04 - machine learning basics

• $y$ needs to be predicted from some input $x$, so the model is some function:

where, $x$ is the input, $y$ is the actual output, $\hat{y}$ is the prediction, and $\theta$ are model parameters

• eg: for Hooke's law, $\hat{F}=kx$, $x$ is the extension, $y=F_{measured}$, $\hat{y}=F_{predicted}$, and $\theta =k$, where $k$ was previously determined from experiment

• the features is denoted as $x$, which can more than one number, so it is written as a vector:

regression and classification

• regression involves predicting a continuous numerical quantity

• eg: weather measurements $\to$ temperature tomorrow

• classification involves predicting a category

• eg: detector measurements $\to$ normal/faulty

• they often produce probabilities such as : $P(\text{normal}|x)$ and $P(\text{faulty}|x)$

• they can be binary or multiclass

learning

• considering the simple model:

• here, the model has some adjustable parameter, $\theta$
• defining the loss function:

• so mathematically, learning can be formulated as:

  $${\theta }^{\ast }=\arg \underset{\theta }{min}L(\theta )$$

• hence, it is simply an optimisation problem

• now considering a more realistic model:

• the model parameters are:

• training involves finding good values for these parameters
• conceptually, the same thing happens for a neural network, where there may be:

• a billion-parameter model roughly means a model with a billion adjustable parameters
• the architecture is vastly more sophisticated but the principle remains: data + objective + optimisation $\to$ learned parameters

parameters and hyperparameters

• a parameter is learned during the training

• eg: slope of a regression model, intercept, decision-tree values, neural-network weights, neural-network biases

• a hyperparameter is chosen/configured by the user

• eg: learning rate, number of trees, max tree depth, batch size, number of neural-network layers, regularisation strength

model = RandomForestClassifier(
	n_estimators=200, 
	max_depth=10
)

model.fit(X_train, y_train)

• here, n_estimators and max_depth are hyperparameters
• the internal properties learned by those 200 trees from the training data are parameters

supervised and unsupervised learning

• previous discussions have assumed possession of both $\mathbf{X}$ and $y$

• so, the model can compare $\hat{y}$ against $y$, hence it is supervised learning

• consider there are ${10}^6$ events with measurements for each event, but no labels

• an algorithm can discover groups of events, which is an example of clustering, falling under unsupervised learning

• note that there is no known targe $y$ being supplied during the training

• other unsupervised learning tasks include dimensionality reduction and discovering latent structure

overfitting

• a complicated function can exactly fit 1000 points, but this is not the objective
• the model must learn a relationship that generalised to new data rather than perform perfectly on a given dataset
• an ML model will need a sufficiently flexible family of functions and data, and it will optimize the parameters to approximate it