In the paper, we propose a class of efficient adaptive bilevel methods based on mirror descent for nonconvex bilevel optimization, where its upper-level problem is nonconvex possibly with nonsmooth regularization, and its lower-level problem is also nonconvex while satisfies Polyak-Łojasiewicz (PL) condition. To solve these deterministic bilevel problems, we present an efficient adaptive projection-aid gradient (i.e., AdaPAG) method based on mirror descent, and prove that it obtains the best known gradient complexity of $O(ε^{-1})$ for finding an $ε$-stationary solution of nonconvex bilevel problems. To solve these stochastic bilevel problems, we propose an efficient adaptive stochastic projection-aid gradient (i.e., AdaVSPAG) methods based on mirror descent and variance-reduced techniques, and prove that it obtains the best known gradient complexity of $O(ε^{-3/2})$ for finding an $ε$-stationary solution. Since the PL condition relaxes the strongly convex, our algorithms can be used to nonconvex strongly-convex bilevel optimization. Theoretically, we provide a useful convergence analysis framework for our methods under some mild conditions, and prove that our methods have a fast convergence rate of $O(\frac{1}{T})$, where $T$ denotes the number of iterations.
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.
Bilevel Learning via Inexact Stochastic Gradient Descent
EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath（巴斯大学统计应用数学博士培训中心）；Australian Research Council（澳大利亚研究理事会）；Indian Institute of Technology (IIT), Kharagpur（印度理工学院（Kharagpur分校））；University of Melbourne（墨尔本大学）；Department of Electronics & Electrical Communication Engineering（电子与电气通信工程系）；School of Mathematics and Statistics（数学与统计学学院）
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…
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."
MirrorCBO: A consensus-based optimization method in the spirit of mirror descent
Department of Computing and Mathematical Sciences, Caltech（计算与数学科学系，加州理工学院）；Helmholtz Imaging, Deutsches Elektronen-Synchrotron DESY（海德堡成像，德意志电子同步辐射研究中心（DESY））
University of Edinburgh（爱丁堡大学）；KAIST（韩国科学技术院）；Samsung AI Research, Cambridge（三星剑桥人工智能研究院）
Optimisers are an essential component for training machine learning models, and their design influences learning speed and generalisation. Several studies have attempted to learn more effective gradient-descent optimisers via solving a bi-level optimisation problem where generalisation error is minimised with respect to optimiser parameters. However, most existing optimiser learning methods are intuitively motivated, without clear theoretical support. We take a different perspective starting from mirror descent rather than gradient descent, and meta-learning the corresponding Bregman divergenc…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Stochastic bilevel optimization generalizes the classic stochastic optimization from the minimization of a single objective to the minimization of an objective function that depends the solution of another optimization problem. Recently, stochastic bilevel optimization is regaining popularity in emerging machine learning applications such as hyper-parameter optimization and model-agnostic meta learning. To solve this class of stochastic optimization problems, existing methods require either double-loop or two-timescale updates, which are sometimes less efficient. This paper develops a new opti…
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…
Fully First-Order Methods for Decentralized Bilevel Optimization
The Hong Kong University of Science and Technology（香港科技大学）；University of California Davis（加州大学戴维斯分校）；Rice University（莱斯大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We present a novel regularization approach to train neural networks that enjoys better generalization and test error than standard stochastic gradient descent. Our approach is based on the principles of cross-validation, where a validation set is used to limit the model overfitting. We formulate such principles as a bilevel optimization problem. This formulation allows us to define the optimization of a cost on the validation set subject to another optimization on the training set. The overfitting is controlled by introducing weights on each mini-batch in the training set and by choosing their…
Most of the recent successful applications of neural networks have been based on training with gradient descent updates. However, for some small networks, other mirror descent updates learn provably more efficiently when the target is sparse. We present a general framework for casting a mirror descent update as a gradient descent update on a different set of parameters. In some cases, the mirror descent reparameterization can be described as training a modified network with standard backpropagation. The reparameterization framework is versatile and covers a wide range of mirror descent updates…
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…
Massachusetts Institute of Technology（麻省理工学院）；University of California, San Diego（加利福尼亚大学圣地亚哥分校）
The Polyak-Lojasiewicz (PL) inequality is a sufficient condition for establishing linear convergence of gradient descent, even in non-convex settings. While several recent works use a PL-based analysis to establish linear convergence of stochastic gradient descent methods, the question remains as to whether a similar analysis can be conducted for more general optimization methods. In this work, we present a PL-based analysis for linear convergence of generalized mirror descent (GMD), a generalization of mirror descent with a possibly time-dependent mirror. GMD subsumes popular first order opti…
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…
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…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；KAUST（阿卜杜拉国王科技大学）
In this note, we establish a descent lemma for the population limit Mirrored Stein Variational Gradient Method~(MSVGD). This descent lemma does not rely on the path information of MSVGD but rather on a simple assumption for the mirrored distribution $\nablaΨ_{\#}π\propto\exp(-V)$. Our analysis demonstrates that MSVGD can be applied to a broader class of constrained sampling problems with non-smooth $V$. We also investigate the complexity of the population limit MSVGD in terms of dimension $d$.
