In this work we consider the online control of a known linear dynamic system with adversarial disturbance and adversarial controller cost. The goal in online control is to minimize the regret, defined as the difference between cumulative cost over a period $T$ and the cumulative cost for the best policy from a comparator class. For the setting we consider, we generalize the previously proposed online Disturbance Response Controller (DRC) to the adaptive gradient online Disturbance Response Controller. Using the modified controller, we present novel regret guarantees that improves the established regret guarantees for the same setting. We show that the proposed online learning controller is able to achieve intermediate intermediate regret rates between $\sqrt{T}$ and $\log{T}$ for intermediate convex conditions, while it recovers the previously established regret results for general convex controller cost and strongly convex controller cost.
This paper presents an online adaptive learning solution to optimal synchronization control problem of heterogeneous multi-agent systems via a novel distributed policy iteration approach.
This paper studies the design of feedback controllers to steer a switching linear time-invariant dynamical system towards the solution trajectory of a time-varying convex optimization problem. We propose two types of controllers: (i) a continuous controller inspired by the online gradient descent method, and (ii) a hybrid controller that can be interpreted as an online version of Nesterov's accelerated gradient method with restarts of the state variables. By design, the controllers continuously steer the system towards the time-varying optimizer without requiring knowledge of exogenous disturb…
This paper investigates the problem of tracking solutions of stochastic optimization problems with time-varying costs that depend on random variables with decision-dependent distributions. In this context, we propose the use of an online stochastic gradient descent method to solve the optimization, and we provide explicit bounds in expectation and in high probability for the distance between the optimizers and the points generated by the algorithm. In particular, we show that when the gradient error due to sampling is modeled as a sub-Weibull random variable, then the tracking error is ultimat…
We consider online convex optimization with time-varying stage costs and additional switching costs. Since the switching costs introduce coupling across all stages, multi-step-ahead (long-term) predictions are incorporated to improve the online performance. However, longer-term predictions tend to suffer from lower quality. Thus, a critical question is: how to reduce the impact of long-term prediction errors on the online performance? To address this question, we introduce a gradient-based online algorithm, Receding Horizon Inexact Gradient (RHIG), and analyze its performance by dynamic regret…
We describe a framework for deriving and analyzing online optimization algorithms that incorporate adaptive, data-dependent regularization, also termed preconditioning. Such algorithms have been proven useful in stochastic optimization by reshaping the gradients according to the geometry of the data. Our framework captures and unifies much of the existing literature on adaptive online methods, including the AdaGrad and Online Newton Step algorithms as well as their diagonal versions. As a result, we obtain new convergence proofs for these algorithms that are substantially simpler than previous…
Dynamical systems, for instance in model predictive control, often contain unknown parameters, which must be determined during system operation. Online or on-the-fly parameter identification methods are therefore necessary. The challenge of online methods is that one must continuously estimate parameters as experimental data becomes available. The existing techniques in the context of time-dependent partial differential equations exclude the case where the system depends nonlinearly on the parameters.Based on a model reference adaptive system approach, we present an online parameter identifica…
We consider the problem of estimating a function from $n$ noisy samples whose discrete Total Variation (TV) is bounded by $C_n$. We reveal a deep connection to the seemingly disparate problem of Strongly Adaptive online learning (Daniely et al, 2015) and provide an $O(n \log n)$ time algorithm that attains the near minimax optimal rate of $\tilde O (n^{1/3}C_n^{2/3})$ under squared error loss. The resulting algorithm runs online and optimally adapts to the unknown smoothness parameter $C_n$. This leads to a new and more versatile alternative to wavelets-based methods for (1) adaptively estimat…
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…
We consider non-differentiable dynamic optimization problems such as those arising in robotics and subspace tracking. Given the computational constraints and the time-varying nature of the problem, a low-complexity algorithm is desirable, while the accuracy of the solution may only increase slowly over time. We put forth the proximal online gradient descent (OGD) algorithm for tracking the optimum of a composite objective function comprising of a differentiable loss function and a non-differentiable regularizer. An online learning framework is considered and the gradient of the loss function i…
This paper studies an online optimization problem with a finite prediction window of cost functions and additional switching costs on decisions. We propose two gradient-based online algorithms: Receding Horizon Gradient Descent (RHGD), and Receding Horizon Accelerated Gradient (RHAG). Both algorithms only require a finite number of projected gradient evaluations at each stage. We provide upper bounds on the dynamic regrets of the proposed algorithms and show that the regret upper bounds decay exponentially with the length of the prediction window. Moreover, we study the fundamental lower bound…
Based on the method of FGD, we apply the method of adaptive gradient descent which uses different step length at different epoch. Adaptive gradient descent performs much better than FGD in the tests and keeps the guarantee of convergence speed at the same time.
