In citep{Hazan-2008-extract}, the authors showed that the regret of online linear optimization can be bounded by the total variation of the cost vectors. In this paper, we extend this result to general online convex optimization. We first analyze the limitations of the algorithm in \citep{Hazan-2008-extract} when applied it to online convex optimization. We then present two algorithms for online convex optimization whose regrets are bounded by the variation of cost functions. We finally consider the bandit setting, and present a randomized algorithm for online bandit convex optimization with a variation-based regret bound. We show that the regret bound for online bandit convex optimization is optimal when the variation of cost functions is independent of the number of trials.
Some of the most compelling applications of online convex optimization, including online prediction and classification, are unconstrained: the natural feasible set is R^n. Existing algorithms fail to achieve sub-linear regret in this setting unless constraints on the comparator point x^* are known in advance. We present algorithms that, without such prior knowledge, offer near-optimal regret bounds with respect to any choice of x^*. In particular, regret with respect to x^* = 0 is constant. We then prove lower bounds showing that our guarantees are near-optimal in this setting.
Google AI Princeton（谷歌AI普林斯顿）；Princeton University（普林斯顿大学）
The theory of deep learning focuses almost exclusively on supervised learning, non-convex optimization using stochastic gradient descent, and overparametrized neural networks. It is common belief that the optimizer dynamics, network architecture, initialization procedure, and other factors tie together and are all components of its success. This presents theoretical challenges for analyzing state-based and/or online deep learning. Motivated by applications in control, we give a general black-box reduction from deep learning to online convex optimization. This allows us to decouple optimization…
University of Science and Technology of China（中国科学技术大学）
Follow-the-Regularized-Lead (FTRL) and Online Mirror Descent (OMD) are regret minimization algorithms for Online Convex Optimization (OCO), they are mathematically elegant but less practical in solving Extensive-Form Games (EFGs). Counterfactual Regret Minimization (CFR) is a technique for approximating Nash equilibria in EFGs. CFR and its variants have a fast convergence rate in practice, but their theoretical results are not satisfactory. In recent years, researchers have been trying to link CFRs with OCO algorithms, which may provide new theoretical results and inspire new algorithms. Howev…
In this paper, we investigate the online non-convex optimization problem which generalizes the classic {online convex optimization problem by relaxing the convexity assumption on the cost function. For this type of problem, the classic exponential weighting online algorithm has recently been shown to attain a sub-linear regret of $O(\sqrt{T\log T})$. In this paper, we introduce a novel recursive structure to the online algorithm to define a recursive exponential weighting algorithm that attains a regret of $O(\sqrt{T})$, matching the well-known regret lower bound. To the best of our knowledge,…
In online learning an algorithm plays against an environment with losses possibly picked by an adversary at each round. The generality of this framework includes problems that are not adversarial, for example offline optimization, or saddle point problems (i.e. min max optimization). However, online algorithms are typically not designed to leverage additional structure present in non-adversarial problems. Recently, slight modifications to well-known online algorithms such as optimism and adaptive step sizes have been used in several domains to accelerate online learning -- recovering optimal r…
