On the Local Linear Rate of Consensus on the Stiefel Manifold

We study the convergence properties of Riemannian gradient method for solving the consensus problem (for an undirected connected graph) over the Stiefel manifold. The Stiefel manifold is a non-convex set and the standard notion of averaging in the Euclidean space does not work for this problem. We propose Distributed Riemannian Consensus on Stiefel Manifold (DRCS) and prove that it enjoys a local linear convergence rate to global consensus. More importantly, this local rate asymptotically scales with the second largest singular value of the communication matrix, which is on par with the well-known rate in the Euclidean space. To the best of our knowledge, this is the first work showing the equality of the two rates. The main technical challenges include (i) developing a Riemannian restricted secant inequality for convergence analysis, and (ii) to identify the conditions (e.g., suitable step-size and initialization) under which the algorithm always stays in the local region.

Asynchronous Advantage Actor Critic: Non-asymptotic Analysis and Linear Speedup

Asynchronous and parallel implementation of standard reinforcement learning (RL) algorithms is a key enabler of the tremendous success of modern RL. Among many asynchronous RL algorithms, arguably the most popular and effective one is the asynchronous advantage actor-critic (A3C) algorithm. Although A3C is becoming the workhorse of RL, its theoretical properties are still not well-understood, including the non-asymptotic analysis and the performance gain of parallelism (a.k.a. speedup). This paper revisits the A3C algorithm with TD(0) for the critic update, termed A3C-TD(0), with provable convergence guarantees. With linear value function approximation for the TD update, the convergence of A3C-TD(0) is established under both i.i.d. and Markovian sampling. Under i.i.d. sampling, A3C-TD(0) obtains sample complexity of $\mathcal{O}(\epsilon^{-2.5}/N)$ per worker to achieve $\epsilon$ accuracy, where $N$ is the number of workers. Compared to the best-known sample complexity of $\mathcal{O}(\epsilon^{-2.5})$ for two-timescale AC, A3C-TD(0) achieves \emph{linear speedup}, which justifies the advantage of parallelism and asynchrony in AC algorithms theoretically for the first time. Numerical tests on synthetically generated instances and OpenAI Gym environments have been provided to verify our theoretical analysis.

Hybrid Federated Learning: Algorithms and Implementation

Federated learning (FL) is a recently proposed distributed machine learning paradigm dealing with distributed and private data sets. Based on the data partition pattern, FL is often categorized into horizontal, vertical, and hybrid settings. Despite the fact that many works have been developed for the first two approaches, the hybrid FL setting (which deals with partially overlapped feature space and sample space) remains less explored, though this setting is extremely important in practice. In this paper, we first set up a new model-matching-based problem formulation for hybrid FL, then propose an efficient algorithm that can collaboratively train the global and local models to deal with full and partial featured data. We conduct numerical experiments on the multi-view ModelNet40 data set to validate the performance of the proposed algorithm. To the best of our knowledge, this is the first formulation and algorithm developed for the hybrid FL.

Hybrid FL: Algorithms and Implementation

Federated learning (FL) is a recently proposed distributed machine learning paradigm dealing with distributed and private data sets. Based on the data partition pattern, FL is often categorized into horizontal, vertical, and hybrid settings. Despite the fact that many works have been developed for the first two approaches, the hybrid FL setting (which deals with partially overlapped feature space and sample space) remains less explored, though this setting is extremely important in practice. In this paper, we first set up a new model-matching-based problem formulation for hybrid FL, then propose an efficient algorithm that can collaboratively train the global and local models to deal with full and partial featured data. We conduct numerical experiments on the multi-view ModelNet40 data set to validate the performance of the proposed algorithm. To the best of our knowledge, this is the first formulation and algorithm developed for the hybrid FL.

Learning to Continuously Optimize Wireless Resource In Episodically Dynamic Environment

There has been a growing interest in developing data-driven and in particular deep neural network (DNN) based methods for modern communication tasks. For a few popular tasks such as power control, beamforming, and MIMO detection, these methods achieve state-of-the-art performance while requiring less computational efforts, less channel state information (CSI), etc. However, it is often challenging for these approaches to learn in a dynamic environment where parameters such as CSIs keep changing. This work develops a methodology that enables data-driven methods to continuously learn and optimize in a dynamic environment. Specifically, we consider an ``episodically dynamic" setting where the environment changes in ``episodes", and in each episode the environment is stationary. We propose to build the notion of continual learning (CL) into the modeling process of learning wireless systems, so that the learning model can incrementally adapt to the new episodes, {\it without forgetting} knowledge learned from the previous episodes. Our design is based on a novel min-max formulation which ensures certain ``fairness" across different data samples. We demonstrate the effectiveness of the CL approach by customizing it to two popular DNN based models (one for power control and one for beamforming), and testing using both synthetic and real data sets. These numerical results show that the proposed CL approach is not only able to adapt to the new scenarios quickly and seamlessly, but importantly, it maintains high performance over the previously encountered scenarios as well.

