We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time an adversary chooses an input distribution with density function bounded above by $\tfrac{1}σ$ times that of the uniform distribution; nature then samples an input from this distribution. Crucially, our results hold for {\em adaptive} adversaries that can choose an input distribution based on the decisions of the algorithm and the realizations of the inputs in the previous time steps.
This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of adaptive adversaries to the simpler case of oblivious adversaries. We apply this technique to prove strong smoothed guarantees for three problems:
-Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of $σ$-smooth distributions and the hypothesis class has VC dimension $d$. We bound the regret by $\tilde{O}\big(\sqrt{T d\ln(1/σ)} + d\sqrt{\ln(T/σ)}\big)$. This answers open questions of [RST11,Hag18].
-Online discrepancy minimization: We consider the online Komlós problem, where the input is generated from an adaptive sequence of $σ$-smooth and isotropic distributions on the $\ell_2$ unit ball. We bound the $\ell_\infty$ norm of the discrepancy vector by $\tilde{O}\big(\ln^2\!\big( \frac{nT}σ\big) \big)$.
-Dispersion in online optimization: We consider online optimization of piecewise Lipschitz functions where functions with $\ell$ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is $\big( σ/{\sqrt{T\ell}}, \tilde O\big(\sqrt{T\ell} \big)\big)$-dispersed. This matches the parameters of [BDV18] for oblivious adversaries, up to log factors.
In this paper, we study oracle-efficient algorithms for beyond worst-case analysis of online learning. We focus on two settings. First, the smoothed analysis setting of [RST11,HRS22] where an adversary is constrained to generating samples from distributions whose density is upper bounded by $1/σ$ times the uniform density. Second, the setting of $K$-hint transductive learning, where the learner is given access to $K$ hints per time step that are guaranteed to include the true instance. We give the first known oracle-efficient algorithms for both settings that depend only on the pseudo (or VC) …
University of Colorado, Boulder（科罗拉多大学博尔德分校）；University of Virginia（弗吉尼亚大学）
We investigate the sparse linear contextual bandit problem where the parameter $θ$ is sparse. To relieve the sampling inefficiency, we utilize the "perturbed adversary" where the context is generated adversarilly but with small random non-adaptive perturbations. We prove that the simple online Lasso supports sparse linear contextual bandit with regret bound $\mathcal{O}(\sqrt{kT\log d})$ even when $d \gg T$ where $k$ and $d$ are the number of effective and ambient dimension, respectively. Compared to the recent work from Sivakumar et al. (2020), our analysis does not rely on the precondition p…
Practical and pervasive needs for robustness and privacy in algorithms have inspired the design of online adversarial and differentially private learning algorithms. The primary quantity that characterizes learnability in these settings is the Littlestone dimension of the class of hypotheses [Ben-David et al., 2009, Alon et al., 2019]. This characterization is often interpreted as an impossibility result because classes such as linear thresholds and neural networks have infinite Littlestone dimension. In this paper, we apply the framework of smoothed analysis [Spielman and Teng, 2004], in whic…
We study a fundamental question concerning adversarial noise models in statistical problems where the algorithm receives i.i.d. draws from a distribution $\mathcal{D}$. The definitions of these adversaries specify the type of allowable corruptions (noise model) as well as when these corruptions can be made (adaptivity); the latter differentiates between oblivious adversaries that can only corrupt the distribution $\mathcal{D}$ and adaptive adversaries that can have their corruptions depend on the specific sample $S$ that is drawn from $\mathcal{D}$. In this work, we investigate whether oblivio…
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Facebook Reality Labs（Facebook现实实验室）；University of Oxford（牛津大学）
Deep neural networks are vulnerable to small input perturbations known as adversarial attacks. Inspired by the fact that these adversaries are constructed by iteratively minimizing the confidence of a network for the true class label, we propose the anti-adversary layer, aimed at countering this effect. In particular, our layer generates an input perturbation in the opposite direction of the adversarial one and feeds the classifier a perturbed version of the input. Our approach is training-free and theoretically supported. We verify the effectiveness of our approach by combining our layer with…
House of Energy Markets and Finance（能源市场与金融研究院）；University of Duisburg-Essen（杜伊斯堡-埃森大学）
We present a winning method of the IEEE DataPort Competition on Day-Ahead Electricity Demand Forecasting: Post-COVID Paradigm. The day-ahead load forecasting approach is based on online forecast combination of multiple point prediction models. It contains four steps: i) data cleaning and preprocessing, ii) a holiday adjustment procedure, iii) training of individual forecasting models, iv) forecast combination by smoothed Bernstein Online Aggregation (BOA). The approach is flexible and can quickly adopt to new energy system situations as they occurred during and after COVID-19 shutdowns. The po…
KAIST（韩国科学技术院）；Vector Institute for Artificial Intelligence（向量人工智能研究所）；Kakao Enterprise；Inha University（仁荷大学）
Randomized smoothing is currently a state-of-the-art method to construct a certifiably robust classifier from neural networks against $\ell_2$-adversarial perturbations. Under the paradigm, the robustness of a classifier is aligned with the prediction confidence, i.e., the higher confidence from a smoothed classifier implies the better robustness. This motivates us to rethink the fundamental trade-off between accuracy and robustness in terms of calibrating confidences of a smoothed classifier. In this paper, we propose a simple training scheme, coined SmoothMix, to control the robustness of sm…
Nanjing University（南京大学）；the University of Iowa（爱荷华大学）
In this paper, we investigate an online prediction strategy named as Discounted-Normal-Predictor (Kapralov and Panigrahy, 2010) for smoothed online convex optimization (SOCO), in which the learner needs to minimize not only the hitting cost but also the switching cost. In the setting of learning with expert advice, Daniely and Mansour (2019) demonstrate that Discounted-Normal-Predictor can be utilized to yield nearly optimal regret bounds over any interval, even in the presence of switching costs. Inspired by their results, we develop a simple algorithm for SOCO: Combining online gradient desc…
Columbia University（哥伦比亚大学）；UC Berkeley（加州大学伯克利分校）；Google（谷歌）
We provide the first sub-linear space and sub-linear regret algorithm for online learning with expert advice (against an oblivious adversary), addressing an open question raised recently by Srinivas, Woodruff, Xu and Zhou (STOC 2022). We also demonstrate a separation between oblivious and (strong) adaptive adversaries by proving a linear memory lower bound of any sub-linear regret algorithm against an adaptive adversary. Our algorithm is based on a novel pool selection procedure that bypasses the traditional wisdom of leader selection for online learning, and a generic reduction that transform…
Georgia Institute of Technology（佐治亚理工学院）；California Institute of Technology（加州理工学院）
We examine the problem of smoothed online optimization, where a decision maker must sequentially choose points in a normed vector space to minimize the sum of per-round, non-convex hitting costs and the costs of switching decisions between rounds. The decision maker has access to a black-box oracle, such as a machine learning model, that provides untrusted and potentially inaccurate predictions of the optimal decision in each round. The goal of the decision maker is to exploit the predictions if they are accurate, while guaranteeing performance that is not much worse than the hindsight optimal…
Constrained Reinforcement Learning with Smoothed Log Barrier Function
University of Freiburg（弗莱堡大学）；BrainLinks-BrainTools, University of Freiburg（弗莱堡大学BrainLinks-BrainTools）；Fraunhofer Institute for Solar Energy Systems（弗劳恩霍夫太阳能系统研究所）
