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.
Dynamic discrete choice models are used to estimate the intertemporal preferences of an agent as described by a reward function based upon observable histories of states and implemented actions. However, in many applications, such as reliability and healthcare, the system state is partially observable or hidden (e.g., the level of deterioration of an engine, the condition of a disease), and the decision maker only has access to information imperfectly correlated with the true value of the hidden state. In this paper, we consider the estimation of a dynamic discrete choice model with state variables and system dynamics that are hidden (or partially observed) to both the agent and the modeler, thus generalizing Rust's model to partially observable cases. We analyze the structural properties of the model and prove that this model is still identifiable if the cardinality of the state space, the discount factor, the distribution of random shocks, and the rewards for a given (reference) action are given. We analyze both theoretically and numerically the potential mis-specification errors that may be incurred when Rust's model is improperly used in partially observable settings. We further apply the developed model to a subset of Rust's dataset for bus engine mileage and replacement decisions. The results show that our model can improve model fit as measured by the $\log$-likelihood function by $17.7\%$ and the $\log$-likelihood ratio test shows that our model statistically outperforms Rust's model. Interestingly, our hidden state model also reveals an economically meaningful route assignment behavior in the dataset which was hitherto ignored, i.e. routes with lower mileage are assigned to buses believed to be in worse condition.
The stochastic subgradient method is a widely-used algorithm for solving large-scale optimization problems arising in machine learning. Often these problems are neither smooth nor convex. Recently, Davis et al. [1-2] characterized the convergence of the stochastic subgradient method for the weakly convex case, which encompasses many important applications (e.g., robust phase retrieval, blind deconvolution, biconvex compressive sensing, and dictionary learning). In practice, distributed implementations of the projected stochastic subgradient method (stoDPSM) are used to speed-up risk minimization. In this paper, we propose a distributed implementation of the stochastic subgradient method with a theoretical guarantee. Specifically, we show the global convergence of stoDPSM using the Moreau envelope stationarity measure. Furthermore, under a so-called sharpness condition, we show that deterministic DPSM (with a proper initialization) converges linearly to the sharp minima, using geometrically diminishing step-size. We provide numerical experiments to support our theoretical analysis.
We propose a new method for distributed estimation of a linear model by a network of local learners with heterogeneously distributed datasets. Unlike other ensemble learning methods, in the proposed method, model averaging is done continuously over time in a distributed and asynchronous manner. To ensure robust estimation, a network regularization term which penalizes models with high local variability is used. We provide a finite-time characterization of convergence of the weighted ensemble average and compare this result to centralized estimation. We illustrate the general applicability of the method in two examples: estimation of a Markov random field using wireless sensor networks and modeling prey escape behavior of birds based on a real-world dataset.
We study a distributed framework for stochastic optimization which is inspired by models of collective motion found in nature (e.g., swarming) with mild communication requirements. Specifically, we analyze a scheme in which each one of $N > 1$ independent threads, implements in a distributed and unsynchronized fashion, a stochastic gradient-descent algorithm which is perturbed by a swarming potential. Assuming the overhead caused by synchronization is not negligible, we show the swarming-based approach exhibits better performance than a centralized algorithm (based upon the average of $N$ observations) in terms of (real-time) convergence speed. We also derive an error bound that is monotone decreasing in network size and connectivity. We characterize the scheme's finite-time performances for both convex and non-convex objective functions.
In this paper we consider distributed optimization problems over a multi-agent network, where each agent can only partially evaluate the objective function, and it is allowed to exchange messages with its immediate neighbors. Differently from all existing works on distributed optimization, our focus is given to optimizing a class of difficult non-convex problems, and under the challenging setting where each agent can only access the zeroth-order information (i.e., the functional values) of its local functions. For different types of network topologies such as undirected connected networks or star networks, we develop efficient distributed algorithms and rigorously analyze their convergence and rate of convergence (to the set of stationary solutions). Numerical results are provided to demonstrate the efficiency of the proposed algorithms.