arXiv subjects

Jinlong Lei

Publications and source records attributed to Jinlong Lei.

35 records · Page 2Linked to original sources

arXiv2022

No-regret learning for repeated non-cooperative games with lossy bandits

This paper considers no-regret learning for repeated continuous-kernel games with lossy bandit feedback. Since it is difficult to give the explicit model of the utility functions in dynamic environments, the players' action can only be learned with bandit feedback. Moreover, because of unreliable communication channels or privacy protection, the bandit feedback may be lost or dropped at random. Therefore, we study the asynchronous online learning strategy of the players to adaptively adjust the next actions for minimizing the long-term regret loss. The paper provides a novel no-regret learning algorithm, called Online Gradient Descent with lossy bandits (OGD-lb). We first give the regret analysis for concave games with differentiable and Lipschitz utilities. Then we show that the action profile converges to a Nash equilibrium with probability 1 when the game is also strictly monotone. We further provide the mean square convergence rate $\mathcal{O}\left(k^{-2\min\{β, 1/6\}}\right)$ when the game is $β-$ strongly monotone. In addition, we extend the algorithm to the case when the loss probability of the bandit feedback is unknown, and prove its almost sure convergence to Nash equilibrium for strictly monotone games. Finally, we take the resource management in fog computing as an application example, and carry out numerical experiments to empirically demonstrate the algorithm performance.

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arXiv2022

Distributed coordination for seeking the optimal Nash equilibrium of aggregative games

This paper aims to design a distributed coordination algorithm for solving a multi-agent decision problem with a hierarchical structure. The primary goal is to search the Nash equilibrium of a noncooperative game such that each player has no incentive to deviate from the equilibrium under its private objective. Meanwhile, the agents can coordinate to optimize the social cost within the set of Nash equilibria of the underlying game. Such an optimal Nash equilibrium problem can be modeled as a distributed optimization problem with variational inequality constraints. We consider the scenario where the objective functions of both the underlying game and social cost optimization problem have a special aggregation structure. Since each player only has access to its local objectives while cannot know all players' decisions, a distributed algorithm is highly desirable. By utilizing the Tikhonov regularization and dynamical averaging tracking technique, we propose a distributed coordination algorithm by introducing an incentive term in addition to the gradient-based Nash equilibrium seeking, so as to intervene players' decisions to improve the system efficiency. We prove its convergence to the optimal Nash equilibrium of a monotone aggregative game with simulation studies.

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arXiv2022

Distributed Variable Sample-size Stochastic Optimization with Fixed Step-sizes

The paper considers distributed stochastic optimization over randomly switching networks, where agents collaboratively minimize the average of all agents' local expectation-valued convex cost functions. Due to the stochasticity in gradient observations, distributedness of local functions, and randomness of communication topologies, distributed algorithms with a convergence guarantee under fixed step-sizes have not been achieved yet. This work incorporates variance reduction scheme into the distributed stochastic gradient tracking algorithm, where local gradients are estimated by averaging across a variable number of sampled gradients. With an identically and independently distributed (i.i.d.) random network, we show that all agents' iterates converge almost surely to the same optimal solution under fixed step-sizes. When the global cost function is strongly convex and the sample size increases at a geometric rate, we prove that the iterates geometrically converge to the unique optimal solution, and establish the iteration, oracle, and communication complexity. The algorithm performance including rate and complexity analysis are further investigated with constant step-sizes and a polynomially increasing sample size. Finally, the empirical algorithm performance are illustrated with numerical examples.

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arXiv2022

Dynamical Primal-Dual Accelerated Method with Applications to Network Optimization

This paper develops a continuous-time primal-dual accelerated method with an increasing damping coefficient for a class of convex optimization problems with affine equality constraints. This paper analyzes critical values for parameters in the proposed method and prove that the rate of convergence in terms of the duality gap function is $O(\tfrac{1}{t^2})$ by choosing suitable parameters. As far as we know, this is the first continuous-time primal-dual accelerated method that can obtain the optimal rate. Then this work applies the proposed method to two network optimization problems, a distributed optimization problem with consensus constraints and a distributed extended monotropic optimization problem, and obtains two variant distributed algorithms. Finally, numerical simulations are given to demonstrate the efficacy of the proposed method.

