We study Online Convex Optimization (OCO) with adversarial constraints, where an online algorithm must make sequential decisions to minimize both convex loss functions and cumulative constraint violations. We focus on a setting where the algorithm has access to predictions of the loss and constraint functions. Our results show that we can improve the current best bounds of $ O(\sqrt{T}) $ regret and $ \tilde{O}(\sqrt{T}) $ cumulative constraint violations to $ O(\sqrt{E_T(f)}) $ and $ \tilde{O}(\sqrt{E_T(g^+)}) $, respectively, where $ E_T(f) $ and $E_T(g^+)$ represent the cumulative prediction errors of the loss and constraint functions. In the worst case, where $E_T(f) = O(T) $ and $ E_T(g^+) = O(T) $ (assuming bounded gradients of the loss and constraint functions), our rates match the prior $ O(\sqrt{T}) $ results. However, when the loss and constraint predictions are accurate, our approach yields significantly smaller regret and cumulative constraint violations. Finally, we apply this to the setting of adversarial contextual bandits with sequential risk constraints, obtaining optimistic bounds $O (\sqrt{E_T(f)} T^{1/3})$ regret and $O(\sqrt{E_T(g^+)} T^{1/3})$ constraints violation, yielding better performance than existing results when prediction quality is sufficiently high.
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…
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…
China University of Petroleum-Beijing（中国石油大学（北京））；College of Information Science and Engineering（信息科学与工程学院）
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…
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 …
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）
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…
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…
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…
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'.
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…
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…
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 …
University of Amsterdam（阿姆斯特丹大学）；Korteweg-de Vries Institute for Mathematics（科尔特韦格-德弗里斯数学研究所）；University of Twente（特文特大学）；Pontificia Universidad Católica de Chile（智利天主教大学）
Stochastic and adversarial data are two widely studied settings in online learning. But many optimization tasks are neither i.i.d. nor fully adversarial, which makes it of fundamental interest to get a better theoretical understanding of the world between these extremes. In this work we establish novel regret bounds for online convex optimization in a setting that interpolates between stochastic i.i.d. and fully adversarial losses. By exploiting smoothness of the expected losses, these bounds replace a dependence on the maximum gradient length by the variance of the gradients, which was previo…
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 …
We study the problem of agnostically learning homogeneous halfspaces in the distribution-specific PAC model. For a broad family of structured distributions, including log-concave distributions, we show that non-convex SGD efficiently converges to a solution with misclassification error $O(\opt)+\eps$, where $\opt$ is the misclassification error of the best-fitting halfspace. In sharp contrast, we show that optimizing any convex surrogate inherently leads to misclassification error of $ω(\opt)$, even under Gaussian marginals.
Carnegie Mellon University（卡内基梅隆大学）；Conversant LLC（Conversant有限责任公司）；California Institute of Technology（加州理工学院）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；University of Pennsylvania（宾夕法尼亚大学）
We propose a robust adversarial prediction framework for general multiclass classification. Our method seeks predictive distributions that robustly optimize non-convex and non-continuous multiclass loss metrics against the worst-case conditional label distributions (the adversarial distributions) that (approximately) match the statistics of the training data. Although the optimized loss metrics are non-convex and non-continuous, the dual formulation of the framework is a convex optimization problem that can be recast as a risk minimization model with a prescribed convex surrogate loss we call …
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We consider online learning in episodic loop-free Markov decision processes (MDPs), where the loss function can change arbitrarily between episodes, and the transition function is not known to the learner. We show $\tilde{O}(L|X|\sqrt{|A|T})$ regret bound, where $T$ is the number of episodes, $X$ is the state space, $A$ is the action space, and $L$ is the length of each episode. Our online algorithm is implemented using entropic regularization methodology, which allows to extend the original adversarial MDP model to handle convex performance criteria (different ways to aggregate the losses of …
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…
We consider the use of no-regret algorithms to compute equilibria for particular classes of convex-concave games. While standard regret bounds would lead to convergence rates on the order of $O(T^{-1/2})$, recent work \citep{RS13,SALS15} has established $O(1/T)$ rates by taking advantage of a particular class of optimistic prediction algorithms. In this work we go further, showing that for a particular class of games one achieves a $O(1/T^2)$ rate, and we show how this applies to the Frank-Wolfe method and recovers a similar bound \citep{D15}. We also show that such no-regret techniques can ev…
