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TASK : develop code below Policy Optimization with Dual Function Approximation (end to end we have develop)  in great details using torch module only from stratch 


3 Policy Optimization with Dual Function Approximation
Recall the setting of MDP in Section 2.1. In the tabular case, the policy mirror descent (PMD) method
[Shani et al., 2020a, Lan, 2023, Xiao, 2022] takes the form of (9):
π
(k+1)
s = arg min
πs∈∆(A)
n
ηk


Qb(k)
s
, πs


+ DΦ(πs, π(k)
s
)
o
, s ∈ S, (13)
where Qb(k)
is some approximation of Q(k)
. We note that Q(k)
is not the gradient of the value function
V
π
ρ
at π
(k)
, which is given in (4); rather, it is a preconditioned gradient [Kakade, 2001].
When the size of the state-action space becomes large (possibly infinite), we have to resort to function
approximation. Specifically, let π
θ be a differentiable mapping from the set of parameters Θ ⊂ R
n to
the set of stochastic policies. The parameter update step corresponding to (13) becomes
θ
(k+1) = arg min
θ∈Θ
Es∼d
(k)
ρ
h
Ea∼πθ
s

Qb(k)
s,a
+DΦ(π
θ
s
, π(k)
s
)
i
, (14)
where π
(k) means π
θ
(k)
, and d
(k)
ρ and Qb(k)
s,a are simple notations for d
π
(k)
ρ
and Qbπ
(k)
s,a respectively. This
approach is adopted by, e.g., Tomar et al. [2020] and Vaswani et al. [2021]. However, the optimization
problem is no longer convex in θ, and its convergence analysis becomes more challenging.
Alfano et al. [2024] introduced Approximate Mirror Policy Optimization (AMPO), a framework that
incorporates general parametrization into PMD with convergence guarantees. A key instrument they
introduced is the Bregman projected policy class. The idea is to use a parametrized function f
θ
:
S×A → R to approximate the dual update in (7), which in the context of PMD is ∇Φ(π
(k)
)−ηkQb(k)
.
Then follow the second step in MD to define the policy class

π
θ
: π
θ
s = projΦ
∆(A)

∇Φ
∗
(f
θ
s
)

, s ∈ S	
, θ ∈ Θ.
For example, using the negative-entropy (Example 2.2 or 2.3), it leads to the softmax policy class:
π
θ
s,a = exp(f
θ
s,a)/∥ exp(f
θ
s
)∥1, (s, a) ∈ S × A. (15)
While such policy classes are widely used in both theory and practice, recognizing them as the composition of a Bregman projection, a conjugate mirror map and a generic function approximation f
θ
(such as neural networks) allows more structured and sharper convergence analysis.
Facilitated with the Bregman projected policy class, extending PMD with function approximation
rests upon how we approximate ∇Φ(π
(k)
) − ηkQb(k)
(existing in the dual space) using f
θ
. AMPO
[Alfano et al., 2024] proposes to minimize the expected L2-distance between them, i.e.,
min
θ
Es∼d
(k)
ρ
h

f
θ
s −

∇Φ(π
(k)
s
) − ηkQb(k)
s


2
2
i
. (16)
On the other hand, Lan [2022] tries to minimize the expected (in state distribution) L∞-norm of the
difference between f
θ
s
and ∇Φ(π
(k)
s ) − ηQb(k)
s .
In contrast, we propose to use the corresponding dual Bregman divergence DΦ∗ to measure their
similarity in the dual space. In particular, Our method finds θ
(k+1) by (approximately) solving
min
θ∈Θ
Es∼d
(k)
ρ
h
DΦ∗

∇Φ(π
(k)
s
) − ηkQb(k)
s
, f θ
s

i
, (1
where d
(k)
ρ can be replaced with other distributions to accommodate the scenario of off-policy training.
Here, we can see that the similarity between the two dual vectors f
θ
s
and ∇Φ(π
(k)
s ) − ηQb(k)
s are
measured by the Bregman divergence of Φ
∗
, which naturally lives in the dual space. Together with
the Bregman divergence of Φ used in policy projection, they form a complete duality framework.
A complete description of our method is given as Algorithm 1, and we call it Dual Approximation
Policy Optimization (DAPO).


 Comparison with AMPO, MDPO and FMA-PG
AMPO [Alfano et al., 2024] replaces the minimization problem in Line 4 of Algorithm 1 by (16)
regardless of the mirror map used in policy projection. More concretely, let Φ1 be the negative
entropy on R
n
+ and Φ2 be the squared L2 norm. Then AMPO’s approximation loss can be written as
Es∼d
(k)
ρ
h
DΦ∗
2

∇Φ1(π
(k)
s
) − ηkQb(k)
s
, f θ
s
i , (22)
and Φ1 is again used in the policy projection step. In theory, as long as the approximation error is
small, it is possible to establish convergence of the method [Alfano et al., 2024]. However, such a
mismatch, or inconsistency, between approximations in primal and dual spaces may cause problems
when the approximation error cannot be made sufficiently small. This is precisely the case in practice,
where we can only afford to run at most a few steps of the stochastic gradient method to reduce the
approximation error. The importance of the consistency between the two mirror maps has also been
pointed out by Tomar et al. [2020].
In Section 5, we demonstrate that on standard benchmarks DAPO-KL obtains state-of-the-art performance with only one step of stochastic gradient method in reducing the approximation loss (17),
comparable to SAC [Haarnoja et al., 2018b]. On the other hand, we could not get AMPO competitive
with many numbers of stochastic gradient steps.
The Mirror Descent Policy Optimization (MDPO) method of Tomar et al. [2020] is based on
minimizing over θ directly in the formulation (14). If π
θ belongs to the softmax class of (15) and
DΦ is the KL-divergence, then it is equivalent to DAPO-KL. Therefore, our convergence analysis in
Section 4.1 directly applies to MDPO, which is not provided by Tomar et al. [2020].
The Functional Mirror Ascent (FMA-PG) framework of Vaswani et al. [2021] also takes the form (14).
However, similar to MDPO, Vaswani et al. [2021] did not exploit any composition structure of the
parametrization π
θ or the MDP structure. Rather, they conducted convergence analysis based on the
general theory for smooth, non-convex optimization, which leads to considerably weaker results.