In this paper, we study the optimistic online convex optimization problem in dynamic environments. Existing works have shown that Ader enjoys an $O\left(\sqrt{\left(1+P_T\right)T}\right)$ dynamic regret upper bound, where $T$ is the number of rounds, and $P_T$ is the path length of the reference strategy sequence. However, Ader is not environment-adaptive. Based on the fact that optimism provides a framework for implementing environment-adaptive, we replace Greedy Projection (GP) and Normalized Exponentiated Subgradient (NES) in Ader with Optimistic-GP and Optimistic-NES respectively, and name the corresponding algorithm ONES-OGP. We also extend the doubling trick to the adaptive trick, and introduce three characteristic terms naturally arise from optimism, namely $M_T$, $\widetilde{M}_T$ and $V_T+1_{L^2ρ\left(ρ+2 P_T\right)\leqslant\varrho^2 V_T}D_T$, to replace the dependence of the dynamic regret upper bound on $T$. We elaborate ONES-OGP with adaptive trick and its subgradient variation version, all of which are environment-adaptive.
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；The Hong Kong Polytechnic University（香港理工大学）；Imperial College Business School（帝国理工学院商学院）
A fundamental problem arising in many areas of machine learning is the evaluation of the likelihood of a given observation under different nominal distributions. Frequently, these nominal distributions are themselves estimated from data, which makes them susceptible to estimation errors. We thus propose to replace each nominal distribution with an ambiguity set containing all distributions in its vicinity and to evaluate an \emph{optimistic likelihood}, that is, the maximum of the likelihood over all distributions in the ambiguity set. When the proximity of distributions is quantified by the F…
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 …
Delft University of Technology（代尔夫特理工大学）；Trinity College Dublin（都柏林圣三一学院）
The design of effective online caching policies is an increasingly important problem for content distribution networks, online social networks and edge computing services, among other areas. This paper proposes a new algorithmic toolbox for tackling this problem through the lens of optimistic online learning. We build upon the Follow-the-Regularized-Leader (FTRL) framework which is developed further here to include predictions for the file requests, and we design online caching algorithms for bipartite networks with fixed-size caches or elastic leased caches subject to time-average budget cons…
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 study the subject of universal online learning with non-i.i.d. processes for bounded losses. The notion of an universally consistent learning was defined by Hanneke in an effort to study learning theory under minimal assumptions, where the objective is to obtain low long-run average loss for any target function. We are interested in characterizing processes for which learning is possible and whether there exist learning rules guaranteed to be universally consistent given the only assumption that such learning is possible. The case of unbounded losses is very restrictive, since the learnable…
Delft University of Technology（代尔夫特理工大学）；Trinity College Dublin（都柏林圣三一学院）
The design of effective online caching policies is an increasingly important problem for content distribution networks, online social networks and edge computing services, among other areas. This paper proposes a new algorithmic toolbox for tackling this problem through the lens of \emph{optimistic} online learning. We build upon the Follow-the-Regularized-Leader (FTRL) framework, which is developed further here to include predictions for the file requests, and we design online caching algorithms for bipartite networks with pre-reserved or dynamic storage subject to time-average budget constra…
Department of Electrical and Computer Engineering, University of Texas at Austin（德克萨斯大学奥斯汀分校电气与计算机工程系）；Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology（麻省理工学院电气工程与计算机科学系）
We study the iteration complexity of the optimistic gradient descent-ascent (OGDA) method and the extra-gradient (EG) method for finding a saddle point of a convex-concave unconstrained min-max problem. To do so, we first show that both OGDA and EG can be interpreted as approximate variants of the proximal point method. This is similar to the approach taken in [Nemirovski, 2004] which analyzes EG as an approximation of the `conceptual mirror prox'. In this paper, we highlight how gradients used in OGDA and EG try to approximate the gradient of the Proximal Point method. We then exploit this in…
The monotone variational inequality is a central problem in mathematical programming that unifies and generalizes many important settings such as smooth convex optimization, two-player zero-sum games, convex-concave saddle point problems, etc. The extragradient algorithm by Korpelevich [1976] and the optimistic gradient descent-ascent algorithm by Popov [1980] are arguably the two most classical and popular methods for solving monotone variational inequalities. Despite their long histories, the following major problem remains open. What is the last-iterate convergence rate of the extragradient…
Massachusetts Institute of Technology（麻省理工学院）；Uppsala University（乌普萨拉大学）
Recent work by Mania et al. has proved that certainty equivalent control achieves nearly optimal regret for linear systems with quadratic costs. However, when parameter uncertainty is large, certainty equivalence cannot be relied upon to stabilize the true, unknown system. In this paper, we present a dual control strategy that attempts to combine the performance of certainty equivalence, with the practical utility of robustness. The formulation preserves structure in the representation of parametric uncertainty, which allows the controller to target reduction of uncertainty in the parameters t…
