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- W4297662211 abstract "Empirical risk minimization (ERM) is a fundamental learning rule for statistical learning problems where the data is generated according to some unknown distribution $mathsf{P}$ and returns a hypothesis $f$ chosen from a fixed class $mathcal{F}$ with small loss $ell$. In the parametric setting, depending upon $(ell, mathcal{F},mathsf{P})$ ERM can have slow $(1/sqrt{n})$ or fast $(1/n)$ rates of convergence of the excess risk as a function of the sample size $n$. There exist several results that give sufficient conditions for fast rates in terms of joint properties of $ell$, $mathcal{F}$, and $mathsf{P}$, such as the margin condition and the Bernstein condition. In the non-statistical prediction with expert advice setting, there is an analogous slow and fast rate phenomenon, and it is entirely characterized in terms of the mixability of the loss $ell$ (there being no role there for $mathcal{F}$ or $mathsf{P}$). The notion of stochastic mixability builds a bridge between these two models of learning, reducing to classical mixability in a special case. The present paper presents a direct proof of fast rates for ERM in terms of stochastic mixability of $(ell,mathcal{F}, mathsf{P})$, and in so doing provides new insight into the fast-rates phenomenon. The proof exploits an old result of Kemperman on the solution to the general moment problem. We also show a partial converse that suggests a characterization of fast rates for ERM in terms of stochastic mixability is possible." @default.
- W4297662211 created "2022-09-30" @default.
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- W4297662211 date "2014-06-14" @default.
- W4297662211 modified "2023-09-27" @default.
- W4297662211 title "From Stochastic Mixability to Fast Rates" @default.
- W4297662211 doi "https://doi.org/10.48550/arxiv.1406.3781" @default.
- W4297662211 hasPublicationYear "2014" @default.
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