We resolve an open question from (Christiano, 2014b) posed in COLT'14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework the algorithm is shown a pair of items at each step, chosen from a set of $n$ items. The learner then predicts a label for each item, from a label set of size $L$ and receives a real valued payoff. This is a natural framework which captures many interesting scenarios such as collaborative filtering, online gambling, and online max cut among others. (Christiano, 2014a) designed an efficient onlin…
Improved Approximate Regret for Decentralized Online Continuous Submodular Maximization via Reductions
School of Software Technology, Zhejiang University, Ningbo, China（浙江大学宁波校区软件学院）；State Key Laboratory of Blockchain and Data Security, Zhejiang University, Hangzhou, China（浙江大学区块链与数据安全国家重点实验室）；School of Artificial Intelligence, Nanjing University, Nanjing, China（南京大学人工智能学院）；Department of Computer Science, City University of Hong Kong, Hong Kong, China（香港城市大学计算机科学系）
In this paper, we improve the regret bound for online kernel selection under bandit feedback. Previous algorithm enjoys a $O((\Vert f\Vert^2_{\mathcal{H}_i}+1)K^{\frac{1}{3}}T^{\frac{2}{3}})$ expected bound for Lipschitz loss functions. We prove two types of regret bounds improving the previous bound. For smooth loss functions, we propose an algorithm with a $O(U^{\frac{2}{3}}K^{-\frac{1}{3}}(\sum^K_{i=1}L_T(f^\ast_i))^{\frac{2}{3}})$ expected bound where $L_T(f^\ast_i)$ is the cumulative losses of optimal hypothesis in $\mathbb{H}_{i}=\{f\in\mathcal{H}_i:\Vert f\Vert_{\mathcal{H}_i}\leq U\}$.…
Microsoft Research（微软研究院）；Google Research India（谷歌印度研究院）
We study online learning with bandit feedback (i.e. learner has access to only zeroth-order oracle) where cost/reward functions $\f_t$ admit a "pseudo-1d" structure, i.e. $\f_t(\w) = \loss_t(\pred_t(\w))$ where the output of $\pred_t$ is one-dimensional. At each round, the learner observes context $\x_t$, plays prediction $\pred_t(\w_t; \x_t)$ (e.g. $\pred_t(\cdot)=\langle \x_t, \cdot\rangle$) for some $\w_t \in \mathbb{R}^d$ and observes loss $\loss_t(\pred_t(\w_t))$ where $\loss_t$ is a convex Lipschitz-continuous function. The goal is to minimize the standard regret metric. This pseudo-1d b…
Nanjing University（南京大学）；Microsoft Research Asia（微软亚洲研究院）
We investigate online convex optimization in changing environments, and choose the adaptive regret as the performance measure. The goal is to achieve a small regret over every interval so that the comparator is allowed to change over time. Different from previous works that only utilize the convexity condition, this paper further exploits smoothness to improve the adaptive regret. To this end, we develop novel adaptive algorithms for convex and smooth functions, and establish problem-dependent regret bounds over any interval. Our regret bounds are comparable to existing results in the worst ca…
University of Chicago（芝加哥大学）；Microsoft Research India（微软印度研究院）；University of Michigan（密歇根大学）
This paper considers the stability of online learning algorithms and its implications for learnability (bounded regret). We introduce a novel quantity called {\em forward regret} that intuitively measures how good an online learning algorithm is if it is allowed a one-step look-ahead into the future. We show that given stability, bounded forward regret is equivalent to bounded regret. We also show that the existence of an algorithm with bounded regret implies the existence of a stable algorithm with bounded regret and bounded forward regret. The equivalence results apply to general, possibly n…
We analyze and evaluate an online gradient descent algorithm with adaptive per-coordinate adjustment of learning rates. Our algorithm can be thought of as an online version of batch gradient descent with a diagonal preconditioner. This approach leads to regret bounds that are stronger than those of standard online gradient descent for general online convex optimization problems. Experimentally, we show that our algorithm is competitive with state-of-the-art algorithms for large scale machine learning problems.
