Applied Probability Notes

Temporal Difference Learning – Linear Function Approximation

For a Markov chain $\hat{x} = (\hat x_t : t\in\mathbb Z_+)$ , consider the reward function

$\label{TD:Reward} R(x) := \mathbb E_{x} \left[ \sum_{t=0}^\infty r(\hat x_t) \right]$

associated with rewards given by $r = (r(x) : x\in\mathcal X)$ . We approximate the reward function $R(x)$ with a linear approximation,

$R(x;\bm w) = \bm w^\top \bm \phi (x) = \sum_{j\in \mathcal J} w_j \phi_j(x) .$

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Mixing Times

In loose terms, the mixing time is the amount of time to wait before you can expect a Markov chain to be close to its stationary distribution. We give an upper bound for this.

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Bayesian Online Learning

We briefly describe an Online Bayesian Framework which is sometimes referred to as Assumed Density Filer (ADF). And we review a heuristic proof of its convergence in the Gaussian case.

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Stochastic Linear Regression

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

$y_n = \beta_1 x_{n1} + ... + \beta_p x_{np} + \epsilon_n$

for $n=1,2,...$ where $\epsilon_n$ are unobservable random errors and $\beta_1,...,\beta_p$ are unknown parameters.

Typically for a regression problem, it is assumed that inputs $x_{1},...,x_{n}$ are given and errors are IID random variables. However, we now want to consider a setting where we sequentially choose inputs $x_i$ and then get observations $y_i$ , and errors $\epsilon_i$ are a martingale difference sequence with respect to the filtration $\mathcal F_i$ generated by $\{ x_j, y_{j-1} : j\leq i \}$ .

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Q-learning

Q-learning is an algorithm, that contains many of the basic structures required for reinforcement learning and acts as the basis for many more sophisticated algorithms. The Q-learning algorithm can be seen as an (asynchronous) implementation of the Robbins-Monro procedure for finding fixed points. For this reason we will require results from Robbins-Monro when proving convergence.

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Robbins-Monro

We review a method for finding fixed points then extend it to slightly more general, modern proofs. This is a much more developed version of an earlier post. We now cover the basic Robbin-Monro proof, Robbins-Siegmund Theorem, Stochastic Gradient Descent and Asynchronous update (as is required for Q-learning).

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Merton Portfolio Optimization

HJB equation for Merton Problem; CRRA utility solution; Proof of Optimality.
Multiple Assets; Dual Value function Approach.

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