L INEAR C ONVERGENCE OF N ATURAL P OLICY G RADIENT M ETHODS WITH L OG -L INEAR P OLICIES
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A BSTRACT
We consider inﬁnite-horizon discounted Markov decision processes and study the convergence rates of the natural policy gradient (NPG) and the Q-NPG methods with the log-linear policy class.
Using the compatible function approximation framework, both methods with log-linear policies can be written as inexact versions of the policy mirror descent (PMD) method.
We show that both methods attain linear convergence rates and ˜ O (1 /(cid:15) 2 ) sample complexities using a simple, non-adaptive geometrically increasing step size, without resorting to entropy or other strongly convex regularization.
Lastly, as a byproduct, we obtain sublinear convergence rates for both methods with arbitrary constant step size
I NTRODUCTION
Policy gradient (PG) methods have emerged as a popular class of algorithms for reinforcement learning.
Unlike classical methods based on (approximate) dynamic programming (e.g., Puterman, 1994; Sutton & Barto, 2018), PG methods update directly the policy and its parametrization along the gradient direction of the value function (e.g., Williams, 1992; Sutton et al., 2000; Baxter & Bartlett, 2001).
An important variant of PG is the natural policy gradient (NPG) method (Kakade, 2001).
NPG uses the Fisher information matrix of the policy distribution as a preconditioner to improve the policy gradient direction, similar to quasi-Newton methods in classical optimization.
Variants of NPG with policy parametrization through deep neural networks were shown to have impressive empirical successes (Schulman et al., 2015; Lillicrap et al., 2016; Mnih et al., 2016; Schulman et al., 2017).
Motivated by the success of NPG in practice, there is now a concerted effort to develop convergence theories for the NPG method.
Neu et al.
(2017) provide the ﬁrst interpretation of NPG as a mirror descent (MD) method (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003).
By leveraging different techniques for analyzing MD, it has been established that NPG converges to the global optimum in the tabular case (Agarwal et al., 2021; Khodadadian et al., 2021b; Xiao, 2022) and some more general settings (Shani et al., 2020; Vaswani et al., 2022; Grudzien et al., 2022).
In order to get a fast linear convergence rate for NPG, several recent works consider the regularized NPG methods, such as the entropy-regularized NPG (Cen et al., 2021) and other convex regularized NPG methods (Lan, 2022;
Zhan et al., 2021).
By designing appropriate step sizes, Khodadadian et al.
(2021b)
and Xiao (2022) obtain linear convergence of NPG without regularization.
However, all these linear convergence results are limited in the tabular setting with a direct parametrization.
It remains unclear whether this same linear convergence rate can be established in function approximation settings.
In this paper we provide an afﬁrmative answer to this question for the log-linear policy class.
Our approach is based on the framework of compatible function approximation (Sutton et al., 2000;
Kakade, 2001), which was extensively developed by Agarwal et al. (2021).
Using this framework, variants of NPG with log-linear policies can be written as policy mirror descent (PMD) methods with inexact evaluations of the advantage function or Q-function (giving rise to NPG or Q-NPG respectively).
Then by extending a recent analysis of PMD (Xiao, 2022), we obtain a non-asymptotic linear convergence of both NPG and Q-NPG with log-linear policies.
A distinctive feature of this approach is the use of a simple, non-adaptive geometrically increasing step size, without resorting to entropy or other (strongly) convex regularization.
The extensions are highly nontrivial and require quite different techniques.
This linear convergence leads to the ˜ O (1 /(cid:15) 2 ) sample complexities for both methods.
In particular, our sample complexity analysis also ﬁxes errors of previous work.
Lastly, as a byproduct, we obtain sublinear convergence rates for both methods with arbitrary constant step size.
See
Appendix A for a thorough review.
In particular, Table 1 provides a complete overview of our results.
P RELIMINARIES ON M ARKOV D ECISION P ROCESSES
We consider an MDP denoted as M = {S , A , P , c, γ } , where S is a ﬁnite state space, A is a ﬁnite action space, P : S ×
A → S is a Markovian transition model with P ( s (cid:48) | s, a ) being the transition probability from state s to s (cid:48) under action a , c is a cost function with c ( s, a ) ∈ [0 , 1] for all ( s, a ) ∈ S × A , and γ ∈ [0 , 1) is a discounted factor.
Here we use cost instead of reward to better align with the minimization convention in the optimization literature.
The agent’s behavior is modeled as a stochastic policy π ∈ ∆( A ) |S| , where π s ∈ ∆( A ) is the probability distribution over actions A in state s ∈ S .
At each time t , the agent takes an action a t ∈
A given the current state s t ∈ S , following the policy π , i.e., a t ∼ π s t .
Then the MDP transitions into the next state s t +1 with probability P ( s t +1 | s t , a t ) and the agent encounters the cost c t =
c ( s t , a t ) .
Thus, a policy induces a distribution over trajectories { s t , a t , c t } t ≥ 0 .
In the inﬁnite-horizon discounted setting, the cost function of π with an initial state s is deﬁned as
[Equation]
Given an initial state distribution ρ ∈ ∆( S ) , the goal of the agent is to ﬁnd a policy π that minimizes
[Equation]
A more granular characterization of the performance of a policy is the state-action cost function (Q-function).
For any pair ( s, a ) ∈ S × A , it is deﬁned as
[Equation]
Let Q s ∈ R |A| denote the vector [ Q s,a ] a ∈A .
