Constrained Optimization $$\min_{x \in R^n} f(x)$$ subject to: $$c_i (x) = 0, i \in E \text{(equality constraints)}$$ $$c_i (x) \geq 0, i \in I \text{(inequality constraints)}$$ Feasible Set $\Omega = { x_i | c_i(x) = 0, i \in E \text{ and } c_i(x) \geq 0, i \in I}$. Case 1 $\min_{x \in R^n} f(x)$ subject to $c_1 (x) = 0$ x is local minimum if x + s $\notin \Omega$ or f(x+s) $\geq$ f(x).……

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Gradient Descent A simple algorithm to go “downward” against the gradient of the function. Algebrically: $$w_{t+1} = w_t - \eta \nabla f(w_t)$$ where $\eta$ is called learning rate or step size. Step Size $\eta$ too small, slow convergence $\eta$ too large, solution will bounce around In practice: Set $\eta$ to be a smalle constant Backtracking line search (work when $\nabla f$ is continuous) Parameter $\bar{\alpha}, c \in (0,1), \rho \in (0,1)$.……

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Gradient && Hessian The gradient of a f, d x 1, can be represented as follow $$\nabla f(x) = \begin{bmatrix} \frac {\partial f(x)} {\partial x_1} \newline …\newline \frac {\partial f(x)} {\partial x_d} \end{bmatrix}$$ and the Hessian, d x d, can be represented as $$\nabla^2 f(x) = \begin{bmatrix} \frac {\partial^2 f(x)} {\partial x_1^2} & \frac {\partial^2 f(x)} {\partial x_1x_2} & \frac {\partial^2 f(x)} {\partial x_1x_d} \newline … & … & …\newline \frac {\partial^2 f(x)} {\partial x_d^2} & \frac {\partial^2 f(x)} {\partial x_dx_2} & \frac {\partial^2 f(x)} {\partial x_dx_d} \newline \end{bmatrix}$$ ……

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Uncertainty in Prediction Related to Linear Regression. The available features x do not contain enough information to perfectly predict y, such as x = medical record for patients at risk for a disease y = will he contact disease in next 5 years Model We still going to use linear model for conditional probability estmation $$w_1x_1 + w_2x_2 + … + w_dx_d + b = w \cdot x + b$$ ……

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Basic Idea Fit a line to a bunch of points. Example Without extra information, we will predict the mean 2.47. Average squared error = $\mathbb{E} [(studentGPA - predictedGPA)^2]$ = Variance If we have SAT scores, then we can fit a line. Now if we predict based on this line, the MSE drops to 0.43. This is a regression problem with: Predictor variable: SAT score Response variable: College GPA Formula For $\mathbb{R}$ $$y = ax + b$$ ……

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Background Marginal Distribution Three ways to sample from P Draw (x,y) Draw y according to its marginal distribution, then x according to the conditional distribution of x | y Draw X according to its marginal distribution, then Y according to the conditional distribution of y | x Define: $\mu$: distribution on $X$ $\eta$: conditional distribution y|x Classifier Normal Classifier $h : x \rightarrow y$ $R(h) = Pr_{(x,y) \in p} (h(x) \neq y)$, where R = risk……

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Families of Distance Function $l_p$ norm The most common one is $l_2$ norm (Euclidean distance): $$||x - z||_2 = \sqrt{\sum_{i=1}^{m}(x_i - z_i)^2}$$ Notes: sometime 2 is dropped. For $p \geq 1$, the $l_p$ distance: $$||x - z||_p = (\sum_{i=1}^{m}(x_i - z_i)^p)^{1/p}$$ Special case: $l_1$ distance: $$||x - z||_1 = \sum_{i=1}^{m}|x_i - z_i|$$ $l_\infty$ distance: $$||x - z||_1 = max_i |x_i - z_i|$$ Metric space Let $X$ be the space that data lie.……

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Nearest Neighbor Classification Procedures Assemble a data set (training set) How to classify a new image x? find its closest neighbor y, and label it the same Notes: training set of 60000 images test set of 10000 images How do we determine if two data (images) are closest? With 28 x 28 image, we can strech it to become a vector of 784.……

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Z-Score vs T-Score Z-Score Link Z-Score’s formula $$z = \frac{X - \mu}{\sigma}$$ where X = sample mean, $\mu$ = population means, $\sigma$ = population standard deviation. Also, we use Z Score when sample size >= 30 or we know the population’s mean dna SD. Z Table T-Score T-Score T-Score’s formula $$T = \frac{X - \mu}{s/ \sqrt{n}}$$ where X = sample mean, $\mu$ = population mean, s = sample standard deviation, and n = sample size.……

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