An open problem in optimization with noisy information is the computation of an exact minimizer that is independent of the amount of noise. A standard practice in stochastic approximation algorithms is to use a decreasing step-size. This however leads to a slower convergence. A second alternative is to use a fixed step-size and run independent replicas of the algorithm and average these. A third option is to run replicas of the algorithm and allow them to interact. It is unclear which of these options works best. To address this question, we reduce the problem of the computation of an exact minimizer with noisy gradient information to the study of stochastic mirror descent with interacting particles. We study the convergence of stochastic mirror descent and make explicit the tradeoffs between communication and variance reduction. We provide theoretical and numerical evidence to suggest that interaction helps to improve convergence and reduce the variance of the estimate.
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."
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Information Transmission Problems（信息传输问题研究所）
We consider the problem of multi-class classification and a stochastic opti- mization approach to it. We derive risk bounds for stochastic mirror descent algorithm and provide examples of set geometries that make the use of the algorithm efficient in terms of error in k.
We generalize stochastic subgradient descent methods to situations in which we do not receive independent samples from the distribution over which we optimize, but instead receive samples that are coupled over time. We show that as long as the source of randomness is suitably ergodic---it converges quickly enough to a stationary distribution---the method enjoys strong convergence guarantees, both in expectation and with high probability. This result has implications for stochastic optimization in high-dimensional spaces, peer-to-peer distributed optimization schemes, decision problems with dep…
We establish a connection between the stability of mirror descent and the information ratio by Russo and Van Roy [2014]. Our analysis shows that mirror descent with suitable loss estimators and exploratory distributions enjoys the same bound on the adversarial regret as the bounds on the Bayesian regret for information-directed sampling. Along the way, we develop the theory for information-directed sampling and provide an efficient algorithm for adversarial bandits for which the regret upper bound matches exactly the best known information-theoretic upper bound.
We consider the problem of learning the optimal policy for infinite-horizon Markov decision processes (MDPs). For this purpose, some variant of Stochastic Mirror Descent is proposed for convex programming problems with Lipschitz-continuous functionals. An important detail is the ability to use inexact values of functional constraints and compute the value of dual variables. We analyze this algorithm in a general case and obtain an estimate of the convergence rate that does not accumulate errors during the operation of the method. Using this algorithm, we get the first parallel algorithm for mi…
We formulate and study a general family of (continuous-time) stochastic dynamics for accelerated first-order minimization of smooth convex functions. Building on an averaging formulation of accelerated mirror descent, we propose a stochastic variant in which the gradient is contaminated by noise, and study the resulting stochastic differential equation. We prove a bound on the rate of change of an energy function associated with the problem, then use it to derive estimates of convergence rates of the function values, (a.s. and in expectation) both for persistent and asymptotically vanishing no…
This work addresses distributed optimization, where a network of agents wants to minimize a global strongly convex objective function. The global function can be written as a sum of local convex functions, each of which is associated with an agent. We propose a continuous-time distributed mirror descent algorithm that uses purely local information to converge to the global optimum. Unlike previous work on distributed mirror descent, we incorporate an integral feedback in the update, allowing the algorithm to converge with a constant step-size when discretized. We establish the asymptotic conve…
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…
We consider the problem of minimizing the sum of an average function of a large number of smooth convex components and a general, possibly non-differentiable, convex function. Although many methods have been proposed to solve this problem with the assumption that the sum is strongly convex, few methods support the non-strongly convex case. Adding a small quadratic regularization is a common devise used to tackle non-strongly convex problems; however, it may cause loss of sparsity of solutions or weaken the performance of the algorithms. Avoiding this devise, we propose an accelerated randomize…
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.
In this work, we investigate the idea of variance reduction by studying its properties with general adaptive mirror descent algorithms in nonsmooth nonconvex finite-sum optimization problems. We propose a simple yet generalized framework for variance reduced adaptive mirror descent algorithms named SVRAMD and provide its convergence analysis in both the nonsmooth nonconvex problem and the P-L conditioned problem. We prove that variance reduction reduces the SFO complexity of adaptive mirror descent algorithms and thus accelerates their convergence. In particular, our general theory implies tha…
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…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
We present a new perspective on the celebrated Sinkhorn algorithm by showing that is a special case of incremental/stochastic mirror descent. In order to see this, one should simply plug Kullback-Leibler divergence in both mirror map and the objective function. Since the problem has unbounded domain, the objective function is neither smooth nor it has bounded gradients. However, one can still approach the problem using the notion of relative smoothness, obtaining that the stochastic objective is 1-relative smooth. The discovered equivalence allows us to propose 1) new methods for optimal trans…
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.
Many problems in machine learning can be formulated as optimizing a convex functional over a vector space of measures. This paper studies the convergence of the mirror descent algorithm in this infinite-dimensional setting. Defining Bregman divergences through directional derivatives, we derive the convergence of the scheme for relatively smooth and convex pairs of functionals. Such assumptions allow to handle non-smooth functionals such as the Kullback--Leibler (KL) divergence. Applying our result to joint distributions and KL, we show that Sinkhorn's primal iterations for entropic optimal tr…
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…
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…
The logarithmic divergence is an extension of the Bregman divergence motivated by optimal transport and a generalized convex duality, and satisfies many remarkable properties. Using the geometry induced by the logarithmic divergence, we introduce a generalization of continuous time mirror descent that we term the conformal mirror descent. We derive its dynamics under a generalized mirror map, and show that it is a time change of a corresponding Hessian gradient flow. We also prove convergence results in continuous time. We apply the conformal mirror descent to online estimation of a generalize…
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…
This note considers the problem of minimizing interacting free energy. Motivated by the mirror descent algorithm, for a given interacting free energy, we propose a descent dynamics with a novel metric that takes into consideration the reference measure and the interacting term. This metric naturally suggests a monotone reparameterization of the probability measure. By discretizing the reparameterized descent dynamics with the explicit Euler method, we arrive at a new mirror-descent-type algorithm for minimizing interacting free energy. Numerical results are included to demonstrate the efficien…
Inspired by the recent paper (L. Ying, Mirror descent algorithms for minimizing interacting free energy, Journal of Scientific Computing, 84 (2020), pp. 1-14),we explore the relationship between the mirror descent and the variable metric method. When the metric in the mirror decent is induced by a convex function, whose Hessian is close to the Hessian of the objective function, this method enjoys both robustness from the mirror descent and superlinear convergence for Newton type methods. When applied to a linearly constrained minimization problem, we prove the global and local convergence, bot…
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…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；University of California, Davis（加州大学戴维斯分校）
We study stochastic convex optimization under infinite noise variance. Specifically, when the stochastic gradient is unbiased and has uniformly bounded $(1+κ)$-th moment, for some $κ\in (0,1]$, we quantify the convergence rate of the Stochastic Mirror Descent algorithm with a particular class of uniformly convex mirror maps, in terms of the number of iterations, dimensionality and related geometric parameters of the optimization problem. Interestingly this algorithm does not require any explicit gradient clipping or normalization, which have been extensively used in several recent empirical an…
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…
California Institute of Technology（加州理工学院）；Google（谷歌）
The stochastic mirror descent (SMD) algorithm is a general class of training algorithms, which includes the celebrated stochastic gradient descent (SGD), as a special case. It utilizes a mirror potential to influence the implicit bias of the training algorithm. In this paper we explore the performance of the SMD iterates on mean-field ensemble models. Our results generalize earlier ones obtained for SGD on such models. The evolution of the distribution of parameters is mapped to a continuous time process in the space of probability distributions. Our main result gives a nonlinear partial diffe…
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.
