## Oracle-Efficient Online Learning for Smoothed Adversaries

### Nika Haghtalab · Yanjun Han · Abhishek Shetty · Kunhe Yang

##### Hall J #824

Keywords: [ Smoothed Analysis ] [ Computational Efficiency ] [ Online Learning ]

[ Abstract ]
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Thu 1 Dec 2 p.m. PST — 4 p.m. PST

Abstract: We study the design of computationally efficient online learning algorithms under smoothed analysis. In this setting, at every step, an adversary generates a sample from an adaptively chosen distribution whose density is upper bounded by $1/\sigma$ times the uniform density. Given access to an offline optimization (ERM) oracle, we give the first computationally efficient online algorithms whose sublinear regret depends only on the pseudo/VC dimension $d$ of the class and the smoothness parameter $\sigma$. In particular, we achieve \emph{oracle-efficient} regret bounds of $O ( \sqrt{T d\sigma^{-1}} )$ for learning real-valued functions and $O ( \sqrt{T d\sigma^{-\frac{1}{2}} } )$ for learning binary-valued functions. Our results establish that online learning is computationally as easy as offline learning, under the smoothed analysis framework. This contrasts the computational separation between online learning with worst-case adversaries and offline learning established by [HK16].Our algorithms also achieve improved bounds for some settings with binary-valued functions and worst-case adversaries. These include an oracle-efficient algorithm with $O ( \sqrt{T(d |\mathcal{X}|)^{1/2} })$ regret that refines the earlier $O ( \sqrt{T|\mathcal{X}|})$ bound of [DS16] for finite domains, and an oracle-efficient algorithm with $O(T^{3/4} d^{1/2})$ regret for the transductive setting.

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