Online optimisation facilitates the solution of dynamic inverse problems, such as image stabilisation, fluid flow monitoring, and dynamic medical imaging. In this paper, we improve upon previous work on predictive online primal-dual methods on two fronts. Firstly, we provide a more concise analysis that symmetrises previously unsymmetric regret bounds, and relaxes previous restrictive conditions on the dual predictor. Secondly, based on the latter, we develop several improved dual predictors. We numerically demonstrate their efficacy in image stabilisation and dynamic positron emission tomography.
Otto von Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）；Human Technopole（人类技术园）；German Centre for Neurodegenerative Disease（德国神经退行性疾病中心）；Centre for Behavioural Brain Sciences（行为脑科学中心）
Computed tomography and magnetic resonance imaging are two widely used clinical imaging modalities for non-invasive diagnosis. However, both of these modalities come with certain problems. CT uses harmful ionising radiation, and MRI suffers from slow acquisition speed. Both problems can be tackled by undersampling, such as sparse sampling. However, such undersampled data leads to lower resolution and introduces artefacts. Several techniques, including deep learning based methods, have been proposed to reconstruct such data. However, the undersampled reconstruction problem for these two modalit…
Stochastic approximation techniques have been used in various contexts in data science. We propose a stochastic version of the forward-backward algorithm for minimizing the sum of two convex functions, one of which is not necessarily smooth. Our framework can handle stochastic approximations of the gradient of the smooth function and allows for stochastic errors in the evaluation of the proximity operator of the nonsmooth function. The almost sure convergence of the iterates generated by the algorithm to a minimizer is established under relatively mild assumptions. We also propose a stochastic…
This paper studies the online stochastic resource allocation problem (RAP) with chance constraints and conditional expectation constraints. The online RAP is an integer linear programming problem where resource consumption coefficients are revealed column by column along with the corresponding revenue coefficients. When a column is revealed, the corresponding decision variables are determined instantaneously without future information. In online applications, the resource consumption coefficients are often obtained by prediction. An application for such scenario rises from the online order ful…
We consider the problem of finding the minimization of the sum of a convex function and the composition of another convex function with a continuous linear operator from the view of fixed point algorithms based on proximity operators. We design a primal-dual fixed point algorithm with dynamic stepsize based on the proximity operator and obtain a scheme with a closed form solution for each iteration. Based on Modified Mann iteration and the firmly nonexpansive properties of the proximity operator, we achieve the convergence of the proposed algorithm.
This paper addresses the design and analysis of feedback-based online algorithms to control systems or networked systems based on performance objectives and engineering constraints that may evolve over time. The emerging time-varying convex optimization formalism is leveraged to model optimal operational trajectories of the systems, as well as explicit local and network-level operational constraints. Departing from existing batch and feed-forward optimization approaches, the design of the algorithms capitalizes on an online implementation of primal-dual projected-gradient methods; the gradient…
CMAP, Ecole Polytechnique, CNRS（应用数学中心，巴黎综合理工学院，法国国家科学研究中心）；Department for Applied Mathematics and Theoretical Physics, University of Cambridge（应用数学与理论物理系，剑桥大学）；Visual Computing Center & Extreme Computing Research Center, KAUST（视觉计算中心与极端计算研究中心，阿卜杜拉国王科技大学）；School of Mathematics, University of Edinburgh（数学学院，爱丁堡大学）；The Alan Turing Institute（阿兰·图灵研究所）
We propose a stochastic extension of the primal-dual hybrid gradient algorithm studied by Chambolle and Pock in 2011 to solve saddle point problems that are separable in the dual variable. The analysis is carried out for general convex-concave saddle point problems and problems that are either partially smooth / strongly convex or fully smooth / strongly convex. We perform the analysis for arbitrary samplings of dual variables, and obtain known deterministic results as a special case. Several variants of our stochastic method significantly outperform the deterministic variant on a variety of i…
Modern Primal-Dual Frameworks for Prior-Free Online Resource Allocation
University of Chicago Booth School of Business（芝加哥大学布斯商学院）；Department of Industrial Engineering & Operations Research, University of California, Berkeley（加州大学伯克利分校工业工程与运筹学系）
Finite-Time Complexity of Online Primal-Dual Natural Actor-Critic Algorithm for Constrained Markov Decision Processes
School of Electrical and Computer Engineering, Georgia Institute of Technology（佐治亚理工学院电气与计算机工程学院）；Bradley Department of Electrical and Computer Engineering, Virginia Tech（弗吉尼亚理工大学布拉德利电气与计算机工程学院）
