436 lines
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HTML
436 lines
25 KiB
HTML
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<title>Week 36: Linear Regression and Gradient descent</title>
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{'highest level': 2,
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'sections': [('Plans for week 36', 2, None, 'plans-for-week-36'),
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('Material for lecture Monday September 2',
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2,
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None,
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'material-for-lecture-monday-september-2'),
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('Mathematical Interpretation of Ordinary Least Squares',
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2,
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None,
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('Linear Regression Problems',
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('Fixing the singularity', 2, None, 'fixing-the-singularity'),
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None,
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('Deriving the Ridge Regression Equations',
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('Basic math of the SVD', 2, None, 'basic-math-of-the-svd'),
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('The SVD, a Fantastic Algorithm',
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2,
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'the-svd-a-fantastic-algorithm'),
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('Economy-size SVD', 2, None, 'economy-size-svd'),
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('Codes for the SVD', 2, None, 'codes-for-the-svd'),
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('Note about SVD Calculations',
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2,
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None,
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'note-about-svd-calculations'),
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('Mathematics of the SVD and implications',
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2,
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None,
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'mathematics-of-the-svd-and-implications'),
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('Example Matrix', 2, None, 'example-matrix'),
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('Setting up the Matrix to be inverted',
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2,
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None,
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'setting-up-the-matrix-to-be-inverted'),
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('Further properties (important for our analyses later)',
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2,
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None,
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'further-properties-important-for-our-analyses-later'),
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('Back to Ridge and LASSO Regression',
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2,
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None,
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'back-to-ridge-and-lasso-regression'),
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('Interpreting the Ridge results',
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2,
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None,
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'interpreting-the-ridge-results'),
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('More interpretations', 2, None, 'more-interpretations'),
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('Deriving the Lasso Regression Equations',
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2,
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None,
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'deriving-the-lasso-regression-equations'),
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('Optimization and gradient descent, the central part of any '
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'Machine Learning algortithm',
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2,
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None,
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'optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm'),
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("Reminder on Newton-Raphson's method",
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2,
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None,
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'reminder-on-newton-raphson-s-method'),
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('The equations', 2, None, 'the-equations'),
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('Simple geometric interpretation',
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2,
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None,
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'simple-geometric-interpretation'),
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('Extending to more than one variable',
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2,
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None,
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'extending-to-more-than-one-variable'),
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('Steepest descent', 2, None, 'steepest-descent'),
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('More on Steepest descent', 2, None, 'more-on-steepest-descent'),
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('The ideal', 2, None, 'the-ideal'),
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('The sensitiveness of the gradient descent',
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2,
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None,
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'the-sensitiveness-of-the-gradient-descent'),
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('Convex functions', 2, None, 'convex-functions'),
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('Convex function', 2, None, 'convex-function'),
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('Conditions on convex functions',
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2,
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None,
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'conditions-on-convex-functions'),
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('More on convex functions', 2, None, 'more-on-convex-functions'),
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('Some simple problems', 2, None, 'some-simple-problems'),
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('Revisiting Ordinary Least Squares',
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2,
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None,
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'revisiting-ordinary-least-squares'),
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('Gradient descent example', 2, None, 'gradient-descent-example'),
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('The derivative of the cost/loss function',
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2,
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None,
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'the-derivative-of-the-cost-loss-function'),
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('The Hessian matrix', 2, None, 'the-hessian-matrix'),
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('Simple program', 2, None, 'simple-program'),
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('Gradient Descent Example', 2, None, 'gradient-descent-example'),
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('Gradient descent and Ridge',
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2,
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None,
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'gradient-descent-and-ridge'),
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('The Hessian matrix for Ridge Regression',
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2,
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None,
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'the-hessian-matrix-for-ridge-regression'),
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('Program example for gradient descent with Ridge Regression',
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2,
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None,
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'program-example-for-gradient-descent-with-ridge-regression'),
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('Using gradient descent methods, limitations',
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2,
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None,