BiLO: Bilevel Local Operator Learning for PDE Inverse Problems
Department of Mathematics, University of California, Irvine（加州大学伊文斯分校数学系）；Department of Computer Science, University of California, Irvine（加州大学伊文斯分校计算机科学系）；Department of Biomedical Engineering, University of California, Irvine（加州大学伊文斯分校生物医学工程系）
Boston University（波士顿大学）；Northeastern University（东北大学）
In this work, we describe a generic approach to show convergence with high probability for stochastic convex optimization. In previous works, either the convergence is only in expectation or the bound depends on the diameter of the domain. Instead, we show high probability convergence with bounds depending on the initial distance to the optimal solution as opposed to the domain diameter. The algorithms use step sizes analogous to the standard settings and are universal to Lipschitz functions, smooth functions, and their linear combinations.
We study discrete-time mirror descent applied to the unregularized empirical risk in matrix sensing. In both the general case of rectangular matrices and the particular case of positive semidefinite matrices, a simple potential-based analysis in terms of the Bregman divergence allows us to establish convergence of mirror descent -- with different choices of the mirror maps -- to a matrix that, among all global minimizers of the empirical risk, minimizes a quantity explicitly related to the nuclear norm, the Frobenius norm, and the von Neumann entropy. In both cases, this characterization impli…
We analyze continuous-time mirror descent applied to sparse phase retrieval, which is the problem of recovering sparse signals from a set of magnitude-only measurements. We apply mirror descent to the unconstrained empirical risk minimization problem (batch setting), using the square loss and square measurements. We provide a convergence analysis of the algorithm in this non-convex setting and prove that, with the hypentropy mirror map, mirror descent recovers any $k$-sparse vector $\mathbf{x}^\star\in\mathbb{R}^n$ with minimum (in modulus) non-zero entry on the order of $\| \mathbf{x}^\star \…
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…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Duke University（杜克大学）
Information geometry applies concepts in differential geometry to probability and statistics and is especially useful for parameter estimation in exponential families where parameters are known to lie on a Riemannian manifold. Connections between the geometric properties of the induced manifold and statistical properties of the estimation problem are well-established. However developing first-order methods that scale to larger problems has been less of a focus in the information geometry community. The best known algorithm that incorporates manifold structure is the second-order natural gradie…
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…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；The Technion（以色列理工学院）；Università degli Studi di Milano（米兰大学）
Online learning algorithms are fast, memory-efficient, easy to implement, and applicable to many prediction problems, including classification, regression, and ranking. Several online algorithms were proposed in the past few decades, some based on additive updates, like the Perceptron, and some on multiplicative updates, like Winnow. A unifying perspective on the design and the analysis of online algorithms is provided by online mirror descent, a general prediction strategy from which most first-order algorithms can be obtained as special cases. We generalize online mirror descent to time-vary…
Sequential QCQP for Bilevel Optimization with Line Search
Department of Electrical and Computer Engineering at Johns Hopkins University（约翰霍普金斯大学电气与计算机工程系）；Department of Systems and Industrial Engineering, The University of Arizona（亚利桑那大学系统与工业工程系）
Recently there has been a surge of interest in understanding implicit regularization properties of iterative gradient-based optimization algorithms. In this paper, we study the statistical guarantees on the excess risk achieved by early-stopped unconstrained mirror descent algorithms applied to the unregularized empirical risk with the squared loss for linear models and kernel methods. By completing an inequality that characterizes convexity for the squared loss, we identify an intrinsic link between offset Rademacher complexities and potential-based convergence analysis of mirror descent meth…
