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Register a new userS-ADDOPT: Decentralized stochastic first-order optimization over directed graphs
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Added by
Usman Khan
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this report, we study decentralized stochastic optimization to minimize a sum of smooth and strongly convex cost functions when the functions are distributed over a directed network of nodes. In contrast to the existing work, we use gradient tracking to improve certain aspects of the resulting algorithm. In particular, we propose the~textbf{texttt{S-ADDOPT}} algorithm that assumes a stochastic first-order oracle at each node and show that for a constant step-size~$alpha$, each node converges linearly inside an error ball around the optimal solution, the size of which is controlled by~$alpha$. For decaying step-sizes~$mathcal{O}(1/k)$, we show that~textbf{texttt{S-ADDOPT}} reaches the exact solution sublinearly at~$mathcal{O}(1/k)$ and its convergence is asymptotically network-independent. Thus the asymptotic behavior of~textbf{texttt{S-ADDOPT}} is comparable to the centralized stochastic gradient descent. Numerical experiments over both strongly convex and non-convex problems illustrate the convergence behavior and the performance comparison of the proposed algorithm.
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In this paper, we propose Push-SAGA, a decentralized stochastic first-order method for finite-sum minimization over a directed network of nodes. Push-SAGA combines node-level variance reduction to remove the uncertainty caused by stochastic gradients, network-level gradient tracking to address the distributed nature of the data, and push-sum consensus to tackle the challenge of directed communication links. We show that Push-SAGA achieves linear convergence to the exact solution for smooth and strongly convex problems and is thus the first linearly-convergent stochastic algorithm over arbitrary strongly connected directed graphs. We also characterize the regimes in which Push-SAGA achieves a linear speed-up compared to its centralized counterpart and achieves a network-independent convergence rate. We illustrate the behavior and convergence properties of Push-SAGA with the help of numerical experiments on strongly convex and non-convex problems.
This article focuses on multi-agent distributed optimization problems with a common decision variable, a global linear equality constraint, and local set constraints over directed interconnection topologies. We propose a novel ADMM based distributed algorithm to solve the above problem. During every iteration of the algorithm, each agent solves a local convex optimization problem and utilizes a finite-time ``approximate consensus protocol to update its local estimate of the optimal solution. The proposed algorithm is the first ADMM based algorithm with convergence guarantees to solve distributed multi-agent optimization problems where the interconnection topology is directed. We establish two strong explicit convergence rate estimates for the proposed algorithm to the optimal solution under two different sets of assumptions on the problem data. Further, we evaluate our proposed algorithm by solving two non-linear and non-differentiable constrained distributed optimization problems over directed graphs. Additionally, we provide a numerical comparison of the proposed algorithm with other state-of-the-art algorithms to show its efficacy over the existing methods in the literature.
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Trajectory optimization with contact-rich behaviors has recently gained attention for generating diverse locomotion behaviors without pre-specified ground contact sequences. However, these approaches rely on precise models of robot dynamics and the terrain and are susceptible to uncertainty. Recent works have attempted to handle uncertainties in the system model, but few have investigated uncertainty in contact dynamics. In this study, we model uncertainty stemming from the terrain and design corresponding risk-sensitive objectives under the framework of contact-implicit trajectory optimization. In particular, we parameterize uncertainties from the terrain contact distance and friction coefficients using probability distributions and propose a corresponding expected residual minimization cost design approach. We evaluate our method in three simple robotic examples, including a legged hopping robot, and we benchmark one of our examples in simulation against a robust worst-case solution. We show that our risk-sensitive method produces contact-averse trajectories that are robust to terrain perturbations. Moreover, we demonstrate that the resulting trajectories converge to those generated by a traditional, non-robust method as the terrain model becomes more certain. Our study marks an important step towards a fully robust, contact-implicit approach suitable for deploying robots on real-world terrain.
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