Then we have V s ( π ) =
E a ∼ π s [ Q s,a ( π )]
= (cid:104) π s , Q s ( π ) (cid:105) .
The advantage function 1 is a centered version of the Q-function:
A s,a ( π ) def = Q s,a ( π ) − V s ( π ) , which satisﬁes E a ∼ π s [ A s,a ( π )]
= 0 for all s ∈ S .
Visitation probabilities.
Given a starting state distribution ρ ∈ ∆( S ) , we deﬁne the state visitation distribution d π ( ρ ) ∈ ∆( S ) , induced by a policy π , as
[Equation]
where Pr π ( s t = s | s 0 )
is the probability that s t is equal to s by following the trajectory generated by π starting from s 0 .
Intuitively, the state visitation distribution measures the probability of being at state s across the entire trajectory.
We deﬁne the state-action visitation distribution ¯ d π ( ρ ) ∈ ∆( S × A ) as
[Equation]
In addition, we extend the deﬁnition of ¯ d π ( ρ ) by specifying the initial state-action distribution ν , i.e.,
[Equation]
The difference in the last two deﬁnitions is that for the former, the initial action a 0 is sampled directly from π , whereas for the latter, it is prescribed by the initial state-action distribution ν .
We use ˜ d compared to ¯ d to better distinguish the cases with ν and ρ .
Without speciﬁcation, we even omit the argument ν or ρ throughout the paper to simplify the presentation as they are self-evident.
From these deﬁnitions, we have for all ( s, a ) ∈ S × A ,
[Equation]
Policy parametrization.
In practice, both the state and action spaces S and A can be large and some form of function approximation is needed to reduce the dimensions and make the computation feasible.
In particular, the policy π is often parametrized as π ( θ ) with θ ∈ R m , where m is much smaller than |S| and |A| .
In this paper, we focus on the log-linear policy class.
Indeed, we assume that for each state-action pair ( s, a ) , there is a feature mapping φ s,a ∈ R m and the policy takes the form
[Equation]
This setting is important since it is the simplest instantiation of the widely-used neural policy parametrization.
To simplify notations in the rest , we use the shorthand V ρ ( θ ) for V ρ ( π ( θ )) and similarly Q s,a ( θ ) for Q s,a ( π ( θ )) , A s,a ( θ ) for A s,a ( π ( θ )) , d θs for d π ( θ ) s , ¯ d θs,a for ¯ d π ( θ ) s,a , and ˜ d θs,a for ˜ d π ( θ ) s,a .
Natural Policy Gradient (NPG) Method.
Using the notations deﬁned above, the parametrized policy optimization problem is to minimize the function V ρ ( θ ) over θ ∈ R m .
The policy gradient is given by (see, e.g., Williams, 1992; Sutton et al., 2000)
[Equation]
For parametrizations that are differentiable and satisfy (cid:80) a ∈A π s,a ( θ ) = 1 , including the log-linear class deﬁned in (6), we can replace Q s,a ( θ ) by A s,a ( θ ) in the above expression (Agarwal et al., 2021).
The NPG method (Kakade, 2001) takes the form
[Equation]
where η k > 0 is a scalar step size, F ρ ( θ ) is the Fisher information matrix
[Equation]
9) and F ρ ( θ ) † denotes the Moore-Penrose pseudoinverse of F ρ ( θ ) .
NPG WITH C OMPATIBLE F UNCTION A PPROXIMATION
The parametrized value function V ρ ( θ ) is non-convex in general (see, e.g., Agarwal et al., 2021).
Despite being a non-convex optimization problem, there is still additional structure we can leverage to ensure convergence.
Following Agarwal et al.
(2021), we adopt the framework of compatible function approximation (Sutton et al., 2000; Kakade, 2001), which exploits the MDP structure and leads to tight convergence rate analysis.
For any w, θ ∈ R m and state-action distribution ζ , deﬁne compatible function approximation error as
[Equation]
Kakade (2001) showed that the NPG update (8) is equivalent to (up to a constant scaling of η k )
[Equation]
where ¯ d ( k ) is a shorthand for the state-action visitation distribution ¯ d π ( θ ( k ) )
( ρ ) deﬁned in (3).
A derivation of (11) is provided in Appendix B (Lemma 1) for completeness.
In other words, w ( k ) (cid:63) is the solution to a regression problem that tries to approximate A s,a ( θ ( k ) ) using ∇ θ log π s,a ( θ ( k ) ) as features.
This is where the term "compatible function approximation error" comes from.
For the log-linear policy class deﬁned in (6), we have
[Equation]
where ¯ φ s,a ( θ ) are called centered features vectors .
In practice, we cannot minimize L A exactly; instead, a sample-based regression problem is solved to obtain an approximate solution w ( k ) .
This leads to the following inexact NPG update rule:
[Equation]
The inexact NPG updates require samples of unbiased estimates of A s,a ( θ ) , the corresponding sampling procedure is given in Algorithm 4, and a sample-based regression solver to minimize L A is given in Algorithm 5 in the Appendix.
Alternatively, as proposed by Agarwal et al.
(2021), we can deﬁne the compatible function approximation error as
[Equation]
and use it to derive a variant of the inexact NPG update called Q-NPG :
[Equation]
For Q-NPG, the sampling procedure for estimating Q s,a ( θ ) is given in Algorithm 3 and a samplebased regression solver for w ( k ) is proposed in Algorithm 6 in the Appendix.