We consider online convex optimization with stochastic constraints where the objective functions are arbitrarily time-varying and the constraint functions are independent and identically distributed (i.i.d.) over time. Both the objective and constraint functions are revealed after the decision is made at each time slot. The best known expected regret for solving such a problem is $\mathcal{O}(\sqrt{T})$, with a coefficient that is polynomial in the dimension of the decision variable and relies on the Slater condition (i.e. the existence of interior point assumption), which is restrictive and i…
Primal-dual methods in online optimization give several of the state-of-the art results in both of the most common models: adversarial and stochastic/random order. Here we try to provide a more unified analysis of primal-dual algorithms to better understand the mechanisms behind this important method. With this we are able of recover and extend in one goal several results of the literature. In particular, we obtain robust online algorithm for fairly general online convex problems: we consider the MIXED model where in some of the time steps the data is stochastic and in the others the data is a…
This paper introduces consensus-based primal-dual methods for distributed online optimization where the time-varying system objective function $f_t(\mathbf{x})$ is given as the sum of local agents' objective functions, i.e., $f_t(\mathbf{x}) = \sum_i f_{i,t}(\mathbf{x}_i)$, and the system constraint function $\mathbf{g}(\mathbf{x})$ is given as the sum of local agents' constraint functions, i.e., $\mathbf{g}(\mathbf{x}) = \sum_i \mathbf{g}_i (\mathbf{x}_i) \preceq \mathbf{0}$. At each stage, each agent commits to an adaptive decision pertaining only to the past and locally available informatio…
We propose a new type of efficient deep-unrolling networks for solving imaging inverse problems. Conventional deep-unrolling methods require full forward operator and its adjoint across each layer, and hence can be significantly more expensive computationally as compared with other end-to-end methods that are based on post-processing of model-based reconstructions, especially for 3D image reconstruction tasks. We develop a stochastic (ordered-subsets) variant of the classical learned primal-dual (LPD), which is a state-of-the-art unrolling network for tomographic image reconstruction. The prop…
We consider the problem of distributed online optimization, with a group of learners connected via a dynamic communication graph. The goal of the learners is to track the global minimizer of a sum of time-varying loss functions in a distributed manner. We propose a novel algorithm, termed Distributed Online Mirror Descent with Multiple Averaging Decision and Gradient Consensus (DOMD-MADGC), which is based on mirror descent but incorporates multiple consensus averaging iterations over local gradients as well as local decisions. The key idea is to allow the local learners to collect a sufficient…
University of Pennsylvania（宾夕法尼亚大学）；University of California San Diego（加利福尼亚大学圣迭戈分校）
This work presents an explicit-implicit procedure to compute a model predictive control (MPC) law with guarantees on recursive feasibility and asymptotic stability. The approach combines an offline-trained fully-connected neural network with an online primal active set solver. The neural network provides a control input initialization while the primal active set method ensures recursive feasibility and asymptotic stability. The neural network is trained with a primal-dual loss function, aiming to generate control sequences that are primal feasible and meet a desired level of suboptimality. Sin…
We study the online constrained ranking problem motivated by an application to web-traffic shaping: an online stream of sessions arrive in which, within each session, we are asked to rank items. The challenge involves optimizing the ranking in each session so that local vs. global objectives are controlled: within each session one wishes to maximize a reward (local) while satisfying certain constraints over the entire set of sessions (global). A typical application of this setup is that of page optimization in a web portal. We wish to rank items so that not only is user engagement maximized in…
In this paper we provide an algorithm for solving constrained composite primal-dual monotone inclusions, i.e., monotone inclusions in which a priori information on primal-dual solutions is represented via closed convex sets. The proposed algorithm incorporates a projection step onto the a priori information sets and generalizes the method proposed in [Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38, 667-681 (2013)]. Moreover, under the presence of strong monotonicity, we derive an accelerated scheme inspired on [Chambolle, A.; …
