Discounted regret measures the performance of an online learner by assigning exponentially decaying weights to past losses, reflecting the greater importance of recent losses in non-stationary online learning. While previous work obtains uniform discounted regret bounds over a continuous range of discount factors using a specialized meta-algorithm, its results are limited to convex functions. In this paper, we demonstrate that strongly adaptive regret bounds can directly imply uniform discounted regret guarantees for convex, strongly convex, and exp-concave loss functions. Our key observation is that discounted regret against a fixed comparator admits an exact decomposition into a convex combination of interval regrets. For convex functions, this reduction yields an $O(\sqrt{\log T/(1-λ)})$ discounted regret bound for all discount factors $λ\in(0,1)$, matching the previous rate over a broader range of discount factors. For strongly convex and exp-concave functions, we provide regret bounds of $O\left(\log T\left(1+\log\frac{1}{1-λ}\right)\right)$.
Leo Betthauser, Yingqi Liu, Uday Shah, Jishang Wei, Sainyam Galhotra, Liang Gou, Vedant Dharnidharka, Hao Yang
Many log-analysis tasks involve choosing among known options, such as detecting anomalous sessions or identifying attack techniques, and benefit from calibrated probabilities. We present Lev-3B, an open 3B-parameter decision model built on Granite-4.1-3B-Base with a shared pointer readout, and study its training recipe. The recipe has two key parts. First, training follows a curriculum: the model learns bounded natural-language decisions, using no log data, before it sees any logs. Second, the log stage mixes the target tasks with general log data spanning 35 auxiliary tasks unrelated to either evaluation task. We evaluate one shared checkpoint on sources seen in training and on sources excluded from training, calibration and model selection. The two-stage recipe transfers substantially better than training on log data alone. Adding general log data improves performance on excluded sources. At a fraction of their serving cost, Lev-3B outperforms prompted frontier models on anomaly detection, including on an excluded logging system, but trails them on attack-technique classification. Its probabilities are well calibrated on familiar sources, but on the excluded logging system it confidently misses anomalies. We release the checkpoint, recipe, data manifests, and prediction-level artifacts.
In this paper, we investigate decentralized online convex optimization with compressed communication, where $n$ learners over a network jointly minimize a sequence of global loss functions based only on their local losses and compressed messages from their neighbors. The best known regret bounds for this problem are $O(ω^{-1/2}nρ^{-1}\sqrt{T\ln n})$ and $O(ω^{-1}nρ^{-2}\ln n\ln T)$ for convex and strongly convex loss functions, respectively, where $ρ\in(0,1)$ is the spectral gap of the communication matrix and $ω\in(0,1]$ is the compression quality factor. Compared with the corresponding lower bounds of $Ω(ω^{-1/2}nρ^{-1/4}\sqrt{T})$ and $Ω(ω^{-1}nρ^{-1/2}\ln T)$, these upper bounds exhibit polynomial gaps in their dependence on $ρ^{-1}$. To close these gaps, we first develop Ascent-gossip, which combines an online compression strategy with a Chebyshev recursion and attains a faster consensus rate than existing compressed gossip methods. Furthermore, by adopting Ascent-gossip, we propose an online algorithm that achieves optimal regret bounds of $\widetilde O(ω^{-1/2}nρ^{-1/4}\sqrt{T})$ and $\widetilde O(ω^{-1}nρ^{-1/2}\ln T)$ for convex and strongly convex loss functions, respectively. Up to logarithmic factors in $n$ and $ρ^{-1}$, these results match the lower bounds in terms of $T$, $n$, $ρ$, and $ω$.