In this paper, we consider an online distributed composite optimization problem over a time-varying multi-agent network that consists of multiple interacting nodes, where the objective function of each node consists of two parts: a loss function that changes over time and a regularization function. This problem naturally arises in many real-world applications ranging from wireless sensor networks to signal processing. We propose a class of online distributed optimization algorithms that are based on approximate mirror descent, which utilize the Bregman divergence as distance-measuring function that includes the Euclidean distances as a special case. We consider two standard information feedback models when designing the algorithms, that is, full-information feedback and bandit feedback. For the full-information feedback model, the first algorithm attains an average regularized regret of order $\mathcal{O}(1/\sqrt{T})$ with the total number of rounds $T$. The second algorithm, which only requires the information of the values of the loss function at two predicted points instead of the gradient information, achieves the same average regularized regret as that of the first algorithm. Simulation results of a distributed online regularized linear regression problem are provided to illustrate the performance of the proposed algorithms.
We propose two Polyak-type step sizes for mirror descent and prove their convergences for minimizing convex locally Lipschitz functions. Both step sizes, unlike the original Polyak step size, do not need the optimal value of the objective function.
Mirror Descent (MD) is a well-known method of solving non-smooth convex optimization problems. This paper analyzes the stochastic variant of MD with adaptive stepsizes. Its convergence on average is shown to be faster than with the fixed stepsizes and optimal in terms of lower bounds.
We investigate different randomizations for mirror descent method. We try to propose such a randomization that allows us to use sparsity of the problem as much as it possible. In the paper one can also find a generalization of randomizaed mirror descent for the convex optimization problems with functional restrictions.
We propose primal-dual stochastic mirror descent for the convex optimization problems with functional constraints. We obtain the rate of convergence in terms of probability of large deviations.
We consider distributed optimization with smooth convex objective functions defined on an undirected connected graph. Inspired by mirror descent mehod and RLC circuits, we propose a novel distributed mirror descent method. Compared with mirror-prox method, our algorithm achieves the same \(\mathcal{O}(1/k)\) iteration complexity with only half the computation cost per iteration. We further extend our results to cases where a) gradients are corrupted by stochastic noise, and b) objective function is composed of both smooth and non-smooth terms. We demonstrate our theoretical results via numeric…
The paper is devoted to a special Mirror Descent algorithm for problems of convex minimization with functional constraints. The objective function may not satisfy the Lipschitz condition, but it must necessarily have the Lipshitz-continuous gradient. We assume, that the functional constraint can be non-smooth, but satisfying the Lipschitz condition. In particular, such functionals appear in the well-known Truss Topology Design problem. Also we have applied the technique of restarts in the mentioned version of Mirror Descent for strongly convex problems. Some estimations for a rate of convergen…
In the paper we consider an application of mirror descent (dual averaging) to the stochastic online convex optimization problems. We compare classical mirror descent (Nemirovski-Yudin, 1979) with dual averaging (Nesterov, 2005) and Grigoriadis-Khachiyan algorithm (1995). Grigoriadis-Khachiyan algorithm has just proved to be a randomized mirror descent (dual averaging) with randomization in KL-projection of (sub)gradient to a unit simplex. We found out that this randomization is an optimal way of solving sparse matrix games and some other problems arising in convex optimization and experts weig…
To solve distributed optimization efficiently with various constraints and nonsmooth functions, we propose a distributed mirror descent algorithm with embedded Bregman damping, as a generalization of conventional distributed projection-based algorithms. In fact, our continuous-time algorithm well inherits good capabilities of mirror descent approaches to rapidly compute explicit solutions to the problems with some specific constraint structures. Moreover, we rigorously prove the convergence of our algorithm, along with the boundedness of the trajectory and the accuracy of the solution.
This paper seeks to address how to solve non-smooth convex and strongly convex optimization problems with functional constraints. The introduced Mirror Descent (MD) method with adaptive stepsizes is shown to have a better convergence rate than MD with fixed stepsizes due to the improved constant. For certain types of constraints, the method is proved to generate dual solution. For the strongly convex case, the restart technique is applied.
Existing convergence guarantees for the mirror descent algorithm require the objective function to have a bounded gradient or be smooth relative to a Legendre function. The bounded gradient and relative smoothness conditions, however, may not hold in important applications, such as quantum state tomography and portfolio selection. In this paper, we propose a local version of the relative smoothness condition as a generalization of its existing global version, and prove that under this local relative smoothness condition, the mirror descent algorithm with Armijo line search always converges. Nu…
We consider centralized and distributed mirror descent algorithms over a finite-dimensional Hilbert space, and prove that the problem variables converge to an optimizer of a possibly nonsmooth function when the step sizes are square summable but not summable. Prior literature has focused on the convergence of the function value to its optimum. However, applications from distributed optimization and learning in games require the convergence of the variables to an optimizer, which is generally not guaranteed without assuming strong convexity of the objective function. We provide numerical simula…
In this paper we describe how imitative logit dynamic (rather popular in population games theory) can be interpreted in terms of mirror descent method (well known in convex optimization). We demonstrate the connection of this two type dynamics on the Shortest Path Problem.