Princeton University（普林斯顿大学）；Google AI Princeton（谷歌人工智能普林斯顿（Google AI Princeton））
Boosting is a widely used machine learning approach based on the idea of aggregating weak learning rules. While in statistical learning numerous boosting methods exist both in the realizable and agnostic settings, in online learning they exist only in the realizable case. In this work we provide the first agnostic online boosting algorithm; that is, given a weak learner with only marginally-better-than-trivial regret guarantees, our algorithm boosts it to a strong learner with sublinear regret. Our algorithm is based on an abstract (and simple) reduction to online convex optimization, which ef…
A natural goal when designing online learning algorithms for non-stationary environments is to bound the regret of the algorithm in terms of the temporal variation of the input sequence. Intuitively, when the variation is small, it should be easier for the algorithm to achieve low regret, since past observations are predictive of future inputs. Such data-dependent "pathlength" regret bounds have recently been obtained for a wide variety of online learning problems, including OCO and bandits. We obtain the first pathlength regret bounds for online control and estimation (e.g. Kalman filtering) …
We analyze the minimax regret of the adversarial bandit convex optimization problem. Focusing on the one-dimensional case, we prove that the minimax regret is $\widetildeΘ(\sqrt{T})$ and partially resolve a decade-old open problem. Our analysis is non-constructive, as we do not present a concrete algorithm that attains this regret rate. Instead, we use minimax duality to reduce the problem to a Bayesian setting, where the convex loss functions are drawn from a worst-case distribution, and then we solve the Bayesian version of the problem with a variant of Thompson Sampling. Our analysis featur…
Università degli Studi di Milano（米兰大学）；ETH Zürich（苏黎世联邦理工学院）
We consider prediction with expert advice for strongly convex and bounded losses, and investigate trade-offs between regret and "variance" (i.e., squared difference of learner's predictions and best expert predictions). With $K$ experts, the Exponentially Weighted Average (EWA) algorithm is known to achieve $O(\log K)$ regret. We prove that a variant of EWA either achieves a negative regret (i.e., the algorithm outperforms the best expert), or guarantees a $O(\log K)$ bound on both variance and regret. Building on this result, we show several examples of how variance of predictions can be expl…
This paper studies online optimization from a high-level unified theoretical perspective. We not only generalize both Optimistic-DA and Optimistic-MD in normed vector space, but also unify their analysis methods for dynamic regret. Regret bounds are the tightest possible due to the introduction of $ϕ$-convex. As instantiations, regret bounds of normalized exponentiated subgradient and greedy/lazy projection are better than the currently known optimal results. By replacing losses of online game with monotone operators, and extending the definition of regret, namely regret$^n$, we extend online …
Johnson Controls（江森自控）；Iowa State University（爱荷华州立大学）；New York University（纽约大学）
Regret has been widely adopted as the metric of choice for evaluating the performance of online optimization algorithms for distributed, multi-agent systems. However, data/model variations associated with agents can significantly impact decisions and requires consensus among agents. Moreover, most existing works have focused on developing approaches for (either strongly or non-strongly) convex losses, and very few results have been obtained regarding regret bounds in distributed online optimization for general non-convex losses. To address these two issues, we propose a novel composite regret …
Google AI Princeton（谷歌AI普林斯顿）；Princeton University（普林斯顿大学）
We study optimal regret bounds for control in linear dynamical systems under adversarially changing strongly convex cost functions, given the knowledge of transition dynamics. This includes several well studied and fundamental frameworks such as the Kalman filter and the linear quadratic regulator. State of the art methods achieve regret which scales as $O(\sqrt{T})$, where $T$ is the time horizon. We show that the optimal regret in this setting can be significantly smaller, scaling as $O(\text{poly}(\log T))$. This regret bound is achieved by two different efficient iterative methods, online …
National Key Laboratory for Novel Software Technology, Nanjing University（南京大学新型软件技术国家重点实验室）；Paradigm Inc.（第四范式有限公司）
To deal with changing environments, a new performance measure -- adaptive regret, defined as the maximum static regret over any interval, was proposed in online learning. Under the setting of online convex optimization, several algorithms have been successfully developed to minimize the adaptive regret. However, existing algorithms lack universality in the sense that they can only handle one type of convex functions and need apriori knowledge of parameters. By contrast, there exist universal algorithms, such as MetaGrad, that attain optimal static regret for multiple types of convex functions …
We consider online optimization procedures in the context of logistic regression, focusing on the Extended Kalman Filter (EKF). We introduce a second-order algorithm close to the EKF, named Semi-Online Step (SOS), for which we prove a O(log(n)) regret in the adversarial setting, paving the way to similar results for the EKF. This regret bound on SOS is the first for such parameter-free algorithm in the adversarial logistic regression. We prove for the EKF in constant dynamics a O(log(n)) regret in expectation and in the well-specified logistic regression model.