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'using-gradient-descent-methods-limitations'),
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('Material for lab sessions sessions Tuesday and Wednesday',
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2,
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None,
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'material-for-lab-sessions-sessions-tuesday-and-wednesday'),
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('Linear Regression and the SVD',
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2,
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None,
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'linear-regression-and-the-svd'),
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('What does it mean?', 2, None, 'what-does-it-mean'),
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('Ridge and LASSO Regression',
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2,
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None,
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'ridge-and-lasso-regression'),
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('From OLS to Ridge and Lasso',
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2,
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None,
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('Deriving the Ridge Regression Equations',
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2,
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None,
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'deriving-the-ridge-regression-equations'),
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('Note on Scikit-Learn', 2, None, 'note-on-scikit-learn'),
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('Comparison with OLS', 2, None, 'comparison-with-ols'),
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('SVD analysis', 2, None, 'svd-analysis'),
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('Interpreting the Ridge results',
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2,
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None,
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'interpreting-the-ridge-results'),
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('More interpretations', 2, None, 'more-interpretations'),
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('Deriving the Lasso Regression Equations',
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2,
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None,
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'deriving-the-lasso-regression-equations'),
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('Simple example to illustrate Ordinary Least Squares, Ridge and '
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'Lasso Regression',
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2,
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None,
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'simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression'),
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('Ridge Regression', 2, None, 'ridge-regression'),
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('Lasso Regression', 2, None, 'lasso-regression'),
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('Yet another Example', 2, None, 'yet-another-example'),
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('The OLS case', 2, None, 'the-ols-case'),
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('The Ridge case', 2, None, 'the-ridge-case'),
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('Writing the Cost Function',
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2,
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None,
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'writing-the-cost-function'),
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('Lasso case', 2, None, 'lasso-case'),
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('The first Case', 2, None, 'the-first-case'),
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('Simple code for solving the above problem',
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2,
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None,
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'simple-code-for-solving-the-above-problem'),
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('With Lasso Regression', 2, None, 'with-lasso-regression'),
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('Another Example, now with a polynomial fit',
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2,
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None,
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'another-example-now-with-a-polynomial-fit')]}
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end of tocinfo -->
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week36-bs001.html#plans-for-week-36" style="font-size: 80%;">Plans for week 36</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs002.html#material-for-lecture-monday-september-2" style="font-size: 80%;">Material for lecture Monday September 2</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs003.html#mathematical-interpretation-of-ordinary-least-squares" style="font-size: 80%;">Mathematical Interpretation of Ordinary Least Squares</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs004.html#residual-error" style="font-size: 80%;">Residual Error</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs005.html#simple-case" style="font-size: 80%;">Simple case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs006.html#the-singular-value-decomposition" style="font-size: 80%;">The singular value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs007.html#linear-regression-problems" style="font-size: 80%;">Linear Regression Problems</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs008.html#fixing-the-singularity" style="font-size: 80%;">Fixing the singularity</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs051.html#ridge-and-lasso-regression" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs053.html#deriving-the-ridge-regression-equations" style="font-size: 80%;">Deriving the Ridge Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs011.html#basic-math-of-the-svd" style="font-size: 80%;">Basic math of the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs012.html#the-svd-a-fantastic-algorithm" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs013.html#economy-size-svd" style="font-size: 80%;">Economy-size SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs014.html#codes-for-the-svd" style="font-size: 80%;">Codes for the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs015.html#note-about-svd-calculations" style="font-size: 80%;">Note about SVD Calculations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs016.html#mathematics-of-the-svd-and-implications" style="font-size: 80%;">Mathematics of the SVD and implications</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs017.html#example-matrix" style="font-size: 80%;">Example Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs018.html#setting-up-the-matrix-to-be-inverted" style="font-size: 80%;">Setting up the Matrix to be inverted</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs019.html#further-properties-important-for-our-analyses-later" style="font-size: 80%;">Further properties (important for our analyses later)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs020.html#back-to-ridge-and-lasso-regression" style="font-size: 80%;">Back to Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs057.html#interpreting-the-ridge-results" style="font-size: 80%;">Interpreting the Ridge results</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs058.