The sampling procedure and the regression solver of NPG are less efﬁcient than those of Q-NPG.
Indeed, the sampling procedure for A s,a ( θ ) in Algorithm 4 not only estimates Q s,a ( θ ) , but also requires an additional estimation of V s ( θ ) , and thus doubles the amount of samples as compared to
Algorithm 3.
Furthermore, the stochastic gradient estimator of L Q in Algorithm 6 only computes on a single action of the feature map φ s,a .
Whereas the one of L A in Algorithm 5 computes on the centered feature map ¯ φ s,a ( θ ) deﬁned in (12), which needs to go through the entire action space, thus is |A| times more expensive to run.
See Appendix C for more details.
Following Agarwal et al.
(2021), we consider slightly different variants of NPG and Q-NPG, where ¯ d ( k ) in (13) and (15) is replaced by a more general state-action visitation distribution ˜ d ( k ) =
˜
d π ( θ ( k ) )
( ν ) deﬁned in (4) with ν ∈ ∆( S × A ) .
The advantage of using ˜ d ( k ) is that it allows better exploration than ¯ d ( k ) as ν can be chosen to be independent to the policy π ( θ ( k ) ) .
For example, it can be seen from (5) that the lower bound of ˜ d π is independent to π , which is not the case for ¯ d π .
This property is crucial in the forthcoming convergence analysis.
3. F ORMULATION AS I NEXACT P OLICY M IRROR D ESCENT
Given an approximate solution w ( k ) for minimizing L Q (cid:0) w, θ ( k ) , ˜ d ( k ) (cid:1) , the Q-NPG update rule θ ( k +1)
= θ ( k ) − η k w ( k ) , when plugged in the log-linear parametrization (6), results in a new policy
[Equation]
where π ( k ) is a shorthand for π s,a ( θ ( k ) ) and Z ( k ) s is a normalization factor to ensure (cid:80) a ∈A π ( k +1)
s,a = 1 , for each s ∈ S .
We note that the above π ( k +1) can also be obtained by a mirror descent update:
[Equation]
where Φ s ∈ R |A|× m is a matrix with rows ( φ s,a ) (cid:62)
∈ R 1 × m for a ∈ A , and D ( p, q ) denotes the Kullback-Leibler (KL) divergence between two distributions p, q ∈ ∆( A ) , i.e., D ( p, q ) def = (cid:80) a ∈A p a log (cid:0) p a q a (cid:1) .
A derivation of (16) is provided in Appendix B (Lemma 2) for completeness.
If we replace Φ s w ( k ) in (16) by the vector (cid:2) Q s,a ( π ( k ) ) (cid:3) a ∈A ∈ R |A| , then it becomes the policy mirror descent (PMD) method in the tabular setting studied by, for example, Shani et al.
Lan (2022) and Xiao (2022).
In fact, the update rule (16) can be viewed as an inexact PMD method where Q s ( π ( k ) ) is linearly approximated by Φ s w ( k ) through compatible function approximation (14).
Similarly, we can write the inexact NPG update rule as
[Equation]
where w ( k ) is an inexact solution for minimizing L A (cid:0) w, θ ( k ) , ˜ d ( k ) (cid:1) deﬁned in (10), and ¯Φ ( k ) s ∈
R |A|× m is a matrix whose rows consist of the centered feature maps (cid:0) ¯ φ s,a ( θ ( k ) ) (cid:1) (cid:62) , as deﬁned in (12).
Reformulating Q-NPG and NPG into the mirror descent forms (16) and (17), respectively, allows us to adapt the analysis of PMD method developed in Xiao (2022) to obtain sharp convergence rates.
In particular, we show that with an increasing step size η k ∝ γ k , both NPG and Q-NPG with log-linear policy parametrization converge linearly up to an error ﬂoor determined by the quality of the compatible function approximation.
A NALYSIS OF Q-NPG WITH L OG -L INEAR P OLICIES
In this section, we provide the convergence analysis of the following inexact Q-NPG method
[Equation]
where ˜ d ( k ) is shorthand for ˜
d π ( θ ( k ) )
( ν ) and ν ∈ ∆( S × A ) is an arbitrary state-action distribution that does not depend on ρ .
The exact minimizer is denoted as w ( k ) (cid:63)
[Equation]
Following Agarwal et al.
(2021), the compatible function approximation error can be decomposed as
[Equation]
The statistical error measures how accurate is our solution to the regression problem, i.e., how good w ( k ) is compared with w ( k ) (cid:63) .
The approximation error measures the best possible solution for approximating Q s,a ( θ ( k ) ) using φ s,a as features in the regression problem (modeling error).
One way to proceed with the analysis is to assume that both the statistical error and the approximation error are bounded for all iterations, which is the approach we take in Section 4.2 and is also the approach we take later in Section 5 for the analysis of the NPG method.
However, in Section 4.1, we ﬁrst take an alternative approach proposed by Agarwal et al.
(2021), where the assumption of bounded approximation error is replaced by a bounded transfer error .
The transfer error refers to L Q (cid:0) w ( k ) (cid:63) , θ ( k ) , ˜ d ∗ (cid:1) , where the iteration-dependent visitation distribution ˜ d ( k ) is shifted to a ﬁxed one ˜ d ∗ (deﬁned in Section 4.1).