Learning to Beamform in Heterogeneous Massive MIMO Networks

It is well-known that the problem of finding the optimal beamformers in massive multiple-input multiple-output (MIMO) networks is challenging because of its non-convexity, and conventional optimization based algorithms suffer from high computational costs. While computationally efficient deep learning based methods have been proposed, their complexity heavily relies upon system parameters such as the number of transmit antennas, and therefore these methods typically do not generalize well when deployed in heterogeneous scenarios where the base stations (BSs) are equipped with different numbers of transmit antennas and have different inter-BS distances. This paper proposes a novel deep learning based beamforming algorithm to address the above challenges. Specifically, we consider the weighted sum rate (WSR) maximization problem in multi-input and single-output (MISO) interference channels, and propose a deep neural network architecture by unfolding a parallel gradient projection algorithm. Somewhat surprisingly, by leveraging the low-dimensional structures of the optimal beamforming solution, our constructed neural network can be made independent of the numbers of transmit antennas and BSs. Moreover, such a design can be further extended to a cooperative multicell network. Numerical results based on both synthetic and ray-tracing channel models show that the proposed neural network can achieve high WSRs with significantly reduced runtime, while exhibiting favorable generalization capability with respect to the antenna number, BS number and the inter-BS distance.

A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic

This paper analyzes a two-timescale stochastic algorithm for a class of bilevel optimization problems with applications such as policy optimization in reinforcement learning, hyperparameter optimization, among others. We consider a case when the inner problem is unconstrained and strongly convex, and the outer problem is either strongly convex, convex or weakly convex. We propose a nonlinear two-timescale stochastic approximation (TTSA) algorithm for tackling the bilevel optimization. In the algorithm, a stochastic (semi)gradient update with a larger step size (faster timescale) is used for the inner problem, while a stochastic mirror descent update with a smaller step size (slower timescale) is used for the outer problem. When the outer problem is strongly convex (resp. weakly convex), the TTSA algorithm finds an $\mathcal{O}(K^{-1/2})$-optimal (resp. $\mathcal{O}(K^{-2/5})$-stationary) solution, where $K$ is the iteration number. To our best knowledge, these are the first convergence rate results for using nonlinear TTSA algorithms on the concerned class of bilevel optimization problems. Lastly, specific to the application of policy optimization, we show that a two-timescale actor-critic proximal policy optimization algorithm can be viewed as a special case of our framework. The actor-critic algorithm converges at $\mathcal{O}(K^{-1/4})$ in terms of the gap in objective value to a globally optimal policy.

Understanding Gradient Clipping in Private SGD: A Geometric Perspective

Deep learning models are increasingly popular in many machine learning applications where the training data may contain sensitive information. To provide formal and rigorous privacy guarantee, many learning systems now incorporate differential privacy by training their models with (differentially) private SGD. A key step in each private SGD update is gradient clipping that shrinks the gradient of an individual example whenever its L2 norm exceeds some threshold. We first demonstrate how gradient clipping can prevent SGD from converging to stationary point. We then provide a theoretical analysis that fully quantifies the clipping bias on convergence with a disparity measure between the gradient distribution and a geometrically symmetric distribution. Our empirical evaluation further suggests that the gradient distributions along the trajectory of private SGD indeed exhibit symmetric structure that favors convergence. Together, our results provide an explanation why private SGD with gradient clipping remains effective in practice despite its potential clipping bias. Finally, we develop a new perturbation-based technique that can provably correct the clipping bias even for instances with highly asymmetric gradient distributions.

Private Stochastic Non-Convex Optimization: Adaptive Algorithms and Tighter Generalization Bounds

We study differentially private (DP) algorithms for stochastic non-convex optimization. In this problem, the goal is to minimize the population loss over a $p$-dimensional space given $n$ i.i.d. samples drawn from a distribution. We improve upon the population gradient bound of ${\sqrt{p}}/{\sqrt{n}}$ from prior work and obtain a sharper rate of $\sqrt[4]{p}/\sqrt{n}$. We obtain this rate by providing the first analyses on a collection of private gradient-based methods, including adaptive algorithms DP RMSProp and DP Adam. Our proof technique leverages the connection between differential privacy and adaptive data analysis to bound gradient estimation error at every iterate, which circumvents the worse generalization bound from the standard uniform convergence argument. Finally, we evaluate the proposed algorithms on two popular deep learning tasks and demonstrate the empirical advantages of DP adaptive gradient methods over standard DP SGD.

On the Divergence of Decentralized Non-Convex Optimization

We study a generic class of decentralized algorithms in which $N$ agents jointly optimize the non-convex objective $f(u):=1/N\sum_{i=1}^{N}f_i(u)$, while only communicating with their neighbors. This class of problems has become popular in modeling many signal processing and machine learning applications, and many efficient algorithms have been proposed. However, by constructing some counter-examples, we show that when certain local Lipschitz conditions (LLC) on the local function gradient $\nabla f_i$'s are not satisfied, most of the existing decentralized algorithms diverge, even if the global Lipschitz condition (GLC) is satisfied, where the sum function $f$ has Lipschitz gradient. This observation raises an important open question: How to design decentralized algorithms when the LLC, or even the GLC, is not satisfied? To address the above question, we design a first-order algorithm called Multi-stage gradient tracking algorithm (MAGENTA), which is capable of computing stationary solutions with neither the LLC nor the GLC. In particular, we show that the proposed algorithm converges sublinearly to certain $\epsilon$-stationary solution, where the precise rate depends on various algorithmic and problem parameters. In particular, if the local function $f_i$'s are $Q$th order polynomials, then the rate becomes $\mathcal{O}(1/\epsilon^{Q-1})$. Such a rate is tight for the special case of $Q=2$ where each $f_i$ satisfies LLC. To our knowledge, this is the first attempt that studies decentralized non-convex optimization problems with neither the LLC nor the GLC.