Abstract
Most traditional online learning algorithms are based on variants of mirror descent or follow-the-leader. In this chapter, we present an online algorithm based on a completely different approach, tailored for transductive settingsTransductive setting—( Transductive online learning—(, which combines “random playout” and randomized rounding of loss subgradients. As an application of our approach, we present the first computationally efficient online algorithm for collaborative filtering with trace-norm constrained matrices. As a second application, we solve an open question linking batch learning and transductive online learning.
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Notes
- 1.
Specifically, we divide the rounds into r consecutive epochs, such that epoch i consists of 2i rounds, and use Theorem 16.3 with confidence \(\delta \prime =\delta /{2}^{i+1}\), and a union bound, to get a regret bound of \(\mathcal{O}(\mathcal{R}_{{2}^{i}}(\mathcal{F}) + \sqrt{\left (i +\log (1/\delta ) \right ) {2}^{i}})\) over any epoch i. In the typical case where \(\mathcal{R}_{T}(\mathcal{F}) = \mathcal{O}(\sqrt{T})\), summing over i = 1, …, r where \(r =\log _{2}(T + 1) - 1\) yields a total regret bound of order \(\mathcal{O}(\sqrt{\log (T/\delta )T})\). Up to log factors, this is the same bound as if T were known in advance.
- 2.
Formally, at each step t: (1) the adversary chooses and reveals the next element π t of the permutation; (2) the forecaster chooses \(p_{t} \in \mathcal{P}\) and simultaneously the adversary chooses \(y_{t} \in \mathcal{Y}\).
- 3.
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The first author acknowledges partial support by the PASCAL2 NoE under EC grant FP7-216886.
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Cesa-Bianchi, N., Shamir, O. (2013). Efficient Transductive Online Learning via Randomized Rounding. In: Schölkopf, B., Luo, Z., Vovk, V. (eds) Empirical Inference. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41136-6_16
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