  In this paper, we study the online learning of real-valued functions where
the hidden function is known to have certain smoothness properties.
Specifically, for $q \ge 1$, let $\mathcal F_q$ be the class of absolutely
continuous functions $f: [0,1] \to \mathbb R$ such that $\|f'\|_q \le 1$. For
$q \ge 1$ and $d \in \mathbb Z^+$, let $\mathcal F_{q,d}$ be the class of
functions $f: [0,1]^d \to \mathbb R$ such that any function $g: [0,1] \to
\mathbb R$ formed by fixing all but one parameter of $f$ is in $\mathcal F_q$.
For any class of real-valued functions $\mathcal F$ and $p>0$, let
$\text{opt}_p(\mathcal F)$ be the best upper bound on the sum of
$p^{\text{th}}$ powers of absolute prediction errors that a learner can
guarantee in the worst case. In the single-variable setup, we find new bounds
for $\text{opt}_p(\mathcal F_q)$ that are sharp up to a constant factor. We
show for all $\varepsilon \in (0, 1)$ that
$\text{opt}_{1+\varepsilon}(\mathcal{F}_{\infty}) =
\Theta(\varepsilon^{-\frac{1}{2}})$ and
$\text{opt}_{1+\varepsilon}(\mathcal{F}_q) =
\Theta(\varepsilon^{-\frac{1}{2}})$ for all $q \ge 2$. We also show for
$\varepsilon \in (0,1)$ that $\text{opt}_2(\mathcal
F_{1+\varepsilon})=\Theta(\varepsilon^{-1})$. In addition, we obtain new exact
results by proving that $\text{opt}_p(\mathcal F_q)=1$ for $q \in (1,2)$ and $p
\ge 2+\frac{1}{q-1}$. In the multi-variable setup, we establish inequalities
relating $\text{opt}_p(\mathcal F_{q,d})$ to $\text{opt}_p(\mathcal F_q)$ and
show that $\text{opt}_p(\mathcal F_{\infty,d})$ is infinite when $p<d$ and
finite when $p>d$. We also obtain sharp bounds on learning $\mathcal
F_{\infty,d}$ for $p < d$ when the number of trials is bounded.
