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07 February 2025
 
  » arxiv » 2301.01434

 Article overview



Online Learning of Smooth Functions
Jesse Geneson ; Ethan Zhou ;
Date 4 Jan 2023
AbstractIn 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 $f: [0,1] o mathbb R$ 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 $f: [0,1]^d o mathbb R$ such that any function $g: [0,1] o mathbb R$ 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 $ ext{opt}_p(mathcal F)$ be the best upper bound on the sum of $p^{ ext{th}}$ powers of absolute prediction errors that a learner can guarantee in the worst case. In the single-variable setup, we find new bounds for $ ext{opt}_p(mathcal F_q)$ that are sharp up to a constant factor. We show for all $varepsilon in (0, 1)$ that $ ext{opt}_{1+varepsilon}(mathcal{F}_{infty}) = Theta(varepsilon^{-frac{1}{2}})$ and $ ext{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 $ ext{opt}_2(mathcal F_{1+varepsilon})=Theta(varepsilon^{-1})$. In addition, we obtain new exact results by proving that $ ext{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 $ ext{opt}_p(mathcal F_{q,d})$ to $ ext{opt}_p(mathcal F_q)$ and show that $ ext{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.
Source arXiv, 2301.01434
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