Given a sequence of $n$ independent random variables with common continuous distribution, we propose a simple adaptive online policy that selects a monotone increasing subsequence. We show that the expected number of monotone increasing selections made by such a policy is within $O(\log n)$ of optimal. Our construction provides a direct and natural way for proving the $O(\log n)$-optimality gap. An earlier proof of the same result made crucial use of a key inequality of Bruss and Delbaen (2001) and of de-Poissonization.
We present an approach towards convex optimization that relies on a novel scheme which converts online adaptive algorithms into offline methods. In the offline optimization setting, our derived methods are shown to obtain favourable adaptive guarantees which depend on the harmonic sum of the queried gradients. We further show that our methods implicitly adapt to the objective's structure: in the smooth case fast convergence rates are ensured without any prior knowledge of the smoothness parameter, while still maintaining guarantees in the non-smooth setting. Our approach has a natural extensio…
In this paper, online convex optimization is applied to the problem of controlling linear dynamical systems. An algorithm similar to online gradient descent, which can handle time-varying and unknown cost functions, is proposed. Then, performance guarantees are derived in terms of regret analysis. We show that the proposed control scheme achieves sublinear regret if the variation of the cost functions is sublinear. In addition, as a special case, the system converges to the optimal equilibrium if the cost functions are invariant after some finite time. Finally, the performance of the resulting…
We propose an online parametric estimation method of stochastic differential equations with discrete observations and misspecified modelling based on online gradient descent. Our study provides uniform upper bounds for the risks of the estimators over a family of stochastic differential equations. The derivation of the bounds involves three underlying theoretical results: the analysis of the stochastic mirror descent algorithm based on dependent and biased subgradients, the simultaneous exponential ergodicity of classes of diffusion processes, and the proposal of loss functions whose approxima…
Multi-agent systems are being increasingly deployed in challenging environments for performing complex tasks such as multi-target tracking, search-and-rescue, and intrusion detection. Notwithstanding the computational limitations of individual robots, such systems rely on collaboration to sense and react to the environment. This paper formulates the generic target tracking problem as a time-varying optimization problem and puts forth an inexact online gradient descent method for solving it sequentially. The performance of the proposed algorithm is studied by characterizing its dynamic regret, …
In this paper we propose several adaptive gradient methods for stochastic optimization. Unlike AdaGrad-type of methods, our algorithms are based on Armijo-type line search and they simultaneously adapt to the unknown Lipschitz constant of the gradient and variance of the stochastic approximation for the gradient. We consider an accelerated and non-accelerated gradient descent for convex problems and gradient descent for non-convex problems. In the experiments we demonstrate superiority of our methods to existing adaptive methods, e.g. AdaGrad and Adam.
