Finite-time 4-expert prediction problem

Bayraktar, Erhan; Ekren, Ibrahim; Zhang, Xin

doi:10.1080/03605302.2020.1712418

Cited by 9 publications

(13 citation statements)

References 12 publications

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“…[Zhu14] first derived such a PDE to characterize the continuous-time limit of LEA, whose arguments were streamlined in [DK20b]. Exact solutions were obtained in special cases [BEZ20a,BEZ20b,DK20b], and more generally, approximate solutions were derived in [Rok17,KKW20a,KKW20b]. Follow-up works considered history-dependent experts [DC20,DK20a] and malicious experts [BPZ20,BEZ21].…”

Section: Differential Equations For Online Learningmentioning

confidence: 99%

PDE-Based Optimal Strategy for Unconstrained Online Learning

Zhang¹,

Cutkosky²,

Paschalidis³

2022

Preprint

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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 such potential functions is ad hoc and heavily relies on guessing. In this paper, we present a framework that generates time-varying potential functions by solving a Partial Differential Equation (PDE). Our framework recovers some classical potentials, and more importantly provides a systematic approach to design new ones.The power of our framework is demonstrated through a concrete example. When losses are 1-Lipschitz, we design a novel OLO algorithm with anytime regret upper boundwhere C is a user-specified constant and u is any comparator whose norm is unknown and unbounded a priori. By constructing a matching lower bound, we further show that the leading order term, including the constant multiplier √ 2, is tight. To our knowledge, this is the first parameter-free algorithm with optimal leading constant. The obtained theoretical benefits are validated by experiments.

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Section: Differential Equations For Online Learningmentioning

confidence: 99%

PDE-Based Optimal Strategy for Unconstrained Online Learning

Zhang¹,

Cutkosky²,

Paschalidis³

2022

Preprint

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“…The comb adversary a c is then defined by Comb adversary a c : At each t ∈ [T ], the adversary assigns probability 1 2 to each of q c and −q c where (q c ) (i) = 1 if i is odd and (q c ) (i) = −1 if i is even. Gravin et al (2016) suggested that a c might be optimal asymptotically in T for any fixed N and Abbasi-Yadkori et al ( 2017) and Bayraktar et al (2019) showed that to be the case for N = 3 and 4 respectively.…”

Section: Heat Potentialsmentioning

confidence: 99%

“…Drenska and Kohn (2020) showed that, for any fixed N , the value function, in the scaling limit, is the unique solution of an associated nonlinear PDE. Bayraktar et al (2019) determined the closed-form solutions of the PDEs for N = 3 and 4.…”

Section: Introductionmentioning

confidence: 99%

New Potential-Based Bounds for Prediction with Expert Advice

Kobzar,

Kohn,

Wang

2019

Preprint

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This work addresses the classic machine learning problem of online prediction with expert advice. We consider the finite-horizon version of this zero-sum, two-person game.Using verification arguments from optimal control theory, we view the task of finding better lower and upper bounds on the value of the game (regret) as the problem of finding better sub-and supersolutions of certain partial differential equations (PDEs). These sub-and supersolutions serve as the potentials for player and adversary strategies, which lead to the corresponding bounds. Our techniques extend in a nonasymptotic setting the recent work of Drenska and Kohn (J. Nonlinear Sci. 2020), which showed that the asymptotically optimal value function is the unique solution of an associated nonlinear PDE.To get explicit bounds, we use closed-form solutions of specific PDEs. Our bounds hold for any fixed number of experts and any time-horizon; in certain regimes (which we identify) they improve upon the previous state-of-the-art.For up to three experts, our bounds provide the asymptotically optimal leading order term. Therefore, in this setting, we provide a continuum perspective on recent work on optimal strategies.

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“…In the adversarial setting, the advice of experts is chosen by an adversary so as to maximize the regret of the forecaster, and therefore the problem can be viewed as a zero-sum game between the forecaster and the adversary (see e.g. [12] [9] [8] [5] [4]). In the stochastic setting, the losses of each expert are drawn independent and identically distributed (i.i.d.)…”

Section: Introductionmentioning

confidence: 99%

Malicious Experts versus the multiplicative weights algorithm in online prediction

Bayraktar

Poor

Zhang

2020

Preprint

Self Cite

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We consider a prediction problem with two experts and a forecaster. We assume that one of the experts is honest and makes correct prediction with probability µ at each round. The other one is malicious, who knows true outcomes at each round and makes predictions in order to maximize the loss of the forecaster. Assuming the forecaster adopts the classical multiplicative weights algorithm, we find an upper bound (3.12) for the value function of the malicious expert, and also a lower bound (4.1). Our results imply that the multiplicative weights algorithm cannot resist the corruption of malicious experts. We also show that an adaptive multiplicative weights algorithm is asymptotically optimal for the forecaster, and hence more resistant to the corruption of malicious experts.

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Finite-time 4-expert prediction problem

Cited by 9 publications

References 12 publications

PDE-Based Optimal Strategy for Unconstrained Online Learning

PDE-Based Optimal Strategy for Unconstrained Online Learning

New Potential-Based Bounds for Prediction with Expert Advice

Malicious Experts versus the multiplicative weights algorithm in online prediction

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