These two approaches require different additional assumptions and result in slightly different convergence rates.
Here we ﬁrst state the common assumption on the bounded statistical error.
Assumption 1 (Bounded statistical error, Assumption 6.1.1 in Agarwal et al.
There exists (cid:15) stat > 0 such that for all iterations k ≥ 0 of the Q-NPG method (18) , we have
[Equation]
By solving the regression problem with sampling based approaches, we can expect (cid:15) stat =
O (1 / √ T ) (Agarwal et al., 2021) or (cid:15) stat =
O (1 /T )
(see Corollary 1) where T is the number of iterations used to ﬁnd the inexact solution w ( k ) .
4. A NALYSIS WITH B OUNDED T RANSFER E RROR
Here we introduce some additional notations.
For any state distributions p, q ∈ ∆( S ) , we deﬁne the distribution mismatch coefﬁcient of p relative to q as (cid:107) p/q (cid:107) ∞ def =
max s ∈S p s /q s .
Let π ∗ be an arbitrary comparator policy , which is not necessarily an optimal policy and does not need to belong to the log-linear policy class.
Fix a state distribution ρ ∈ ∆( S ) .
We denote d π ∗ ( ρ ) as d ∗ and d π ( θ ( k ) )
( ρ ) as d ( k ) , and deﬁne the following distribution mismatch coefﬁcients:
[Equation]
and
[Equation]
Thus, for all k ≥ 0 , we have ϑ k ≤ ϑ ρ .
We assume that ϑ ρ < ∞ , which is the case, for example, if ρ s > 0 for all s ∈ S .
This is commonly used in the literature on policy gradient methods (e.g., Zhang et al., 2020; Wang et al., 2020) and the NPG convergence analysis (e.g., Cayci et al., 2021; Xiao, 2022).
We further relax this condition in Appendix H.
Given a state distribution ρ and a comparator policy π ∗ , we deﬁne a state-action measure ˜ d ∗ as
[Equation]
(21) and use it to express the transfer error as L Q (cid:0) w ( k ) (cid:63) , θ ( k ) , ˜ d ∗ (cid:1) .
Assumption 2 (Bounded transfer error, Assumption 6.1.2 in Agarwal et al.
There exists (cid:15) bias > 0 such that for all iterations k ≥ 0 of the Q-NPG method (18) , we have
[Equation]
The (cid:15) bias is often referred to as the transfer error, since it is the error due to replacing the relevant distribution ˜ d ( k ) by ˜ d ∗ .
This transfer error bound characterizes how well the Q-values can be linearly approximated by the feature maps φ s,a .
It can be shown that (cid:15) bias = 0 when π ( k ) is the softmax tabular policy (Agarwal et al., 2021) or the MDP has a certain low-rank structure (Jiang et al., 2017;
Yang & Wang, 2019; Jin et al., 2020).
For rich neural parametrizations, (cid:15) bias can be made small (Wang et al., 2020).
The next assumption concerns the relative condition number between two covariance matrices of φ s,a deﬁned under different state-action distributions.
Assumption 3 (Bounded relative condition number, Assumption 6.2 in Agarwal et al.
Fix a state distribution ρ , a state-action distribution ν and a comparator policy π ∗ .
Let
[Equation]
and
[Equation]
23) where ˜ d ∗ is speciﬁed in (21) .
We deﬁne the relative condition number between Σ ˜ d ∗ and Σ ν as
[Equation]
and assume that κ ν is ﬁnite.
The κ ν is referred to as the relative condition number, since the ratio is between two different matrix induced norm.
Assumption 3 beneﬁts from the use of ν .
In fact, it is shown in Agarwal et al.
Remark 22 and Lemma 23) that κ ν can be reasonably small (e.g., κ ν ≤ m is always possible) and independent to the size of the state space by controlling ν .
Our analysis also needs the following assumption, which does not appear in Agarwal et al.
Assumption 4 (Concentrability coefﬁcient for state visitation) .
There exists a ﬁnite C ρ > 0 such that for all iterations k ≥ 0 of the Q-NPG method (18) , it holds that
[Equation]
The concentrability coefﬁcient is studied in the analysis of approximate dynamic programming algorithms (Munos, 2003; 2005; Munos & Szepesvári, 2008).
It measures how much ρ can get ampliﬁed in k steps as compared to the reference distribution d ∗ s .
Let ρ min =
min s ∈S ρ s .
A sufﬁcient condition for Assumption 4 to hold is that ρ min > 0 .
Indeed,
[Equation]
In reality, (cid:112) C ρ can be much smaller than the pessimistic bound shown above.
This is especially the case if we choose π ∗ to be the optimal policy and d ( k ) → d ∗ .
We further replace C ρ by C ν deﬁned in Section 4.2 that is independent to ρ and thus is more easily satisﬁed.
Now we present our ﬁrst main result.
Theorem 1.
Fix a state distribution ρ , an state-action distribution ν and a comparator policy π ∗ .
We consider the Q-NPG method (18) with the uniform initial policy 2 and the step sizes satisfying η 0 ≥ 1 −
γγ log |A| and η k +1 ≥ 1 γ η k .
Suppose that Assumptions 1, 2, 3 and 4 all hold.
Then we have for all k ≥ 0 ,
[Equation]
The main differences between our Theorem 1 and Theorem 20 of Agarwal et al.