In this paper, we develop a unified framework for analyzing the tracking error and dynamic regret of inexact online optimization methods under a variety of settings. Specifically, we leverage the quadratic constraint approach from control theory to formulate sequential semidefinite programs (SDPs) whose feasible points naturally correspond to tracking error bounds of various inexact online optimization methods including the inexact online gradient descent (OGD) method, the online gradient descent-ascent method, the online stochastic gradient method, and the inexact proximal online gradient met…
We present a novel method for convex unconstrained optimization that, without any modifications, ensures: (i) accelerated convergence rate for smooth objectives, (ii) standard convergence rate in the general (non-smooth) setting, and (iii) standard convergence rate in the stochastic optimization setting. To the best of our knowledge, this is the first method that simultaneously applies to all of the above settings. At the heart of our method is an adaptive learning rate rule that employs importance weights, in the spirit of adaptive online learning algorithms (Duchi et al., 2011; Levy, 2017), …
In this paper, we describe a stochastic adaptive fast gradient descent method based on the mirror variant of similar triangles method. To our knowledge, this is the first attempt to use adaptivity in stochastic method. Additionally, a main result was proved in terms of probabilities of large deviations.
We cast Amari's natural gradient in statistical learning as a specific case of Kalman filtering. Namely, applying an extended Kalman filter to estimate a fixed unknown parameter of a probabilistic model from a series of observations, is rigorously equivalent to estimating this parameter via an online stochastic natural gradient descent on the log-likelihood of the observations. In the i.i.d. case, this relation is a consequence of the "information filter" phrasing of the extended Kalman filter. In the recurrent (state space, non-i.i.d.) case, we prove that the joint Kalman filter over states a…
In this paper, we present a heuristic adaptive fast gradient method. We show that in practice our method has a better convergence rate than popular today optimization methods. Moreover, we justify our method and point out some problems that do not allow us to obtain theoretical estimates.
Amirkabir University of Technology（阿米尔卡比尔理工大学）；University of Florida（佛罗里达大学）
Adaptive gradient-based optimization methods such as \textsc{Adagrad}, \textsc{Rmsprop}, and \textsc{Adam} are widely used in solving large-scale machine learning problems including deep learning. A number of schemes have been proposed in the literature aiming at parallelizing them, based on communications of peripheral nodes with a central node, but incur high communications cost. To address this issue, we develop a novel consensus-based distributed adaptive moment estimation method (\textsc{Dadam}) for online optimization over a decentralized network that enables data parallelization, as wel…
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…
This work considered an online distributed optimization problem, with a group of agents whose local objective functions vary with time. Moreover, the value of the objective function is revealed to the corresponding agent after the decision is executed per time-step. Thus, each agent can only update the decision variable based on the revealed value and information collected from the neighbors, without the knowledge on the explicit expression of the objective function. To solve this problem, an online gradient-free distributed projected gradient descent (DPGD) algorithm is presented, where each …
Online Convex Optimization with Switching Cost with Only One Single Gradient Evaluation
Indian Institute of Technology Bombay (IITB)（印度理工学院班加罗尔分校）；Indian Institute of Technology Delhi (IITD)（印度理工学院德里分校）；Tata Institute of Fundamental Research (TIFR)（塔塔基础研究机构）
Gradient Methods with Online Scaling Part II. Practical Aspects
Department of Mathematics, Stanford University（斯坦福大学数学系）；ICME, Stanford University（斯坦福大学工业工程与运营管理系）；Department of Management Science and Engineering, Stanford University（斯坦福大学管理科学与工程系）
Zhejiang University（浙江大学）；University of Pennsylvania（宾夕法尼亚大学）；Tencent AI Lab（腾讯人工智能实验室）；University of Western Ontario（西安大略大学）
This paper focuses on projection-free methods for solving smooth Online Convex Optimization (OCO) problems. Existing projection-free methods either achieve suboptimal regret bounds or have high per-iteration computational costs. To fill this gap, two efficient projection-free online methods called ORGFW and MORGFW are proposed for solving stochastic and adversarial OCO problems, respectively. By employing a recursive gradient estimator, our methods achieve optimal regret bounds (up to a logarithmic factor) while possessing low per-iteration computational costs. Experimental results demonstrate…
University of Southern California（南加州大学）；Google Brain（谷歌大脑）；University of British Columbia（不列颠哥伦比亚大学）；Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）
