Parameter-Free Online Convex Optimization with Sub-Exponential Noise

Kwang Sung Jun, Francesco Orabona

Research output: Contribution to journalConference articlepeer-review

15 Scopus citations


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 around the lower bound by allowing the observed subgradients to be unbounded via stochastic noise. However, the presence of unbounded noise in unconstrained OCO is challenging; existing algorithms do not provide near-optimal regret bounds or fail to have a guarantee. So, we design a novel parameter-free OCO algorithm for Banach space, which we call BANCO, via a reduction to betting on noisy coins. We show that BANCO achieves the optimal regret rate in our problem. Finally, we show the application of our results to obtain a parameter-free locally private stochastic subgradient descent algorithm, and the connection to the law of iterated logarithms.

Original languageEnglish (US)
Pages (from-to)1802-1823
Number of pages22
JournalProceedings of Machine Learning Research
StatePublished - 2019
Externally publishedYes
Event32nd Conference on Learning Theory, COLT 2019 - Phoenix, United States
Duration: Jun 25 2019Jun 28 2019


  • differentially-private stochastic subgradient descent
  • online convex optimization
  • Parameter-free
  • unconstrained

ASJC Scopus subject areas

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability


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