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On p.87 in this online Understanding Machine Learning book, the authors wrote:

Unlike the ERM paradigm discussed in previous chapters, we no longer just care about the empirical risk, $L_S(h)$, but we are willing to trade some of our bias toward low empirical risk with a bias toward classes for which $\epsilon_{n(h)}(m, w(n(h))\cdot\epsilon)$ is smaller, for the sake of a smaller estimation error.

With the output $h$ of SRM defined to be:

Output: $h\in\underset{h\in\mathcal{H}}{argmin}\ L_S(h)+\epsilon_{n(h)}(m, w(n(h))\cdot\delta)$.

Where $\epsilon_n(m, \delta) := min\ \lbrace\epsilon\in (0, 1): m_\mathcal{H_n}^{UC}(\epsilon, \delta)\le m \rbrace$, $w(n)$ is a function from $n\in\mathbb{N}\to[0, 1]$ such that $\sum_{n=1}^{\infty}w(n)\le1$ and $n(h):=min\lbrace n: h\in\mathcal{H}_n\rbrace$

That is the output learner $h$ will minimize $L_S(h)+\epsilon_{n(h)}(m, w(n(h))\cdot\delta)$.

However, $\epsilon_n$ as defined above is the lowest possible upper bound on the gap between $L_S(h)-L_\mathcal{D}(h)$ achieved by using a sample of $m$ examples from the hypothesis class $\mathcal{H}_n$ , while the estimation error is defined to be:

$$\epsilon_{est}:=L_\mathcal{D}(h)-\underset{h\in\mathcal{H}}{min} L_\mathcal{D}(h).$$

Besides, the Theorem 7.4. in the book shows that:

$$ \mathbb{P}\lbrace S\sim\mathcal{D}^m: (\forall h\in\mathcal{H}) (L_\mathcal{D}(h)\le L_S(h) + \epsilon_{n(h)}(m, w(n(h))\cdot\delta)) \rbrace \ge 1-\delta, $$

so minimizing $L_S(h)+\epsilon_{n(h)}(m, w(n(h))\cdot\delta)$ will, with at least $1-\delta$ probability, minimize $L_\mathcal{D}(h)$ but it's not the same as to minimize $\epsilon_{est}$?

And so I couldn't find the connection between the minimization of $L_S(h)+\epsilon_{n(h)}(m, w(n(h))\cdot\delta)$ and the smaller estimation error.

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1 Answer 1

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The answer is right under my noise but I'd failed to see it.

Theorem 7.4 in the book says that:

$$ \mathbb{P}\lbrace S\sim\mathcal{D}^m: (\forall h\in\mathcal{H}) (L_\mathcal{D}(h)\le L_S(h) + \epsilon_{n(h)}(m, w(n(h))\cdot\delta)) \rbrace \ge 1-\delta. $$

So when $L_S(h) + \epsilon_{n(h)}(m, w(n(h))\cdot\epsilon)$ is minimized, we have with probability of at least $1-\epsilon$, the upper bound of $L_\mathcal{D}(h)$ is also minimized. This implies the upper bound $\epsilon_{est}:=L_\mathcal{D}(h)-\min_{h\in\mathcal{H}} L_\mathcal{H}(h)$ is minimized also (with probability $\ge1-\epsilon$) because $\min_{h\in\mathcal{H}} L_\mathcal{H}(h)$ is constant for some hypothesis class $\mathcal{H}$.

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