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arXiv2021

No-regret distributed learning in subnetwork zero-sum games

In this paper, we consider a distributed learning problem in a subnetwork zero-sum game, where agents are competing in different subnetworks. These agents are connected through time-varying graphs where each agent has its own cost function and can receive information from its neighbors. We propose a distributed mirror descent algorithm for computing a Nash equilibrium and establish a sublinear regret bound on the sequence of iterates when the graphs are uniformly strongly connected and the cost functions are convex-concave. Moreover, we prove its convergence with suitably selected diminishing stepsizes for a strictly convex-concave cost function. We also consider a constant step-size variant of the algorithm and establish an asymptotic error bound between the cost function values of running average actions and a Nash equilibrium. In addition, we apply the algorithm to compute a mixed-strategy Nash equilibrium in subnetwork zero-sum finite-strategy games, which have merely convex-concave (to be specific, multilinear) cost functions, and obtain a final-iteration convergence result and an ergodic convergence result, respectively, under different assumptions.

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arXiv2021

Distributed Linear Equations over Random Networks

Distributed linear algebraic equation over networks, where nodes hold a part of problem data and cooperatively solve the equation via node-to-node communications, is a basic distributed computation task receiving an increasing research attention. Communications over a network have a stochastic nature, with both temporal and spatial dependence due to link failures, packet dropouts or node recreation, etc. In this paper, we study the convergence and convergence rate of distributed linear equation protocols over a $\ast$-mixing random network, where the temporal and spatial dependencies between the node-to-node communications are allowed. When the network linear equation admits exact solutions, we prove the mean-squared exponential convergence rate of the distributed projection consensus algorithm, while the lower and upper bound estimations of the convergence rate are also given for independent and identically distributed (i.i.d.) random graphs. Motivated by the randomized Kaczmarz algorithm, we also propose a distributed randomized projection consensus algorithm, where each node randomly selects one row of local linear equations for projection per iteration, and establish an exponential convergence rate. When the network linear equation admits no exact solution, we prove that a distributed gradient-descent-like algorithm with diminishing step-sizes can drive all nodes' states to a least-squares solution at a sublinear rate. These results collectively illustrate that distributed computations may overcome communication correlations if the prototype algorithms enjoy certain contractive properties or are designed with suitable parameters.

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arXiv2020

Distributed Variable Sample-Size Gradient-response and Best-response Schemes for Stochastic Nash Equilibrium Problems over Graphs

This paper considers a stochastic Nash game in which each player minimizes an expectation valued composite objective. We make the following contributions. (I) Under suitable monotonicity assumptions on the concatenated gradient map, we derive optimal rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme when the sample-size increases at a geometric rate. If the sample-size increases at a polynomial rate of degree $v > 0$, the mean-squared errordecays at a corresponding polynomial rate while the iteration and oracle complexities to obtain an $ε$-NE are $\mathcal{O}(1/ε^{1/v})$ and $\mathcal{O}(1/ε^{1+1/v})$, respectively. (II) We then overlay (VS-PGR) with a consensus phase with a view towards developing distributed protocols for aggregative stochastic Nash games. In the resulting scheme, when the sample-size and the consensus steps grow at a geometric and linear rate, computing an $ε$-NE requires similar iteration and oracle complexities to (VS-PGR) with a communication complexity of $\mathcal{O}(\ln^2(1/ε))$; (III) Under a suitable contractive property associated with the proximal best-response (BR) map, we design a variable sample-size proximal BR (VS-PBR) scheme, where each player solves a sample-average BR problem. Akin to (I), we also give the rate statements, oracle and iteration complexity bounds. (IV) Akin to (II), the distributed variant achieves similar iteration and oracle complexities to the centralized (VS-PBR) with a communication complexity of $\mathcal{O}(\ln^2(1/ε))$ when the communication rounds per iteration increase at a linear rate. Finally, we present some preliminary numerics to provide empirical support for the rate and complexity statements.

