Answers.Body: <p>Strictly speaking Hoeffding’s inequality as a non-asymptotic probabilistic bound alone cannot guarantee this, you also need additional assumptions such as all in-samples are <em>i.i.d.</em> from their population and the said fixed hypothesis has to be <em>independent</em> of the in-sample data. Otherwise you would've never needed to consider the <em>union bound</em> for finite hypothesis class or infinite hypothesis class via its possible finite VC dimension to account for the worst-case selected hypothesis through empirical risk minimization out of the hypothesis class in PAC learning.</p>
  <p>For a simple example, if you just select a perfectly over-fitted hypothesis from a fixed <span class="math-container">$N$</span> training data points such that the in-sample error is almost <span class="math-container">$0$</span>, then the out-of-sample error cannot be guaranteed to be arbitrarily small as <span class="math-container">$\epsilon$</span> with arbitrarily high probability <span class="math-container">$\delta$</span> via any means including Hoeffding’s inequality.</p>
  
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Body: <p>Is this True or False:</p>
  <p>&quot;For a hypothesis with a small in-sample error on a large dataset, Hoeffding’s inequality guarantees that the out-of-sample error will also be small with high probability.&quot;</p>
  
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Comments.Text: Can you please formulate the title as a specific question? Also, we've asked your many times to put more effort into your posts. This post is very lazily written. This also looks like a homework problem and you should disclose that.
Comments.UserDisplayName: nbro
Tags: <machine-learning><generalization>
Title: Hoeffding’s inequality
UserDisplayName: DSPinfinity
ThreadText: # Hoeffding’s inequality
  
  (Asked by: username_0 on 2024-12-06)
  
  Is this True or False:
  
  "For a hypothesis with a small in-sample error on a large dataset, Hoeffding’s inequality guarantees that the out-of-sample error will also be small with high probability."
  
  **Tags:** machine-learning, generalization
  
  > **Comments:**
  > **username_1**: Can you please formulate the title as a specific question? Also, we've asked your many times to put more effort into your posts. This post is very lazily written. This also looks like a homework problem and you should disclose that.  
  
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  ## 1 answers
  
  ### Answer by username_2 on 2024-12-06. Score: 0 | Accepted answer
  
  Strictly speaking Hoeffding’s inequality as a non-asymptotic probabilistic bound alone cannot guarantee this, you also need additional assumptions such as all in-samples are *i.i.d.* from their population and the said fixed hypothesis has to be *independent* of the in-sample data. Otherwise you would've never needed to consider the *union bound* for finite hypothesis class or infinite hypothesis class via its possible finite VC dimension to account for the worst-case selected hypothesis through empirical risk minimization out of the hypothesis class in PAC learning.
  
  For a simple example, if you just select a perfectly over-fitted hypothesis from a fixed $N$ training data points such that the in-sample error is almost $0$, then the out-of-sample error cannot be guaranteed to be arbitrarily small as $\epsilon$ with arbitrarily high probability $\delta$ via any means including Hoeffding’s inequality.
  
  
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