Online learning of smooth functions on $\mathbb{R}$
We study adversarial online learning of real-valued functions on $\mathbb{R}$. In each round the learner is queried at $x_t\in\mathbb{R}$, predicts $\hat y_t$, and then observes the true value $f(x_t)$; performance is measured by cumulative $p$-loss $\sum_{t\ge 1}|\hat y_t-f(x_t)|^p$. For the class \[ \mathcal{G}_q=\Bigl\{f:\mathbb{R}\to\mathbb{R}\ \text{absolutely continuous}:\ \int_{\mathbb{R}}|f'(x)|^q\,dx\le 1\Bigr\}, \] we show that the standard model becomes ill-posed on $\mathbb{R}$: for every $p\ge 1$ and $q>1$, an adversary can force infinite loss. Motivated by this obstruction, we analyze three modified learning scenarios that limit the influence of queries that are far from pre...
| Authors | Jesse Geneson, Kuldeep Singh, Alexander Wang |
|---|---|
| Primary category | cs.LG |
| Published | 4 April 2026 |
| Last updated at source | 4 April 2026 |
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