We introduce a new algorithm for online linear-quadratic control in a known system subject to adversarial disturbances. Existing regret bounds for this setting scale as $\sqrt{T}$ unless strong stochastic assumptions are imposed on the disturbance process. We give the first algorithm with logarithmic regret for arbitrary adversarial disturbance sequences, provided the state and control costs are given by known quadratic functions. Our algorithm and analysis use a characterization for the optimal offline control law to reduce the online control problem to (delayed) online learning with approxim…
National University of Defense Technology（国防科技大学）；Xinjiang University（新疆大学）；Beijing Jiaotong University（北京交通大学）；Dongguan University of Technology（东莞理工学院）
As a metric to measure the performance of an online method, dynamic regret with switching cost has drawn much attention for online decision making problems. Although the sublinear regret has been provided in many previous researches, we still have little knowledge about the relation between the dynamic regret and the switching cost. In the paper, we investigate the relation for two classic online settings: Online Algorithms (OA) and Online Convex Optimization (OCO). We provide a new theoretical analysis framework, which shows an interesting observation, that is, the relation between the switch…
We revisit the problem of online learning with sleeping experts/bandits: in each time step, only a subset of the actions are available for the algorithm to choose from (and learn about). The work of Kleinberg et al. (2010) showed that there exist no-regret algorithms which perform no worse than the best ranking of actions asymptotically. Unfortunately, achieving this regret bound appears computationally hard: Kanade and Steinke (2014) showed that achieving this no-regret performance is at least as hard as PAC-learning DNFs, a notoriously difficult problem. In the present work, we relax the ori…
University of California, Santa Barbara（加州大学圣巴巴拉分校）；Amazon Research（亚马逊研究院）
We consider the framework of non-stationary Online Convex Optimization where a learner seeks to control its dynamic regret against an arbitrary sequence of comparators. When the loss functions are strongly convex or exp-concave, we demonstrate that Strongly Adaptive (SA) algorithms can be viewed as a principled way of controlling dynamic regret in terms of path variation $V_T$ of the comparator sequence. Specifically, we show that SA algorithms enjoy $\tilde O(\sqrt{TV_T} \vee \log T)$ and $\tilde O(\sqrt{dTV_T} \vee d\log T)$ dynamic regret for strongly convex and exp-concave losses respectiv…
This paper considers a variant of the online paging problem, where the online algorithm has access to multiple predictors, each producing a sequence of predictions for the page arrival times. The predictors may have occasional prediction errors and it is assumed that at least one of them makes a sublinear number of prediction errors in total. Our main result states that this assumption suffices for the design of a randomized online algorithm whose time-average regret with respect to the optimal offline algorithm tends to zero as the time tends to infinity. This holds (with different regret bou…
This paper describes a new online convex optimization method which incorporates a family of candidate dynamical models and establishes novel tracking regret bounds that scale with the comparator's deviation from the best dynamical model in this family. Previous online optimization methods are designed to have a total accumulated loss comparable to that of the best comparator sequence, and existing tracking or shifting regret bounds scale with the overall variation of the comparator sequence. In many practical scenarios, however, the environment is nonstationary and comparator sequences with sm…
Adaptive gradient algorithms such as ADAGRAD and its variants have gained popularity in the training of deep neural networks. While many works as for adaptive methods have focused on the static regret as a performance metric to achieve a good regret guarantee, the dynamic regret analyses of these methods remain unclear. As opposed to the static regret, dynamic regret is considered to be a stronger concept of performance measurement in the sense that it explicitly elucidates the non-stationarity of the environment. In this paper, we go through a variant of ADAGRAD (referred to as M-ADAGRAD ) in…
Recursive least-squares algorithms often use forgetting factors as a heuristic to adapt to non-stationary data streams. The first contribution of this paper rigorously characterizes the effect of forgetting factors for a class of online Newton algorithms. For exp-concave and strongly convex objectives, the algorithms achieve the dynamic regret of $\max\{O(\log T),O(\sqrt{TV})\}$, where $V$ is a bound on the path length of the comparison sequence. In particular, we show how classic recursive least-squares with a forgetting factor achieves this dynamic regret bound. By varying $V$, we obtain a t…
