This paper considers online convex optimization with time-varying constraint functions. Specifically, we have a sequence of convex objective functions $\{f_t(x)\}_{t=0}^{\infty}$ and convex constraint functions $\{g_{t,i}(x)\}_{t=0}^{\infty}$ for $i \in \{1, ..., k\}$. The functions are gradually revealed over time. For a given $ε>0$, the goal is to choose points $x_t$ every step $t$, without knowing the $f_t$ and $g_{t,i}$ functions on that step, to achieve a time average at most $ε$ worse than the best fixed-decision that could be chosen with hindsight, subject to the time average of the constraint functions being nonpositive. It is known that this goal is generally impossible. This paper develops an online algorithm that solves the problem with $O(1/ε^2)$ convergence time in the special case when all constraint functions are nonpositive over a common subset of $\mathbb{R}^n$. Similar performance is shown in an expected sense when the common subset assumption is removed but the constraint functions are assumed to vary according to a random process that is independent and identically distributed (i.i.d.) over time slots $t \in \{0, 1, 2, \ldots\}$. Finally, in the special case when both the constraint and objective functions are i.i.d. over time slots $t$, the algorithm is shown to come within $ε$ of optimality with respect to the best (possibly time-varying) causal policy that knows the full probability distribution.
The online optimization problem with non-convex loss functions over a closed convex set, coupled with a set of inequality (possibly non-convex) constraints is a challenging online learning problem. A proximal method of multipliers with quadratic approximations (named as OPMM) is presented to solve this online non-convex optimization with long term constraints. Regrets of the violation of Karush-Kuhn-Tucker conditions of OPMM for solving online non-convex optimization problems are analyzed. Under mild conditions, it is shown that this algorithm exhibits ${\cO}(T^{-1/8})$ Lagrangian gradient vio…
In this paper we propose a new approach to obtain mixing least square regression estimate by means of stochastic online mirror descent in non-euclidian set-up.
Recently, there were introduced important classes of relatively smooth, relatively continuous, and relatively strongly convex optimization problems. These concepts have significantly expanded the class of problems for which optimal complexity estimates of gradient-type methods in high-dimensional spaces take place. Basing on some recent works devoted to online optimization (regret minimization) problems with both relatively Lipschitz continuous and relatively strongly convex objective function, we introduce algorithms for solving the strongly convex optimization problem with inequality constra…
Gradient-tracking (GT) based decentralized methods have emerged as an effective and viable alternative method to decentralized (stochastic) gradient descent (DSGD) when solving distributed online stochastic optimization problems. Initial studies of GT methods implied that GT methods have worse network dependent rate than DSGD, contradicting experimental results. This dilemma has recently been resolved, and tighter rates for GT methods have been established, which improves upon DSGD. In this work, we establish more enhanced rates for GT methods under the online stochastic convex settings. We pr…
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…
Devising efficient algorithms that track the optimizers of continuously varying convex optimization problems is key in many applications. A possible strategy is to sample the time-varying problem at constant rate and solve the resulting time-invariant problem. This can be too computationally burdensome in many scenarios. An alternative strategy is to set up an iterative algorithm that generates a sequence of approximate optimizers, which are refined every time a new sampled time-invariant problem is available by one iteration of the algorithm. These types of algorithms are called running. A ma…
This paper considers distributed online convex constrained optimization, in which various agents in a multi-agent system cooperate to minimize a global cost function through communicating with neighbors over a time-varying network. When the constraint set of optimization problem is high-dimensional and complicated, the computational cost of the projection operation often becomes prohibitive. To handle this problem, we develop a distributed online Frank-Wolfe optimization algorithm combining with gradient tracking technique. We rigorously establish the dynamic regret bound of the proposed optim…
Many techniques for online optimization problems involve making decisions based solely on presently available information: fewer works take advantage of potential predictions. In this paper, we discuss the problem of online convex optimization for parametrizable objectives, i.e. optimization problems that depend solely on the value of a parameter at a given time. We introduce a new regularity for dynamic regret based on the accuracy of predicted values of the parameters and show that, under mild assumptions, accurate prediction can yield tighter bounds on dynamic regret. Inspired by recent adv…
