Introduction

• Definition

  • A computer program is said to learn from experience E with respect to some class of tasks T, and performance measure P,
  • If its performance at tasks in T, as measured by P, improves with experience E.

• Supervised Learning

  • Task is to learn a mapping function from inputs to outputs
  • Inputs are vectors called features, covariates, predictors
  • Output is called as label, target, response

• Classification

  • Output is a set of unordered, mutually exclusive labels
  • Goal is to design a model with minimum misclassification rate
  • $L(\theta) = {1 \over N} \sum I{y \ne f(x, \theta)}$
  • Empirical Risk is the average loss on training set
  • Model fitting minimizes empirical risk (ERM)
  • Overall goal is generalization

• Uncertainty

  • Model cannot map inputs to outputs with 100% certainty
  • Model Uncertainty due to lack of knowledge between input and output
  • Data Uncertainty due to stochasticity in labels
  • Good model assigns high probability to true output for each input
  • Intuition for minimizing NLL (negative log-likelihood)
  • $NLL(\theta) = {1 \over N} \sum p(y | f(x, \theta))$
  • Optimal parameters give the MLE estimate

• Regression

  • Output is a real valued quantity
  • Model fitting often involves minimizing the quadratic loss or MSE
  • $L(\theta) = {1 \over N} \sum (y - f(x, \theta))^2$
  • Data uncertainty. For example: if the output distribution is Gaussian
  • $p(y | x, \theta) = \mathcal N(y | f(x, \theta), \ \sigma^2)$
  • $NLL(\theta) \propto MSE(\theta)$
  • Linear Regression is an affine transformation between inputs and outputs
  • Polynomial Regression improves the fit by considering higher order interactions
  • FNNs do feature extraction by nesting the functions

• Overfitting and Generalization

  • A model that perfectly fits training data but is too complex suffers from overfitting
  • Population risk the theoretical expected loss on the true data generating process
  • Generalization gap is the difference between empirical risk and population risk
  • High generalization gap is a sign of overfitting
  • Population risk is hard to estimate. Approximate using test risk. Expected error on unseen data points.
  • Test error has U-shaped curve wrt model's degree of freedom

• No Free Lunch Theorem: No single best model that works optimally for all kinds of problems

• Unsupervised Learning

  • Learn an unconditional model of the data $p(x)$ rather than $p(y | x)$
  • In clustering, the goal is to partition the input space into regions with homogenous points.
  • Dimensionality reduction projects input data from high dimension to lower dimension subspace.
  • Self-Supervised Learning involves creating proxy supervised tasks from unlabled data
  • Evaluation is done by measuring the probability assigned by the model to unseen data
  • This treats the problem as one of density estimation
  • Unsupervised learning also aims to increase sample efficiency in downstream supervised learning tasks

• Reinforcement Learning

  • A system or an agent learns how to interact with its environment
  • Goal is learn a policy that specifies optimal action given a state
  • Unlike, Supervised Learning, the reward signal is occasional and delayed.
  • Learning with critic vs learning with teacher.

• Data Preprocessing

  • Text Data

    • Bag of Words transforms a document to a term frequency matrix
    • Frequent words have undue influence (pareto distribution)
    • Log of counts assuage some of the problems
    • Inverse Document Frequency: $\text{IDF}_i = \log {N \over 1 + \text{DF}_i}$
    • N is the total number of documents and DF is the documents with term i
    • $\text{TF-IDF} = \log(1 + TF) \times IDF$
    • Word embeddings map sparse vector representation of word to lower dimension dense vector
    • UNK token can help capture OOV words
    • Sub-word units or word pieces created using byte-pair encoding perform better than UNK token and help in reducing the vocabulary.

  • Missing Data

    • MCAR: missing completely at random
    • MAR: missing at random
    • NMAR: not missing at random
    • Handling of the missing values depends to the type