Recall, from conditional plan evaluation, we had that:

\begin{equation} U^{\pi}(b) = \sum_{s}^{} b(s) U^{\pi}(s) \end{equation}

let’s write it as:

\begin{equation} U^{\pi}(b) = \sum_{s}^{} b(s) U^{\pi}(s) = {\alpha_{\pi}}^{\top} b \end{equation}

where \(\U_{\pi}(s)\) is the conditional plan evaluation starting at each of the initial states.

\begin{equation} \alpha_{\pi} = \qty[ U^{\pi}(s_1), U^{\pi}(s_2) ] \end{equation}

You will notice, then the utility of \(b\) is linear on \(b\) for different policies \(\alpha_{\pi}\):

At every belief \(b\), there is a policy which has the highest \(U(b)\) at that \(b\) given be the alpha vector formulation.

see also in programming string (C)

• an alphabet \(\Sigma\) is a finite set
• a finite-sequence of elements in \(\Sigma\) is called a string
• the set of all strings in \(\Sigma\) is called \(\Sigma^{*}\), which includes the empty string
• for a particular string \(x\), the length of it is \(|x|\)
• the string of length zero is called \(\varepsilon\)
• a language is a subset of \(\Sigma^{*}\), meaning its a set of strings

Omer seems to call strings “words” sometimes.

languages are boolean function over strings

For every language \(L\) over \(\Sigma\) corresponds to a unique function \(f: \Sigma^{*} \to \{0,1\}\), whereby if \(f(x) = 1\), then \(x \in L\); otherwise, if \(f(x) = 0\), \(x \not \in L\).

Alternating Least Squares is a method to Factoring a matrix into two components:

\begin{equation} \mathcal{M}( R) \approx \mathcal{M}(U) \cdot \mathcal{M}(P) \end{equation}

where, we want to come up matricies \(U\) and \(P\) with a certain side length \(k\) that we exdogenously come up with

To perform Alternating Least Squares, we fix the values of either \(U\) or \(P\), then perform the least-squares optimization on

(This is proven best-fit for “non-pathological matricies”)

Problem: current ambulance routing don’t optimize significantly on the contextual cases for stroke patients

Stroke hospitals: PSC is smaller than a CSC.

Previous work

Routing methods—

• route all patient to nearest PSC, which is worse than
• route high risk patient to CSC, which is worse than
• always route to CSC

This is counter-intuitive. How do we solve, given a stroke condition, available PSC/CSC locations, traffic, etc., for where and how to route a patient?

Hello! Welcome to the series of guided code-along labs to introduce you to the basis of using the PyTorch library and its friends to create a neural network! We will dive deeply into Torch, focusing on how practically it can be used to build Neural Networks, as well as taking sideroads into how it works under the hood.

Getting Started

To get started, let’s open a colab and import Torch!

import torch
import torch.nn as nn

The top line here import PyTorch generally, and the bottom line imports the Neural Network libraries. We will need both for today and into the future!