We study the problem of networked online convex optimization, where each agent individually decides on an action at every time step and agents cooperatively seek to minimize the total global cost over a finite horizon. The global cost is made up of three types of local costs: convex node costs, temporal interaction costs, and spatial interaction costs. In deciding their individual action at each time, an agent has access to predictions of local cost functions for the next $k$ time steps in an $r$-hop neighborhood. Our work proposes a novel online algorithm, Localized Predictive Control (LPC), …
We consider cooperative multi-agent consensus optimization problems over both static and time-varying communication networks, where only local communications are allowed. The objective is to minimize the sum of agent-specific possibly non-smooth composite convex functions over agent-specific private conic constraint sets; hence, the optimal consensus decision should lie in the intersection of these private sets. Assuming the sum function is strongly convex, we provide convergence rates in suboptimality, infeasibility and consensus violation; examine the effect of underlying network topology on…
We present the online Newton's method, a single-step second-order method for online nonconvex optimization. We analyze its performance and obtain a dynamic regret bound that is linear in the cumulative variation between round optima. We show that if the variation between round optima is limited, the method leads to a constant regret bound. In the general case, the online Newton's method outperforms online convex optimization algorithms for convex functions and performs similarly to a specialized algorithm for strongly convex functions. We simulate the performance of the online Newton's method …
This note studies the distributed non-convex optimization problem with non-smooth regularization, which has wide applications in decentralized learning, estimation and control. The objective function is the sum of different local objective functions, which consist of differentiable (possibly non-convex) cost functions and non-smooth convex functions. This paper presents a distributed proximal gradient algorithm for the non-smooth non-convex optimization problem over time-varying multi-agent networks. Each agent updates local variable estimate by the multi-step consensus operator and the proxim…
In this paper, a time-varying distributed convex optimization problem is studied for continuous-time multi-agent systems. Control algorithms are designed for the cases of single-integrator and double-integrator dynamics. Two discontinuous algorithms based on the signum function are proposed to solve the problem in each case. Then in the case of double-integrator dynamics, two continuous algorithms based on, respectively, a time-varying and a fixed boundary layer are proposed as continuous approximations of the signum function. Also, to account for inter-agent collision for physical agents, a d…
In this paper, we consider the problem of distributed online convex optimization, where a group of agents collaborate to track the global minimizers of a sum of time-varying objective functions in an online manner. Specifically, we propose a novel distributed online gradient descent algorithm that relies on an online adaptation of the gradient tracking technique used in static optimization. We show that the dynamic regret bound of this algorithm has no explicit dependence on the time horizon and, therefore, can be tighter than existing bounds especially for problems with long horizons. Our bou…
We propose a data-driven online convex optimization algorithm for controlling dynamical systems. In particular, the control scheme makes use of an initially measured input-output trajectory and behavioral systems theory which enable it to handle unknown discrete-time linear time-invariant systems as well as a priori unknown time-varying cost functions. Further, only output feedback instead of full state measurements is required for the proposed approach. Analysis of the closed loop's performance reveals that the algorithm achieves sublinear regret if the variation of the cost functions is subl…
This paper considers a distributed convex optimization problem over a time-varying multi-agent network, where each agent has its own decision variables that should be set so as to minimize its individual objective subject to local constraints and global coupling equality constraints. Over directed graphs, a distributed algorithm is proposed that incorporates the push-sum protocol into dual subgradient methods. Under the convexity assumption, the optimality of primal and dual variables, and constraint violations is first established. Then the explicit convergence rates of the proposed algorithm…
This paper studies the problem of controlling linear dynamical systems subject to point-wise-in-time constraints. We present an algorithm similar to online gradient descent, that can handle time-varying and a priori unknown convex cost functions while restraining the system states and inputs to polytopic constraint sets. Analysis of the algorithm's performance, measured by dynamic regret, reveals that sublinear regret is achieved if the variation of the cost functions is sublinear in time. Finally, we present a simple example to illustrate implementation details as well as the algorithm's perf…
