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Article overview
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Online Learning of Smooth Functions | Jesse Geneson
; Ethan Zhou
; | Date: |
4 Jan 2023 | Abstract: | 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 $f: [0,1] o mathbb R$ 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 | Services: | Forum | Review | PDF | Favorites |
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