Most stochastic optimization methods use gradients once before discarding them. While variance reduction methods have shown that reusing past gradients can be beneficial when there is a finite number of datapoints, they do not easily extend to the online setting. One issue is the staleness due to using past gradients. We propose to correct this staleness using the idea of implicit gradient transport (IGT) which transforms gradients computed at previous iterates into gradients evaluated at the current iterate without using the Hessian explicitly. In addition to reducing the variance and bias of…
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…
This paper presents an adaptive online distributed optimal control approach that is applicable to optimal planning for very-large-scale robotics systems in highly uncertain environments. This approach is developed based on the optimal mass transport theory. It is also viewed as an online reinforcement learning and approximate dynamic programming approach in the Wasserstein-GMM space, where a novel value functional is defined based on the probability density functions of robots and the time-varying obstacle map functions describing the changing environmental information. The proposed approach i…
The online portfolio selection (OLPS) problem differs from classical portfolio model problems, as it involves making sequential investment decisions. Many OLPS strategies described in the literature capture market movement based on various beliefs and are shown to be profitable. In this paper, we propose a robust optimization (RO)-based strategy that takes transaction costs into account. Moreover, unlike existing studies that calibrate model parameters from benchmark data sets, we develop a novel adaptive scheme that decides the parameters sequentially. With a wide range of parameters as input…
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…
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…
Adaptive gradient methods have shown excellent performances for solving many machine learning problems. Although multiple adaptive gradient methods were recently studied, they mainly focus on either empirical or theoretical aspects and also only work for specific problems by using some specific adaptive learning rates. Thus, it is desired to design a universal framework for practical algorithms of adaptive gradients with theoretical guarantee to solve general problems. To fill this gap, we propose a faster and universal framework of adaptive gradients (i.e., SUPER-ADAM) by introducing a univer…
Linear optimization is many times algorithmically simpler than non-linear convex optimization. Linear optimization over matroid polytopes, matching polytopes and path polytopes are example of problems for which we have simple and efficient combinatorial algorithms, but whose non-linear convex counterpart is harder and admits significantly less efficient algorithms. This motivates the computational model of convex optimization, including the offline, online and stochastic settings, using a linear optimization oracle. In this computational model we give several new results that improve over the …
We present an adaptive online gradient descent algorithm to solve online convex optimization problems with long-term constraints , which are constraints that need to be satisfied when accumulated over a finite number of rounds T , but can be violated in intermediate rounds. For some user-defined trade-off parameter $β$ $\in$ (0, 1), the proposed algorithm achieves cumulative regret bounds of O(T^max{$β$,1--$β$}) and O(T^(1--$β$/2)) for the loss and the constraint violations respectively. Our results hold for convex losses and can handle arbitrary convex constraints without requiring knowledge …
On the Convergence of Adaptive Gradient Methods for Nonconvex Optimization
Indiana University（印第安纳大学）；The Pennsylvania State University（宾夕法尼亚州立大学）；The University of Hong Kong（香港大学）；Rice University（莱斯大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
University of Utah（犹他大学）；University of Kentucky（肯塔基大学）
Momentum plays a crucial role in stochastic gradient-based optimization algorithms for accelerating or improving training deep neural networks (DNNs). In deep learning practice, the momentum is usually weighted by a well-calibrated constant. However, tuning hyperparameters for momentum can be a significant computational burden. In this paper, we propose a novel \emph{adaptive momentum} for improving DNNs training; this adaptive momentum, with no momentum related hyperparameter required, is motivated by the nonlinear conjugate gradient (NCG) method. Stochastic gradient descent (SGD) with this n…
Sharif University of Technology（谢里夫理工大学）；Amirkabir University of Technology（阿米尔卡比尔理工大学）
As application demands for online convex optimization accelerate, the need for designing new methods that simultaneously cover a large class of convex functions and impose the lowest possible regret is highly rising. Known online optimization methods usually perform well only in specific settings, and their performance depends highly on the geometry of the decision space and cost functions. However, in practice, lack of such geometric information leads to confusion in using the appropriate algorithm. To address this issue, some adaptive methods have been proposed that focus on adaptively learn…
