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-specified constant and $u$ is any comparator unknown and unbounded a priori. Such a bound attains an optimal loss-regret trade-off without the impractical doubling trick. Moreover, a matching lower bound shows that the leading order term, including the constant multiplier $\sqrt{2}$, is tight. To our knowledge, the proposed algorithm is the first to achieve such optimalities.
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.
We study algorithms for online linear optimization in Hilbert spaces, focusing on the case where the player is unconstrained. We develop a novel characterization of a large class of minimax algorithms, recovering, and even improving, several previous results as immediate corollaries. Moreover, using our tools, we develop an algorithm that provides a regret bound of $\mathcal{O}\Big(U \sqrt{T \log(U \sqrt{T} \log^2 T +1)}\Big)$, where $U$ is the $L_2$ norm of an arbitrary comparator and both $T$ and $U$ are unknown to the player. This bound is optimal up to $\sqrt{\log \log T}$ terms. When $T$ …
We propose an online convex optimization algorithm (RescaledExp) that achieves optimal regret in the unconstrained setting without prior knowledge of any bounds on the loss functions. We prove a lower bound showing an exponential separation between the regret of existing algorithms that require a known bound on the loss functions and any algorithm that does not require such knowledge. RescaledExp matches this lower bound asymptotically in the number of iterations. RescaledExp is naturally hyperparameter-free and we demonstrate empirically that it matches prior optimization algorithms that requ…
We consider a variant of online convex optimization in which both the instances (input vectors) and the comparator (weight vector) are unconstrained. We exploit a natural scale invariance symmetry in our unconstrained setting: the predictions of the optimal comparator are invariant under any linear transformation of the instances. Our goal is to design online algorithms which also enjoy this property, i.e. are scale-invariant. We start with the case of coordinate-wise invariance, in which the individual coordinates (features) can be arbitrarily rescaled. We give an algorithm, which achieves es…
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 provide algorithms that guarantee regret $R_T(u)\le \tilde O(G\|u\|^3 + G(\|u\|+1)\sqrt{T})$ or $R_T(u)\le \tilde O(G\|u\|^3T^{1/3} + GT^{1/3}+ G\|u\|\sqrt{T})$ for online convex optimization with $G$-Lipschitz losses for any comparison point $u$ without prior knowledge of either $G$ or $\|u\|$. Previous algorithms dispense with the $O(\|u\|^3)$ term at the expense of knowledge of one or both of these parameters, while a lower bound shows that some additional penalty term over $G\|u\|\sqrt{T}$ is necessary. Previous penalties were exponential while our bounds are polynomial in all quantitie…
University of California, Berkeley（加州大学伯克利分校）；Microsoft Research（微软研究院）
Motivated by applications to online learning in sparse estimation and Bayesian optimization, we consider the problem of online unconstrained nonsubmodular minimization with delayed costs in both full information and bandit feedback settings. In contrast to previous works on online unconstrained submodular minimization, we focus on a class of nonsubmodular functions with special structure, and prove regret guarantees for several variants of the online and approximate online bandit gradient descent algorithms in static and delayed scenarios. We derive bounds for the agent's regret in the full in…
We design and analyze minimax-optimal algorithms for online linear optimization games where the player's choice is unconstrained. The player strives to minimize regret, the difference between his loss and the loss of a post-hoc benchmark strategy. The standard benchmark is the loss of the best strategy chosen from a bounded comparator set. When the the comparison set and the adversary's gradients satisfy L_infinity bounds, we give the value of the game in closed form and prove it approaches sqrt(2T/pi) as T -> infinity. Interesting algorithms result when we consider soft constraints on the com…
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…
This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the $\ell_0$-pseudo-norm, instead finding that directly applying its proximal ope…
We introduce several new black-box reductions that significantly improve the design of adaptive and parameter-free online learning algorithms by simplifying analysis, improving regret guarantees, and sometimes even improving runtime. We reduce parameter-free online learning to online exp-concave optimization, we reduce optimization in a Banach space to one-dimensional optimization, and we reduce optimization over a constrained domain to unconstrained optimization. All of our reductions run as fast as online gradient descent. We use our new techniques to improve upon the previously best regret …
