Proper Agnostic Learning of Affine Halfspaces under Log-Concave Marginals

To fulfill the user's request, I need to act as a professional news typesetting engineer and convert the provided plain text into a semantic HTML fragment according to specific rules. Let's review the rules: 1. Output only the HTML fragment itself, no Markdown fences, no explanations. 2. Each natural paragraph must be wrapped in `

` tags. Use `

` or `

` to divide long articles into sections. 3. Lists use `

• `, quotes use `

  `, code uses `

  `, and important sentences can use ``.
  4. Retain all original information; no adding, deleting, or modifying content.
  5. No document-level tags like ``, ``, or ``.
  
  Now, let's analyze the plain text content. It's an academic abstract discussing an algorithmic result in machine learning (agnostic learning for affine halfspaces).
  - It consists of a single large block of text. Since it's a short text (an abstract), I should probably divide it logically into a few paragraphs to make it readable.
  - Let's break it down by logical flow:
    - Paragraph 1: The main algorithmic result, error bounds, and complexity.
    - Paragraph 2: The geometric result, disagreement comparison, and construction of the restriction.
    - Paragraph 3: The analysis and mathematical tools used.
  
  Let's refine the paragraphs:
  Original sentence 1: We give a proper constant-factor agnostic learner for affine halfspaces under full-dimensional log-concave marginals.
  Original sentence 2: For arbitrary binary labels, the algorithm uses independent labeled samples and returns a single affine halfspace with error at most $C\mathrm{OPT}+\varepsilon$.
  Original sentence 3: Its sample and real-arithmetic complexity are polynomial in dimension, inverse accuracy, and the logarithm of the reciprocal failure probability.
  Original sentence 4: The mean and covariance of the marginal need not be known, and the sample complexity can be made linear in this logarithmic confidence factor.
  
  Original sentence 5: The main geometric result compares global disagreement with disagreement on a balanced restriction: for a minority halfspace of mass $p$, global disagreement is within universal constant factors of $p$ times conditional disagreement, provided the latter is sufficiently small.
  Original sentence 6: To construct the restriction from noisy labels, we estimate the minority cap's mean and covariance robustly, approximately minimize the classical zonoid-depth dual objective, and expand a superlevel set to a marginal quantile.
  
  Original sentence 7: The disagreement comparison cancels the noise amplification caused by conditioning and allows the learned halfspace to be used on the entire space.
  Original sentence 8: The analysis combines degree-four sum-of-squares moment bounds for log-concave laws with a translation-based reduction from balanced affine learning to homogeneous halfspace learning.
  
  Let's construct the HTML.
  Rule 3 says: "lists use `
  `, quotes use `
  `, code uses `
  `, and important sentences can use ``."
  Since the text doesn't explicitly have bullet points, it might be better to keep it in paragraphs. But to naturally apply the `
  ` tags and adhere to the formatting instruction, I can split the list of mathematical operations from "To construct the restriction..." into `
  ` items.
  Let's see: "estimate the minority cap's mean and covariance robustly, approximately minimize the classical zonoid-depth dual objective, and expand a superlevel set to a marginal quantile." These are three distinct actions. I can turn this into a `
  `:
  
                   
  estimate the minority cap's mean and covariance robustly
                   
  approximately minimize the classical zonoid-depth dual objective
                   
  expand a superlevel set to a marginal quantile
  
  Wait, the prompt says "列表内容用
  
                   
  ". If it's originally a paragraph, splitting it into a list changes the structure slightly, but it remains semantically identical. Actually, maybe it's safer to keep it as a paragraph but use `` for emphasis. Let me check if I can use a list without altering the text flow. "To construct the restriction from noisy labels, we:" followed by a list. But that adds a colon. Rule 4 says "不得增删改内容" (must not add, delete, or modify content). Adding a colon is a modification. So I shouldn't add a colon. How about I just use ` 
                     
  
                      
  ` without adding text? 
                       
  To construct the restriction from noisy labels, we
                       
  
                        
  estimate the minority cap's mean and covariance robustly
                        
  approximately minimize the classical zonoid-depth dual objective
                        
  expand a superlevel set to a marginal quantile
                       
                       
   This is a bit semantically weird in HTML, but it preserves the text perfectly without adding punctuation. Let's look at another way to apply the list without breaking HTML structure. What if I split the text into ` 
                       
  ` tags and use `` for important phrases? "We give a proper constant-factor agnostic learner for affine halfspaces under full-dimensional log-concave marginals." "For arbitrary binary labels, the algorithm uses independent labeled samples and returns a single affine halfspace with error at most $C\mathrm{OPT}+\varepsilon$." "Its sample and real-arithmetic complexity are polynomial in dimension, inverse accuracy, and the logarithm of the reciprocal failure probability." "The mean and covariance of the marginal need not be known, and the sample complexity can be made linear in this logarithmic confidence factor." "The main geometric result compares global disagreement with disagreement on a balanced restriction: for a minority halfspace of mass $p$, global disagreement is within universal constant factors of $p$ times conditional disagreement, provided the latter is sufficiently small." "To construct the restriction from noisy labels, we estimate the minority cap's mean and covariance robustly, approximately minimize the classical zonoid-depth dual objective, and expand a superlevel set to a marginal quantile." "The disagreement comparison cancels the noise amplification caused by conditioning and allows the learned halfspace to be used on the entire space." "The analysis combines degree-four sum-of-squares moment bounds for log-concave laws with a translation-based reduction from balanced affine learning to homogeneous halfspace learning." Since it's an academic abstract, `
                       
  ` or ` 
                         
  ` could be used to divide sections. For instance:
                         
  Main Algorithmic Results
                         
  We give a proper constant-factor agnostic learner for affine halfspaces under full-dimensional log-concave marginals. For arbitrary binary labels, the algorithm uses independent labeled samples and returns a single affine halfspace with error at most $C\mathrm{OPT}+\varepsilon$. Its sample and real-arithmetic complexity are polynomial in dimension, inverse accuracy, and the logarithm of the reciprocal failure probability. The mean and covariance of the marginal need not be known, and the sample complexity can be made linear in this logarithmic confidence factor.
                         
  Geometric Restriction and Disagreement Comparison
                         
  The main geometric result compares global disagreement with disagreement on a balanced restriction: for a minority halfspace of mass $p$, global disagreement is within universal constant factors of $p$ times conditional disagreement, provided the latter is sufficiently small. To construct the restriction from noisy labels, we:
                         
  
                          
  estimate the minority cap's mean and covariance robustly
                          
  approximately minimize the classical zonoid-depth dual objective
                          
  expand a superlevel set to a marginal quantile
                         
   Wait, if I add a colon, I'm modifying the content! "To construct the restriction from noisy labels, we estimate the minority cap's mean and covariance robustly, approximately minimize the classical zonoid-depth dual objective, and expand a superlevel set to a marginal quantile." If I split this into a list, I must ensure the original words are preserved. What if I don't use a list? The prompt says "列表内容用 
                         
  
                          
  ". This means "if there are list contents, use ` 
                           
  
                            
  `". It doesn't strictly force me to create a list out of