10: Empirical Risk Minimization

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 $z\to -\mathrm{\infty }$?
Some additional notes on loss functions:
3. 1. As $z\to -\mathrm{\infty }$, the log-loss and the hinge loss become increasingly parallel.
4. 2. The exponential loss and the hinge loss are both upper bounds of the zero-one loss. (For the exponential loss, this is an important aspect in Adaboost, which we will cover later.)
5. 3. Zero-one loss is zero when the prediction is correct, and one when incorrect.

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 Also known as Ordinary Least Squares (OLS) 🙂 Differentiable everywhere 😡 Somewhat sensitive to outliers/noise
Absolute Loss $|h({\mathbf{x}}_i)-y_i|$						Also a very popular loss function Estimates Median Label 🙂 Less sensitive to noise 😡 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 "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 $log(cosh(h({\mathbf{x}}_i)-y_i))$, $cosh(x)=\frac{e^x+e^{-x}}{2}$						🙂 Similar to Huber Loss, but twice differentiable everywhere 😡 More expensive to compute

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 investigate regularizers it helps to change the formulation of the optimization problem from an unconstrained to a constraint formulation, 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 above are equivalent, and vice versa. In previous sections, we have already seen the $l_2$-regularizer in the context of SVMs, Ridge Regression, or Logistic Regression. 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 $r(\mathbf{w})={\mathbf{w}}^{\mathrm{\top }}\mathbf{w}=\parallel \mathbf{w}{\parallel }_2^2$						🙂 Strictly Convex 🙂 Differentiable 😡 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 $r(\mathbf{w})=|\mathbf{w}{|}_1$						Convex (but not strictly) 😡 Not differentiable at $0$ (the point which minimization is intended to bring us to Effect: Sparse (i.e. not Dense) Solutions
$l_p$-Norm $\parallel \mathbf{w}{\parallel }_p=(\sum _{i=1}^d|v_i{|}^p{)}^{1/p}$						😡 Non-convex 🙂 Very sparse solutions (if $0<p<1$ ) 😡 Not differentiable, Initialization dependent

Table 4.3: Most popular 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 $\underset{\mathbf{w},b}{min}\frac{1}{n}\sum _{i=1}^n({\mathbf{w}}^{\mathrm{\top }}{x}_i+b-y_i{)}^2+\lambda \parallel \mathbf{w}{\parallel }_2^2$						Squared Loss $l_2$-Regularization $\mathbf{w}=(\mathbf{X}{\mathbf{X}}^{\mathrm{\top }}+\lambda \mathbb{I}{)}^{-1}\mathbf{X}{\mathbf{y}}^{\mathrm{\top }}$
Lasso $\underset{\mathbf{w},b}{min}\frac{1}{n}\sum _{i=1}^n({\mathbf{w}}^{\mathrm{\top }}{\mathbf{x}}_i+b-y_i{)}^2+\lambda |\mathbf{w}{|}_1$						🙂 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 $\underset{\mathbf{w},b}{min}\frac{1}{n}\sum _{i=1}^n({\mathbf{w}}^{\mathrm{\top }}{\mathbf{x}}_i+b-y_i{)}^2$ $+\alpha |\mathbf{w}{|}_1+(1-\alpha )\parallel \mathbf{w}{\parallel }_2^2$ $\alpha \in (0,1)$						🙂 Strictly convex (i.e. unique solution) 🙂 sparsity inducing (good for feature selection) 🙂 Dual of squared-loss SVM, see SVEN 😡 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 and can be solved in closed form if the data isn't too high dimensional (in just 1 line of code.)
2. 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.