Morten Hjorth-Jensen
414 lines
30 KiB
HTML
414 lines
30 KiB
HTML
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<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
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{'highest level': 2,
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'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
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('Lecture Monday September 30, 2024',
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2,
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None,
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'lecture-monday-september-30-2024'),
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('Suggested readings and videos',
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2,
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None,
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('Lab sessions Tuesday and Wednesday',
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2,
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None,
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'lab-sessions-tuesday-and-wednesday'),
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('Summary from last week, using gradient descent methods, '
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'limitations',
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2,
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None,
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'summary-from-last-week-using-gradient-descent-methods-limitations'),
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('Simple implementation of GD for OLS, Ridge and Lasso',
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2,
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None,
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'simple-implementation-of-gd-for-ols-ridge-and-lasso'),
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("But none of these can compete with Newton's method",
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2,
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None,
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'but-none-of-these-can-compete-with-newton-s-method'),
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('Gradient descent and Logistic regression',
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2,
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None,
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('Overview video on Stochastic Gradient Descent',
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2,
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('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
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('Stochastic Gradient Descent (SGD)',
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2,
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None,
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'stochastic-gradient-descent-sgd'),
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('Stochastic Gradient Descent',
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2,
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None,
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'stochastic-gradient-descent'),
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('Computation of gradients', 2, None, 'computation-of-gradients'),
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('SGD example', 2, None, 'sgd-example'),
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('The gradient step', 2, None, 'the-gradient-step'),
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('Simple example code', 2, None, 'simple-example-code'),
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('When do we stop?', 2, None, 'when-do-we-stop'),
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('Slightly different approach',
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2,
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None,
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'slightly-different-approach'),
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None,
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'code-with-a-number-of-minibatches-which-varies'),
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('Replace or not', 2, None, 'replace-or-not'),
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('Momentum based GD', 2, None, 'momentum-based-gd'),
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('More on momentum based approaches',
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2,
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None,
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'more-on-momentum-based-approaches'),
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('Momentum parameter', 2, None, 'momentum-parameter'),
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('Second moment of the gradient',
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2,
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None,
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'second-moment-of-the-gradient'),
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('RMS prop', 2, None, 'rms-prop'),
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('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
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2,
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None,
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'adam-optimizer-https-arxiv-org-abs-1412-6980'),
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('Algorithms and codes for Adagrad, RMSprop and Adam',
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2,
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None,
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'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
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('AdaGrad algorithm, taken from "Goodfellow et '
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'al":"https://www.deeplearningbook.org/contents/optimization.html"',
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2,
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None,
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'adagrad-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html'),
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('RMSProp algorithm, taken from "Goodfellow et '
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'al":"https://www.deeplearningbook.org/contents/optimization.html"',
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2,
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None,
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'rmsprop-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html'),
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('ADAM algorithm, taken from "Goodfellow et '
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'al":"https://www.deeplearningbook.org/contents/optimization.html"',
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2,
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None,
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'adam-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html'),
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('Practical tips', 2, None, 'practical-tips'),
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('Automatic differentiation',
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2,
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None,
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'automatic-differentiation'),
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('Using autograd', 2, None, 'using-autograd'),
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('Autograd with more complicated functions',
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2,
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None,
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'autograd-with-more-complicated-functions'),
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('More complicated functions using the elements of their '
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'arguments directly',
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2,
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None,
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'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
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('Functions using mathematical functions from Numpy',
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2,
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None,
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'functions-using-mathematical-functions-from-numpy'),
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('More autograd', 2, None, 'more-autograd'),
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('And with loops', 2, None, 'and-with-loops'),
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('Using recursion', 2, None, 'using-recursion'),
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('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
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('Same code but now with momentum gradient descent',
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2,
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None,
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'same-code-but-now-with-momentum-gradient-descent'),
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('Including Stochastic Gradient Descent with Autograd',
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2,
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('Same code but now with momentum gradient descent',
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2,
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'same-code-but-now-with-momentum-gradient-descent'),
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('Similar (second order function now) problem but now with '
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'AdaGrad',
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2,
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None,
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'similar-second-order-function-now-problem-but-now-with-adagrad'),
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('RMSprop for adaptive learning rate with Stochastic Gradient '
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'Descent',
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2,
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None,
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'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
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('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
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2,
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None,
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'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
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('And Logistic Regression', 2, None, 'and-logistic-regression'),
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('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
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2,
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None,
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'introducing-jax-https-jax-readthedocs-io-en-latest'),
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('Getting started with Jax, note the way we import numpy',
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3,