We develop an online data-enabled predictive (ODeePC) control method for optimal control of unknown systems, building on the recently proposed DeePC [1]. Our proposed ODeePC method leverages a primal-dual algorithm with real-time measurement feedback to iteratively compute the corresponding real-time optimal control policy as system conditions change. The proposed ODeePC conceptual-wise resembles standard adaptive system identification and model predictive control (MPC), but it provides a new alternative for the standard methods. ODeePC is enabled by computationally efficient methods that expl…
University of Helsinki（赫尔辛基大学）；Escuela Politécnica Nacional（国立理工学院）
Online optimisation revolves around new data being introduced into a problem while it is still being solved; think of deep learning as more training samples become available. We adapt the idea to dynamic inverse problems such as video processing with optical flow. We introduce a corresponding predictive online primal-dual proximal splitting method. The video frames now exactly correspond to the algorithm iterations. A user-prescribed predictor describes the evolution of the primal variable. To prove convergence we need a predictor for the dual variable based on (proximal) gradient flow. This a…
We describe convergence acceleration schemes for multistep optimization algorithms. The extrapolated solution is written as a nonlinear average of the iterates produced by the original optimization method. Our analysis does not need the underlying fixed-point operator to be symmetric, hence handles e.g. algorithms with momentum terms such as Nesterov's accelerated method, or primal-dual methods. The weights are computed via a simple linear system and we analyze performance in both online and offline modes. We use Crouzeix's conjecture to show that acceleration performance is controlled by the …
This paper examines the problem of real-time optimization of networked systems and develops online algorithms that steer the system towards the optimal trajectory without explicit knowledge of the system model. The problem is modeled as a dynamic optimization problem with time-varying performance objectives and engineering constraints. The design of the algorithms leverages the online zero-order primal-dual projected-gradient method. In particular, the primal step that involves the gradient of the objective function (and hence requires networked systems model) is replaced by its zero-order app…
Online optimisation for dynamic electrical impedance tomography
School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学学院）；Department of Technical Physics, University of Eastern Finland（东芬兰大学技术物理系）
The Stochastic Primal-Dual Hybrid Gradient (SPDHG) was proposed by Chambolle et al. (2018) and is an efficient algorithm to solve some nonsmooth large-scale optimization problems. In this paper we prove its almost sure convergence for convex but not necessarily strongly convex functionals. We also look into its application to parallel Magnetic Resonance Imaging reconstruction in order to test performance of SPDHG. Our numerical results show that for a range of settings SPDHG converges significantly faster than its deterministic counterpart.
Block coordinate descent (BCD) methods and their variants have been widely used in coping with large-scale nonconstrained optimization problems in many fields such as imaging processing, machine learning, compress sensing and so on. For problem with coupling constraints, Nonlinear convex cone programs (NCCP) are important problems with many practical applications, but these problems are hard to solve by using existing block coordinate type methods. This paper introduces a stochastic primal-dual coordinate (SPDC) method for solving large-scale NCCP. In this method, we randomly choose a block of…
In this work, we consider a distributed online convex optimization problem, with time-varying (potentially adversarial) constraints. A set of nodes, jointly aim to minimize a global objective function, which is the sum of local convex functions. The objective and constraint functions are revealed locally to the nodes, at each time, after taking an action. Naturally, the constraints cannot be instantaneously satisfied. Therefore, we reformulate the problem to satisfy these constraints in the long term. To this end, we propose a distributed primal-dual mirror descent based approach, in which the…
Isotonic regression (IR) is a non-parametric calibration method used in supervised learning. For performing large-scale IR, we propose a primal-dual active-set (PDAS) algorithm which, in contrast to the state-of-the-art Pool Adjacent Violators (PAV) algorithm, can be parallized and is easily warm-started thus well-suited in the online settings. We prove that, like the PAV algorithm, our PDAS algorithm for IR is convergent and has a work complexity of O(n), though our numerical experiments suggest that our PDAS algorithm is often faster than PAV. In addition, we propose PDAS variants (with safe…
University of Michigan（密歇根大学）；University of Southern California（南加州大学）；Princeton University（普林斯顿大学）；Northwestern University（西北大学）