This paper investigates distributed online convex optimization in the presence of an aggregative variable without any global/central coordinators over a multi-agent network, where each individual agent is only able to access partial information of time-varying global loss functions, thus requiring local information exchanges between neighboring agents. Motivated by many applications in reality, the considered local loss functions depend not only on their own decision variables, but also on an aggregative variable, such as the average of all decision variables. To handle this problem, an Online…
Google AI Princeton（谷歌AI普林斯顿）；Princeton University（普林斯顿大学）
We study an algorithmic equivalence technique between non-convex gradient descent and convex mirror descent. We start by looking at a harder problem of regret minimization in online non-convex optimization. We show that under certain geometric and smoothness conditions, online gradient descent applied to non-convex functions is an approximation of online mirror descent applied to convex functions under reparameterization. In continuous time, the gradient flow with this reparameterization was shown to be exactly equivalent to continuous-time mirror descent by Amid and Warmuth 2020, but theory f…
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…
This paper studies the online convex optimization problem by using an Online Continuous-Time Nesterov Accelerated Gradient method (OCT-NAG). We show that the continuous-time dynamics generated by the online version of the Bregman Lagrangian achieves a constant static regret $\frac{c}σ$ independent of $T$, provided that some boundedness assumptions on the objective functions and optimal solutions hold. To the best of the authors' knowledge, this is the lowest static regret in the literature (lower than $O(\text{log}(T))$). We further show that under the same assumptions, the dynamic regret of t…
In the large collection of existing distributed algorithms for convex multi-agent optimization, only a handful of them provide convergence rate guarantees on agent networks with time-varying topologies, which, however, restrict the problem to be unconstrained. Motivated by this, we develop a family of distributed Fenchel dual gradient methods for solving constrained, strongly convex but not necessarily smooth multi-agent optimization problems over time-varying undirected networks. The proposed algorithms are constructed based on the application of weighted gradient methods to the Fenchel dual …
In this paper, we develop an interior-point method for solving a class of convex optimization problems with time-varying objective and constraint functions. Using log-barrier penalty functions, we propose a continuous-time dynamical system for tracking the (time-varying) optimal solution with an asymptotically vanishing error. This dynamical system is composed of two terms: (i) a correction term consisting of a continuous-time version of Newton's method, and (ii) a prediction term able to track the drift of the optimal solution by taking into account the time-varying nature of the objective an…
This paper studies a distributed online constrained optimization problem over time-varying unbalanced digraphs without explicit subgradients. In sharp contrast to the existing algorithms, we design a novel consensus-based distributed online algorithm with a local randomized zeroth-order oracle and then rescale the oracle by constructing row-stochastic matrices, which aims to address the unbalancedness of time-varying digraphs. Under mild conditions, the average dynamic regret over a time horizon is shown to asymptotically converge at a sublinear rate provided that the accumulated variation gro…
Devising efficient algorithms to solve continuously-varying strongly convex optimization programs is key in many applications, from control systems to signal processing and machine learning. In this context, solving means to find and track the optimizer trajectory of the continuously-varying convex optimization program. Recently, a novel prediction-correction methodology has been put forward to set up iterative algorithms that sample the continuously-varying optimization program at discrete time steps and perform a limited amount of computations to correct their approximate optimizer with the …
University of California, Berkeley（加州大学伯克利分校）；University of Melbourne（墨尔本大学）；University of Toronto（多伦多大学）
We incorporate future information in the form of the estimated value of future gradients in online convex optimization. This is motivated by demand response in power systems, where forecasts about the current round, e.g., the weather or the loads' behavior, can be used to improve on predictions made with only past observations. Specifically, we introduce an additional predictive step that follows the standard online convex optimization step when certain conditions on the estimated gradient and descent direction are met. We show that under these conditions and without any assumptions on the pre…
The development of online algorithms to track time-varying systems has drawn a lot of attention in the last years, in particular in the framework of online convex optimization. Meanwhile, sparse time-varying optimization has emerged as a powerful tool to deal with widespread applications, ranging from dynamic compressed sensing to parsimonious system identification. In most of the literature on sparse time-varying problems, some prior information on the system's evolution is assumed to be available. In contrast, in this paper, we propose an online learning approach, which does not employ a giv…