(2021), which is their corresponding result on the inexact Q-NPG method, are summarized as follows.
• The convergence rate of Agarwal et al.
(2021, Theorem 20) is O (1 / √ k ) up to an error ﬂoor determined by (cid:15) stat and (cid:15) bias .
We have linear convergence up to an error ﬂoor that also depends on (cid:15) stat and (cid:15) bias .
However, the magnitude of our error ﬂoor is worse (larger) by a factor of ϑ ρ (cid:112) C ρ , due to the concentrability and the distribution mismatch coefﬁcients used in our proof.
A very pessimistic bound on this factor is as large as |S| 2 / (1 − γ ) 2 .
In terms of required conditions, both results use Assumptions 1, 2 and 3.
Agarwal et al.
(2021, Theorem 20) further assume that the norms of the feature maps φ s,a are uniformly bounded and w ( k ) has a bounded norm (e.g., obtained by a projected stochastic gradient descent).
Due to different analysis techniques referred next, we avoid such boundedness assumptions but rely on the concentrability coefﬁcient C ρ deﬁned in Assumption 4.
• Agarwal et al.
(2021, Theorem 20) uses a diminishing step size η ∝ 1 / √ k where k is the total number of iterations, but we use a geometrically increasing step size η k ∝ γ k for all k ≥ 0 .
This discrepancy reﬂects the different analysis techniques adopted.
The key analysis tool in Agarwal et al.
(2021) is a NPG Regret Lemma (their Lemma 34) which relies on the smoothness of the functions log π s,a ( θ ) (thus the boundedness of (cid:107) φ s,a (cid:107) ) and the boundedness of (cid:107) w ( k ) (cid:107) , and thus the classical O (1 / √ k ) diminishing step size in the optimization literature.
Our analysis exploits
the three-point descent lemma (Chen & Teboulle, 1993) and the performance difference lemma (Kakade & Langford, 2002), without reliance on smoothness parameters.
As a consequence, we can take advantage of exponentially growing step sizes and avoid assuming the boundedness of (cid:107) φ s,a (cid:107) or (cid:107) w ( k ) (cid:107) .
The reason of using increasing step size can be interpreted as approximate policy iteration.
See
Appendix H the connection with policy iteration for more details.
As a by product, we also obtain a sublinear O (1 /k ) convergence result while using arbitrary constant step size.
Theorem 2.
Fix a state distribution ρ , an state-action distribution ν and an optimal policy π ∗ .
We consider the Q-NPG method (18) with the uniform initial policy and any constant step size η k = η > 0 .
Suppose that Assumptions 1, 2, 3 and 4 all hold.
Then we have for all k ≥ 0 ,
[Equation]
4. A NALYSIS WITH B OUNDED A PPROXIMATION E RROR
In this section, instead of assuming bounded transfer error, we provide a convergence analysis based on the usual notion of approximation error and a weaker concentrability coefﬁcient.
Assumption 5 (Bounded approximation error) .
There exists (cid:15) approx > 0 such that for all iterations k ≥ 0 of the Q-NPG method (18) , it holds that
[Equation]
[Equation] [Equation]
As mentioned in Agarwal et al.
[Equation]
Assumption 5 is stronger than Assumption 2 (bounded transfer error).
Indeed,
[Equation]
Assumption 6 (Concentrability coefﬁcient for state-action visitation) .
There exists C ν < ∞ such that for all iterations of the Q-NPG method (18)
[Equation]
we have
[Equation]
where h ( k ) s
[Equation]
a represents all of the following quantities:
[Equation]
Since we are free to choose ν independently to ρ , we can choose ν s,a > 0 for all ( s, a ) ∈ S × A for
Assumption 6 to hold.
Indeed, with ν min denoting min ( s,a ) ∈S×A ν s,a , we have
[Equation]
where the upper bound can be smaller than that in (26) if ρ min is smaller than ν min .
Theorem 3.
Fix a state distribution ρ , an state-action distribution ν and a comparator policy π ∗ .
We consider Q-NPG method (18) with uniform initial policy and step sizes satisfying η 0 ≥ 1 −
γγ log |A| and η k +1 ≥ 1 γ η k .
Suppose that Assumptions 1, 5 and 6 hold.
Then we have for all k ≥ 0 ,
[Equation]
Compared to Theorem 1, while the approximation error assumption is stronger than the transfer error assumption, we do not require the assumption on relative condition number κ ν and the error ﬂoor does not depends on κ ν nor explicitly on |A| .
Besides, we can always choose ν so that C ν is ﬁnite even if C ρ is unbounded.
However, it is not clear if Theorem 3 is better than Theorem 1.
Remark 1.
Note that Theorems 1, 2 and 3 beneﬁt from using the visitation distribution ˜ d ( k ) instead of ¯ d ( k ) (i.e., beneﬁt from using ν instead of ρ ).
In particular, from (5) , ˜ d ( k ) has a lower bound that is independent to the policy π ( k ) or ρ .
This property allows us to deﬁne a weak notion of relative condition number (Assumption 3) that is independent to the iterates, and also allows us to get a ﬁnite upper bound of C ν
(Assumption 6 and (30) ) that is independent to ρ .
4. S AMPLE C OMPLEXITY OF Q-NPG
Here we establish the sample complexity results, i.e., total number of samples of single-step interaction with the environment, of a sample-based Q-NPG method (Algorithm 2 in Appendix C).