University of Milan（米兰大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；University College London（伦敦大学学院）
We introduce and analyze MT-OMD, a multitask generalization of Online Mirror Descent (OMD) which operates by sharing updates between tasks. We prove that the regret of MT-OMD is of order $\sqrt{1 + σ^2(N-1)}\sqrt{T}$, where $σ^2$ is the task variance according to the geometry induced by the regularizer, $N$ is the number of tasks, and $T$ is the time horizon. Whenever tasks are similar, that is $σ^2 \le 1$, our method improves upon the $\sqrt{NT}$ bound obtained by running independent OMDs on each task. We further provide a matching lower bound, and show that our multitask extensions of Online…
This paper studies early-stopped mirror descent applied to noisy sparse phase retrieval, which is the problem of recovering a $k$-sparse signal $\mathbf{x}^\star\in\mathbb{R}^n$ from a set of quadratic Gaussian measurements corrupted by sub-exponential noise. We consider the (non-convex) unregularized empirical risk minimization problem and show that early-stopped mirror descent, when equipped with the hyperbolic entropy mirror map and proper initialization, achieves a nearly minimax-optimal rate of convergence, provided the sample size is at least of order $k^2$ (modulo logarithmic term) and …
Yale University（耶鲁大学）；Georgia Institute of Technology（佐治亚理工学院）；Singapore University of Technology and Design（新加坡科技设计大学）
In this paper we study two-player bilinear zero-sum games with constrained strategy spaces. An instance of natural occurrences of such constraints is when mixed strategies are used, which correspond to a probability simplex constraint. We propose and analyze the alternating mirror descent algorithm, in which each player takes turns to take action following the mirror descent algorithm for constrained optimization. We interpret alternating mirror descent as an alternating discretization of a skew-gradient flow in the dual space, and use tools from convex optimization and modified energy functio…
Michigan State University（密歇根州立大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
We present a method for supervised learning of sparsity-promoting regularizers, a key ingredient in many modern signal reconstruction problems. The parameters of the regularizer are learned to minimize the mean squared error of reconstruction on a training set of ground truth signal and measurement pairs. Training involves solving a challenging bilevel optimization problem with a nonsmooth lower-level objective. We derive an expression for the gradient of the training loss using the implicit closed-form solution of the lower-level variational problem given by its dual problem, and provide an a…
In this paper, we focus on the question of the extent to which online learning can benefit from distributed computing. We focus on the setting in which $N$ agents online-learn cooperatively, where each agent only has access to its own data. We propose a generic data-distributed online learning meta-algorithm. We then introduce the Distributed Weighted Majority and Distributed Online Mirror Descent algorithms, as special cases. We show, using both theoretical analysis and experiments, that compared to a single agent: given the same computation time, these distributed algorithms achieve smaller …
In machine learning and statistical data analysis, we often run into objective function that is a summation: the number of terms in the summation possibly is equal to the sample size, which can be enormous. In such a setting, the stochastic mirror descent (SMD) algorithm is a numerically efficient method -- each iteration involving a very small subset of the data. The variance reduction version of SMD (VRSMD) can further improve SMD by inducing faster convergence. On the other hand, algorithms such as gradient descent and stochastic gradient descent have the implicit regularization property th…