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…
Recent work has shown that the training of a one-hidden-layer, scalar-output fully-connected ReLU neural network can be reformulated as a finite-dimensional convex program. Unfortunately, the scale of such a convex program grows exponentially in data size. In this work, we prove that a stochastic procedure with a linear complexity well approximates the exact formulation. Moreover, we derive a convex optimization approach to efficiently solve the "adversarial training" problem, which trains neural networks that are robust to adversarial input perturbations. Our method can be applied to binary c…
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 …
We consider the problem of strongly-convex online optimization in presence of adversarial delays; in a T-iteration online game, the feedback of the player's query at time t is arbitrarily delayed by an adversary for d_t rounds and delivered before the game ends, at iteration t+d_t-1. Specifically for \algo{online-gradient-descent} algorithm we show it has a simple regret bound of \Oh{\sum_{t=1}^T \log (1+ \frac{d_t}{t})}. This gives a clear and simple bound without resorting any distributional and limiting assumptions on the delays. We further show how this result encompasses and generalizes s…
University of Alberta（阿尔伯塔大学）；Imperial College London（伦敦帝国学院）
Recently, much work has been done on extending the scope of online learning and incremental stochastic optimization algorithms. In this paper we contribute to this effort in two ways: First, based on a new regret decomposition and a generalization of Bregman divergences, we provide a self-contained, modular analysis of the two workhorses of online learning: (general) adaptive versions of Mirror Descent (MD) and the Follow-the-Regularized-Leader (FTRL) algorithms. The analysis is done with extra care so as not to introduce assumptions not needed in the proofs and allows to combine, in a straigh…
In statistical learning theory, convex surrogates of the 0-1 loss are highly preferred because of the computational and theoretical virtues that convexity brings in. This is of more importance if we consider smooth surrogates as witnessed by the fact that the smoothness is further beneficial both computationally- by attaining an {\it optimal} convergence rate for optimization, and in a statistical sense- by providing an improved {\it optimistic} rate for generalization bound. In this paper we investigate the smoothness property from the viewpoint of statistical consistency and show how it affe…
Distributed Online Convex Optimization with Adversarial Constraints: Reduced Cumulative Constraint Violation Bounds under Slater's Condition
Massachusetts Institute of Technology（麻省理工学院）；Tongji University（同济大学）；Shanghai Research Institute for Intelligent Autonomous Systems（上海智能自主系统研究院）；Northeastern University（东北大学）；Nanyang Technological University（南洋理工大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）；Digital Futures（数字未来）
University of Southern California（南加州大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We study high-probability regret bounds for adversarial $K$-armed bandits with time-varying feedback graphs over $T$ rounds. For general strongly observable graphs, we develop an algorithm that achieves the optimal regret $\widetilde{\mathcal{O}}((\sum_{t=1}^Tα_t)^{1/2}+\max_{t\in[T]}α_t)$ with high probability, where $α_t$ is the independence number of the feedback graph at round $t$. Compared to the best existing result [Neu, 2015] which only considers graphs with self-loops for all nodes, our result not only holds more generally, but importantly also removes any $\text{poly}(K)$ dependence …
We develop a novel and generic algorithm for the adversarial multi-armed bandit problem (or more generally the combinatorial semi-bandit problem). When instantiated differently, our algorithm achieves various new data-dependent regret bounds improving previous work. Examples include: 1) a regret bound depending on the variance of only the best arm; 2) a regret bound depending on the first-order path-length of only the best arm; 3) a regret bound depending on the sum of first-order path-lengths of all arms as well as an important negative term, which together lead to faster convergence rates fo…
Learning Brenier Potentials with Convex Generative Adversarial Neural Networks
University of Wuppertal, School of Mathematics and Natural Sciences, IMACM & IZMD（乌珀塔尔大学数学与自然科学学院，IMACM与IZMD）；Faculty II - Mathematics and Naural Sciences, Technical University Berlin & Math+（柏林技术大学第二学院——数学与自然科学学院，Math+）
In this work we consider adversarial contextual bandits with risk constraints. At each round, nature prepares a context, a cost for each arm, and additionally a risk for each arm. The learner leverages the context to pull an arm and then receives the corresponding cost and risk associated with the pulled arm. In addition to minimizing the cumulative cost, the learner also needs to satisfy long-term risk constraints -- the average of the cumulative risk from all pulled arms should not be larger than a pre-defined threshold. To address this problem, we first study the full information setting wh…
