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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).
3.1 Instantiations of DAPO
We give three instantiations of DAPO using the three mirror maps given in Examples 2.1-2.3. In
deriving these instantiations as well as implementing the algorithms, instead of directly using the
dual Bregman divergence DΦ∗ , it is often more convenient to use the following identity:
DΦ∗

∇Φ(π
(k)
s
) − ηkQb(k)
s
, f θ
s

= DΦ

∇Φ
∗
(f
θ
s
), ∇Φ
∗
(∇Φ(π
(k)
s
) − ηkQb(k)
s
)

. (18)
See Corollary B.4 for a proof. This identity will also facilitate our convergence analysis later.
DAPO-L2. With Φ being the squared L2-norm mirror map described in Example 2.1, the approximation problem in (17) (same as line 4 in Algorithm 1) becomes
min
θ
Es∼d
(k)
ρ
h

f
θ
s − π
(k)
s + ηkQb(k)
s


2
2
i
, (19)
and Line 5 of Algorithm 1 is the Euclidean projection
π
(k+1)
s = arg min
π∈∆(A)


π − f
(k+1)
s


2
2
.
Here we have used the simpler notation f
(k+1)
s for f
θ
(k+1)
s
.
DAPO-KL∗
. With Φ being the negative entropy defined on R
|A|
+ (see Example 2.2), we have
∇Φ
∗
(f
(k+1)
s
) = exp(f
θ
s,a),
∇Φ
∗

∇Φ(π
(k)
) − ηkQb(k)

= π
(k)
s
exp
−ηkQb(k)
s

.
Using the identity (18), we can write the loss in the approximation problem (17) as
Es∼d
(k)
ρ
h
DKL 
exp(f
θ
s
)


π
(k)
s
exp
−ηkQb(k)
s

i , (20)
where DKL is given by (10). Policy projection as in (15) is necessary to obtain π
(k+1) because
∇Φ
∗
(f
(k+1)
s ) is not in the simplex in general. DAPO-KL∗ has disadvantages in both theory and
practice compared with its close variant DAPO-KL, which we will explain next.
DAPO-KL. With Φ being the negative entropy restricted on ∆(A) (see Example 2.3), the range of
∇Φ
∗
is ∆(A), thus the projection step (Line 5 of Algorithm 1) becomes redundant. Here, we have
π
(k+1)
s = ∇Φ
∗
(f
(k+1)
s
) = exp(f
θ
s,a)/∥ exp(f
θ
s
)∥1,
∇Φ
∗

∇Φ(π
(k)
) − ηkQb(k)

= π
(k)
s
exp
−ηkQb(k)
s

/Z(k)
s
,
where Z
(k)
s =


π
(k)
s exp
−ηkQb(k)
s


1
. Again using (18), the loss in (17) becomes
Es∼d
(k)
ρ
h
DKL 
π
θ
s



 π
(k)
s
exp
−ηkQb(k)
s

/Z(k)
s
i , (21)
where DKL is given by (12). There are several distinctions between DAPO-KL and DAPO-KL∗
.
• The approximation loss in (21) is in terms of the full policy parametrization π
θ
(normalized
over the simplex), matching the implementation of several popular algorithms [Tomar et al.,
2020, Vaswani et al., 2021]. In contrast, the loss in (20) is in terms of the unnormalized
entity exp(f
θ
), which will suffer additional loss after policy projection.
• In theory,we are able to provide a competitive convergence analysis of DAPO-KL (see
Section 4.1) thanks to the fact that the two arguments in DKL in (21) are both on the simplex,
which is not the case in (20).
For these reasons, we will only consider DAPO-KL from now on. However, we think it is necessary
to expose the subtleties between the two variants, because many works on policy mirror descent
methods [e.g. Alfano et al., 2024], assumes ∇Φ
∗ = (∇Φ)−1
. We demonstrate that the more nuanced
extension of the MD theory (Lemma B.2) is crucial for developing and analyzing practical al