John A. Paulson School of Engineering and Applied Sciences（约翰·A·保尔森工程与应用科学学院）；Harvard University（哈佛大学）
We analyze the dynamics of an online algorithm for independent component analysis in the high-dimensional scaling limit. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measure of the target feature vector and the estimates provided by the algorithm will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numerical solutions of this PDE, which involves two spatial variables and one time variable, can be efficiently obtained. These solutions provide…
We consider the problem of unconstrained online convex optimization (OCO) with sub-exponential noise, a strictly more general problem than the standard OCO. In this setting, the learner receives a subgradient of the loss functions corrupted by sub-exponential noise and strives to achieve optimal regret guarantee, without knowledge of the competitor norm, i.e., in a parameter-free way. Recently, Cutkosky and Boahen (COLT 2017) proved that, given unbounded subgradients, it is impossible to guarantee a sublinear regret due to an exponential penalty. This paper shows that it is possible to go arou…
Texas Tech University（德州理工大学）；North Dakota State University（北达科他州立大学）；New Jersey Institute of Technology（新泽西理工学院）
Stochastic gradient descent (SGD) and projected stochastic gradient descent (PSGD) are scalable algorithms to compute model parameters in unconstrained and constrained optimization problems. In comparison with SGD, PSGD forces its iterative values into the constrained parameter space via projection. From a statistical point of view, this paper studies the limiting distribution of PSGD-based estimate when the true parameters satisfy some linear-equality constraints. Our theoretical findings reveal the role of projection played in the uncertainty of the PSGD-based estimate. As a byproduct, we pr…
Optimizing over the stationary distribution of stochastic differential equations (SDEs) is computationally challenging. A new forward propagation algorithm has been recently proposed for the online optimization of SDEs. The algorithm solves an SDE, derived using forward differentiation, which provides a stochastic estimate for the gradient. The algorithm continuously updates the SDE model's parameters and the gradient estimate simultaneously. This paper studies the convergence of the forward propagation algorithm for nonlinear dissipative SDEs. We leverage the ergodicity of this class of nonli…
We present a novel method for convex unconstrained optimization that, without any modifications, ensures: (i) accelerated convergence rate for smooth objectives, (ii) standard convergence rate in the general (non-smooth) setting, and (iii) standard convergence rate in the stochastic optimization setting. To the best of our knowledge, this is the first method that simultaneously applies to all of the above settings. At the heart of our method is an adaptive learning rate rule that employs importance weights, in the spirit of adaptive online learning algorithms (Duchi et al., 2011; Levy, 2017), …
Safe and Efficient Online Convex Optimization with Linear Budget Constraints and Partial Feedback
Research Institute of China Electronics Technology Group（中国电子科技集团公司研究所）；School of Information Science and Technology（信息科学与技术学院）；ShanghaiTech University（上海科技大学）
MIT（麻省理工学院）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；ENSAE（法国国立统计与经济管理学院）；IP Paris（巴黎理工学院）；CREST（经济与统计研究中心）；IBM Research（IBM研究院）
Sampling from a target measure whose density is only known up to a normalization constant is a fundamental problem in computational statistics and machine learning. In this paper, we present a new optimization-based method for sampling called mollified interaction energy descent (MIED). MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs). These energies rely on mollifier functions -- smooth approximations of the Dirac delta originated from PDE theory. We show that as the mollifier approaches the Dirac delta, the MIE converges to the chi-s…
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…
We consider a sequence of repeated prediction games and formally pass to the limit. The supersolutions of the resulting non-linear parabolic partial differential equation are closely related to the potential functions in the sense of N.\,Cesa-Bianci, G.\,Lugosi (2003). Any such supersolution gives an upper bound for forecaster's regret and suggests a potential-based prediction strategy, satisfying the Blackwell condition. A conventional upper bound for the worst-case regret is justified by a simple verification argument.
