We consider an online learning problem on a continuum. A decision maker is given a compact feasible set $S$, and is faced with the following sequential problem: at iteration~$t$, the decision maker chooses a distribution $x^{(t)} \in Δ(S)$, then a loss function $\ell^{(t)} : S \to \mathbb{R}_+$ is revealed, and the decision maker incurs expected loss $\langle \ell^{(t)}, x^{(t)} \rangle = \mathbb{E}_{s \sim x^{(t)}} \ell^{(t)}(s)$. We view the problem as an online convex optimization problem on the space $Δ(S)$ of Lebesgue-continnuous distributions on $S$. We prove a general regret bound for the Dual Averaging method on $L^2(S)$, then prove that dual averaging with $ω$-potentials (a class of strongly convex regularizers) achieves sublinear regret when $S$ is uniformly fat (a condition weaker than convexity).
Criteo AI Lab；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Grenoble INP（格勒诺布尔理工学院）
We propose a hierarchical version of dual averaging for zeroth-order online non-convex optimization - i.e., learning processes where, at each stage, the optimizer is facing an unknown non-convex loss function and only receives the incurred loss as feedback. The proposed class of policies relies on the construction of an online model that aggregates loss information as it arrives, and it consists of two principal components: (a) a regularizer adapted to the Fisher information metric (as opposed to the metric norm of the ambient space); and (b) a principled exploration of the problem's state spa…
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…
Online mirror descent (OMD) and dual averaging (DA) -- two fundamental algorithms for online convex optimization -- are known to have very similar (and sometimes identical) performance guarantees when used with a fixed learning rate. Under dynamic learning rates, however, OMD is provably inferior to DA and suffers a linear regret, even in common settings such as prediction with expert advice. We modify the OMD algorithm through a simple technique that we call stabilization. We give essentially the same abstract regret bound for OMD with stabilization and for DA by modifying the classical OMD c…
We study a general version of the adversarial online learning problem. We are given a decision set $\mathcal{X}$ in a reflexive Banach space $X$ and a sequence of reward vectors in the dual space of $X$. At each iteration, we choose an action from $\mathcal{X}$, based on the observed sequence of previous rewards. Our goal is to minimize regret, defined as the gap between the realized reward and the reward of the best fixed action in hindsight. Using results from infinite dimensional convex analysis, we generalize the method of Dual Averaging (or Follow the Regularized Leader) to our setting an…
Criteo AI Lab（Criteo人工智能实验室）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；LIG（格勒诺布尔信息学实验室）
We consider the problem of online learning with non-convex losses. In terms of feedback, we assume that the learner observes - or otherwise constructs - an inexact model for the loss function encountered at each stage, and we propose a mixed-strategy learning policy based on dual averaging. In this general context, we derive a series of tight regret minimization guarantees, both for the learner's static (external) regret, as well as the regret incurred against the best dynamic policy in hindsight. Subsequently, we apply this general template to the case where the learner only has access to the…
We present tools for the analysis of Follow-The-Regularized-Leader (FTRL), Dual Averaging, and Mirror Descent algorithms when the regularizer (equivalently, prox-function or learning rate schedule) is chosen adaptively based on the data. Adaptivity can be used to prove regret bounds that hold on every round, and also allows for data-dependent regret bounds as in AdaGrad-style algorithms (e.g., Online Gradient Descent with adaptive per-coordinate learning rates). We present results from a large number of prior works in a unified manner, using a modular and tight analysis that isolates the key a…
We consider optimization of generalized performance metrics for binary classification by means of surrogate losses. We focus on a class of metrics, which are linear-fractional functions of the false positive and false negative rates (examples of which include $F_β$-measure, Jaccard similarity coefficient, AM measure, and many others). Our analysis concerns the following two-step procedure. First, a real-valued function $f$ is learned by minimizing a surrogate loss for binary classification on the training sample. It is assumed that the surrogate loss is a strongly proper composite loss functio…
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 three families of online convex optimization algorithms: follow-the-proximally-regularized-leader (FTRL-Proximal), regularized dual averaging (RDA), and composite-objective mirror descent. We first prove equivalence theorems that show all of these algorithms are instantiations of a general FTRL update. This provides theoretical insight on previous experimental observations. In particular, even though the FOBOS composite mirror descent algorithm handles L1 regularization explicitly, it has been observed that RDA is even more effective at producing sparsity. Our results demonstrate that…