In this paper we consider solving saddle point problems using two variants of Gradient Descent-Ascent algorithms, Extra-gradient (EG) and Optimistic Gradient Descent Ascent (OGDA) methods. We show that both of these algorithms admit a unified analysis as approximations of the classical proximal point method for solving saddle point problems. This viewpoint enables us to develop a new framework for analyzing EG and OGDA for bilinear and strongly convex-strongly concave settings. Moreover, we use the proximal point approximation interpretation to generalize the results for OGDA for a wide range …
We study the connection between gradient-based meta-learning and convex op-timisation. We observe that gradient descent with momentum is a special case of meta-gradients, and building on recent results in optimisation, we prove convergence rates for meta-learning in the single task setting. While a meta-learned update rule can yield faster convergence up to constant factor, it is not sufficient for acceleration. Instead, some form of optimism is required. We show that optimism in meta-learning can be captured through Bootstrapped Meta-Gradients (Flennerhag et al., 2022), providing deeper insig…
Optimistic Online Mirror Descent for Bridging Stochastic and Adversarial Online Convex Optimization
National Key Laboratory for Novel Software Technology, Nanjing University（南京大学计算机软件新技术国家重点实验室）；The University of Tokyo（东京大学）；Paradigm Inc.（第四范式公司）；School of Artificial Intelligence, Nanjing University（南京大学人工智能学院）
Massachusetts Institute of Technology（麻省理工学院）；Purdue University（普渡大学）
We resolve an open problem of Hanneke on the subject of universally consistent online learning with non-i.i.d. processes and unbounded losses. The notion of an optimistically universal learning rule was defined by Hanneke in an effort to study learning theory under minimal assumptions. A given learning rule is said to be optimistically universal if it achieves a low long-run average loss whenever the data generating process makes this goal achievable by some learning rule. Hanneke posed as an open problem whether, for every unbounded loss, the family of processes admitting universal learning a…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；The Hong Kong Polytechnic University（香港理工大学）；Imperial College Business School（帝国理工商学院）
The likelihood function is a fundamental component in Bayesian statistics. However, evaluating the likelihood of an observation is computationally intractable in many applications. In this paper, we propose a non-parametric approximation of the likelihood that identifies a probability measure which lies in the neighborhood of the nominal measure and that maximizes the probability of observing the given sample point. We show that when the neighborhood is constructed by the Kullback-Leibler divergence, by moment conditions or by the Wasserstein distance, then our \textit{optimistic likelihood} c…
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…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
We establish an excess risk bound of O(H R_n^2 + R_n \sqrt{H L*}) for empirical risk minimization with an H-smooth loss function and a hypothesis class with Rademacher complexity R_n, where L* is the best risk achievable by the hypothesis class. For typical hypothesis classes where R_n = \sqrt{R/n}, this translates to a learning rate of O(RH/n) in the separable (L*=0) case and O(RH/n + \sqrt{L^* RH/n}) more generally. We also provide similar guarantees for online and stochastic convex optimization with a smooth non-negative objective.
Online Convex Optimization plays a key role in large scale machine learning. Early approaches to this problem were conservative, in which the main focus was protection against the worst case scenario. But recently several algorithms have been developed for tightening the regret bounds in easy data instances such as sparsity, predictable sequences, and curved losses. In this work we unify some of these existing techniques to obtain new update rules for the cases when these easy instances occur together. First we analyse an adaptive and optimistic update rule which achieves tighter regret bound …
Delft University of Technology（代尔夫特理工大学）；Tata Institute of Fundamental Research（塔塔基础研究所）；Amazon（亚马逊）
We take a systematic look at the problem of storing whole files in a cache with limited capacity in the context of optimistic learning, where the caching policy has access to a prediction oracle (provided by, e.g., a Neural Network). The successive file requests are assumed to be generated by an adversary, and no assumption is made on the accuracy of the oracle. In this setting, we provide a universal lower bound for prediction-assisted online caching and proceed to design a suite of policies with a range of performance-complexity trade-offs. All proposed policies offer sublinear regret bounds…
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…
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…
University of Michigan, Ann Arbor（密歇根大学安娜堡分校）；University of Rhode Island（罗德岛大学）；Purdue University（普渡大学）
This paper considers constrained online dispatching with unknown arrival, reward and constraint distributions. We propose a novel online dispatching algorithm, named POND, standing for Pessimistic-Optimistic oNline Dispatching, which achieves $O(\sqrt{T})$ regret and $O(1)$ constraint violation. Both bounds are sharp. Our experiments on synthetic and real datasets show that POND achieves low regret with minimal constraint violations.
