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# CS代写|强化学习代写Reinforcement learning代考|CSE546 Fixed-Size Empirical Representations

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## CS代写|强化学习代写Reinforcement learning代考|Fixed-Size Empirical Representations

The empirical representation is expressive because it can use more particles to describe more complex probability distributions. This “blank cheque” approach to memory and computation, however, results in an intractable algorithm. On the other hand, the simple normal distribution is rarely sufficient to give a good approximation of the return distribution. A good middle ground is to preserve the form of the empirical representation while imposing a limit on its expressivity. Our approach is to fix the number and type of particles used to represent probability distributions.

Definition 5.11. The $m$-quantile representation parametrises the location of $m$ equally-weighted particles. That is,
$$\mathscr{F}{\mathrm{Q}, m}=\left{\frac{1}{m} \sum{i=1}^m \delta_{\theta_i}: \theta_i \in \mathbb{R}\right} .$$
Definition 5.12. Given a collection of $m$ evenly-spaced locations $\theta_1<\cdots<\theta_m$, the $m$-categorical representation parametrises the probability of $m$ particles at these fixed locations:
$$\mathscr{F}{\mathrm{C}, m}=\left{\sum{i=1}^m p_i \delta_{\theta_i}: p_i \geq 0, \sum_{i=1}^m p_i=1\right}$$

## CS代写|强化学习代写Reinforcement learning代考|The Projection Step

We now describe projection operators for the categorical and quantile representations, correspondingly called categorical projection and quantile projection. In both cases, these operators can be seen as finding the best approximation to a given probability distribution, as measured according to a specific probability metric.

To begin, recall that for a probability metric $d, \mathscr{P}d(\mathbb{R}) \subseteq \mathscr{P}(\mathbb{R})$ is the set of probability distributions with finite mean and finite distance from the reference distribution $\delta_0$ (Equation 4.26). For a representation $\mathscr{F} \subseteq \mathscr{P}_d(\mathbb{R})$, a d-projection of $\nu \in \mathscr{P}_d(\mathbb{R})$ onto $\mathscr{F}$ is a function $\Pi{\mathscr{F}, d}: \mathscr{P}d(\mathbb{R}) \rightarrow \mathscr{F}$ that finds a distribution $\hat{\nu} \in \mathscr{F}$ that is $d$-closest to $\nu$ : $$\Pi{\mathscr{F}, d} \nu \in \underset{\hat{\nu} \in \mathscr{F}}{\arg \min } d(\nu, \hat{\nu}) .$$
Although both the categorical and quantile projections that we present here satisfy this definition, it is worth noting that in the most general setting neither the existence nor uniqueness of a $d$-projection $\Pi_{\mathscr{F}, d}$ is actually guaranteed (see Remark 5.3). We lift the notion of a $d$-projection to return-distribution functions in our usual manner; the $\bar{d}$-projection of $\eta \in \mathscr{P}d(\mathbb{R})^{\mathcal{X}}$ onto $\mathscr{F}^{\mathcal{X}}$ is $$\left(\Pi{\mathscr{F} X,}, \eta\right)(x)=\Pi_{\mathscr{F}, d}(\eta(x)) .$$
When unambiguous, we overload notation and write $\Pi_{\mathscr{F}, d} \eta$ to denote the projection onto $\mathscr{F}^{\mathcal{X}}$

It is natural to think of the $\bar{d}$-projection of the return-distribution function $\eta$ onto $\mathscr{F} \mathcal{X}$ as the best achievable approximation within this representation, measured in terms of $d$. We thus call $\Pi_{\mathscr{F}, d} \nu$ and $\Pi_{\mathscr{F} X, \bar{d}} \eta$ the (d, $\mathscr{F}$ )-optimal approximations to $\nu \in \mathscr{P}(\mathbb{R})$ and $\eta$, respectively.

## CS代写|强化学习代写Reinforcement learning代考|Fixed-Size Empirical Representations

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## CS代写|强化学习代写Reinforcement learning代考|The Projection Step

$\Pi \mathscr{F}, d \nu \in \underset{\hat{\nu} \in \mathscr{F}}{\arg \min } d(\nu, \hat{\nu})$.

$(\Pi \mathscr{F} X,, \eta)(x)=\Pi_{\mathscr{F}, d}(\eta(x))$

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