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We consider the online linear optimization problem, where at every step the algorithm plays a point x_t in the unit ball, and suffers loss for some cost vector c_t that is then revealed to the algorithm. Recent work showed that if an algorithm receives a "hint" h_t that has non-trivial correlation with c_t before it plays x_t, then it can achieve a logarithmic regret guarantee, improving on the classical sqrt(T) bound. In this work, we study the question of whether an algorithm really requires a hint at every time step. Somewhat surprisingly, we show that an algorithm can obtain logarithmic regret with just O(sqrt(T)) hints under a natural query model. We give two applications of our result, to the well-studied setting of optimistic regret bounds and to the problem of online learning with abstention.
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This content will become publicly available on December 22, 2026
Logarithmic Regret and Polynomial Scaling in Online Multi-Step-Ahead Prediction
This letter studies the problem of online multi-step-ahead prediction for unknown linear stochastic systems. Using conditional distribution theory, we derive an optimal parameterization of the prediction policy as a linear function of future inputs, past inputs, and past outputs. Based on this characterization, we propose an online least-squares algorithm to learn the policy and analyze its regret relative to the optimal model-based predictor. We show that the online algorithm achieves logarithmic regret with respect to the optimal Kalman filter in the multi-step setting. Furthermore, with new proof techniques, we establish an almost-sure regret bound that does not rely on fixed failure probabilities for sufficiently large horizons N. Finally, our analysis also reveals that, while the regret remains logarithmic in N, its constant factor grows polynomially with the prediction horizon H, with the polynomial order set by the largest Jordan block of eigenvalue 1 in the system matrix.
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- PAR ID:
- 10677014
- Publisher / Repository:
- IEEE
- Date Published:
- Journal Name:
- IEEE Control Systems Letters
- Volume:
- 9
- ISSN:
- 2475-1456
- Page Range / eLocation ID:
- 2981 to 2986
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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