Study Notes

Preliminaries It is usually the case that we have a dataset $\mathcal{D} = {x_1, \cdots, x_N}$ and a parametrized family of distributions $p_\theta (x)$. We would like to find the parameters that best describe the data. This is typically done using [[MLE and MAP|maximum likelihood estimation (MLE)]]. In this method, the optimal parameters are those that maximize the log likelihood of the data. Mathematically speaking, $$ \hat{\theta}_\mathrm{MLE} = \arg\max_\theta \frac{1}{N}\sum_{i=1}^{N}\log p_{\theta}(x_i)....

This is a short note about the einsum functionality that is present in numpy, jax, etc. Understanding what it does is a bit tricky -naturally, because it can do the job of many other functions- but it is also very useful and can help a lot with linear algebraic computations. I will use numpy’s np.einsum() notation, but the underlaying concepts are the same regardless of syntactic differences in other libraries....

MAP-Elites is an elegant algorithm for solving general optimization problems. To be more accurate, it is an illumination algorithm that tries to find high-performing and diverse solutions in a search space. At its core, it is a simple algorithm, both conceptually and to implement. Here, I briefly introduce the main idea behind the algorithm and its components. I will also discuss its merits and demerits compared to other approaches. This note is based on Illuminating Search Spaces by Mapping Elites....

(Based on a lecture by professor Coralia Cartis, University of Oxford) (I don’t currently plan to extend it, but may expand and add more details to some of the later chapters in the future. I also like to eventually add some useful resources (books, talks, notes, etc.) about optimization) This brief note is about optimization problems. Though the main focus is on the general non-convex optimization problem, a lot of the methods borrow some ideas from convex optimization, so there are a lot of similarities....