We investigate the problem of online convex optimization with unknown delays, in which the feedback of a decision arrives with an arbitrary delay. Previous studies have presented a delayed variant of online gradient descent (OGD), and achieved the regret bound of $O(\sqrt{T+D})$ by only utilizing the convexity condition, where $D$ is the sum of delays over $T$ rounds. In this paper, we further exploit the strong convexity to improve the regret bound. Specifically, we first extend the delayed variant of OGD for strongly convex functions, and establish a better regret bound of $O(d\log T)$, wher…
In this paper, we consider the problem of distributed online convex optimization, where a network of local agents aim to jointly optimize a convex function over a period of multiple time steps. The agents do not have any information about the future. Existing algorithms have established dynamic regret bounds that have explicit dependence on the number of time steps. In this work, we show that we can remove this dependence assuming that the local objective functions are strongly convex. More precisely, we propose a gradient tracking algorithm where agents jointly communicate and descend based o…
We extend the regret analysis of the online distributed weighted dual averaging (DWDA) algorithm [1] to the dynamic setting and provide the tightest dynamic regret bound known to date with respect to the time horizon for a distributed online convex optimization (OCO) algorithm. Our bound is linear in the cumulative difference between consecutive optima and does not depend explicitly on the time horizon. We use dynamic-online DWDA (D-ODWDA) and formulate a performance-guaranteed distributed online demand response approach for heating, ventilation, and air-conditioning (HVAC) systems of commerci…
It is known that the curvature of the feasible set in convex optimization allows for algorithms with better convergence rates, and there has been renewed interest in this topic both for offline as well as online problems. In this paper, leveraging results on geometry and convex analysis, we further our understanding of the role of curvature in optimization: - We first show the equivalence of two notions of curvature, namely strong convexity and gauge bodies, proving a conjecture of Abernethy et al. As a consequence, this show that the Frank-Wolfe-type method of Wang and Abernethy has accelerat…
In this paper, we consider online convex optimization (OCO) with time-varying loss and constraint functions. Specifically, the decision maker chooses sequential decisions based only on past information, meantime the loss and constraint functions are revealed over time. We first develop a class of model-based augmented Lagrangian methods (MALM) for time-varying functional constrained OCO (without feedback delay). Under standard assumptions, we establish sublinear regret and sublinear constraint violation of MALM. Furthermore, we extend MALM to deal with time-varying functional constrained OCO w…
Technion - Israel Institute of Technology（以色列理工学院）
Hoffman's classical result gives a bound on the distance of a point from a convex and compact polytope in terms of the magnitude of violation of the constraints. Recently, several results showed that Hoffman's bound can be used to derive strongly-convex-like rates for first-order methods for \textit{offline} convex optimization of curved, though not strongly convex, functions, over polyhedral sets. In this work, we use this classical result for the first time to obtain faster rates for \textit{online convex optimization} over polyhedral sets with curved convex, though not strongly convex, loss…
We study the online saddle point problem, an online learning problem where at each iteration a pair of actions need to be chosen without knowledge of the current and future (convex-concave) payoff functions. The objective is to minimize the gap between the cumulative payoffs and the saddle point value of the aggregate payoff function, which we measure using a metric called "SP-Regret". The problem generalizes the online convex optimization framework but here we must ensure both players incur cumulative payoffs close to that of the Nash equilibrium of the sum of the games. We propose an algorit…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We consider the problem of controlling an unknown linear dynamical system under adversarially changing convex costs and full feedback of both the state and cost function. We present the first computationally-efficient algorithm that attains an optimal $\smash{\sqrt{T}}$-regret rate compared to the best stabilizing linear controller in hindsight, while avoiding stringent assumptions on the costs such as strong convexity. Our approach is based on a careful design of non-convex lower confidence bounds for the online costs, and uses a novel technique for computationally-efficient regret minimizati…
University of Colorado Boulder（科罗拉多大学博尔德分校）；IBM Research Ireland（IBM爱尔兰研究院）
This paper considers an online proximal-gradient method to track the minimizers of a composite convex function that may continuously evolve over time. The online proximal-gradient method is inexact, in the sense that: (i) it relies on an approximate first-order information of the smooth component of the cost; and, (ii) the proximal operator (with respect to the non-smooth term) may be computed only up to a certain precision. Under suitable assumptions, convergence of the error iterates is established for strongly convex cost functions. On the other hand, the dynamic regret is investigated when…