We consider risk-averse convex stochastic programs expressed in terms of extended polyhedral risk measures. We derive computable confidence intervals on the optimal value of such stochastic programs using the Robust Stochastic Approximation and the Stochastic Mirror Descent (SMD) algorithms. When the objective functions are uniformly convex, we also propose a multistep extension of the Stochastic Mirror Descent algorithm and obtain confidence intervals on both the optimal values and optimal solutions. Numerical simulations show that our confidence intervals are much less conservative and are q…
We consider the problem of minimization of a convex function on a simple set with convex non-smooth inequality constraint and describe first-order methods to solve such problems in different situations: smooth or non-smooth objective function; convex or strongly convex objective and constraint; deterministic or randomized information about the objective and constraint. We hope that it is convenient for a reader to have all the methods for different settings in one place. Described methods are based on Mirror Descent algorithm and switching subgradient scheme. One of our focus is to propose, fo…
In this paper, we present a new stochastic algorithm, namely the stochastic block mirror descent (SBMD) method for solving large-scale nonsmooth and stochastic optimization problems. The basic idea of this algorithm is to incorporate the block-coordinate decomposition and an incremental block averaging scheme into the classic (stochastic) mirror-descent method, in order to significantly reduce the cost per iteration of the latter algorithm. We establish the rate of convergence of the SBMD method along with its associated large-deviation results for solving general nonsmooth and stochastic opti…
We provide new insight into a {\em generalized conditional subgradient} algorithm and a {\em generalized mirror descent} algorithm for the convex minimization problem \[ \min_x \; \{f(Ax) + h(x)\}.\] As Bach showed in [{\em SIAM J. Optim.}, 25 (2015), pp. 115--129], applying either of these two algorithms to this problem is equivalent to applying the other one to its Fenchel dual. We leverage this duality relationship to develop new upper bounds and convergence results for the gap between the primal and dual iterates generated by these two algorithms. We also propose a new {\em primal-dual hyb…
Microsoft Research（微软研究院）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We present a primal only derivation of Mirror Descent as a "partial" discretization of gradient flow on a Riemannian manifold where the metric tensor is the Hessian of the Mirror Descent potential. We contrast this discretization to Natural Gradient Descent, which is obtained by a "full" forward Euler discretization. This view helps shed light on the relationship between the methods and allows generalizing Mirror Descent to general Riemannian geometries, even when the metric tensor is {\em not} a Hessian, and thus there is no "dual."
The goal is to modify the known method of mirror descent (MD), proposed by A.S. Nemirovsky and D.B. Yudin in 1979. The paper shows the idea of a new, so-called inertial MD method with the example of a deterministic optimization problem in continuous time. In particular, in the Euclidean case, the heavy ball method by B.T. Polyak is realized. It is noted that the new method does not use additional averaging. A discrete algorithm of inertial MD is described. The theorem on the upper bound on the error in the objective function is proved.
Theoretical estimates of the convergence rate of many well-known gradient-type optimization methods are based on quadratic interpolation, provided that the Lipschitz condition for the gradient is satisfied. In this article we obtain a possibility of constructing an analogue of such interpolation in the class of locally Lipschitz quasi-convex functionals with the special conditions of non-smoothness (Lipshitz-continuous subgradient) introduced in this paper. As an application, estimates are obtained for the rate of convergence of the previously proposed adaptive mirror descent method for the pr…
We consider the following class of online optimization problems with functional constraints. Assume, that a finite set of convex Lipschitz-continuous non-smooth functionals are given on a closed set of $n$-dimensional vector space. The problem is to minimize the arithmetic mean of functionals with a convex Lipschitz-continuous non-smooth constraint. In addition, it is allowed to calculate the (sub)gradient of each functional only once. Using some recently proposed adaptive methods of Mirror Descent the method is suggested to solve the mentioned constrained online optimization problem with opti…
We examine a class of stochastic mirror descent dynamics in the context of monotone variational inequalities (including Nash equilibrium and saddle-point problems). The dynamics under study are formulated as a stochastic differential equation driven by a (single-valued) monotone operator and perturbed by a Brownian motion. The system's controllable parameters are two variable weight sequences that respectively pre- and post-multiply the driver of the process. By carefully tuning these parameters, we obtain global convergence in the ergodic sense, and we estimate the average rate of convergence…
We present and analyse a backtracking strategy for a general Fast Iterative Shrinkage/Thresholding Algorithm which has been recently proposed in (Chambolle, Pock, 2016) for strongly convex objective functions. Differently from classical Armijo-type line searching, our backtracking rule allows for local increasing and decreasing of the descent step size (i.e. proximal parameter) along the iterations. For such strategy accelerated convergence rates are proved and numerical results are shown for some exemplar imaging problems.
