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Register a new userDistributed Derivative-free Learning Method for Stochastic Optimization over a Network with Sparse Activity
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Added by
Wenjie Li
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper addresses a distributed optimization problem in a communication network where nodes are active sporadically. Each active node applies some learning method to control its action to maximize the global utility function, which is defined as the sum of the local utility functions of active nodes. We deal with stochastic optimization problem with the setting that utility functions are disturbed by some non-additive stochastic process. We consider a more challenging situation where the learning method has to be performed only based on a scalar approximation of the utility function, rather than its closed-form expression, so that the typical gradient descent method cannot be applied. This setting is quite realistic when the network is affected by some stochastic and time-varying process, and that each node cannot have the full knowledge of the network states. We propose a distributed optimization algorithm and prove its almost surely convergence to the optimum. Convergence rate is also derived with an additional assumption that the objective function is strongly concave. Numerical results are also presented to justify our claim.
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In this paper, we propose a new method based on the Sliding Algorithm from Lan(2016, 2019) for the convex composite optimization problem that includes two terms: smooth one and non-smooth one. Our method uses the stochastic noised zeroth-order oracle for the non-smooth part and the first-order oracle for the smooth part. To the best of our knowledge, this is the first method in the literature that uses such a mixed oracle for the composite optimization. We prove the convergence rate for the new method that matches the corresponding rate for the first-order method up to a factor proportional to the dimension of the space or, in some cases, its squared logarithm. We apply this method for the decentralized distributed optimization and derive upper bounds for the number of communication rounds for this method that matches known lower bounds. Moreover, our bound for the number of zeroth-order oracle calls per node matches the similar state-of-the-art bound for the first-order decentralized distributed optimization up to to the factor proportional to the dimension of the space or, in some cases, even its squared logarithm.
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We propose and analyze a new stochastic gradient method, which we call Stochastic Unbiased Curvature-aided Gradient (SUCAG), for finite sum optimization problems. SUCAG constitutes an unbiased total gradient tracking technique that uses Hessian information to accelerate con- vergence. We analyze our method under the general asynchronous model of computation, in which each function is selected infinitely often with possibly unbounded (but sublinear) delay. For strongly convex problems, we establish linear convergence for the SUCAG method. When the initialization point is sufficiently close to the optimal solution, the established convergence rate is only dependent on the condition number of the problem, making it strictly faster than the known rate for the SAGA method. Furthermore, we describe a Markov-driven approach of implementing the SUCAG method in a distributed asynchronous multi-agent setting, via gossiping along a random walk on an undirected communication graph. We show that our analysis applies as long as the graph is connected and, notably, establishes an asymptotic linear convergence rate that is robust to the graph topology. Numerical results demonstrate the merits of our algorithm over existing methods.
One of the most widely used methods for solving large-scale stochastic optimization problems is distributed asynchronous stochastic gradient descent (DASGD), a family of algorithms that result from parallelizing stochastic gradient descent on distributed computing architectures (possibly) asychronously. However, a key obstacle in the efficient implementation of DASGD is the issue of delays: when a computing node contributes a gradient update, the global model parameter may have already been updated by other nodes several times over, thereby rendering this gradient information stale. These delays can quickly add up if the computational throughput of a node is saturated, so the convergence of DASGD may be compromised in the presence of large delays. Our first contribution is that, by carefully tuning the algorithms step-size, convergence to the critical set is still achieved in mean square, even if the delays grow unbounded at a polynomial rate. We also establish finer results in a broad class of structured optimization problems (called variationally coherent), where we show that DASGD converges to a global optimum with probability $1$ under the same delay assumptions. Together, these results contribute to the broad landscape of large-scale non-convex stochastic optimization by offering state-of-the-art theoretical guarantees and providing insights for algorithm design.
We consider continuous-time dynamics for distributed optimization with set constraints in the note. To handle the computational complexity of projection-based dynamics due to solving a general quadratic optimization subproblem with projection, we propose a distributed projection-free dynamics by employing the Frank-Wolfe method, also known as the conditional gradient algorithm. The process searches a feasible descent direction with solving an alternative linear optimization instead of a quadratic one. To make the algorithm implementable over weight-balanced digraphs, we design one dynamics for the consensus of local decision variables and another dynamics of auxiliary variables to track the global gradient. Then we prove the convergence of the dynamical systems to the optimal solution, and provide detailed numerical comparisons with both projection-based dynamics and other distributed projection-free algorithms.
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