In this paper, we consider non-convex stochastic bilevel optimization (SBO) problems that have many applications in machine learning. Although numerous studies have proposed stochastic algorithms for solving these problems, they are limited in two perspectives: (i) their sample complexities are high, which do not match the state-of-the-art result for non-convex stochastic optimization; (ii) their algorithms are tailored to problems with only one lower-level problem. When there are many lower-level problems, it could be prohibitive to process all these lower-level problems at each iteration. To address these limitations, this paper proposes fast randomized stochastic algorithms for non-convex SBO problems. First, we present a stochastic method for non-convex SBO with only one lower problem and establish its sample complexity of $O(1/ε^3)$ for finding an $ε$-stationary point under Lipschitz continuous conditions of stochastic oracles, matching the lower bound for stochastic smooth non-convex optimization. Second, we present a randomized stochastic method for non-convex SBO with $m>1$ lower level problems (multi-task SBO) by processing a constant number of lower problems at each iteration, and establish its sample complexity no worse than $O(m/ε^3)$, which could be a better complexity than that of simply processing all $m$ lower problems at each iteration. Lastly, we establish even faster convergence results for gradient-dominant functions. To the best of our knowledge, this is the first work considering multi-task SBO and developing state-of-the-art sample complexity results.
University of Michigan（密歇根大学）；University of California, Riverside（加州大学河滨分校）；University of British Columbia（不列颠哥伦比亚大学）
Standard federated optimization methods successfully apply to stochastic problems with single-level structure. However, many contemporary ML problems -- including adversarial robustness, hyperparameter tuning, and actor-critic -- fall under nested bilevel programming that subsumes minimax and compositional optimization. In this work, we propose \fedblo: A federated alternating stochastic gradient method to address general nested problems. We establish provable convergence rates for \fedblo in the presence of heterogeneous data and introduce variations for bilevel, minimax, and compositional op…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
Bilevel optimization is a popular hierarchical model in machine learning, and has been widely applied to many machine learning tasks such as meta learning, hyperparameter learning and policy optimization. Although many bilevel optimization algorithms recently have been developed, few adaptive algorithm focuses on the bilevel optimization under the distributed setting. It is well known that the adaptive gradient methods show superior performances on both distributed and non-distributed optimization. In the paper, thus, we propose a novel adaptive federated bilevel optimization algorithm (i.e.,A…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；University of Pittsburgh（匹兹堡大学）
Bilevel optimization recently has attracted increased interest in machine learning due to its many applications such as hyper-parameter optimization and meta learning. Although many bilevel methods recently have been proposed, these methods do not consider using adaptive learning rates. It is well known that adaptive learning rates can accelerate optimization algorithms. To fill this gap, in the paper, we propose a novel fast adaptive bilevel framework to solve stochastic bilevel optimization problems that the outer problem is possibly nonconvex and the inner problem is strongly convex. Our fr…
Texas A&M University（得克萨斯农工大学）；University of Iowa（爱荷华大学）
In this paper, we study multi-block min-max bilevel optimization problems, where the upper level is non-convex strongly-concave minimax objective and the lower level is a strongly convex objective, and there are multiple blocks of dual variables and lower level problems. Due to the intertwined multi-block min-max bilevel structure, the computational cost at each iteration could be prohibitively high, especially with a large number of blocks. To tackle this challenge, we present a single-loop randomized stochastic algorithm, which requires updates for only a constant number of blocks at each it…
The Ohio State University（俄亥俄州立大学）；University of Michigan, Ann Arbor（密歇根大学安娜堡分校）
Bilevel optimization has arisen as a powerful tool in modern machine learning. However, due to the nested structure of bilevel optimization, even gradient-based methods require second-order derivative approximations via Jacobian- or/and Hessian-vector computations, which can be costly and unscalable in practice. Recently, Hessian-free bilevel schemes have been proposed to resolve this issue, where the general idea is to use zeroth- or first-order methods to approximate the full hypergradient of the bilevel problem. However, we empirically observe that such approximation can lead to large varia…
The Ohio State University（俄亥俄州立大学）；Iowa State University（爱荷华州立大学）；University of Minnesota（明尼苏达大学）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
In recent years, decentralized bilevel optimization problems have received increasing attention in the networking and machine learning communities thanks to their versatility in modeling decentralized learning problems over peer-to-peer networks (e.g., multi-agent meta-learning, multi-agent reinforcement learning, personalized training, and Byzantine-resilient learning). However, for decentralized bilevel optimization over peer-to-peer networks with limited computation and communication capabilities, how to achieve low sample and communication complexities are two fundamental challenges that r…
Bilevel Optimization has witnessed notable progress recently with new emerging efficient algorithms and has been applied to many machine learning tasks such as data cleaning, few-shot learning, and neural architecture search. However, little attention has been paid to solve the bilevel problems under distributed setting. Federated learning (FL) is an emerging paradigm which solves machine learning tasks over distributed-located data. FL problems are challenging to solve due to the heterogeneity and communication bottleneck. However, it is unclear how these challenges will affect the convergenc…
Georgia Institute of Technology（佐治亚理工学院）；Tsinghua University（清华大学）；IBM Almaden Research Center（IBM阿尔马登研究中心）
We propose a novel randomized incremental gradient algorithm, namely, VAriance-Reduced Accelerated Gradient (Varag), for finite-sum optimization. Equipped with a unified step-size policy that adjusts itself to the value of the condition number, Varag exhibits the unified optimal rates of convergence for solving smooth convex finite-sum problems directly regardless of their strong convexity. Moreover, Varag is the first accelerated randomized incremental gradient method that benefits from the strong convexity of the data-fidelity term to achieve the optimal linear convergence. It also establish…