Tadpole: Autoencoders as Foundation Models for 3D PDEs with Online Learning
TUM School of Computation, Information and Technology（慕尼黑技术大学计算、信息与技术学院）；Technical University of Munich, Garching, Germany（慕尼黑技术大学，慕尼黑，德国）；MCML, Munich Center for Machine Learning, Munich, Germany（慕尼黑机器学习中心，慕尼黑，德国）
We consider a basic problem at the interface of two fundamental fields: submodular optimization and online learning. In the online unconstrained submodular maximization (online USM) problem, there is a universe $[n]=\{1,2,...,n\}$ and a sequence of $T$ nonnegative (not necessarily monotone) submodular functions arrive over time. The goal is to design a computationally efficient online algorithm, which chooses a subset of $[n]$ at each time step as a function only of the past, such that the accumulated value of the chosen subsets is as close as possible to the maximum total value of a fixed sub…
TU Darmstadt（达姆施塔特工业大学）；University of British Columbia（不列颠哥伦比亚大学）
We consider online convex optimization with a zero-order oracle feedback. In particular, the decision maker does not know the explicit representation of the time-varying cost functions, or their gradients. At each time step, she observes the value of the corresponding cost function evaluated at her chosen action (zero-order oracle). The objective is to minimize the regret, that is, the difference between the sum of the costs she accumulates and that of a static optimal action had she known the sequence of cost functions a priori. We present a novel algorithm to minimize regret in unconstrained…
CP-PINNs: Data-Driven Changepoints Detection in PDEs Using Online Optimized Physics-Informed Neural Networks
Stony Brook University（石溪大学）；Center of Excellence in Wireless and Information Technology（无线与信息技术卓越中心）；Institute for Advanced Computational Science（高级计算科学研究所）
We study the Extended Kalman Filter in constant dynamics, offering a bayesian perspective of stochastic optimization. We obtain high probability bounds on the cumulative excess risk in an unconstrained setting. In order to avoid any projection step we propose a two-phase analysis. First, for linear and logistic regressions, we prove that the algorithm enters a local phase where the estimate stays in a small region around the optimum. We provide explicit bounds with high probability on this convergence time. Second, for generalized linear regressions, we provide a martingale analysis of the exc…
State University of New York at Albany（纽约州立大学奥尔巴尼分校）；City University of Hong Kong（香港城市大学）
Pairwise learning usually refers to a learning task which involves a loss function depending on pairs of examples, among which most notable ones include ranking, metric learning and AUC maximization. In this paper, we study an online algorithm for pairwise learning with a least-square loss function in an unconstrained setting of a reproducing kernel Hilbert space (RKHS), which we refer to as the Online Pairwise lEaRning Algorithm (OPERA). In contrast to existing works \cite{Kar,Wang} which require that the iterates are restricted to a bounded domain or the loss function is strongly-convex, OPE…
We show how to take any two parameter-free online learning algorithms with different regret guarantees and obtain a single algorithm whose regret is the minimum of the two base algorithms. Our method is embarrassingly simple: just add the iterates. This trick can generate efficient algorithms that adapt to many norms simultaneously, as well as providing diagonal-style algorithms that still maintain dimension-free guarantees. We then proceed to show how a variant on this idea yields a black-box procedure for generating optimistic online learning algorithms. This yields the first optimistic regr…
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）；Department of Mathematics, Duke University（杜克大学数学系）
We present a framework for analyzing the exact dynamics of a class of online learning algorithms in the high-dimensional scaling limit. Our results are applied to two concrete examples: online regularized linear regression and principal component analysis. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measures of the target feature vector and its estimates provided by the algorithms will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numeric…
University of Minnesota（明尼苏达大学）；Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）
This work addresses a classic problem of online prediction with expert advice. We assume an adversarial opponent, and we consider both the finite-horizon and random-stopping versions of this zero-sum, two-person game. Focusing on an appropriate continuum limit and using methods from optimal control, we characterize the value of the game as the viscosity solution of a certain nonlinear partial differential equation. The analysis also reveals the predictor's and the opponent's minimax optimal strategies. Our work provides, in particular, a continuum perspective on recent work of Gravin, Peres, a…
Unsupervised video object segmentation aims to automatically segment moving objects over an unconstrained video without any user annotation. So far, only few unsupervised online methods have been reported in literature and their performance is still far from satisfactory, because the complementary information from future frames cannot be processed under online setting. To solve this challenging problem, in this paper, we propose a novel Unsupervised Online Video Object Segmentation (UOVOS) framework by construing the motion property to mean moving in concurrence with a generic object for segme…
EvoPINN: Agentic Discovery of Executable Algorithms for Physics-Informed Neural Networks
School of Advanced Interdisciplinary Sciences, University of Chinese Academy of Sciences（中国科学院大学先进交叉科学学院）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）
Tiny machine learning (TinyML) in IoT systems exploits MCUs as edge devices for data processing. However, traditional TinyML methods can only perform inference, limited to static environments or classes. Real case scenarios usually work in dynamic environments, thus drifting the context where the original neural model is no more suitable. For this reason, pre-trained models reduce accuracy and reliability during their lifetime because the data recorded slowly becomes obsolete or new patterns appear. Continual learning strategies maintain the model up to date, with runtime fine-tuning of the pa…
University of Twente（特文特大学）；University of Oxford（牛津大学）；Bocconi University（博科尼大学）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；IN2P3 Computing Center / CNRS（法国国家核物理与粒子物理研究所计算中心/法国国家科学研究中心）
