The Blahut-Arimoto (BA) algorithm has played a fundamental role in the numerical computation of rate-distortion (RD) functions. This algorithm possesses a desirable monotonic convergence property by alternatively minimizing its Lagrangian with a fixed multiplier. In this paper, we propose a novel modification of the BA algorithm, letting the multiplier be updated in each iteration via a one-dimensional root-finding step with respect to a monotonic univariate function, which can be efficiently implemented by Newton's method. This allows the multiplier to be updated in a flexible and efficient manner, overcoming a major drawback of the original BA algorithm wherein the multiplier is fixed throughout iterations. Consequently, the modified algorithm is capable of directly computing the RD function for a given target distortion, without exploring the entire RD curve as in the original BA algorithm. A theoretical analysis shows that the modified algorithm still converges to the RD function and the convergence rate is $\Theta(1/n)$, where $n$ denotes the number of iterations. Numerical experiments demonstrate that the modified algorithm directly computes the RD function with a given target distortion, and it significantly accelerates the original BA algorithm.

Multimodal learning has seen great success mining data features from multiple modalities with remarkable model performance improvement. Meanwhile, federated learning (FL) addresses the data sharing problem, enabling privacy-preserved collaborative training to provide sufficient precious data. Great potential, therefore, arises with the confluence of them, known as multimodal federated learning. However, limitation lies in the predominant approaches as they often assume that each local dataset records samples from all modalities. In this paper, we aim to bridge this gap by proposing an Unimodal Training - Multimodal Prediction (UTMP) framework under the context of multimodal federated learning. We design HA-Fedformer, a novel transformer-based model that empowers unimodal training with only a unimodal dataset at the client and multimodal testing by aggregating multiple clients' knowledge for better accuracy. The key advantages are twofold. Firstly, to alleviate the impact of data non-IID, we develop an uncertainty-aware aggregation method for the local encoders with layer-wise Markov Chain Monte Carlo sampling. Secondly, to overcome the challenge of unaligned language sequence, we implement a cross-modal decoder aggregation to capture the hidden signal correlation between decoders trained by data from different modalities. Our experiments on popular sentiment analysis benchmarks, CMU-MOSI and CMU-MOSEI, demonstrate that HA-Fedformer significantly outperforms state-of-the-art multimodal models under the UTMP federated learning frameworks, with 15%-20% improvement on most attributes.

Federated learning (FL) provides a communication-efficient approach to solve machine learning problems concerning distributed data, without sending raw data to a central server. However, existing works on FL only utilize first-order gradient descent (GD) and do not consider the preceding iterations to gradient update which can potentially accelerate convergence. In this paper, we consider momentum term which relates to the last iteration. The proposed momentum federated learning (MFL) uses momentum gradient descent (MGD) in the local update step of FL system. We establish global convergence properties of MFL and derive an upper bound on MFL convergence rate. Comparing the upper bounds on MFL and FL convergence rate, we provide conditions in which MFL accelerates the convergence. For different machine learning models, the convergence performance of MFL is evaluated based on experiments with MNIST dataset. Simulation results comfirm that MFL is globally convergent and further reveal significant convergence improvement over FL.