This paper is concerned with multi-agent optimization problem. A distributed randomized gradient-free mirror descent (DRGFMD) method is developed by introducing a randomized gradient-free oracle in the mirror descent scheme where the non-Euclidean Bregman divergence is used. The classical gradient descent method is generalized without using subgradient information of objective functions. The proposed algorithm is the first distributed non-Euclidean zeroth-order method which achieves an $O(1/\sqrt{T})$ convergence rate, recovering the best known optimal rate of distributed compact constrained c…
Based on the ideas of arXiv:1710.06612, we consider the problem of minimization of the Holder-continuous non-smooth functional $f$ with non-positive convex (generally, non-smooth) Lipschitz-continuous functional constraint. We propose some novel strategies of step-sizes and adaptive stopping rules in Mirror Descent algorithms for the considered class of problems. It is shown that the methods are applicable to the objective functionals of various levels of smoothness. Applying the restart technique to the Mirror Descent Algorithm there was proposed an optimal method to solve optimization proble…
In this paper, we investigate the non-asymptotic stationary convergence behavior of Stochastic Mirror Descent (SMD) for nonconvex optimization. We focus on a general class of nonconvex nonsmooth stochastic optimization problems, in which the objective can be decomposed into a relatively weakly convex function (possibly non-Lipschitz) and a simple non-smooth convex regularizer. We prove that SMD, without the use of mini-batch, is guaranteed to converge to a stationary point in a convergence rate of $ \mathcal{O}(1/\sqrt{t}) $. The efficiency estimate matches with existing results for stochastic…
Learning-to-optimize is an emerging framework that seeks to speed up the solution of certain optimization problems by leveraging training data. Learned optimization solvers have been shown to outperform classical optimization algorithms in terms of convergence speed, especially for convex problems. Many existing data-driven optimization methods are based on parameterizing the update step and learning the optimal parameters (typically scalars) from the available data. We propose a novel functional parameterization approach for learned convex optimization solvers based on the classical mirror de…
In this paper some adaptive mirror descent algorithms for problems of minimization convex objective functional with several convex Lipschitz (generally, non-smooth) functional constraints are considered. It is shown that the methods are applicable to the objective functionals of various level of smoothness: the Lipschitz condition is valid either for the objective functional itself or for its gradient or Hessian (and the functional may not satisfy the Lipschitz condition). By using the restart technique methods for strongly convex minimization problems are proposed. Estimates of the rate of co…
As part of the effort to understand implicit bias of gradient descent in overparametrized models, several results have shown how the training trajectory on the overparametrized model can be understood as mirror descent on a different objective. The main result here is a characterization of this phenomenon under a notion termed commuting parametrization, which encompasses all the previous results in this setting. It is shown that gradient flow with any commuting parametrization is equivalent to continuous mirror descent with a related Legendre function. Conversely, continuous mirror descent wit…
We introduce and analyze a new family of first-order optimization algorithms which generalizes and unifies both mirror descent and dual averaging. Within the framework of this family, we define new algorithms for constrained optimization that combines the advantages of mirror descent and dual averaging. Our preliminary simulation study shows that these new algorithms significantly outperform available methods in some situations.
In this paper we consider the problem of finding the minimizations of the sum of two convex functions and the composition of another convex function with a continuous linear operator. With the idea of coordinate descent, we design a stochastic coordinate descent primal-dual splitting algorithm with dynamic stepsize. Based on randomized Modified Krasnosel'skii-Mann iterations and the firmly nonexpansive properties of the proximity operator, we achieve the convergence of the proposed algorithms. Moreover, we give two applications of our method.
Network utility maximization is the most important problem in network traffic management. Given the growth of modern communication networks, we consider the utility maximization problem in a network with a large number of connections (links) that are used by a huge number of users. To solve this problem an adaptive mirror descent algorithm for many constraints is proposed. The key feature of the algorithm is that it has a dimension-free convergence rate. The convergence of the proposed scheme is proved theoretically. The theoretical analysis is verified with numerical simulations. We compare t…
Columbia University（哥伦比亚大学）；University of Illinois（伊利诺伊大学）；INRIA（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
Mirror descent, introduced by Nemirovski and Yudin in the 1970s, is a primal-dual convex optimization method that can be tailored to the geometry of the optimization problem at hand through the choice of a strongly convex potential function. It arises as a basic primitive in a variety of applications, including large-scale optimization, machine learning, and control. This paper proposes a variational formulation of mirror descent and of its stochastic variant, mirror Langevin dynamics. The main idea, inspired by the classic work of Brezis and Ekeland on variational principles for gradient flow…
Minimax problems, such as generative adversarial network, adversarial training, and fair training, are widely solved by a multi-step gradient descent ascent (MGDA) method in practice. However, its convergence guarantee is limited. In this paper, inspired by the primal-dual hybrid gradient method, we propose a new semi-anchoring (SA) technique for the MGDA method. This makes the MGDA method find a stationary point of a structured nonconvex-nonconcave composite minimax problem; its saddle-subdifferential operator satisfies the weak Minty variational inequality condition. The resulting method, na…
We consider the problem of finding the minimizations of the sum of two convex functions and the composition of another convex function with a continuous linear operator from the view of fixed point algorithms based on proximity operators, which is is inspired by recent results of Chen, Huang and Zhang. With the idea of coordinate descent, we design a stochastic coordinate descent splitting primal- dual fixed point algorithm. Based on randomized krasnosel'skii mann iterations and the firmly nonexpansive properties of the proximity operator, we achieve the convergence of the proposed algorithms.
