Concetti Chiave
The authors propose novel online algorithms, Sword and Sword++, that can achieve problem-dependent dynamic regret bounds in non-stationary environments. The bounds scale with the gradient variation and the cumulative loss of the comparator sequence, which are at most O(T) but could be much smaller in benign environments, thereby outperforming the minimax optimal rate.
Sintesi
The paper investigates online convex optimization in non-stationary environments and focuses on the dynamic regret as the performance measure. The authors introduce two novel online algorithms, Sword and Sword++, that can exploit smoothness and replace the dependence on the time horizon T in dynamic regret with problem-dependent quantities.
Key highlights:
- The authors propose the Sword algorithm that achieves favorable problem-dependent guarantees under the multi-gradient feedback model, where the player can query gradient information multiple times per round.
- The authors then introduce the Sword++ algorithm, which improves upon Sword by requiring only one gradient per iteration, making it suitable for the more challenging one-gradient feedback model.
- The authors establish that their algorithms enjoy an O(√(1 + PT + min{VT, FT})(1 + PT)) dynamic regret, where PT is the path length of the comparator sequence, VT is the gradient variation, and FT is the cumulative loss of the comparator sequence.
- Compared to the minimax optimal rate of O(√T(1 + PT)), the authors' results replace the dependence on T by the problem-dependent quantity PT + min{VT, FT}, leading to much tighter bounds in benign environments while safeguarding the same guarantee in the worst case.
- The authors propose a collaborative online ensemble framework, which is a key technical contribution enabling the algorithms to achieve the desired problem-dependent dynamic regret with only one gradient per iteration.
Statistiche
The path length PT = ∑T
t=2 ∥ut - ut-1∥2 reflects the non-stationarity of the environments.
The gradient variation VT = ∑T
t=2 supx∈X ∥∇ft(x) - ∇ft-1(x)∥2^2 measures the cumulative variation in gradients of the loss functions.
The cumulative loss of the comparator sequence FT = ∑T
t=1 ft(ut).
Citazioni
"We believe the framework can be useful for broader problems."
"Our results are adaptive to the intrinsic difficulty of the problem, since the bounds are tighter than existing results for easy problems and meanwhile safeguard the same rate in the worst case."