Math @ Duke
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Publications [#361427] of Holden Lee
Papers Published
- Ghai, U; Lee, H; Singh, K; Zhang, C; Zhang, Y, No-Regret Prediction in Marginally Stable Systems
(February, 2020)
(last updated on 2022/08/06)
Abstract: We consider the problem of online prediction in a marginally stable linear
dynamical system subject to bounded adversarial or (non-isotropic) stochastic
perturbations. This poses two challenges. Firstly, the system is in general
unidentifiable, so recent and classical results on parameter recovery do not
apply. Secondly, because we allow the system to be marginally stable, the state
can grow polynomially with time; this causes standard regret bounds in online
convex optimization to be vacuous. In spite of these challenges, we show that
the online least-squares algorithm achieves sublinear regret (improvable to
polylogarithmic in the stochastic setting), with polynomial dependence on the
system's parameters. This requires a refined regret analysis, including a
structural lemma showing the current state of the system to be a small linear
combination of past states, even if the state grows polynomially. By applying
our techniques to learning an autoregressive filter, we also achieve
logarithmic regret in the partially observed setting under Gaussian noise, with
polynomial dependence on the memory of the associated Kalman filter.
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