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arXiv2020

Asynchronous Variance-reduced Block Schemes for Composite Nonconvex Stochastic Optimization: Block-specific Steplengths and Adapted Batch-sizes

We consider the minimization of a sum of an expectation-valued coordinate-wise $L_i$-smooth nonconvex function and a nonsmooth block-separable convex regularizer. We propose an asynchronous variance-reduced algorithm, where in each iteration, a single block is randomly chosen to update its estimates by a proximal variable sample-size stochastic gradient scheme, while the remaining blocks are kept invariant. Notably, each block employs a steplength that is in accordance with its block-specific Lipschitz constant while block-specific batch-sizes are random variables updated at a rate that grows either at a geometric or polynomial rate with the (random) number of times that block is selected. We show that every limit point for almost every sample path is a stationary point and establish the ergodic non-asymptotic rate $\mathcal{O}(1/K) $. Iteration and oracle complexity to obtain an $ε$-stationary point are shown to be $\mathcal{O}(1/ε)$ and $\mathcal{O}(1/ε^2)$, respectively. Furthermore, under a $ μ$-proximal Polyak-Łojasiewicz (PL) condition with the batch size increasing at a geometric rate, we prove that the suboptimality diminishes at a {\em geometric} rate, the {\em optimal} deterministic rate while iteration and oracle complexity to obtain an $ε$-optimal solution are proven to be $\mathcal{O}( (L_{\rm max}/μ) \ln(1/ε))$ and $\mathcal{O}\left((L_{\rm ave}/μ) (1/ε)^{1+c} \right)$ with $c\geq 0$, respectively. In pursuit of less aggressive sampling rates, when the batch sizes increase at a polynomial rate of degree $v \geq 1$, suboptimality decays at a corresponding polynomial rate while the iteration and oracle complexity to obtain an $ε-$optimal solution are provably $\mathcal{O} ( v(1/ε)^{1/v})$ and $\mathcal{O} \left(e^v v^{2v+1}(1/ε)^{1+1/v}\right)$, respectively.

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arXiv2020

Linearly Convergent Algorithm with Variance Reduction for Distributed Stochastic Optimization

This paper considers a distributed stochastic strongly convex optimization, where agents connected over a network aim to cooperatively minimize the average of all agents' local cost functions. Due to the stochasticity of gradient estimation and distributedness of local objective, fast linearly convergent distributed algorithms have not been achieved yet. This work proposes a novel distributed stochastic gradient tracking algorithm with variance reduction, where the local gradients are estimated by an increasing batch-size of sampled gradients. With an undirected connected communication graph and a geometrically increasing batch-size, the iterates are shown to converge in mean to the optimal solution at a geometric rate (achieving linear convergence). The iteration, communication, and oracle complexity for obtaining an $ε$-optimal solution are established as well. Particulary, the communication complexity is $\mathcal{O}(\ln (1/ε))$ while the oracle complexity (number of sampled gradients) is $\mathcal{O}(1/ε^2)$, which is of the same order as that of centralized approaches. Hence, the proposed scheme is communication-efficient without requiring extra sampled gradients. Numerical simulations are given to demonstrate the theoretic results.

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arXiv2019

Asynchronous Schemes for Stochastic and Misspecified Potential Games and Nonconvex Optimization

