+
+ Machine Learning
+ Given a set of sample points: discover their defining function
+
+
+
+ Classical Regression
+ given $\{(x_1,y_1), \ldots, (x_K,y_K)\}$, find $f(x) = y$
+
+ - Linear (Fit $m\cdot x + b$)
+ - Polynomial (Fit $m_0\cdot x^0+m_1\cdot x^1+\ldots+m_N\cdot x^N$)
+ - Logarithmic (Fit $a\cdot \log(x+b) + c$)
+ - Multidimensional Variants (etc...)
+
+ Fit data to a predefined equation / family of equations.
+
+
+
+ Limitations
+
+ - Need to select equation(s) to fit upfront.
+ - Difficult to cope with discontinuities.
+ - Simple equations have very few parameters.
+
+
+
+
+ Neural Networks
+ 
+
+ $$z = RELU(v_{z,1}y_1 + ... + v_{z,7}y_7)$$
+ $$y_i = RELU(v_{y,i,1}x_1 + ... + v_{y,i,7}x_7)$$
+ $$x_i = RELU(v_{x,i,1}w_1 + ... + v_{x,i,10}w_10)$$
+
+ $RELU$: Discontinuity Function
+ One big mega-function to fit to the data
+
+
+
+ More complex flavors
+
+
+
- Convolutional NNs
+ - Train pattern-matching kernels for images.
+
+
+
- Recurrent NNs
+ - Feed output of NN back into itself for time-series data.
+
+
+
+ In general: Pre-specify layering structure as a NN workflow.
+
+
+
+
+
+ Sensitivity Analysis
+
+ Given a target point, figure out which of the point's features are most responsible for the classification.
+
+
+
+ “Why Should I Trust You?” Explaining the Predictions of Any Classifier
+ Ribero, Sing, Guestrin
+
+
+
+ Model
+ $$f : \mathbb R^d \rightarrow \mathbb R$$
+
+
+
+ Model
+ Arguments to $f$ are input features. For example:
+
+
+ - Probability that an email is spam
+ - Each known word is a binary (0-1) feature
+ - 0-1 confidence that a greyscale image contains a "3"
+ - The brightness of each pixel is a feature.
+ - Chance of default on a potential loan
+ - Customer characteristics (income, spending, education)
+
+
+ Note: The source model doesn't have to be an NN.
+
+
+
+
+
+ Focusing on similar inputs, learn an explainable model $g$.
+
+ Define similarity by a distance function:
+ $$\pi_x : R^d \times R^d \rightarrow [0,1]$$
+
+
+
+ Desiderata
+
+
+ - $g$ should be simple (complexity function $\Omega(g)$
+ - $g$ should be similar to $f$ on values similar to $x$ (error function $\mathcal L(f, g, \pi_x)$
+
+ Overall Goal: Find a $g$ that minimizes $\mathcal L(f, g, \pi_x) + \Omega(g)$
+
+
+
+ Simpler Model
+ $$g : \mathbb \{0,1\}^d \rightarrow \mathbb R$$
+ $$g(x) = w_1\cdot x_1 + \ldots + w_d\cdot x_d$$
+ Boolean- (not Real-)valued features.
+
+
+
+ Simplification: Thresholding
+ $$x'_i = \begin{cases} 1 & \textbf{if } x_i \geq T \\ 0 & \textbf{otherwise} \end{cases}$$
+
+
+
+ Sampling Around $x$
+
+ - Pick a number of features $\# \sim Uniform(1, |\{x'_i \neq 0\}|)$
+ - Randomly select features $F \subseteq \{i \;|\;x'_i \neq 0\}$ s.t. $|F| = \#$
+ - Pick $z$ (resp., $z'$) s.t. $z_i = \begin{cases} x_i & \textbf{if } i \in F \\ \textit{random} & \textbf{otherwise}\end{cases}$
+ - Repeat $K$ times to get a collection of samples $(z,z') \in \mathcal Z$
+
+
+
+
+ Simple Error Function
+ $$\mathcal L(f,g,\pi_x) = \sum_{(z,z')\in \mathcal Z} \pi_x(z) (f(z) - g(z'))^2$$
+ Pick a $g$ (resp., $\{w_i\}$) that minimizes this!
+
+
+
+ Example
+ Classifying Email Subjects
+
+ Each feature is a word: '1' if the word is present, '0' if not.
($x = x'$)
+
+ Complexity Function: (at most $K$ features)
+ $\Omega(g) = \begin{cases} 0 & \textbf{if }|\{w_i > 0\}| > K \\ \infty & \textbf{otherwise}\end{cases}$
+
+
+ Simplified model: Find the $K$ features most responsible for differentiating between the target class.
+
+
+
+ 
+ “Why Should I Trust You?” Explaining the Predictions of Any Classifier (Ribero et. al.)
+
+
+
+ Example
+ Interpreting Image Classification
+
+ Each feature is a "superpixel" (a contiguous region of similarly colored pixels): $x_i$ is the color of the pixel; $x_i'$ is 1 if the superpixel is identical to the original.
+
+ Complexity Function: (at most $K$ superpixels)
+ "Lasso" (as in the lasso tool) pixels together into a contiguous region of no more than $K$ superpixels.
+
+
+ Simplified model: Find the $K$ superpixels most responsible for differentiating between the target class.
+
+
+
+ 
+ “Why Should I Trust You?” Explaining the Predictions of Any Classifier (Ribero et. al.)
+
+
+
+ Summary
+
+ - Generate a set of "similar" samples
+ - See how those samples behave on the model
+ - Train a simpler model $g$ to pick the most relevant features.
+
+
+ If the model has the features identified by $g$, it is probably of the target class.
+
+
+
+
+
+ 
+ 
+ 
+ 
+