orthogonality of client updates.Furthermore, we theoretically prove that our algorithm can achieve a linear speedup property $\mathcal{O}(\frac{1}{\sqrt{SKT}})$ for non-convex setting under partial client participation, where $S$ is the participated clients number, $K$ is the local interval and $T$ is the total number of communication rounds.Experiments demonstrate that FedMRUR can achieve a new state-of-the-art (SOTA) accuracy with less communication.

Sparse Mixture-of-Experts (MoE) have been widely adopted in recent large language models since it can efficiently scale up the model capability without increasing the inference cost. However, evaluations on broad downstream tasks reveal a consistent suboptimality of the routers in existing MoE LLMs, which results in a severe performance gap (e.g., 10-20% in accuracy) to the optimal routing. In this paper, we show that aligning the manifold of routing weights with that of task embedding can effectively reduce the gap and improve MoE LLMs' generalization performance. Our method, "Routing Manifold Alignment (RoMA)", introduces an additional manifold regularization term in the post-training objective and only requires lightweight finetuning of routers (with other parameters frozen). Specifically, the regularization encourages the routing weights of each sample to be close to those of its successful neighbors (whose routing weights lead to correct answers) in a task embedding space. Consequently, samples targeting similar tasks will share similar expert choices across layers. Building such bindings between tasks and experts over different samples is essential to achieve better generalization. Moreover, RoMA demonstrates the advantage of unifying the task understanding (by embedding models) with solution generation (by MoE LLMs). In experiments, we finetune routers in OLMoE, DeepSeekMoE, and Qwen3-MoE using RoMA. Evaluations on diverse benchmarks and extensive comparisons with baselines show the substantial improvement brought by RoMA.

In many practical machine learning problems, the acquisition of labeled data is often expensive and/or time consuming. This motivates us to study a problem as follows: given a label budget, how to select data points to label such that the learning performance is optimized. We propose a selective labeling method by analyzing the out-of-sample error of Laplacian regularized Least Squares (LapRLS). In particular, we derive a deterministic out-of-sample error bound for LapRLS trained on subsampled data, and propose to select a subset of data points to label by minimizing this upper bound. Since the minimization is a combinational problem, we relax it into continuous domain and solve it by projected gradient descent. Experiments on benchmark datasets show that the proposed method outperforms the state-of-the-art methods.

Federated Learning (FL) is an emerging collaborative machine learning framework where multiple clients train the global model without sharing their own datasets. In FL, the model inconsistency caused by the local data heterogeneity across clients results in the near-orthogonality of client updates, which leads to the global update norm reduction and slows down the convergence. Most previous works focus on eliminating the difference of parameters (or gradients) between the local and global models, which may fail to reflect the model inconsistency due to the complex structure of the machine learning model and the Euclidean space's limitation in meaningful geometric representations.In this paper, we propose FedMRUR by adopting the manifold model fusion scheme and a new global optimizer to alleviate the negative impacts.Concretely, FedMRUR adopts a hyperbolic graph manifold regularizer enforcing the representations of the data in the local and global models are close to each other in a low-dimensional subspace. Because the machine learning model has the graph structure, the distance in hyperbolic space can reflect the model bias better than the Euclidean distance.In this way, FedMRUR exploits the manifold structures of the representations to significantly reduce the model inconsistency.FedMRUR also aggregates the client updates norms as the global update norm, which can appropriately enlarge each client's contribution to the global update, thereby mitigating the norm reduction introduced by the near-orthogonality of client updates.Furthermore, we theoretically prove that our algorithm can achieve a linear speedup property \mathcal{O}(\frac{1}{\sqrt{SKT}}) for non-convex setting under partial client participation, where S is the participated clients number, K is the local interval and T is the total number of communication rounds.Experiments demonstrate that FedMRUR can achieve a new state-of-the-art (SOTA) accuracy with less communication.

In many practical machine learning problems, the acquisition of labeled data is often expensive and/or time consuming. This motivates us to study a problem as follows: given a label budget, how to select data points to label such that the learning performance is optimized. We propose a selective labeling method by analyzing the out-of-sample error of Laplacian regularized Least Squares (LapRLS). In particular, we derive a deterministic out-of-sample error bound for LapRLS trained on subsampled data, and propose to select a subset of data points to label by minimizing this upper bound. Since the minimization is a combinational problem, we relax it into continuous domain and solve it by projected gradient descent. Experiments on benchmark datasets show that the proposed method outperforms the state-of-the-art methods.

One of the prevailing trends in the machine- and deep-learning community is to gravitate towards the use of increasingly larger models in order to keep pushing the state-of-the-art performance envelope. This tendency makes access to the associated technologies more difficult for the average practitioner and runs contrary to the desire to democratize knowledge production in the field. In this paper, we propose a framework for achieving improved memory efficiency in the process of learning traditional neural networks by leveraging inductive-bias-driven network design principles and layer-wise manifold-oriented regularization objectives. Use of the framework results in improved absolute performance and empirical generalization error relative to traditional learning techniques. We provide empirical validation of the framework, including qualitative and quantitative evidence of its effectiveness on two standard image datasets, namely CIFAR-10 and CIFAR-100. The proposed framework can be seamlessly combined with existing network compression methods for further memory savings.

In this paper, we study the manifold regularization for the Sliced Inverse Regression (SIR). The manifold regularization improves the standard SIR in two aspects: 1) it encodes the local geometry for SIR and 2) it enables SIR to deal with transductive and semi-supervised learning problems. We prove that the proposed graph Laplacian based regularization is convergent at rate root-n. The projection directions of the regularized SIR are optimized by using a conjugate gradient method on the Grassmann manifold. Experimental results support our theory.

We present a novel method for multitask learning (MTL) based on {\it manifold regularization}: assume that all task parameters lie on a manifold. This is the generalization of a common assumption made in the existing literature: task parameters share a common {\it linear} subspace. One proposed method uses the projection distance from the manifold to regularize the task parameters. The manifold structure and the task parameters are learned using an alternating optimization framework. When the manifold structure is fixed, our method decomposes across tasks which can be learnt independently.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

This work presents a two-stage framework for progressively developing neural architectures to adapt/ generalize well on a given training data set. In the first stage, a manifold-regularized layerwise sparsifying training approach is adopted where a new layer is added each time and trained independently by freezing parameters in the previous layers. In order to constrain the functions that should be learned by each layer, we employ a sparsity regularization term, manifold regularization term and a physics-informed term. We derive the necessary conditions for trainability of a newly added layer and analyze the role of manifold regularization. In the second stage of the Algorithm, a sequential learning process is adopted where a sequence of small networks is employed to extract information from the residual produced in stage I and thereby making robust and more accurate predictions. Numerical investigations with fully connected network on prototype regression problem, and classification problem demonstrate that the proposed approach can outperform adhoc baseline networks. Further, application to physics-informed neural network problems suggests that the method could be employed for creating interpretable hidden layers in a deep network while outperforming equivalent baseline networks.