Making use of predictions is a crucial, but under-explored, area of online algorithms. This paper studies a class of online optimization problems where we have external noisy predictions available. We propose a stochastic prediction error model that generalizes prior models in the learning and stochastic control communities, incorporates correlation among prediction errors, and captures the fact that predictions improve as time passes. We prove that achieving sublinear regret and constant competitive ratio for online algorithms requires the use of an unbounded prediction window in adversarial …
University of Missouri--Columbia（密苏里大学哥伦比亚分校）；Purdue University（普渡大学）
Excessive computational cost for learning large data and streaming data can be alleviated by using stochastic algorithms, such as stochastic gradient descent and its variants. Recent advances improve stochastic algorithms on convergence speed, adaptivity and structural awareness. However, distributional aspects of these new algorithms are poorly understood, especially for structured parameters. To develop statistical inference in this case, we propose a class of generalized regularized dual averaging (gRDA) algorithms with constant step size, which improves RDA (Xiao, 2010; Flammarion and Bach…
Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；Inria（法国国家信息与自动化研究所）；Criteo AI Lab（Criteo人工智能实验室）
In this paper, we provide a general framework for studying multi-agent online learning problems in the presence of delays and asynchronicities. Specifically, we propose and analyze a class of adaptive dual averaging schemes in which agents only need to accumulate gradient feedback received from the whole system, without requiring any between-agent coordination. In the single-agent case, the adaptivity of the proposed method allows us to extend a range of existing results to problems with potentially unbounded delays between playing an action and receiving the corresponding feedback. In the mul…
The Australian National University（澳大利亚国立大学）；Nanjing University of Science and Technology（南京理工大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）；The University of Sydney（悉尼大学）
We study online linear regression problems in a distributed setting, where the data is spread over a network. In each round, each network node proposes a linear predictor, with the objective of fitting the \emph{network-wide} data. It then updates its predictor for the next round according to the received local feedback and information received from neighboring nodes. The predictions made at a given node are assessed through the notion of regret, defined as the difference between their cumulative network-wide square errors and those of the best off-line network-wide linear predictor. Various s…
Facebook Artificial Intelligence Research (FAIR)（脸书人工智能研究院（FAIR））；LAMSADE（LAMSADE实验室）；CNRS（法国国家科学研究中心）；Université Paris-Dauphine（巴黎多芬大学）；Université PSL（巴黎文理研究大学）
Choosing the right selection rate is a long standing issue in evolutionary computation. In the continuous unconstrained case, we prove mathematically that a single parent $μ=1$ leads to a sub-optimal simple regret in the case of the sphere function. We provide a theoretically-based selection rate $μ/λ$ that leads to better progress rates. With our choice of selection rate, we get a provable regret of order $O(λ^{-1})$ which has to be compared with $O(λ^{-2/d})$ in the case where $μ=1$. We complete our study with experiments to confirm our theoretical claims.
Hanyang University（汉阳大学）；University of Southern California（南加州大学）
In federated learning (FL), it is commonly assumed that all data are placed at clients in the beginning of machine learning (ML) optimization (i.e., offline learning). However, in many real-world applications, it is expected to proceed in an online fashion. To this end, online FL (OFL) has been introduced, which aims at learning a sequence of global models from decentralized streaming data such that the so-called cumulative regret is minimized. Combining online gradient descent and model averaging, in this framework, FedOGD is constructed as the counterpart of FedSGD in FL. While it can enjoy …
In this paper, we consider the problem of learning online to manage Demand Response (DR) resources. A typical DR mechanism requires the DR manager to assign a baseline to the participating consumer, where the baseline is an estimate of the counterfactual consumption of the consumer had it not been called to provide the DR service. A challenge in estimating baseline is the incentive the consumer has to inflate the baseline estimate. We consider the problem of learning online to estimate the baseline and to optimize the operating costs over a period of time under such incentives. We propose an o…
