Empirical Risk Minimization

Old Recorded Lectures

Recap

Commonly Used Binary Classification Loss Functions

Figure 4.1: Plots of Common Classification Loss Functions - x-axis: $h({\mathbf{x}}_i)y_i$, or "correctness" of prediction; y-axis: loss value

1. Which functions are strict upper bounds on the 0/1-loss?
2. What can you say about the hinge-loss and the log-loss as $z\to -\mathrm{\infty }$?

Commonly Used Regression Loss Functions

Regression algorithms (where a prediction can lie anywhere on the real-number line) also have their own host of loss functions:

Loss $\ell (h_{\mathbf{w}}({\mathbf{x}}_i,y_i))$						Comments
Squared Loss $(h({\mathbf{x}}_i)-y_i{)}^2$						Most popular regression loss function Estimates Mean Label ADVANTAGE: Differentiable everywhere DISADVANTAGE: Somewhat sensitive to outliers/noise Also known as Ordinary Least Squares (OLS)
Absolute Loss $|h({\mathbf{x}}_i)-y_i|$						Also a very popular loss function Estimates Median Label ADVANTAGE: Less sensitive to noise DISADVANTAGE: Not differentiable at $0$
Huber Loss $\frac{1}{2}{(h({\mathbf{x}}_i)-y_i)}^2$ if $|h({\mathbf{x}}_i)-y_i|<\delta$, otherwise $\delta (|h({\mathbf{x}}_i)-y_i|-\frac{\delta }{2})$						Also known as Smooth Absolute Loss ADVANTAGE: "Best of Both Worlds" of Squared and Absolute Loss Once-differentiable Takes on behavior of Squared-Loss when loss is small, and Absolute Loss when loss is large.
Log-Cosh Loss $log(cosh(h({\mathbf{x}}_i)-y_i))$, $cosh(x)=\frac{e^x+e^{-x}}{2}$						ADVANTAGE: Similar to Huber Loss, but twice differentiable everywhere

Table 4.2: Loss Functions With Regression, i.e. $y\in \mathbb{R}$
Quiz: What do the loss functions in Table 4.2 look like with respect to $z=h({\mathbf{x}}_i)-y_i$?



Figure 4.2: Plots of Common Regression Loss Functions - x-axis: $h({\mathbf{x}}_i)y_i$, or "error" of prediction; y-axis: loss value

Regularizers

When we look at regularizers it helps to change the formulation of the optimization problem to obtain a better geometric intuition: $$\underset{\mathbf{w},b}{min}\sum _{i=1}^n\ell (h_{\mathbf{w}}(\mathbf{x}),y_i)+\lambda r(\mathbf{w})\iff \underset{\mathbf{w},b}{min}\sum _{i=1}^n\ell (h_{\mathbf{w}}(\mathbf{x}),y_i)\text{ subject to: }r(\mathbf{w})\le B$$ For each $\lambda \ge 0$, there exists $B\ge 0$ such that the two formulations in (4.1) are equivalent, and vice versa. In previous sections, $l_2$-regularizer has been introduced as the component in SVM that reflects the complexity of solutions. Besides the $l_2$-regularizer, other types of useful regularizers and their properties are listed in Table 4.3.

Regularizer $r(\mathbf{w})$						Properties
$l_2$-Regularization $r(\mathbf{w})={\mathbf{w}}^{\mathrm{\top }}\mathbf{w}=\parallel \mathbf{w}{\parallel }_2^2$						ADVANTAGE: Strictly Convex ADVANTAGE: Differentiable DISADVANTAGE: Uses weights on all features, i.e. relies on all features to some degree (ideally we would like to avoid this) - these are known as Dense Solutions.
$l_1$-Regularization $r(\mathbf{w})=\parallel \mathbf{w}{\parallel }_1$						Convex (but not strictly) DISADVANTAGE: Not differentiable at $0$ (the point which minimization is intended to bring us to Effect: Sparse (i.e. not Dense) Solutions
$l_p$-Norm $\parallel \mathbf{w}{\parallel }_p=(\sum _{i=1}^d{v}_i^p{)}^{1/p}$						(often $0<p\le 1$) DISADVANTAGE: Non-convex ADVANTAGE: Very sparse solutions Initialization dependent DISADVANTAGE: Not differentiable

Table 4.3: Types of Regularizers



Figure 4.3: Plots of Common Regularizers

Famous Special Cases

This section includes several special cases that deal with risk minimization, such as Ordinary Least Squares, Ridge Regression, Lasso, and Logistic Regression. Table 4.4 provides information on their loss functions, regularizers, as well as solutions.

