# What is "prediction problem"?

In a context of machine learning, numerous tasks can be seen as prediction problem. For example, this user guide provides solutions for:

For any kinds of prediction problems, we generally provide a set of input-output pairs as:

• Input: Set of features
• e.g., ["1:0.001","4:0.23","35:0.0035",...]
• Output: Target value
• e.g., 1, 0, 0.54, 42.195, ...

Once a prediction model has been constructed based on the samples, the model can make prediction for unforeseen inputs.

Importantly, depending on types of output value, prediction problem can be categorized into regression and classification problem.

# Regression

The goal of regression is to predict real values as shown below:

features (input) target real value (output)
["1:0.001","4:0.23","35:0.0035",...] 21.3
["1:0.2","3:0.1","13:0.005",...] 6.2
["5:1.3","22:0.0.089","77:0.0001",...] 17.1
... ...

In practice, target values could be any of small/large float/int negative/positive values. Our CTR prediction tutorial solves regression problem with small floating point target values in a 0-1 range, for example.

While there are several ways to realize regression by using Hivemall, train_regression() is one of the most flexible functions. This feature is explained in: Regression.

# Classification

In contrast to regression, output for classification problems should be (integer) labels:

features (input) label (output)
["1:0.001","4:0.23","35:0.0035",...] 0
["1:0.2","3:0.1","13:0.005",...] 1
["5:1.3","22:0.0.089","77:0.0001",...] 1
... ...

In case the number of possible labels is 2 (0/1 or -1/1), the problem is binary classification, and Hivemall's train_classifier() function enables you to build binary classifiers. Binary Classification demonstrates how to use the function.

Another type of classification problems is multi-class classification. This task assumes that the number of possible labels is more than 2. We need to use different functions for the multi-class problems, and our news20 and iris tutorials would be helpful.

# Mathematical formulation of generic prediction model

Here, we briefly explain about how prediction model is constructed.

First and foremost, we represent input and output for prediction models as follows:

• Input: a vector $\mathbf{x}$
• Output: a value $y$

For a set of samples $(\mathbf{x}_1, y_1), (\mathbf{x}_2, y_2), \cdots, (\mathbf{x}_n, y_n)$, the goal of prediction algorithms is to find a weight vector (i.e., parameters) $\mathbf{w}$ by minimizing the following error:

$E(\mathbf{w}) := \frac{1}{n} \sum_{i=1}^{n} L(\mathbf{w}; \mathbf{x}_i, y_i) + \lambda R(\mathbf{w})$

In the above formulation, there are two auxiliary functions we have to know:

• $L(\mathbf{w}; \mathbf{x}_i, y_i)$
• Loss function for a single sample $(\mathbf{x}_i, y_i)$ and given $\mathbf{w}$.
• If this function produces small values, it means the parameter $\mathbf{w}$ is successfully learnt.
• $R(\mathbf{w})$
• Regularization function for the current parameter $\mathbf{w}$.
• It prevents failing to a negative condition so-called over-fitting.

($\lambda$ is a small value which controls the effect of regularization function.)

Eventually, minimizing the function $E(\mathbf{w})$ can be implemented by the SGD technique as described before, and $\mathbf{w}$ itself is used as a "model" for future prediction.

Interestingly, depending on a choice of loss and regularization function, prediction model you obtained will behave differently; even if one combination could work as a classifier, another choice might be appropriate for regression.

Below we list possible options for train_regression and train_classifier, and this is the reason why these two functions are the most flexible in Hivemall:

• Loss function: -loss, -loss_function

• For train_regression
• SquaredLoss (synonym: squared)
• QuantileLoss (synonym: quantile)
• EpsilonInsensitiveLoss (synonym: epsilon_intensitive)
• SquaredEpsilonInsensitiveLoss (synonym: squared_epsilon_intensitive)
• HuberLoss (synonym: huber)
• For train_classifier
• HingeLoss (synonym: hinge)
• LogLoss (synonym: log, logistic)
• SquaredHingeLoss (synonym: squared_hinge)
• ModifiedHuberLoss (synonym: modified_huber)
• The following losses are mainly designed for regression but can sometimes be useful in classification as well:
• SquaredLoss (synonym: squared)
• QuantileLoss (synonym: quantile)
• EpsilonInsensitiveLoss (synonym: epsilon_intensitive)
• SquaredEpsilonInsensitiveLoss (synonym: squared_epsilon_intensitive)
• HuberLoss (synonym: huber)
• Regularization function: -reg, -regularization

• L1
• L2
• ElasticNet
• RDA

Additionally, there are several variants of the SGD technique, and it is also configureable as:

• Optimizer -opt, -optimizer
• SGD
Option values are case insensitive and you can use sgd or rda, or huberloss.