Randomized Smoothing (RS) is a promising method for obtaining robustness certificates by evaluating a base model under noise. In this work, we: (i) theoretically motivate why ensembles are a particularly suitable choice as base models for RS, and (ii) empirically confirm this choice, obtaining state-of-the-art results in multiple settings. The key insight of our work is that the reduced variance of ensembles over the perturbations introduced in RS leads to significantly more consistent classifications for a given input. This, in turn, leads to substantially increased certifiable radii for samp…
Motivated by high-dimensional nonlinear optimization problems as well as ill-posed optimization problems arising in image processing, we consider a bilevel optimization model where we seek among the optimal solutions of the inner level problem, a solution that minimizes a secondary metric. Our goal is to address the high-dimensionality of the bilevel problem, and the nondifferentiability of the objective function. Minimal norm gradient, sequential averaging, and iterative regularization are some of the recent schemes developed for addressing the bilevel problem. But none of them address the hi…
Two types of low cost-per-iteration gradient descent methods have been extensively studied in parallel. One is online or stochastic gradient descent (OGD/SGD), and the other is randomzied coordinate descent (RBCD). In this paper, we combine the two types of methods together and propose online randomized block coordinate descent (ORBCD). At each iteration, ORBCD only computes the partial gradient of one block coordinate of one mini-batch samples. ORBCD is well suited for the composite minimization problem where one function is the average of the losses of a large number of samples and the other…
University of Pittsburgh（匹兹堡大学）；Nanjing University of Aeronautics & Astronautics（南京航空航天大学）
Bilevel optimization has been recently used in many machine learning problems such as hyperparameter optimization, policy optimization, and meta learning. Although many bilevel optimization methods have been proposed, they still suffer from the high computational complexities and do not consider the more general bilevel problems with nonsmooth regularization. In the paper, thus, we propose a class of enhanced bilevel optimization methods with using Bregman distance to solve bilevel optimization problems, where the outer subproblem is nonconvex and possibly nonsmooth, and the inner subproblem i…
The Ohio State University（俄亥俄州立大学）；University of Michigan（密歇根大学）
Bilevel optimization has been widely applied in many important machine learning applications such as hyperparameter optimization and meta-learning. Recently, several momentum-based algorithms have been proposed to solve bilevel optimization problems faster. However, those momentum-based algorithms do not achieve provably better computational complexity than $\mathcal{\widetilde O}(ε^{-2})$ of the SGD-based algorithm. In this paper, we propose two new algorithms for bilevel optimization, where the first algorithm adopts momentum-based recursive iterations, and the second algorithm adopts recurs…
Dynamic optimization of mean and variance in Markov decision processes (MDPs) is a long-standing challenge caused by the failure of dynamic programming. In this paper, we propose a new approach to find the globally optimal policy for combined metrics of steady-state mean and variance in an infinite-horizon undiscounted MDP. By introducing the concepts of pseudo mean and pseudo variance, we convert the original problem to a bilevel MDP problem, where the inner one is a standard MDP optimizing pseudo mean-variance and the outer one is a single parameter selection problem optimizing pseudo mean. …
Bilevel Learning with Inexact Stochastic Gradients
Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）；Department of Electronics & Electrical Communication Engineering, Indian Institute of Technology (IIT) Kharagpur（印度理工学院Kharagpur电子与电气通信工程系）；School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学学院）
Randomized Max-Vertex-Coverage Interdiction under Matroid Constraints
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Beijing Normal University-Zhuhai（北京师范大学珠海校区）；Beijing Normal-Hong Kong Baptist University（北京浸会大学）
We analyze convergence rates of stochastic optimization procedures for non-smooth convex optimization problems. By combining randomized smoothing techniques with accelerated gradient methods, we obtain convergence rates of stochastic optimization procedures, both in expectation and with high probability, that have optimal dependence on the variance of the gradient estimates. To the best of our knowledge, these are the first variance-based rates for non-smooth optimization. We give several applications of our results to statistical estimation problems, and provide experimental results that demo…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
We propose a generic variance-reduced algorithm, which we call MUltiple RANdomized Algorithm (MURANA), for minimizing a sum of several smooth functions plus a regularizer, in a sequential or distributed manner. Our method is formulated with general stochastic operators, which allow us to model various strategies for reducing the computational complexity. For example, MURANA supports sparse activation of the gradients, and also reduction of the communication load via compression of the update vectors. This versatility allows MURANA to cover many existing randomization mechanisms within a unifie…
Classical extragradient schemes and their stochastic counterpart represent a cornerstone for resolving monotone variational inequality problems. Yet, such schemes have a per-iteration complexity of two projections onto a convex set and require two evaluations of the map, the former of which could be relatively expensive if $X$ is a complicated set. We consider two related avenues where the per-iteration complexity is significantly reduced: (i) A stochastic projected reflected gradient method requiring a single evaluation of the map and a single projection; and (ii) A stochastic subgradient ext…
Sun Yat-sen University（中山大学）；Tsinghua University（清华大学）
This paper studies the risk-averse mean-variance optimization in infinite-horizon discounted Markov decision processes (MDPs). The involved variance metric concerns reward variability during the whole process, and future deviations are discounted to their present values. This discounted mean-variance optimization yields a reward function dependent on a discounted mean, and this dependency renders traditional dynamic programming methods inapplicable since it suppresses a crucial property -- time consistency. To deal with this unorthodox problem, we introduce a pseudo mean to transform the untre…
In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense the most robust surely optimal martingale under random perturbations. This new insight leads to a novel randomized dual martingale minimization algorithm that doesn't require …
