Lyapunov functions are an extremely convenient device for proving that a dynamical system converges. We cover:

- The Lyapunov argument
- La Salle’s Invariance Principle
- An Alternative argument for Convex Functions
- Exponential Convergence Rates

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# Category: Control

## Lyapunov Functions

## Kalman Filter

## Temporal Difference Learning – Linear Function Approximation

## Stochastic Linear Regression

## Foster-Lyapunov

## Experts and Bandits (non-stochastic)

## More on Merton Portfolio Optimization

Lyapunov functions are an extremely convenient device for proving that a dynamical system converges. We cover:

- The Lyapunov argument
- La Salle’s Invariance Principle
- An Alternative argument for Convex Functions
- Exponential Convergence Rates

Kalman filtering (and filtering in general) considers the following setting: we have a sequence of states , which evolves under random perturbations over time. Unfortunately we cannot observe , we can only observe some noisy function of , namely, . Our task is to find the best estimate of given our observations of . Continue reading “Kalman Filter”

For a Markov chain , consider the reward function

associated with rewards given by . We approximate the reward function with a linear approximation,

Continue reading “Temporal Difference Learning – Linear Function Approximation”

We consider the following formulation of Lai, Robbins and Wei (1979), and Lai and Wei (1982). Consider the following regression problem,

for where are unobservable random errors and are unknown parameters.

Typically for a regression problem, it is assumed that inputs are given and errors are IID random variables. However, we now want to consider a setting where we sequentially choose inputs and then get observations , and errors are a martingale difference sequence with respect to the filtration generated by .

Foster’s Lemma provides a natural condition to prove the positive recurrence of a Markov chain.

- Weighted majority algorithm its variant for Bandit Problems.