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# CS代写|强化学习代写Reinforcement learning代考|CS60077 Dynamics In the Presence of Multiple Optimal Policies

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## CS代写|强化学习代写Reinforcement learning代考|Dynamics In the Presence of Multiple Optimal Policies

In the value-based setting, it does not matter which greedy selection rule is used to represent the optimality operator: By definition, any greedy selection rule must be equivalent to directly maximising over $Q(x, \cdot)$. In the distributional setting, however, different rules usually result in different operators. As a concrete example, compare the rule “among all actions whose expected value is maximal, pick the one with smallest variance” to “assign equal probability to actions whose expected value is maximal”.

Theorem $7.9$ relies on the fact that, when there is a unique optimal policy $\pi^$, we can identify a time after which the distributional optimality operator behaves like a policy evaluation operator. When there are multiple optimal policies, however, the action gap of the optimal value function $Q^$ is zero and the argument cannot be used. To understand why this is problematic, it is useful to write the iterates $\left(\eta_k\right){k \geq 0}$ more explicitly in terms of the policies $\pi_k=\mathcal{G}\left(\eta_k\right)$ : $$\eta{k+1}=\mathcal{T}^{\pi_k} \eta_k=\mathcal{T}^{\pi_k} \mathcal{T}^{\pi_{k-1}} \eta_{k-1}=\mathcal{T}^{\pi_k} \ldots \mathcal{T}^{\pi 0} \eta_0 .$$
When the action gap is zero, the sequence of policies $\pi_k, \pi_{k+1}, \ldots$ may continue to vary over time, depending on the greedy selection rule. Although all optimal actions have the same optimal value (guaranteeing the convergence of the expected values to $Q^*$ ), they may correspond to different distributions. Thus distributional value iteration – even with a mean-preserving projection – may mix together the distribution of different optimal policies. Even if $\left(\pi_k\right){k \geq 0}$ converges, the policy it converges to may depend on initial conditions (Exercise 7.5). In the worst case, the iterates $\left(\eta_k\right){k \geq 0}$ might not even converge, as the following example shows.

## CS代写|强化学习代写Reinforcement learning代考|Risk and Risk-Sensitive Control

Imagine being invited to interview for a desirable position at a prestigious research institute abroad. Your plane tickets and the hotel have been booked weeks in advance. Now the night before an early morning flight, you make arrangements for a cab to pick you up from your house and take you to the airport. How long in advance of your plane’s actual departure do you request the cab for? If someone tells you that, on average, a cab to your local airport takes an hour – is that sufficient information to make the booking? How does your answer change when the flight is scheduled around rush hour, rather than early morning?

Fundamentally, it is often desirable that our choices be informed by the variability in the process that produces outcomes from these choices. In this context, we call this variability risk. Risk may be inherent to the process, or incomplete knowledge about the state of the world (including any potential traffic jams, and the mechanical condition of the hired cab).

In contrast to risk-neutral behaviour, decisions that take risk into account are called risk-sensitive. The language of distributional reinforcement learning is particularly well-suited for this purpose, since it lets us reason about the full spectrum of outcomes, along with their associated probabilities. The rest of the chapter gives an overview of how one may account for risk in the decisionmaking process and of the computational challenges that arise when doing so. Rather than be exhaustive, here we take the much more modest aim of exposing the reader to some of the major themes in risk-sensitive control and their relation to distributional reinforcement learning; references to more extensive surveys are provided in the bibliographical remarks.

## Cs代写|强化学习代写|风险和风险敏感控制

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