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CS 443 RL Programming

创建日期：2024-06-13 16:37:02

所属分类：计算机computer science

标签： CS 443

Programming

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CS 443 RL Homework

• Please submit a single pdf file that includes both the written part and the coding part. Please
convert the iPython notebook (containing both your code and the outputs) into pdf, and append
the pdf to that of your written work.
1. (8 pts) (Programming) See the python file in the assignment attachments.
2. (7pts) Importance Sampling Let s1, a1, r1, s2, a2, r2, . . . , sH , aH , rH be a random trajectory generated
by sampling s1 ∼ d0 and taking actions at ∼ πb(st) for all t ≥ 1, where πb is a stochastic behavior policy.
For simplicity assume that the process always terminates in H steps.
Both d0 and πb are exploratory, in the sense that: (1) d0(s) > 0, ∀s ∈ S. (2) πb(a|s) > 0, ∀s ∈ S, a ∈ A.
All expectations in this problem are w.r.t. the distribution of this random trajectory.
Let V and Q be an arbitrary state-value and Q-value function, respectively. Consider each of the
following objective functions over V or Q. Let π be some other policy, which may assign 0 probability to
some actions. Let ρt =
π(at|st)
πb(at|st) and ρ1:t be the cumulative product.
1. (1 pts) L(V ) = E[(V (s1)−
∑H
t=1 γ
t−1ρ1:trt)2].
2. (2 pts) L(V ) = E[
(
V (s1)− ρ1 ·
∑H
t=1 γ
t−1rt
)2
].
3. (2 pts) L(Q) = E[
(
Q(s1, a1)− ρ2 ·
∑H
t=1 γ
t−1rt
)2
].
4. (2 pts) L(Q) = E[ ρ1 ·
(
Q(s1, a1)−
∑H
t=1 γ
t−1rt
)2
].
Express the V (or Q) that minimizes each objective using symbols from the following list: (“(·)” can be
either π or πb)
s, a, d0, πb, π, V
(·), Q(·), T (·)
If the minimizer is not unique and can take multiple values in a state (or state-action forQ), you can write
“arbitrary”. Provide a brief derivation/justification for your answers, and while the use of mathematical
language is encouraged, you are not required to provide a full proof.
Example Consider the objective function L(V ) = E[
(
V (s1)−
∑H
t=1 γ
t−1rt
)2
]. For illustration purposes
let’s assume for now that d0 is not exploratory, i.e., ∃s, s.t. d0(s) = 0. The answer is:
V (s) =
{
V πb(s), if d0(s) > 0,
arbitrary, if d0(s) = 0.
1
Justification: on any s where d0(s) = 0, the value of V (s) does not affect the objective, so the minimizer
V (s) is arbitrary. Otherwise, the minimizer should be E[
∑H
t=1 γ
t−1rt|s1 = s] = V πb(s).
2
3. (5 pts) Policy Gradient with Softmax Policy Consider the following MDP M = {S = {s1, s2},A =
{a1, a2}, P,R, γ = 0.8}, where s1 is the fixed initial state, P and R is defined below:
s1 s2
(s1, a1) 1 0
(s1, a2) 0 1
(s2, a1) 1 0
(s2, a2) 0 1
Reward
(s1, a1) 0.5
(s1, a2) 0
(s2, a1) 0
(s2, a2) 1.0
Table 1: Transition Matrix (Left) and Reward Function (Right)
The softmax policy is defined as:
πθ(a|s) = exp(θs,a)∑
a′∈A exp(θs,a′)
, ∀s ∈ S, a ∈ A
where the policy is parameterized by θ ∈ R|S|×|A| = R4 and θs,a denotes the component of θ indexed by
s and a.
1. (2 pts) Initialize with θ0 = (0, 0, 0, 0)T , compute J(πθ0) = V
πθ0 (s1) and
∇θJ(πθ)|θ=θ0 =
1
1− γE(s,a)∼dπθ0 [Q
πθ0 (s, a)∇θ log πθ(a|s)|θ=θ0 ]
where dπ is the normalized discounted state action occupancy, Qπ is the Q function. (Hint: you
can use the following fact to compute the occupancy:
dπ(s) = (1− γ)d0(s) + γ
∑
s′∈S,a∈A
π(a|s′)P (s|s′, a)dπ(s′), ∀s ∈ S
where d0(s) is the initial state distribution)
2. (2 pts) Perform one step policy gradient with learning rate α = 1.0 on θs1,a1 and θs1,a2 (while fixing
θs2,a1 and θs2,a2). Is J(πθ) improved after the update? (Hint: you do not need to compute J(π)
exactly after the update to make the comparison. Instead, you may treat x := π(a1|s1) ∈ (0, 1) as a
variable and represent J(π) as a function of x, then examine how J(π) changes after the update, as
a function of x.)
3. (1 pt) It’s easy to see the optimal softmax policy is π(a1|s1) ≈ 0, π(a2|s2) ≈ 1 (note that they cannot
be exactly 0 or 1 because exp(·) > 0). Show that for arbitrary α > 0, after doing one step policy
gradient θ1 = θ0+α∇θJ(πθ)|θ=θ0 , we always have πθ1(a1|s1) > πθ0(a1|s1), i.e. the distance between
πθ1(·|s1) and optimal policy increases.
Provide a brief discussion of why/how this seeming ”contradiction” happens: although the policy
value is improved, the policy at some states become more different from the optimal policy.
3
(Space for Q3)
4
4. (Optional, 3 pts) Deterministic Policy Gradient Theorem. In the class we derived the following
stochastic policy gradient (PG) theorem (stochastic policy has the form πθ(a|s), i.e. π takes a state as
input and maps to a distribution on the action space):
∇θJ(πθ) = 1
1− γE(s,a)∼dπθ [(∇θ log πθ(a|s))Q
πθ(s, a)] .
Here you are asked to prove the deterministic policy gradient theorem (deterministic policy has the
form πθ(s), i.e. π takes a state as input and maps to a single action).
∇θJ(πθ) = 1
1− γEs∼dπθ
[∇θπθ(s)∇aQπθ(s, a)|a=πθ(s)] .
Here, we typically have a continuous action space A, where A is a subset of some Euclidean space
(which is why we can take gradient w.r.t. a,∇a, above). You can also assumeP (s′|s, a),∇aP (s′|s, a), πθ(s),∇θπθ(s),∇aR(s, a), d(s)
is sufficiently smooth in input argument(s) s, a, s′.
Hint: Since J(πθ) =
∫
S d0(s)V
πθ(s)ds, it makes things easier to start with computing ∇θV πθ(s). You
can apply the Bellman equation recursively when calculating this quantity.

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ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 101H1 IB9520 FINM 3008 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