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…
We study the problem of online learning with non-convex losses, where the learner has access to an offline optimization oracle. We show that the classical Follow the Perturbed Leader (FTPL) algorithm achieves optimal regret rate of $O(T^{-1/2})$ in this setting. This improves upon the previous best-known regret rate of $O(T^{-1/3})$ for FTPL. We further show that an optimistic variant of FTPL achieves better regret bounds when the sequence of losses encountered by the learner is `predictable'.
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…
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…
We study the problem of online learning (OL) from revealed preferences: a learner wishes to learn a non-strategic agent's private utility function through observing the agent's utility-maximizing actions in a changing environment. We adopt an online inverse optimization setup, where the learner observes a stream of agent's actions in an online fashion and the learning performance is measured by regret associated with a loss function. We first characterize a special but broad class of agent's utility functions, then utilize this structure in designing a new convex loss function. We establish th…
University of California, Berkeley（加州大学伯克利分校）；University of Washington, Seattle（西雅图华盛顿大学）
Min-max optimization is emerging as a key framework for analyzing problems of robustness to strategically and adversarially generated data. We propose a random reshuffling-based gradient free Optimistic Gradient Descent-Ascent algorithm for solving convex-concave min-max problems with finite sum structure. We prove that the algorithm enjoys the same convergence rate as that of zeroth-order algorithms for convex minimization problems. We further specialize the algorithm to solve distributionally robust, decision-dependent learning problems, where gradient information is not readily available. T…
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…
Online Non-convex Optimization with Long-term Non-convex Constraints
Department of Electrical and Computer Engineering（电气与计算机工程系）；Johns Hopkins University（约翰霍普金斯大学）；International Business School（国际商务学院）；Suzhou Xi’an Jiaotong-Liverpool University（苏州市西安交通大学利物浦大学）；Department of Data and Systems Engineering（数据与系统工程系）；HKU Musketeers Foundation（HKU Musketeers基金会）；Institute of Data Science（数据科学研究所）；The University of Hong Kong（香港大学）
We study the performance of optimistic regret-minimization algorithms for both minimizing regret in, and computing Nash equilibria of, zero-sum extensive-form games. In order to apply these algorithms to extensive-form games, a distance-generating function is needed. We study the use of the dilated entropy and dilated Euclidean distance functions. For the dilated Euclidean distance function we prove the first explicit bounds on the strong-convexity parameter for general treeplexes. Furthermore, we show that the use of dilated distance-generating functions enable us to decompose the mirror desc…
This paper presents competitive algorithms for a novel class of online optimization problems with memory. We consider a setting where the learner seeks to minimize the sum of a hitting cost and a switching cost that depends on the previous $p$ decisions. This setting generalizes Smoothed Online Convex Optimization. The proposed approach, Optimistic Regularized Online Balanced Descent, achieves a constant, dimension-free competitive ratio. Further, we show a connection between online optimization with memory and online control with adversarial disturbances. This connection, in turn, leads to a …
Michigan State University（密歇根州立大学）；NEC Laboratories America（NEC美国实验室）
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…
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,…
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…
National Key Laboratory for Novel Software Technology, Nanjing University（南京大学计算机软件新技术国家重点实验室）
In this paper, we study adaptive online convex optimization, and aim to design a universal algorithm that achieves optimal regret bounds for multiple common types of loss functions. Existing universal methods are limited in the sense that they are optimal for only a subclass of loss functions. To address this limitation, we propose a novel online method, namely Maler, which enjoys the optimal $O(\sqrt{T})$, $O(d\log T)$ and $O(\log T)$ regret bounds for general convex, exponentially concave, and strongly convex functions respectively. The essential idea is to run multiple types of learning alg…
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 …
Stochastic Gradient Descent (SGD) has played a central role in machine learning. However, it requires a carefully hand-picked stepsize for fast convergence, which is notoriously tedious and time-consuming to tune. Over the last several years, a plethora of adaptive gradient-based algorithms have emerged to ameliorate this problem. They have proved efficient in reducing the labor of tuning in practice, but many of them lack theoretic guarantees even in the convex setting. In this paper, we propose new surrogate losses to cast the problem of learning the optimal stepsizes for the stochastic opti…
A standard way to obtain convergence guarantees in stochastic convex optimization is to run an online learning algorithm and then output the average of its iterates: the actual iterates of the online learning algorithm do not come with individual guarantees. We close this gap by introducing a black-box modification to any online learning algorithm whose iterates converge to the optimum in stochastic scenarios. We then consider the case of smooth losses, and show that combining our approach with optimistic online learning algorithms immediately yields a fast convergence rate of $O(L/T^{3/2}+σ/\…
