SU-CS361 MAY072024 — Jemoka Knowledge Base

Generalization Error \begin{equation} \epsilon_{gen} = \mathbb{E}_{x \sim \mathcal{X}} \left[\left(f(x) - \hat{f}(x)\right)^{2}\right] \end{equation} we usually instead of compute it by averaging specific points we measured. Probabilistic Surrogate Models Gaussian Process A Gaussian Process is a Gaussian distribution over functions! Consider a mean function m(x), and a covariance (kernel) function k(x, x’). And, for a set of objective values y_{j} \in \mathbb{R}, which we are trying to infer using m and k.

\begin{equation} \mqty[y_1 \\ \dots \\ y_{m}] \sim \mathcal{N} \left(\mqty[m(x_1) \\ \dots \\ m(x_{m})], \mqty[k(x_1, x_1) & \dots & k(x_1, x_{m}) \&\dots&\\ k(x_{m}, x_{1}) &\dots &k(x_{m}, x_{m})]\right) \end{equation}

The choice of kernel makes or breaks the your ability to model your system. Its the way by which your input values are “smoothed” together to create a probabilistic estimate. Choice of Kernels squared exponential kernel

\begin{equation} k(x,x’) = \exp \left( \frac{-(x-x’)^{2}}{2 l^{2}}\right) \end{equation}

where, l is the parameter controlling the “length scale” (i.e. distance required for the function to change significantly). As l gets larger, there’s more smoothing. Matérn Kernel This is a very common kernel. Look it up. Prediction Given known means and variances of the sampled points from the original system, we can compute:

\begin{equation} P(Y^{*}|Y) \end{equation}

through using conditioning Gaussian distributions. Specifically, with:

\begin{equation} \mqty[\hat{y} \\ y] \sim \mathcal{N} \left(\mqty[m(X^{*}) \\ m(X)], \mqty[K(X^{*}, X^{*}) & K(X^{*},X) \\ K(X, X^{*}) & K(X, X)]\right) \end{equation}

we can compute a new mean and a new covariance using conditioning Gaussian distributions Noisy Measurements We can account for zero-mean noise by adding some noise to your covariance:

\begin{equation} \mqty[\hat{y} \\ y] \sim \mathcal{N} \left(\mqty[m(X^{*}) \\ m(X)], \mqty[K(X^{*}, X^{*}) & K(X^{*},X) \\ K(X, X^{*}) & K(X, X) + v I]\right) \end{equation}

Surrogate Optimization Prediction Based Exploration Given your existing points D, evaluate \mu_{x|D}, and optimize for the next design point that has the smallest \mu_{x|D}. This is all exploitation, no exploration. Error Based Exploration Use the 95% confidence interval from the Gaussian Process, find the areas with with the biggest gap and then lower those. This is *all exploration, no exploitation. Lower Confidence Bound Exploration tradeoff between exploration and exploitation. Try to minimize:

\begin{equation} LB(x) = \hat{\mu}(x) - \alpha \hat{\Sigma}(x) \end{equation}

try to minimize both the LOWER BOUND as well as the optimum. This is a probabilistic generalization of the Shubert-Piyavskill Method—and no Lipschitz Constant needed! Reminder, though, these are probabilistic bounds—unlike Shubert-Piyavskill Method. Probability of Improvement Exploration We define “improvement” as:

\begin{equation} I(y) = \begin{cases} y_{\min} - y, \text{if}\ y < y_{\min} \\ 0, \text{otherwise} \end{cases} \end{equation}

then, we have the “probability of improvement” metric at:

\begin{equation} P(y < y_{\min}) &= \int_{-\infty}^{y_{\min}} \mathcal{N}(y | \hat{\mu}, \hat{\Sigma}) \dd{y} \\ &= \Phi\left( \frac{y_{\min } - \hat{\mu}}{\hat{\Sigma}}\right) \end{equation}

(i.e. we want to find points that are very possible to improve). This could be zero when \hat{\Sigma} = 0, which happens when we are at a noiseless point. You can also do this by the expected value of improvement