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…
Online Learning with Gradient-Variation Interval Regret
State Key Laboratory for Novel Software Technology and the School of Artificial Intelligence, Nanjing University（新型软件技术国家重点实验室和人工智能学院，南京大学）；Institute of Operations Research and Analytics, National University of Singapore（运筹与分析研究所，新加坡国立大学）
University of British Columbia（不列颠哥伦比亚大学）；Amii（阿尔伯塔机器智能研究所）
In online convex optimization (OCO), Lipschitz continuity of the functions is commonly assumed in order to obtain sublinear regret. Moreover, many algorithms have only logarithmic regret when these functions are also strongly convex. Recently, researchers from convex optimization proposed the notions of "relative Lipschitz continuity" and "relative strong convexity". Both of the notions are generalizations of their classical counterparts. It has been shown that subgradient methods in the relative setting have performance analogous to their performance in the classical setting. In this work, we…
The performance of online convex optimization algorithms in a dynamic environment is often expressed in terms of the dynamic regret, which measures the decision maker's performance against a sequence of time-varying comparators. In the analysis of the dynamic regret, prior works often assume Lipschitz continuity or uniform smoothness of the cost functions. However, there are many important cost functions in practice that do not satisfy these conditions. In such cases, prior analyses are not applicable and fail to guarantee the optimization performance. In this letter, we show that it is possib…
The Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Google Research（谷歌研究院）；Tel Aviv University（特拉维夫大学）
We consider online algorithms under both the competitive ratio criteria and the regret minimization one. Our main goal is to build a unified methodology that would be able to guarantee both criteria simultaneously. For a general class of online algorithms, namely any Metrical Task System (MTS), we show that one can simultaneously guarantee the best known competitive ratio and a natural regret bound. For the paging problem we further show an efficient online algorithm (polynomial in the number of pages) with this guarantee. To this end, we extend an existing regret minimization algorithm (speci…
Establishing Linear Surrogate Regret Bounds for Convex Smooth Losses via Convolutional Fenchel-Young Losses
College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算与数据科学学院）；The Institute of Statistical Mathematics（统计数学研究所）；School of Computer Science and Engineering, Southeast University（东南大学计算机科学与工程学院）；Skywork AI
University of Pennsylvania（宾夕法尼亚大学）；Harvard University（哈佛大学）
In this paper, we address tracking of a time-varying parameter with unknown dynamics. We formalize the problem as an instance of online optimization in a dynamic setting. Using online gradient descent, we propose a method that sequentially predicts the value of the parameter and in turn suffers a loss. The objective is to minimize the accumulation of losses over the time horizon, a notion that is termed dynamic regret. While existing methods focus on convex loss functions, we consider strongly convex functions so as to provide better guarantees of performance. We derive a regret bound that cap…
Tata Institute of Fundamental Research（塔塔基础研究所）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
We consider a fair resource allocation problem in the no-regret setting against an unrestricted adversary. The objective is to allocate resources equitably among several agents in an online fashion so that the difference of the aggregate $α$-fair utilities of the agents between an optimal static clairvoyant allocation and that of the online policy grows sub-linearly with time. The problem is challenging due to the non-additive nature of the $α$-fairness function. Previously, it was shown that no online policy can exist for this problem with a sublinear standard regret. In this paper, we propos…
This paper presents a new framework for analyzing and designing no-regret algorithms for dynamic (possibly adversarial) systems. The proposed framework generalizes the popular online convex optimization framework and extends it to its natural limit allowing it to capture a notion of regret that is intuitive for more general problems such as those encountered in game theory and variational inequalities. The framework hinges on a special choice of a system-wide loss function we have developed. Using this framework, we prove that a simple update scheme provides a no-regret algorithm for monotone …
