https://www.chenk.top/en/tags/learning-theory/Recent content in Learning Theory on Chen Kai BlogHugoenTue, 20 Jan 2026 09:00:00 +0000https://www.chenk.top/en/ml-math-derivations/01-introduction-and-mathematical-foundations/Tue, 20 Jan 2026 09:00:00 +0000https://www.chenk.top/en/ml-math-derivations/01-introduction-and-mathematical-foundations/<p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/ml-math-derivations/01-Introduction-and-Mathematical-Foundations/illustration_1.png" alt="ML Math Derivations (1): Introduction and Mathematical Foundations — Chapter overview" loading="lazy" decoding="async" class="content-image"> </figure> </p> <hr> <h2 id="what-this-chapter-does" class="heading-anchor">What this chapter does<a href="#what-this-chapter-does" class="heading-link" aria-label="Permalink to this section" title="Copy link to this section">#</a> </h2><p>In 2005 Google Research showed, on a public benchmark, that a statistical translation model trained on raw bilingual text could outperform decades of carefully engineered linguistic rules. The conclusion was uncomfortable for the experts of the day, but mathematically liberating: <strong>a system that has never been told the rules of a language can still recover them, given enough examples.</strong> Why?</p>