Combined with a simple SGD solver, Q-NPG-SGD in Algorithm 6, the following corollary shows that Algorithm 2 converges globally by further assuming that the feature map is bounded and has non-singular covariance matrix (31).
The explicit constants of the bound can be found in the appendix.
Corollary 1.
Consider the setting of Theorem 3.
Suppose that the sample-based Q-NPG Algorithmis run for K iterations, with T gradient steps of Q-NPG-SGD (Algorithm 6) per iteration.
Furthermore, suppose that for all ( s, a ) ∈ S × A , we have (cid:107) φ s,a (cid:107) ≤ B with B > 0 , and we choose the step size α = 12 B 2 and the initialization w 0 = 0 for Q-NPG-SGD .
If for all θ ∈ R m , the covariance matrix of the feature map followed by the initial state-action distribution ν satisﬁes
[Equation]
(31) where I m ∈ R m × m is the identity matrix and µ > 0 , then
[Equation]
√ (cid:15) approx / (1 − γ ) (cid:1) + O (cid:0) √ m/ (cid:0) (1 − γ ) 3 √ T (cid:1)(cid:1) .
In Q-NPG-SGD , each trajectory has the expected length 1 / (1 − γ ) (see Lemma 4).
Consequently, with K =
O
(log(1 /(cid:15) )
log(1 / (1 − γ ))) and T =
O (cid:0) 1 (1 − γ ) 6 (cid:15) 2 (cid:1)
, Q-NPG requires K ∗ T/ (1 − γ ) =
˜ O (cid:0) 1 (1 − γ )
(cid:15) 2 (cid:1) samples such that E (cid:2)
V ρ ( π ( K ) ) (cid:3) − V ρ
( π ∗ ) ≤
O ( (cid:15) )
+ O (cid:0) √ (cid:15) approx 1 − γ (cid:1) .
The ˜ O (1 /(cid:15) 2 ) sample complexity matches with the one of value-based algorithms such as Q-learning (Li et al., 2020) .
Compared to Agarwal et al.
(2021, Corollary 26) for the sampled based Q-NPG Algorithm 2, their sample complexity is O (cid:0) 1 (1 − γ ) 11 (cid:15) 6 (cid:1) with K = 1 (1 − γ ) 2 (cid:15) 2 and T = 1 (1 − γ ) 8 (cid:15) 4 .
Despite the improvement on the convergence rate for K , they use the optimization results of Shalev-Shwartz & Ben-David (2014, Theorem 14.8) to obtain (cid:15) stat =
O (1 / √ T ) , while we use the one of Bach & Moulines (2013,
Theorem 1) (see Theorem 12 as well) to establish faster (cid:15) stat = O (1 /T ) .
Besides, they use the projected SGD method and require that the stochastic gradient is bounded which is incorrectly veriﬁed in their proof 3 .
In contrast, to apply Theorem 12, the boundedness of the stochastic gradient is not necessary.
Alternatively, we require a different condition (31).
A proof sketch of our corollary and a discussion of the condition (31) are provided in Appendix D.5 for more details
A NALYSIS OF NPG WITH L OG -L INEAR P OLICIES
We now return to the convergence analysis of the inexact NPG method, speciﬁcally,
[Equation]
Again, let w ( k ) (cid:63) ∈ argmin w L A (cid:0) w, θ ( k ) , ˜ d ( k ) (cid:1) denote the minimizer.
Our analysis of NPG is analogous to that of Q-NPG shown in the previous section.
That is, we again exploit the inexact PMD formulation (17) and use techniques developed in Xiao (2022).
The set of assumptions we use for NPG is analogous to the assumptions used in Section 4.2.
In particular, we assume a bounded approximation error instead of transfer error (c.f., Assumption 2) in minimizing L A and do not need the assumption on relative condition number.
Assumption 7 (Bounded statistical error, Assumption 6.5.1 in Agarwal et al.
There exists (cid:15) stat > 0 such that for all iterations k ≥ 0 of the NPG method (32) , we have
[Equation]
Assumption 8 (Bounded approximation error) .
There exists (cid:15) approx > 0 such that for all iterations k ≥ 0 of the NPG method (32) , we have
[Equation] [Equation]
Assumption 9 (Concentrability coefﬁcient for state-action visitation) .
There exists C ν < ∞ such that for all iterations k ≥ 0 of the NPG method (32) , we have
[Equation]
Under the above assumptions, we have the following result.
Theorem 4.
Fix a state distribution ρ , a state-action distribution ν , and a comparator policy π ∗ .
We consider the NPG method (32) with uniform initial policy and step sizes satisfying η 0 ≥ 1 −
γγ log |A| and η k +1 ≥ 1 γ η k .
Suppose that Assumptions 7, 8 and 9 hold.
Then we have for all k ≥ 0 ,
[Equation]
Now we compare Theorem 4 with Theorem 29 in Agarwal et al.
(2021) for the NPG analysis.
The main differences are similar to those for Q-NPG as summarized right after Theorem 1: Their convergence rate is sublinear while ours is linear; they assume uniformly bounded φ s,a and w ( k ) while we require bounded concentrability coefﬁcient C ν due to different proof techniques; they use diminishing step sizes and we use geometrically increasing ones.
Moreover, Theorem 4 requires bounded approximation error, which is a stronger assumption than the bounded transfer error used by their Theorem 29, but we do not need the assumption on bounded relative condition number.
