

Machine Learning Basics

• "Generalization": Perform well on data it hasn't seen before
• Linear regression measures the error on the training data, not on the test data
  • Only works if training and test data sets have the same distribution
• The "i.i.d. assumptions"
  • Assume that all examples are independent, and training and test set are identically distributed
  • Use distribution of one sample to generate all test and training samples
  • Optimizing for the training data will therefore also optimize for the test data
• Overfitting: Reach a low training error, but the test error is much larger
  • i.e. no generalization, only "memorized" the training data
  • Giving it too much capacity
• Capacity: Flexibility the system has
  • e.g. for linear regression, increase capacity by allowing higher-order polynomials

• Check for systems: If you can't get your system to overfit (by giving it lots of capacity), it doesn't learn properly

• Even with a perfect model, with a reasonably complex system, there's always an error

  • e.g. digital communication: noise makes it impossible to always decide correctly (between 1 and 0)

• "No Free Lunch" theorem: No machine learning algorithm is universally better than any other

  • But assumes data of all possible distributions
  • In ML, we always concentrate on a limited range of data
  • i.e. there's no "general" good ML algorithm

Regularization

• $\lambda w^T w$: All elements squared and summed up
  • i.e. small weights are preferable for low cost
• If we pick $\lambda$ to be large, weights will almost be zero to achieve minimal cost (underfitting)
• Also called a "penalty term"
• Goal ist to minimize the generalization error but not the training error

Hyperparameters

• $\lambda$ is a "hyperparameter"
  • parameters that are not learned by the scheme
  • to train these parameters, we use part of the training data as "validation data" to "tune" the algorithm
• After we've used a test set once, it is "tainted" and should theoretically not be used anymore
  • hard in practice to have enough data

Estimators, Bias, Variance

• Estimator: "Schätzer"
• Bias: Difference between Error of "guess" and real data
• Estimator is unbiased: It is on average right ("Erwartungstreu")
• Variance of Estimator
  • "SE": Standard Error, a.k.a $\sqrt{var(x)}$
• The variance of a sample is $\frac{\sigma}{\sqrt m}$
  • i.e. lots of data reduces the variance of the estimator
• Consistency: In the limit, the estimator "guesses the right value"
  • Variance goes towards zero

Maximum Likelihood Estimation

• $\theta$ are free variables
  • Weights of the network in a neural network
• Find $\theta$ such that the likelihood that the model is producing the "correct" values is maximized
• Usually, the log of the likelihood is maximized
• KL divergence: Measures how similar to distributions are
  • Goal: minimize KL divergence

Todo

Chapters 5.7 and 5.8 (supervised vs unsupervised learning)

Stochastic Gradient Descent

• Gradient Descent with a limited sample
• Negative log-likelihood is the cost function
  • Minimizing this cost = maximizing likelihood
• Gradients over all of millions data points would take forever
• Use a select few of the data as an estimate of the gradient
  • This subset is called a "minibatch"
  • Has to be chosen truly randomly from the whole data set
• learning rate: step size for gradient descent

Building a ML algorithm

• Only gets useful once you have a lot of data