We consider online learning for episodic stochastically constrained Markov decision processes (CMDPs), which plays a central role in ensuring the safety of reinforcement learning. Here the loss function can vary arbitrarily across the episodes, and both the loss received and the budget consumption are revealed at the end of each episode. Previous works solve this problem under the restrictive assumption that the transition model of the Markov decision processes (MDPs) is known a priori and establish regret bounds that depend polynomially on the cardinalities of the state space $\mathcal{S}$ an…
We study the online estimation of the optimal policy of a Markov decision process (MDP). We propose a class of Stochastic Primal-Dual (SPD) methods which exploit the inherent minimax duality of Bellman equations. The SPD methods update a few coordinates of the value and policy estimates as a new state transition is observed. These methods use small storage and has low computational complexity per iteration. The SPD methods find an absolute-$ε$-optimal policy, with high probability, using $\mathcal{O}\left(\frac{|\mathcal{S}|^4 |\mathcal{A}|^2σ^2 }{(1-γ)^6ε^2} \right)$ iterations/samples for th…
We consider the problem of solving packing/covering LPs online, when the columns of the constraint matrix are presented in random order. This problem has received much attention and the main focus is to figure out how large the right-hand sides of the LPs have to be (compared to the entries on the left-hand side of the constraints) to allow $(1+ε)$-approximations online. It is known that the right-hand sides have to be $Ω(ε^{-2} \log m)$ times the left-hand sides, where $m$ is the number of constraints. In this paper we give a primal-dual algorithm that achieve this bound for mixed packing/cov…
We extend a primal-dual fixed point algorithm (PDFP) proposed in [5] to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrate how practical and fully decoupled schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also present. We demonstrate how different algorithms can be obtained by splitting the problems in di…
We study a first-order primal-dual subgradient method to optimize risk-constrained risk-penalized optimization problems, where risk is modeled via the popular conditional value at risk (CVaR) measure. The algorithm processes independent and identically distributed samples from the underlying uncertainty in an online fashion, and produces an $η/\sqrt{K}$-approximately feasible and $η/\sqrt{K}$-approximately optimal point within $K$ iterations with constant step-size, where $η$ increases with tunable risk-parameters of CVaR. We find optimized step sizes using our bounds and precisely characteriz…
Primal-dual multigrid methods for nonsmooth optimization
MODEMAT Research Center in Mathematical Modeling and Optimization（数学建模与优化模式研究中心）；Department of Mathematics, Escuela Politécnica Nacional (EPN)（国家理工大学数学系）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
We develop block structure adapted primal-dual algorithms for non-convex non-smooth optimisation problems whose objectives can be written as compositions $G(x)+F(K(x))$ of non-smooth block-separable convex functions $G$ and $F$ with a non-linear Lipschitz-differentiable operator $K$. Our methods are refinements of the non-linear primal-dual proximal splitting method for such problems without the block structure, which itself is based on the primal-dual proximal splitting method of Chambolle and Pock for convex problems. We propose individual step length parameters and acceleration rules for ea…
We consider large-scale Markov decision processes with an unknown cost function and address the problem of learning a policy from a finite set of expert demonstrations. We assume that the learner is not allowed to interact with the expert and has no access to reinforcement signal of any kind. Existing inverse reinforcement learning methods come with strong theoretical guarantees, but are computationally expensive, while state-of-the-art policy optimization algorithms achieve significant empirical success, but are hampered by limited theoretical understanding. To bridge the gap between theory a…
Stochastic Primal-Dual Three Operator Splitting Algorithm with Extension to Equivariant Regularization-by-Denoising
School of Mathematics, University of Birmingham（伯明翰大学数学学院）；Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）；DAMTP, University of Cambridge（剑桥大学应用数学与理论物理系）
本文提出凸三复合优化的随机原始-对偶三算子分裂算法 TOS-SPDHG，证明 O(1/K) 遍历收敛率，并结合 RED 与等变扩展用于成像逆问题。
A primal-dual algorithm for image reconstruction with input-convex neural network regularizers
Department of Mathematical Sciences, University of Bath, UK（英国巴斯大学数学科学系）；Department of Electronics & Electrical Communication Engineering, Indian Institute of Technology (IIT), Kharagpur, India（印度理工学院Kharagpur分校电子与电气通信工程系）
Decentralized online learning for seeking generalized Nash equilibrium (GNE) of noncooperative games in dynamic environments is studied in this paper. Each player aims at selfishly minimizing its own time-varying cost function subject to time-varying coupled constraints and local feasible set constraints. Only local cost functions and local constraints are available to individual players, who can receive their neighbors' information through a fixed and connected graph. In addition, players have no prior knowledge of cost functions and local constraint functions in the future time. In this sett…