In particular, such bounded relative condition number must hold for the covariance matrix of ¯ φ s,a ( θ ( k ) ) for all k ≥ 0 which depends on the iterates θ ( k ) .
This is in contrast to our Assumption 3, where we use a single ﬁxed covariance matrix for Q-NPG that is independent to the iterates, as deﬁned in (23).
In addition, the inequalities in (35) only involve half of the state-action visitation distributions listed in (29), i.e., the ﬁrst and the fourth terms.
Thus, C ν in (35) can share the same upper bound in (30) independent to the use of the algorithm Q-NPG or NPG.
Consequently, our concentrability coefﬁcient
C ν in (35) is weaker than Assumption 2 in Cayci et al.
(2021) which studies the linear convergence of NPG with entropy regularization for the log-linear policy class.
The reason is that the bound on
C ν in (30) does not depend on the policies throughout the iterations thanks to the use of ˜ d ( k ) instead of ¯ d ( k ) (see Remark 1 as well).
See Appendix H for a thorough discussion on the concentrability coefﬁcient C ν .
As a by product, we also obtain a sublinear rate for NPG while using an unconstrained constant step size in Theorem 9 in Appendix F.
By further assuming that the feature map is bounded and the Fisher information matrix (9) is non-singular, we also provide a ˜ O (1 /(cid:15) 2 ) sample complexity result of NPG as for Q-NPG and ﬁx the errors occurred in the NPG sample complexity proofs of Agarwal et al.
(2021) and Liu et al.
(2020) in Appendix G.
A CKNOWLEDGMENTS
We gratefully acknowledge Daniel Russo who pointed out that we did not cite properly Bhandari & Russo (2021) in the literature review in the previous version. We acknowledge the helpful discussion with Yanli Liu on the sample complexity analysis of both Q-NPG and NPG.
We would also like to thank the anonymous reviewers for their helpful comments
Table
of
Contents
A Related Work
B Standard Reinforcement Learning Results
C Algorithms
D
I Standard Optimization Results
Here we provide the related work discussion, the missing proofs from the main paper and some additional noteworthy observations made in the main paper.
A R ELATED W ORK
We provide an extended discussion for the context of our work, including a discussion comparing the technical novelty of the paper to the analysis of NPG with softmax tabular policies in Xiao (2022) and a comparison of the convergence theories of NPG in the literature.
Furthermore, we discuss future work, such as extending our analysis to other RL settings.
A. T ECHNICAL C ONTRIBUTION AND N OVELTY C OMPARED TO X IAO (2022)
Our technical novelty compared to Xiao (2022) is summarized as follows
• Our linear convergence results (i
e., Theorem 1, 3 and 4) are not direct applications of Theorem 10 in Xiao (2022).
Indeed, Xiao (2022) establishes the connection between NPG and a speciﬁc form of policy mirror descent (PMD) with the use of the weighted Bregman divergence for the tabular setting, while we show that this connection can also be established for the function approximation setting via the compatible function approximation framework (10).
We also modify the PMD framework of Xiao (2022) with the linear approximation of the advantage function in (17), inspired from the compatible function approximation framework.
Thus, the approaches of deriving the PMD form update are different.
Without this work of using the compatible function approximation framework to bridge NPG and PMD, it was not clear at all that the analysis of Xiao (2022) could be extended in the log-linear policy setting.
So our work is the ﬁrst step of showing that the proof techniques used in Xiao (2022) can be extended in function approximation regime
In fact, the extension is highly nontrivial and requires signiﬁcant innovation (see details below).
As for future work, one can extend our work to other function approximation setting through a similar compatible function approximation framework.
See Appendix A.3 for more details about the future work.
• Besides, our linear convergence results only consider the inexact NPG update.
Compared to Theorem 14 in Xiao (2022), which is their corresponding result on the inexact PMD method, we improve their analysis by making much weaker assumptions on the accuracy of the estimation Q ( π ) .
Xiao (2022) requires an L ∞ supremum norm bound on the estimation error of Q , i.e., (cid:107) (cid:98) Q ( π ) − Q ( π ) (cid:107) ∞ ≤ (cid:15) stat , whereas our convergence guarantee depends on the expected L 2 error of the estimate,
i
e., Assumption 1 and 7.
For instance, Assumption 1 from equation (62) can be written as E (cid:104) ( φ (cid:62) s,a w ( k ) − φ (cid:62) s,a w ( k ) (cid:63) ) 2 (cid:105) ≤ (cid:15) stat , which can be interpreted as E (cid:104) ( (cid:98) Q ( π ) − Q ( π )) 2 (cid:105) ≤ (cid:15) stat under the linear approximation setting.
The techniques for handling L ∞ and L 2 errors are very different.
Not only our assumption is weaker, it also beneﬁts from the sample complexity analysis that we explain next.
Consequently, when considering the sample complexity results we derived for sample-based (Q)-NPG in Corollary 1 and 3, the difference between our work and Theorem 16 in Xiao (2022), which corresponds to their sample complexity results, is even more signiﬁcant.
Corollary 1 with Algorithm Q-NPG-SGD
(Algorithm 6) satisﬁes Assumption 1 with a number of samples that depends only on the feature dimension m of φ and does not depend on the cardinality of state space |S| or action space |A| .
