Local Linear Convergence of Infeasible Optimization with Orthogonal Constraints

Many classical and modern machine learning algorithms require solving optimization tasks under orthogonality constraints. Solving these tasks with feasible methods requires a gradient descent update followed by a retraction operation on the Stiefel manifold, which can be computationally expensive. Recently, an infeasible retraction-free approach, termed the landing algorithm, was proposed as an efficient alternative. Motivated by the common occurrence of orthogonality constraints in tasks such as principle component analysis and training of deep neural networks, this paper studies the landing algorithm and establishes a novel linear convergence rate for smooth non-convex functions using only a local Riemannian P{\L} condition. Numerical experiments demonstrate that the landing algorithm performs on par with the state-of-the-art retraction-based methods with substantially reduced computational overhead.

Automating Exploratory Proteomics Research via Language Models

With the development of artificial intelligence, its contribution to science is evolving from simulating a complex problem to automating entire research processes and producing novel discoveries. Achieving this advancement requires both specialized general models grounded in real-world scientific data and iterative, exploratory frameworks that mirror human scientific methodologies. In this paper, we present PROTEUS, a fully automated system for scientific discovery from raw proteomics data. PROTEUS uses large language models (LLMs) to perform hierarchical planning, execute specialized bioinformatics tools, and iteratively refine analysis workflows to generate high-quality scientific hypotheses. The system takes proteomics datasets as input and produces a comprehensive set of research objectives, analysis results, and novel biological hypotheses without human intervention. We evaluated PROTEUS on 12 proteomics datasets collected from various biological samples (e.g. immune cells, tumors) and different sample types (single-cell and bulk), generating 191 scientific hypotheses. These were assessed using both automatic LLM-based scoring on 5 metrics and detailed reviews from human experts. Results demonstrate that PROTEUS consistently produces reliable, logically coherent results that align well with existing literature while also proposing novel, evaluable hypotheses. The system's flexible architecture facilitates seamless integration of diverse analysis tools and adaptation to different proteomics data types. By automating complex proteomics analysis workflows and hypothesis generation, PROTEUS has the potential to considerably accelerate the pace of scientific discovery in proteomics research, enabling researchers to efficiently explore large-scale datasets and uncover biological insights.

Global Convergence of Decentralized Retraction-Free Optimization on the Stiefel Manifold

Many classical and modern machine learning algorithms require solving optimization tasks under orthogonal constraints. Solving these tasks often require calculating retraction-based gradient descent updates on the corresponding Riemannian manifold, which can be computationally expensive. Recently Ablin et al. proposed an infeasible retraction-free algorithm, which is significantly more efficient. In this paper, we study the decentralized non-convex optimization task over a network of agents on the Stiefel manifold with retraction-free updates. We propose \textbf{D}ecentralized \textbf{R}etraction-\textbf{F}ree \textbf{G}radient \textbf{T}racking (DRFGT) algorithm, and show that DRFGT exhibits ergodic $\mathcal{O}(1/K)$ convergence rate, the same rate of convergence as the centralized, retraction-based methods. We also provide numerical experiments demonstrating that DRFGT performs on par with the state-of-the-art retraction based methods with substantially reduced computational overhead.

Linear Convergence of Independent Natural Policy Gradient in Games with Entropy Regularization

This work focuses on the entropy-regularized independent natural policy gradient (NPG) algorithm in multi-agent reinforcement learning. In this work, agents are assumed to have access to an oracle with exact policy evaluation and seek to maximize their respective independent rewards. Each individual's reward is assumed to depend on the actions of all the agents in the multi-agent system, leading to a game between agents. We assume all agents make decisions under a policy with bounded rationality, which is enforced by the introduction of entropy regularization. In practice, a smaller regularization implies the agents are more rational and behave closer to Nash policies. On the other hand, agents with larger regularization acts more randomly, which ensures more exploration. We show that, under sufficient entropy regularization, the dynamics of this system converge at a linear rate to the quantal response equilibrium (QRE). Although regularization assumptions prevent the QRE from approximating a Nash equilibrium, our findings apply to a wide range of games, including cooperative, potential, and two-player matrix games. We also provide extensive empirical results on multiple games (including Markov games) as a verification of our theoretical analysis.

