Non-Asymptotic Monte Carlo Approximations

With a little bit of rigor

By Christian | August 02, 2020

$\newcommand{\Expect}[2][]{\mathbb{E}_{#1}\left(#2\right)}$Suppose we are given some region $\Omega \subseteq \mathbb{R}^d$ where we would like to integrate a function $f(\bv{x})$. Suppose there exists a $d$-dimensional hyperrectangle $\mathcal{H}$, also referred to as a $d$-orthotope, such that $\Omega \subseteq \mathcal{H}$ and $\mathcal{H} = \bigtimes_{i=1}^d [s_i, t_i]$. If we define $\bv{1}_{A}(\bv{x})$ as an indicator function that returns 1 if and only if $\bv{x} \in A$, then we can write out integral in the following manner

Read More

Searching Streams for the Unknown

An algorithmic approach

By Christian | July 25, 2020

In the modern climate of big data, it should not surprise anyone that we can be handed a dataset that is much too large to fit onto our personal computer’s hard drive, let alone having the dataset all load into RAM. Yet, these limitations do not stop us from trying to crunch bigger datasets and harvest even more data! 

Read More

Exponential Convergence of Gradient Descent with Lipschitz Smoothness and Strong Convexity

A little bit of theory

By Christian | March 19, 2018

So in the world of practical optimization, especially with respect to applications in things like Machine Learning, it is super common to hear about the use of Gradient Descent. Gradient Descent is a simple recursive scheme that is used to finding critical points (hopefully local optima!) of functions. This scheme takes the following form:

Read More

Randomized Range Estimator

When linear algebra takes ideas from probability

By Christian | November 17, 2017

Hey there reader! It has been quite a while since I wrote a blog post.. but I have had a ton of things on my mind I wanted to write about! I am stoked to be able to write about some of them now!

Read More

Control from Approximate Dynamic Programming Using State-Space Discretization

Recursing through space and time

By Christian | February 04, 2017

In a recent post, principles of Dynamic Programming were used to derive a recursive control algorithm for Deterministic Linear Control systems. The challenges with the approach used in that blog post is that it is only readily useful for Linear Control Systems with linear cost functions. What if, instead, we had a Nonlinear System to control or a cost function with some nonlinear terms? Such a problem would be challenging to solve using the approach described in the former blog post.

Read More

Intro to Dynamic Programming Based Discrete Optimal Control

Recursion strikes back!

By Christian | January 15, 2017

Oh control. Who doesn't enjoy having control of things in life every so often? While many of us probably wish life could be more easily controlled, alas things often have too much chaos to be adequately predicted and in turn controlled. While lack of complete controllability is the case for many things in life, like getting the ultimate job or getting even one for that matter (sorry I'm a Millenial), there are still plenty of things that can be controlled.. And many are controlled by engineers.

Read More