Online Learning of Smooth Functions

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.