Loss and Regularizer						Comments
Ordinary Least Squares $\underset{\mathbf{w}}{min}\frac{1}{n}\sum _{i=1}^n({\mathbf{w}}^{\mathrm{\top }}{x}_i-y_i{)}^2$						Squared Loss No Regularization Closed form solution: $\mathbf{w}=(\mathbf{X}{\mathbf{X}}^{\mathrm{\top }}{)}^{-1}\mathbf{X}{\mathbf{y}}^{\mathrm{\top }}$ $\mathbf{X}=[{\mathbf{x}}_1,...,{\mathbf{x}}_n]$ $\mathbf{y}=[y_1,...,y_n]$
Ridge Regression $\underset{\mathbf{w}}{min}\frac{1}{n}\sum _{i=1}^n({\mathbf{w}}^{\mathrm{\top }}{x}_i-y_i{)}^2+\lambda \parallel w{\parallel }_2^2$						Squared Loss $l_2$-Regularization $\mathbf{w}=(\mathbf{X}{\mathbf{X}}^{\mathrm{\top }}+\lambda \mathbb{I}{)}^{-1}\mathbf{X}{\mathbf{y}}^{\mathrm{\top }}$
Lasso $\underset{\mathbf{w}}{min}\frac{1}{n}\sum _{i=1}^n({\mathbf{w}}^{\mathrm{\top }}{\mathbf{x}}_i-y_i{)}^2+\lambda \parallel \mathbf{w}{\parallel }_1$						+ sparsity inducing (good for feature selection) + Convex - Not strictly convex (no unique solution) - Not differentiable (at 0) Solve with (sub)-gradient descent or SVEN
Elastic Net $\underset{\mathbf{w}}{min}\frac{1}{n}\sum _{i=1}^n({\mathbf{w}}^{\mathrm{\top }}{\mathbf{x}}_i-y_i{)}^2+\alpha \parallel \mathbf{w}{\parallel }_1+(1-\alpha )\parallel \mathbf{w}{\parallel }_2^2$ $\alpha \in [0,1)$						ADVANTAGE: Strictly convex (i.e. unique solution) + sparsity inducing (good for feature selection) + Dual of squared-loss SVM, see SVEN DISADVANTAGE: - Non-differentiable
Logistic Regression $\underset{\mathbf{w},b}{min}\frac{1}{n}\sum _{i=1}^n\log (1+e^{-y_i({\mathbf{w}}^{\mathrm{\top }}{\mathbf{x}}_i+b)})$						Often $l_1$ or $l_2$ Regularized Solve with gradient descent. $Pr(y|x)=\frac{1}{1+e^{-y({\mathbf{w}}^{\mathrm{\top }}x+b)}}$
Linear Support Vector Machine $\underset{\mathbf{w},b}{min}C\sum _{i=1}^{n}max[1-y_i({\mathbf{w}}^{\mathrm{\top }}{\mathbf{x}}_i+b),0]+\parallel \mathbf{w}{\parallel }_2^2$						Typically $l_2$ regularized (sometimes $l_1$). Quadratic program. When kernelized leads to sparse solutions. Kernelized version can be solved very efficiently with specialized algorithms (e.g. SMO)

Table 4.4: Special Cases
Some additional notes on the Special Cases:
1. Ridge Regression is very fast if data isn't too high dimensional.
2. Ridge Regression is just 1 line of Julia / Python.
3. There is an interesting connection between Ordinary Least Squares and the first principal component of PCA (Principal Component Analysis). PCA also minimizes square loss, but looks at perpendicular loss (the horizontal distance between each point and the regression line) instead.
[1] In Bayesian Machine Learning, it is common to optimize $\lambda$, but for the purposes of this class, it is assumed to be fixed.