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None,
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'getting-started-with-jax-note-the-way-we-import-numpy'),
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('A warm-up example', 3, None, 'a-warm-up-example'),
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('A more advanced example', 3, None, 'a-more-advanced-example'),
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('Introduction to Neural networks',
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2,
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None,
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('Artificial neurons', 2, None, 'artificial-neurons'),
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('Neural network types', 2, None, 'neural-network-types'),
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('Feed-forward neural networks',
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2,
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None,
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('Convolutional Neural Network',
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2,
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None,
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('Recurrent neural networks',
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2,
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None,
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'recurrent-neural-networks'),
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('Other types of networks', 2, None, 'other-types-of-networks'),
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('Multilayer perceptrons', 2, None, 'multilayer-perceptrons'),
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('Why multilayer perceptrons?',
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2,
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None,
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'why-multilayer-perceptrons'),
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('Illustration of a single perceptron model and a '
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'multi-perceptron model',
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2,
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None,
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'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
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('Examples of XOR, OR and AND gates',
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2,
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None,
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'examples-of-xor-or-and-and-gates'),
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('Does Logistic Regression do a better Job?',
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2,
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None,
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'does-logistic-regression-do-a-better-job'),
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('Adding Neural Networks', 2, None, 'adding-neural-networks'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Matrix-vector notation', 3, None, 'matrix-vector-notation'),
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('Matrix-vector notation and activation',
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3,
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None,
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'matrix-vector-notation-and-activation'),
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('Activation functions', 3, None, 'activation-functions'),
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('Activation functions, Logistic and Hyperbolic ones',
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3,
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None,
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'activation-functions-logistic-and-hyperbolic-ones'),
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('Relevance', 3, None, 'relevance')]}
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<a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
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</div>
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<ul class="nav navbar-nav navbar-right">
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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="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs002.html#lecture-monday-september-30-2024" style="font-size: 80%;"><b>Lecture Monday September 30, 2024</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs003.html#suggested-readings-and-videos" style="font-size: 80%;"><b>Suggested readings and videos</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs004.html#lab-sessions-tuesday-and-wednesday" style="font-size: 80%;"><b>Lab sessions Tuesday and Wednesday</b></a></li>
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<!-- navigation toc: --> <li><a href="#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs006.html#simple-implementation-of-gd-for-ols-ridge-and-lasso" style="font-size: 80%;"><b>Simple implementation of GD for OLS, Ridge and Lasso</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs007.html#but-none-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But none of these can compete with Newton's method</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs008.html#gradient-descent-and-logistic-regression" style="font-size: 80%;"><b>Gradient descent and Logistic regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs009.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs010.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs011.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs012.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs013.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs014.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs015.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs016.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs017.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs018.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs019.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs020.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs021.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs022.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs023.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs024.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs025.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs026.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs027.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs028.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs028.html#adagrad-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;"><b>AdaGrad algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs028.html#rmsprop-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;"><b>RMSProp algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs028.html#adam-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;"><b>ADAM algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs029.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs030.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs031.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs032.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs033.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs034.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs035.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs036.html#and-with-loops" style="font-size: 80%;"><b>And with loops</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs037.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs038.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs041.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs040.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs041.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs042.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs043.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs044.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs045.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs045.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs045.html#getting-started-with-jax-note-the-way-we-import-numpy" style="font-size: 80%;"> Getting started with Jax, note the way we import numpy</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs045.html#a-warm-up-example" style="font-size: 80%;"> A warm-up example</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs045.html#a-more-advanced-example" style="font-size: 80%;"> A more advanced example</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs048.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs049.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs050.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs051.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs052.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs053.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs054.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs055.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs056.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs057.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs058.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs063.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs063.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs063.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs063.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs063.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs065.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs067.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs068.html#relevance" style="font-size: 80%;"> Relevance</a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0005"></a>
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<!-- !split -->
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<h2 id="summary-from-last-week-using-gradient-descent-methods-limitations" class="anchor">Summary from last week, using gradient descent methods, limitations </h2>
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<ul>
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<li> <b>Gradient descent (GD) finds local minima of our function</b>. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.</li>
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<li> <b>GD is sensitive to initial conditions</b>. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.</li>
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<li> <b>Gradients are computationally expensive to calculate for large datasets</b>. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, \( E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2 \); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over <em>all</em> \( n \) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called "mini batches". This has the added benefit of introducing stochasticity into our algorithm.</li>
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<li> <b>GD is very sensitive to choices of learning rates</b>. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would <em>adaptively</em> choose the learning rates to match the landscape.</li>
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<li> <b>GD treats all directions in parameter space uniformly.</b> Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.</li>
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<li> GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.</li>
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