The distributed computation of equilibria and optima has seen growing interest in a broad collection of networked problems. We consider the computation of equilibria of convex stochastic Nash games characterized by a possibly nonconvex potential function. Our focus is on two classes of stochastic Nash games: (P1): A potential stochastic Nash game, in which each player solves a parameterized stochastic convex program; and (P2): A misspecified generalization, where the player-specific stochastic program is complicated by a parametric misspecification. In both settings, exact proximal BR solutions are generally unavailable in finite time since they necessitate solving parameterized stochastic programs. Consequently, we design two asynchronous inexact proximal BR schemes to solve the problems, where in each iteration a single player is randomly chosen to compute an inexact proximal BR solution with rivals' possibly outdated information. Yet, in the misspecified regime (P2), each player possesses an extra estimate of the misspecified parameter and updates its estimate by a projected stochastic gradient (SG) algorithm. By Since any stationary point of the potential function is a Nash equilibrium of the associated game, we believe this paper is amongst the first ones for stochastic nonconvex (but block convex) optimization problems equipped with almost-sure convergence guarantees. These statements can be extended to allow for accommodating weighted potential games and generalized potential games. Finally, we present preliminary numerics based on applying the proposed schemes to congestion control and Nash-Cournot games.

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arXiv2019

Distributed Stochastic Approximation Algorithm With Expanding Truncations: Algorithm and Applications

In the paper, a novel distributed stochastic approximation algorithm (DSAA) is proposed to seek roots of the sum of local functions, each of which is associated with an agent from the multiple agents connected in a network. At any time, each agent updates its estimate for the root utilizing the observation of its local function and the information derived from the neighboring agents. The key difference of the proposed algorithm from the existing ones consists in the expanding truncations (so it is called as DSAAWET), by which the boundedness of the estimates can be guaranteed without imposing the growth rate constraints on the local functions. The convergence of the estimates generated by DSAAWET to a consensus set belonging to the root set of the sum function is shown under weaker conditions on the local functions and on the observation noise in comparison with the existing results. We illustrate our approach by two applications, one from signal processing and the other one from distributed optimization. Numerical simulation results are also included.

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arXiv2018

Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games: Best-Response Schemes and Distributed Gradient-Response Schemes

This paper considers an $N$-player stochastic Nash game in which the $i$th player minimizes a composite objective $f_i(x) + r_i(x_i)$, where $f_i$ is expectation-valued and $r_i$ has an efficient prox-evaluation. In this context, we make the following contributions. (i) Under a strong monotonicity assumption on the concatenated gradient map, we derive ({\bf optimal}) rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme; (ii) We overlay (VS-PGR) with a consensus phase with a view towards developing distributed protocols for aggregative stochastic Nash games. Notably, when the sample-size and the number of consensus steps at each iteration grow at a suitable rate, a linear rate of convergence can be achieved; (iii) Finally, under a suitable contractive property associated with the proximal best-response (BR) map, we design a variable sample-size proximal BR (VS-PBR) scheme, where the proximal BR is computed by solving a sample-average problem. If the batch-size for computing the sample-average is raised at a suitable rate, we show that the resulting iterates converge at a linear rate and derive the oracle complexity.

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arXiv2018

Data Rates for Network Linear Equations

In this paper, we study network linear equations subject to digital communications with a finite data rate, where each node is associated with one equation from a system of linear equations. Each node holds a dynamic state and interacts with its neighbors through an undirected connected graph, where along each link the pair of nodes share information. Due to the data-rate constraint, each node builds an encoder-decoder pair, with which it produces transmitted message with a zooming-in finite-level uniform quantizer and also generates estimates of its neighbors' states from the received signals. We then propose a distributed quantized algorithm and show that when the network linear equations admit a unique solution, each node's state is driven to that solution exponentially. We further establish the asymptotic rate of convergence, which shows that a larger number of quantization levels leads to a faster convergence rate but is still fundamentally bounded by the inherent network structure and the linear equations. When a unique least-squares solution exists, we show that the algorithm can compute such a solution with a suitably selected time-varying step size inherited from the encoder and zooming-in quantizer dynamics. In both cases, a minimal data rate is shown to be enough for guaranteeing the desired convergence when the step sizes are properly chosen. These results assure the applicability of various network linear equation solvers in the literature when peer-to-peer communication is digital.