We introduce an online convex optimization algorithm which utilizes projected subgradient descent with optimal adaptive learning rates. Our method provides second-order minimax-optimal dynamic regret guarantee (i.e. dependent on the sum of squared subgradient norms) for a sequence of general convex functions, which may not have strong convexity, smoothness, exp-concavity or even Lipschitz-continuity. The regret guarantee is against any comparator decision sequence with bounded path variation (i.e. sum of the distances between successive decisions). We generate the lower bound of the worst-case…
The regret bound of dynamic online learning algorithms is often expressed in terms of the variation in the function sequence ($V_T$) and/or the path-length of the minimizer sequence after $T$ rounds. For strongly convex and smooth functions, , Zhang et al. establish the squared path-length of the minimizer sequence ($C^*_{2,T}$) as a lower bound on regret. They also show that online gradient descent (OGD) achieves this lower bound using multiple gradient queries per round. In this paper, we focus on unconstrained online optimization. We first show that a preconditioned variant of OGD achieves …
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…
Università di Milano（米兰大学）；Boston University（波士顿大学）
In the setting of online learning, Implicit algorithms turn out to be highly successful from a practical standpoint. However, the tightest regret analyses only show marginal improvements over Online Mirror Descent. In this work, we shed light on this behavior carrying out a careful regret analysis. We prove a novel static regret bound that depends on the temporal variability of the sequence of loss functions, a quantity which is often encountered when considering dynamic competitors. We show, for example, that the regret can be constant if the temporal variability is constant and the learning …
Purdue University（普渡大学）；University of Waterloo（滑铁卢大学）
Online learning has become increasingly popular on handling massive data. The sequential nature of online learning, however, requires a centralized learner to store data and update parameters. In this paper, we consider online learning with {\em distributed} data sources. The autonomous learners update local parameters based on local data sources and periodically exchange information with a small subset of neighbors in a communication network. We derive the regret bound for strongly convex functions that generalizes the work by Ram et al. (2010) for convex functions. Most importantly, we show …
We investigate online convex optimization in non-stationary environments and choose the dynamic regret as the performance measure, defined as the difference between cumulative loss incurred by the online algorithm and that of any feasible comparator sequence. Let $T$ be the time horizon and $P_T$ be the path-length that essentially reflects the non-stationarity of environments, the state-of-the-art dynamic regret is $\mathcal{O}(\sqrt{T(1+P_T)})$. Although this bound is proved to be minimax optimal for convex functions, in this paper, we demonstrate that it is possible to further enhance the d…
We consider a decentralized multiplayer game, played over $T$ rounds, with a leader-follower hierarchy described by a directed acyclic graph. For each round, the graph structure dictates the order of the players and how players observe the actions of one another. By the end of each round, all players receive a joint bandit-reward based on their joint action that is used to update the player strategies towards the goal of minimizing the joint pseudo-regret. We present a learning algorithm inspired by the single-player multi-armed bandit problem and show that it achieves sub-linear joint pseudo-…
Center for Advanced Intelligence Project（高级智能研究中心）；RIKEN（理化学研究所）
The dueling bandit is a learning framework wherein the feedback information in the learning process is restricted to a noisy comparison between a pair of actions. In this research, we address a dueling bandit problem based on a cost function over a continuous space. We propose a stochastic mirror descent algorithm and show that the algorithm achieves an $O(\sqrt{T\log T})$-regret bound under strong convexity and smoothness assumptions for the cost function. Subsequently, we clarify the equivalence between regret minimization in dueling bandit and convex optimization for the cost function. More…
Yale University（耶鲁大学）；Georgia Institute of Technology（佐治亚理工学院）；Technion - Israel Institute of Technology（以色列理工学院）
We develop an algorithmic framework for solving convex optimization problems using no-regret game dynamics. By converting the problem of minimizing a convex function into an auxiliary problem of solving a min-max game in a sequential fashion, we can consider a range of strategies for each of the two-players who must select their actions one after the other. A common choice for these strategies are so-called no-regret learning algorithms, and we describe a number of such and prove bounds on their regret. We then show that many classical first-order methods for convex optimization -- including a…