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs059.html#deriving-the-lasso-regression-equations" style="font-size: 80%;">Deriving the Lasso Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs024.html#optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm" style="font-size: 80%;">Optimization and gradient descent, the central part of any Machine Learning algortithm</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs025.html#reminder-on-newton-raphson-s-method" style="font-size: 80%;">Reminder on Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs026.html#the-equations" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs027.html#simple-geometric-interpretation" style="font-size: 80%;">Simple geometric interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs028.html#extending-to-more-than-one-variable" style="font-size: 80%;">Extending to more than one variable</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs029.html#steepest-descent" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs030.html#more-on-steepest-descent" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs031.html#the-ideal" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs032.html#the-sensitiveness-of-the-gradient-descent" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs033.html#convex-functions" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs034.html#convex-function" style="font-size: 80%;">Convex function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs035.html#conditions-on-convex-functions" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs036.html#more-on-convex-functions" style="font-size: 80%;">More on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs037.html#some-simple-problems" style="font-size: 80%;">Some simple problems</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs038.html#revisiting-ordinary-least-squares" style="font-size: 80%;">Revisiting Ordinary Least Squares</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs043.html#gradient-descent-example" style="font-size: 80%;">Gradient descent example</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs040.html#the-derivative-of-the-cost-loss-function" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs041.html#the-hessian-matrix" style="font-size: 80%;">The Hessian matrix</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs042.html#simple-program" style="font-size: 80%;">Simple program</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs043.html#gradient-descent-example" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs044.html#gradient-descent-and-ridge" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs045.html#the-hessian-matrix-for-ridge-regression" style="font-size: 80%;">The Hessian matrix for Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs046.html#program-example-for-gradient-descent-with-ridge-regression" style="font-size: 80%;">Program example for gradient descent with Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs047.html#using-gradient-descent-methods-limitations" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs048.html#material-for-lab-sessions-sessions-tuesday-and-wednesday" style="font-size: 80%;">Material for lab sessions sessions Tuesday and Wednesday</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs049.html#linear-regression-and-the-svd" style="font-size: 80%;">Linear Regression and the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs050.html#what-does-it-mean" style="font-size: 80%;">What does it mean?</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs051.html#ridge-and-lasso-regression" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs052.html#from-ols-to-ridge-and-lasso" style="font-size: 80%;">From OLS to Ridge and Lasso</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs053.html#deriving-the-ridge-regression-equations" style="font-size: 80%;">Deriving the Ridge Regression Equations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs054.html#note-on-scikit-learn" style="font-size: 80%;">Note on Scikit-Learn</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs055.html#comparison-with-ols" style="font-size: 80%;">Comparison with OLS</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs056.html#svd-analysis" style="font-size: 80%;">SVD analysis</a></li>
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|
<!-- navigation toc: --> <li><a href="._week36-bs057.html#interpreting-the-ridge-results" style="font-size: 80%;">Interpreting the Ridge results</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs058.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs059.html#deriving-the-lasso-regression-equations" style="font-size: 80%;">Deriving the Lasso Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs060.html#simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression" style="font-size: 80%;">Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs061.html#ridge-regression" style="font-size: 80%;">Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs062.html#lasso-regression" style="font-size: 80%;">Lasso Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week36-bs063.html#yet-another-example" style="font-size: 80%;">Yet another Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs064.html#the-ols-case" style="font-size: 80%;">The OLS case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs065.html#the-ridge-case" style="font-size: 80%;">The Ridge case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs066.html#writing-the-cost-function" style="font-size: 80%;">Writing the Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs067.html#lasso-case" style="font-size: 80%;">Lasso case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs068.html#the-first-case" style="font-size: 80%;">The first Case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs069.html#simple-code-for-solving-the-above-problem" style="font-size: 80%;">Simple code for solving the above problem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs070.html#with-lasso-regression" style="font-size: 80%;">With Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs071.html#another-example-now-with-a-polynomial-fit" style="font-size: 80%;">Another Example, now with a polynomial fit</a></li>
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</ul>
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</li>
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</ul>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0009"></a>
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<!-- !split -->
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<h2 id="ridge-and-lasso-regression" class="anchor">Ridge and LASSO Regression </h2>
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<p>Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is
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our optimization problem is
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</p>
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$$
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{\displaystyle \min_{\boldsymbol{\theta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)\right\}.
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$$
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<p>or we can state it as</p>
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$$
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{\displaystyle \min_{\boldsymbol{\theta}\in
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{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\vert\vert_2^2,
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$$
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<p>where we have used the definition of a norm-2 vector, that is</p>
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$$
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\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}.
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$$
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<p>By minimizing the above equation with respect to the parameters
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\( \boldsymbol{\theta} \) we could then obtain an analytical expression for the
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parameters \( \boldsymbol{\theta} \). We can add a regularization parameter \( \lambda \) by
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defining a new cost function to be optimized, that is
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</p>
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$$
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{\displaystyle \min_{\boldsymbol{\theta}\in
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{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\theta}\vert\vert_2^2
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$$
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<p>which leads to the Ridge regression minimization problem where we
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require that \( \vert\vert \boldsymbol{\theta}\vert\vert_2^2\le t \), where \( t \) is
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a finite number larger than zero. By defining
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</p>
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$$
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C(\boldsymbol{X},\boldsymbol{\theta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\theta}\vert\vert_1,
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$$
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<p>we have a new optimization equation</p>
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$$
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{\displaystyle \min_{\boldsymbol{\theta}\in
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{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\theta}\vert\vert_1
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$$
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<p>which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. </p>
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<p>Here we have defined the norm-1 as </p>
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$$
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\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert.
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$$
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