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arXiv2018

On Synchronous, Asynchronous, and Randomized Best-Response schemes for computing equilibria in Stochastic Nash games

This work considers a stochastic Nash game in which each player solves a parameterized stochastic optimization problem. In deterministic regimes, best-response schemes have been shown to be convergent under a suitable spectral property associated with the proximal best-response map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization at each iteration. Instead, we propose an inexact generalization in which an inexact solution is computed via an increasing number of projected stochastic gradient steps. Based on this framework, we present three inexact best-response schemes: (i) First, we propose a synchronous scheme where all players simultaneously update their strategies; (ii) Subsequently, we extend this to a randomized setting where a subset of players is randomly chosen to their update strategies while the others keep their strategies invariant; (iii) Finally, we propose an asynchronous scheme, where each player determines its own update frequency and may use outdated rival-specific data in updating its strategy. Under a suitable contractive property of the proximal best-response map, we derive a.s. convergence of the iterates for (i) and (ii) and mean-convergence for (i) -- (iii). In addition, we show that for (i) -- (iii), the iterates converge to the unique equilibrium in mean at a prescribed linear rate. Finally, we establish the overall iteration complexity in terms of projected stochastic gradient steps for computing an $ε-$Nash equilibrium and in all settings, the iteration complexity is ${\cal O}(1/ε^{2(1+c) + δ})$ where $c = 0$ in the context of (i) and represents the positive cost of randomization (in (ii)) and asynchronicity and delay (in (iii)). The schemes are further extended to linear and quadratic recourse-based stochastic Nash games.

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arXiv2016

Distributed Resource Allocation Over Random Networks Based on Stochastic Approximation

In this paper, a stochastic approximation (SA) based distributed algorithm is proposed to solve the resource allocation (RA) with uncertainties. In this problem, a group of agents cooperatively optimize a separable optimization problem with a linear network resource constraint and allocation feasibility constraints, where the global objective function is the sum of agents' local objective functions. Each agent can only get noisy observations of its local function's gradient and its local resource, which cannot be shared by other agents or transmitted to a center. Moreover, there are communication uncertainties such as time-varying topologies (described by random graphs) and additive channel noises. To solve the RA, we propose an SA-based distributed algorithm, and prove that agents can collaboratively achieve the optimal allocation with probability one by virtue of ordinary differential equation (ODE) method for SA. Finally, simulations related to the demand response management in power systems verify the effectiveness of the proposed algorithm.

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arXiv2016

Asymptotic Properties of Primal-Dual Algorithm for Distributed Stochastic Optimization Over Random Networks

This paper studies a distributed stochastic optimization problem over random networks with imperfect communications subject to a global constraint, which is the intersection of local constraint sets assigned to agents. The global cost function is the sum of local cost functions, each of which is the expectation of a random cost function. By incorporating the augmented Lagrange technique with the projection method, a stochastic approximation based distributed primal-dual algorithm is proposed to solve the problem. Each agent updates its estimate by using the local observations and the information derived from neighbors. For the constrained problem, the estimates are first shown to be bounded almost surely (a.s.), and then are proved to converge to the optimal solution set a.s. Furthermore, the asymptotic normality and efficiency of the algorithm are addressed for the unconstrained case. The results demonstrate the influence of random networks, communication noises, and gradient errors on the performance of the algorithm. Finally, numerical simulations demonstrate the theoretic results.

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arXiv2016

Primal-Dual Algorithm for Distributed Constrained Optimization

The paper studies a distributed constrained optimization problem, where multiple agents connected in a network collectively minimize the sum of individual objective functions subject to a global constraint being an intersection of the local constraint sets assigned to the agents. Based on the augmented Lagrange method, a distributed primal-dual algorithm with a projection operation included is proposed to solve the problem. It is shown that with appropriately chosen constant step size, the local estimates derived at all agents asymptotically reach a consensus at an optimal solution. In addition, the value of the cost function at the time-averaged estimate converges with rate $O(\frac{1}{k})$ to the optimal value for the unconstrained problem. By these properties the proposed primal-dual algorithm is distinguished from the existing algorithms for distributed constrained optimization. The theoretical analysis is justified by numerical simulations.

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