Improving LoRA in Privacy-preserving Federated Learning

Low-rank adaptation (LoRA) is one of the most popular task-specific parameter-efficient fine-tuning (PEFT) methods on pre-trained language models for its good performance and computational efficiency. LoRA injects a product of two trainable rank decomposition matrices over the top of each frozen pre-trained model module. However, when applied in the setting of privacy-preserving federated learning (FL), LoRA may become unstable due to the following facts: 1) the effects of data heterogeneity and multi-step local updates are non-negligible, 2) additive noise enforced on updating gradients to guarantee differential privacy (DP) can be amplified and 3) the final performance is susceptible to hyper-parameters. A key factor leading to these phenomena is the discordance between jointly optimizing the two low-rank matrices by local clients and separately aggregating them by the central server. Thus, this paper proposes an efficient and effective version of LoRA, Federated Freeze A LoRA (FFA-LoRA), to alleviate these challenges and further halve the communication cost of federated fine-tuning LLMs. The core idea of FFA-LoRA is to fix the randomly initialized non-zero matrices and only fine-tune the zero-initialized matrices. Compared to LoRA, FFA-LoRA is motivated by practical and theoretical benefits in privacy-preserved FL. Our experiments demonstrate that FFA-LoRA provides more consistent performance with better computational efficiency over vanilla LoRA in various FL tasks.

Provably Fast Convergence of Independent Natural Policy Gradient for Markov Potential Games

This work studies an independent natural policy gradient (NPG) algorithm for the multi-agent reinforcement learning problem in Markov potential games. It is shown that, under mild technical assumptions and the introduction of the \textit{suboptimality gap}, the independent NPG method with an oracle providing exact policy evaluation asymptotically reaches an $\epsilon$-Nash Equilibrium (NE) within $\mathcal{O}(1/\epsilon)$ iterations. This improves upon the previous best result of $\mathcal{O}(1/\epsilon^2)$ iterations and is of the same order, $\mathcal{O}(1/\epsilon)$, that is achievable for the single-agent case. Empirical results for a synthetic potential game and a congestion game are presented to verify the theoretical bounds.

On the Stability Analysis of Open Federated Learning Systems

We consider the open federated learning (FL) systems, where clients may join and/or leave the system during the FL process. Given the variability of the number of present clients, convergence to a fixed model cannot be guaranteed in open systems. Instead, we resort to a new performance metric that we term the stability of open FL systems, which quantifies the magnitude of the learned model in open systems. Under the assumption that local clients' functions are strongly convex and smooth, we theoretically quantify the radius of stability for two FL algorithms, namely local SGD and local Adam. We observe that this radius relies on several key parameters, including the function condition number as well as the variance of the stochastic gradient. Our theoretical results are further verified by numerical simulations on both synthetic and real-world benchmark data-sets.

On Centralized and Distributed Mirror Descent: Exponential Convergence Analysis Using Quadratic Constraints

Mirror descent (MD) is a powerful first-order optimization technique that subsumes several optimization algorithms including gradient descent (GD). In this work, we study the exact convergence rate of MD in both centralized and distributed cases for strongly convex and smooth problems. We view MD with a dynamical system lens and leverage quadratic constraints (QCs) to provide convergence guarantees based on the Lyapunov stability. For centralized MD, we establish a semi-definite programming (SDP) that certifies exponentially fast convergence of MD subject to a linear matrix inequality (LMI). We prove that the SDP always has a feasible solution that recovers the optimal GD rate. Next, we analyze the exponential convergence of distributed MD and characterize the rate using two LMIs. To the best of our knowledge, the exact (exponential) rate of distributed MD has not been previously explored in the literature. We present numerical results as a verification of our theory and observe that the richness of the Lyapunov function entails better (worst-case) convergence rates compared to existing works on distributed GD.

Linear Convergence of Distributed Mirror Descent with Integral Feedback for Strongly Convex Problems

Distributed optimization often requires finding the minimum of a global objective function written as a sum of local functions. A group of agents work collectively to minimize the global function. We study a continuous-time decentralized mirror descent algorithm that uses purely local gradient information to converge to the global optimal solution. The algorithm enforces consensus among agents using the idea of integral feedback. Recently, Sun and Shahrampour (2020) studied the asymptotic convergence of this algorithm for when the global function is strongly convex but local functions are convex. Using control theory tools, in this work, we prove that the algorithm indeed achieves (local) exponential convergence. We also provide a numerical experiment on a real data-set as a validation of the convergence speed of our algorithm.

Distributed Mirror Descent with Integral Feedback: Asymptotic Convergence Analysis of Continuous-time Dynamics

This work addresses distributed optimization, where a network of agents wants to minimize a global strongly convex objective function. The global function can be written as a sum of local convex functions, each of which is associated with an agent. We propose a continuous-time distributed mirror descent algorithm that uses purely local information to converge to the global optimum. Unlike previous work on distributed mirror descent, we incorporate an integral feedback in the update, allowing the algorithm to converge with a constant step-size when discretized. We establish the asymptotic convergence of the algorithm using Lyapunov stability analysis. We further illustrate numerical experiments that verify the advantage of adopting integral feedback for improving the convergence rate of distributed mirror descent.