# 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

# 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!

# 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:

## When the approximate gradient has bounded error

##### By Christian | March 15, 2018

Hello! So this semester has been a fairly busy one for me and so I have not made much time to get anything new written.. until now!

# 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!

# 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.