We study Online Convex Optimization in the unbounded setting where neither predictions nor gradient are constrained. The goal is to simultaneously adapt to both the sequence of gradients and the comparator. We first develop parameter-free and scale-free algorithms for a simplified setting with hints. We present two versions: the first adapts to the squared norms of both comparator and gradients separately using $O(d)$ time per round, the second adapts to their squared inner products (which measure variance only in the comparator direction) in time $O(d^3)$ per round. We then generalize two prior reductions to the unbounded setting; one to not need hints, and a second to deal with the range ratio problem (which already arises in prior work). We discuss their optimality in light of prior and new lower bounds. We apply our methods to obtain sharper regret bounds for scale-invariant online prediction with linear models.
Leiden University（莱顿大学）；Boston University（波士顿大学）；University of Southern California（南加州大学）
We study bandit convex optimization methods that adapt to the norm of the comparator, a topic that has only been studied before for its full-information counterpart. Specifically, we develop convex bandit algorithms with regret bounds that are small whenever the norm of the comparator is small. We first use techniques from the full-information setting to develop comparator-adaptive algorithms for linear bandits. Then, we extend the ideas to convex bandits with Lipschitz or smooth loss functions, using a new single-point gradient estimator and carefully designed surrogate losses.
Practical online learning tasks are often naturally defined on unconstrained domains, where optimal algorithms for general convex losses are characterized by the notion of comparator adaptivity. In this paper, we design such algorithms in the presence of switching cost - the latter penalizes the typical optimism in adaptive algorithms, leading to a delicate design trade-off. Based on a novel dual space scaling strategy discovered by a continuous-time analysis, we propose a simple algorithm that improves the existing comparator adaptive regret bound [ZCP22a] to the optimal rate. The obtained be…
The Australian National University（澳大利亚国立大学）；Data61（Data61研究所）；Centrum Wiskunde & Informatica（荷兰数学与计算机科学研究所）；Leiden University（莱顿大学）
We aim to design adaptive online learning algorithms that take advantage of any special structure that might be present in the learning task at hand, with as little manual tuning by the user as possible. A fundamental obstacle that comes up in the design of such adaptive algorithms is to calibrate a so-called step-size or learning rate hyperparameter depending on variance, gradient norms, etc. A recent technique promises to overcome this difficulty by maintaining multiple learning rates in parallel. This technique has been applied in the MetaGrad algorithm for online convex optimization and th…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Linköping University（林雪平大学）
We consider the problem of finding a saddle point for the convex-concave objective $\min_x \max_y f(x) + \langle Ax, y\rangle - g^*(y)$, where $f$ is a convex function with locally Lipschitz gradient and $g$ is convex and possibly non-smooth. We propose an adaptive version of the Condat-Vũ algorithm, which alternates between primal gradient steps and dual proximal steps. The method achieves stepsize adaptivity through a simple rule involving $\|A\|$ and the norm of recently computed gradients of $f$. Under standard assumptions, we prove an $\mathcal{O}(k^{-1})$ ergodic convergence rate. Furthe…
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…
One of the main strengths of online algorithms is their ability to adapt to arbitrary data sequences. This is especially important in nonparametric settings, where performance is measured against rich classes of comparator functions that are able to fit complex environments. Although such hard comparators and complex environments may exhibit local regularities, efficient algorithms, which can provably take advantage of these local patterns, are hardly known. We fill this gap by introducing efficient online algorithms (based on a single versatile master algorithm) each adapting to one of the fo…
Parameter-free Algorithms for the Stochastically Extended Adversarial Model
Institute of Operations Research and Analytics, National University of Singapore（新加坡国立大学运营研究与分析研究所）；Department of Computer Science and Engineering, Indian Institute of Technology Delhi（印度理工学院德里分校计算机科学与工程系）；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）；School of Artificial Intelligence, Nanjing University（南京大学人工智能学院）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）；Department of Electrical and Computer Engineering, National University of Singapore（新加坡国立大学电气与计算机工程系）
In this paper we study the limitations of parallelization in convex optimization. A convenient approach to study parallelization is through the prism of \emph{adaptivity} which is an information theoretic measure of the parallel runtime of an algorithm [BS18]. Informally, adaptivity is the number of sequential rounds an algorithm needs to make when it can execute polynomially-many queries in parallel at every round. For combinatorial optimization with black-box oracle access, the study of adaptivity has recently led to exponential accelerations in parallel runtime and the natural question is w…
We provide a theoretical explanation for the effectiveness of gradient clipping in training deep neural networks. The key ingredient is a new smoothness condition derived from practical neural network training examples. We observe that gradient smoothness, a concept central to the analysis of first-order optimization algorithms that is often assumed to be a constant, demonstrates significant variability along the training trajectory of deep neural networks. Further, this smoothness positively correlates with the gradient norm, and contrary to standard assumptions in the literature, it can grow…
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）
Bandit Convex Optimization with Gradient Prediction Adaptivity
Department of Mathematics, National University of Singapore, Singapore（新加坡国立大学数学系）；Department of Computer Science and Engineering, Indian Institute of Technology Delhi, India（印度理工学院德里分校计算机科学与工程系）；Department of Electrical and Computer Engineering, National University of Singapore, Singapore（新加坡国立大学电子与计算机工程系）
Unconstrained Online Linear Optimization (OLO) is a practical problem setting to study the training of machine learning models. Existing works proposed a number of potential-based algorithms, but in general the design of these potential functions relies heavily on guessing. To streamline this workflow, we present a framework that generates new potential functions by solving a Partial Differential Equation (PDE). Specifically, when losses are 1-Lipschitz, our framework produces a novel algorithm with anytime regret bound $C\sqrt{T}+||u||\sqrt{2T}[\sqrt{\log(1+||u||/C)}+2]$, where $C$ is a user-…
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…
Korteweg-de Vries Institute for Mathematics（科尔特韦格-德弗里斯数学研究所）；University of Amsterdam（阿姆斯特丹大学）
A sequence of works in unconstrained online convex optimisation have investigated the possibility of adapting simultaneously to the norm $U$ of the comparator and the maximum norm $G$ of the gradients. In full generality, matching upper and lower bounds are known which show that this comes at the unavoidable cost of an additive $G U^3$, which is not needed when either $G$ or $U$ is known in advance. Surprisingly, recent results by Kempka et al. (2019) show that no such price for adaptivity is needed in the specific case of $1$-Lipschitz losses like the hinge loss. We follow up on this observat…
