# CS代考 How utilities are calculated – cscodehelp代写

How utilities are calculated
So far we have assumed that utilities are summed along a run. ‚ Nottheonlyway.
In general we need to compute Urprs0, s1, . . . , snsq for general Urp ̈q. That is, the utility of a run.
Before Urp ̈q was just the sum of rewards in every state. Can consider finite and infinite horizons.
‚ Isit“gameover”atsomepoint?
Turns out that infinite horizons are mostly easier to deal with.
‚ Thatiswhatwewilluse.
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How utilities are calculated
Also have to consider whether utilities are stationary or non-stationary.
‚ Think of: does the same state always have the same value?
‚ E.g., in Pacman when you pick up a fruit, there is a large reward for that tile. That
changes after you picked up the fruit. Example:
‚ Normallywepreferonestatetoanother.
‚ PassingtheAImoduletofailingit
‚ Inthiscasewhentheexamis,todayornextweek,isirrelevant.
We assume utilities are stationary.
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But are they?
Not clear that utilities are always stationary.
In truth, I don’t always most want to eat cherry pie. Despite this, we will assume that utilities are stationary.
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How utilities are calculated
With stationary utilities, there are two ways to establish Ur prs0 , s1 , . . . , sn sq from Rpsq.
Urprs0, s1, . . . , snsq “ Rps0q ` Rps1q ` . . . ` Rpsnq
as above. Discounted rewards:
Urprs0, s1, . . . , snsq “ Rps0q ` γRps1q ` . . . ` γnRpsnq where the discount factor γ is a number between 0 and 1.
The discount factor models the preference of the agent for current over future rewards.
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How utilities are calculated
There is an issue with infinite sequences with additive, undiscounted rewards. ‚ Whatwilltheutilityofapolicybe?
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How utilities are calculated
There is an issue with infinite sequences with additive, undiscounted rewards. ‚ Whatwilltheutilityofapolicybe?
Unbounded
8 or ́8.
This is problematic if we want to compare policies.
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How utilities are calculated
Some solutions are (definitions follow): ‚ Properpolicies
‚ Averagereward
‚ Discountedrewards
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How utilities are calculated
Proper policies always end up in a terminal state eventually. Thus they have a finite expected utility.
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How utilities are calculated
We can compute the average reward per time step. Even for an infinite policy this will (usually) be finite.
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How utilities are calculated
Assume: 0 ď γ ă 1 and rewards are bounded by Rmax
With discounted rewards the utility of an infinite sequence is finite:
Urprs0, s1, . . . , snsq
n

“ γtRpstq t“0
8

ď γtRpstq t“0
8

ď γtRmax t“0
Rmax p1 ́γq

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Optimal policies
With discounted rewards we compare policies by computing their expected values. The expected utility of executing π starting in s is given by:
«ff
8

Uπpsq “ E
where St is the state the agent gets to at time t.
γtRpStq
St is a random variable and we compute the probability of all its values by looking
at all the runs which end up there after t steps.
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t“0

Optimal policies
The optimal policy is then:
π ̊ “argmaxUπpsq π
It turns out that this is independent of the state the agent starts in.
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Optimal policies
Here we have the values of states if the agent executes an optimal policy
Uπ ̊psq
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Optimal policies
Here we have the values of states if the agent executes an optimal policy
Uπ ̊psq
What should the agent do if it is in (3, 1)?
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Example