In contrast, the assumption (cid:107) (cid:98) Q ( π ) − Q ( π ) (cid:107) ∞ ≤ (cid:15) stat with the L ∞ norm in Xiao (2022, Theorem 16) causes the sample complexity to depend on |S||A| .
Furthermore, Xiao (2022) uses a Monte-Carlo approach with multiple independent rollouts per iteration, while our sample-based (Q)-NPG uses one single rollout (Algorithm 3 and 4) combined with regression solvers; Xiao (2022) derives a high probability sample complexity result, while we derive the convergence of the optimality gap E (cid:2) V ρ ( π ( K ) ) (cid:3) − V ρ ( π ∗ ) which can guarantee that the variance of V ρ ( π ( K ) ) converges to zero.
Thus, our sample-based algorithms had not been considered in Xiao (2022) and our proofs of Corollary 1 andrequire a different approach.
In particular, our sample complexity analysis regarding to the policy evaluation is novel.
Although our sample-based algorithms had been considered previously in Agarwal et al.
(2021) and Liu et al.
(2020), none of their analysis on the sample complexity was correct.
Indeed, Agarwal et al.
(2021) required the boundedness of the stochastic gradient estimator, which might not hold as we extensively discussed in Appendix D.5.
We ﬁxed this by showing that E (cid:104) (cid:98) Q s,a ( θ ) 2 (cid:105) is bounded.
See Appendix D.5 for all the subtleties, including a proof sketch of Corollary 1.
Liu et al.
(2020) also incorrectly used an inequality where the random variables are correlated.
See the detailed explanation (Footnote 7) in Appendix
G.1.
We ﬁxed this error with a careful conditional expectation argument.
Please refer to
Appendix G.1 for all the details, including a proof sketch of Corollary 3.
These dimensions are where an important part of the technical work was done.
Therefore, outside of the tabular setting, and considering NPG methods that make use of a regression solver, our complexity analysis is currently the only analysis that is entirely correct that we are aware of.
• Finally we not only extend the work of Xiao (2022) to NPG for log-linear policy, but also consider the Q-NPG method and establish its linear convergence analysis.
This is a method that is unique to log-linear policy and again had not been considered in Xiao (2022).
A. F INITE -T IME A NALYSIS OF THE N ATURAL P OLICY G RADIENT
NPG for the softmax tabular policies.
For the softmax tabular policies, Shani et al.
(2020) show that the unregularized NPG has a O (1 / √ k ) convergence rate and the regularized NPG has a faster
O (1 /k ) convergence rate by using a decaying step size.
Agarwal et al.
(2021) improve the convergence rate of the unregularized NPG to O (1 /k ) with constant step sizes.
Further, Khodadadian et al.
(2021a) also achieves O (1 /k ) convergence rate for the off-policy natural actor-critic (NAC), and a slower sublinear result is established by Khodadadian et al.
(2022a) for the two-time-scale NAC.
By using the entropy regularization, Cen et al.
(2021) achieve a linear convergence rate for NPG.
A similar linear convergence result has been obtained by rewriting the NPG update under the PMD framework with the Kullback–Leibler (KL) divergence (Lan, 2022) or with a more general convex regularizer (Zhan et al., 2021).
Such approach is also applied in the averaged MDP setting to achieve linear convergence for NPG (Li et al., 2022a).
However, adding regularization might induce bias for the solution.
Thus, Lan (2022) considers exponentially diminishing regularization to guarantee unbiased solution.
Furthermore, by considering both the KL divergence and the diminishing entropy regularization, Li et al.
(2022b) establish the linear convergence rate not only for the optimality gap but also for the policy.
That is, the policy will converge to the ﬁxed high entropy optimal policy.
Consequently, Li et al.
(2022b) show a local super-linear convergence of both the policy and optimality gap, as discussed in Xiao (2022, Section 4.3).
Recently, Bhandari & Russo (2021), Khodadadian et al.
(2021b; 2022b) and Xiao (2022) show that regularization is unnecessary for obtaining linear convergence, and it sufﬁces to use appropriate step sizes for NPG.
In particular, Bhandari & Russo (2021) propose to use an exact line search for the step size (Theorem 1 (a)) or to choose an adaptive step size (Theorem 1 (c)).
Similar adaptive step size is proposed by Khodadadian et al.
(2021b; 2022b).
Notice that such adaptive step size requires complete knowledge about the environmental model.
Instead, a sufﬁciently large step size might be enough.
In this paper, we extend the results of Xiao (2022) from the tabular setting to the log-linear policies, using non-adaptive geometrically increasing step size and obtaining a linear convergence rate for NPG without regularization.
NPG with function approximation.
In the function approximation regime, there have been many works investigating the convergence rate of the NPG or NAC algorithms from different perspectives.
Wang et al.
(2020) establish the O (1 / √ k ) convergence rate for two-layer neural NAC with a projection step.
The sublinear convergence results are also established by Zanette et al.
(2021) and Hu et al.
for the linear MDP (Jin et al., 2020).
Agarwal et al.
(2021) obtain the same O (1 / √ k ) convergence rate for both projected NPG with smooth policies and projected Q-NPG with log-linear policies.
This was later improved to O (1 /k ) by Liu et al.
(2020) by replacing the projection step with a strong regularity condition on the Fisher information matrix,
and it was also improved to O (1 /k ) by Xu et al.
(2020) with NAC under Markovian sampling.
The same O (1 /k ) convergence rate is established for log-linear policies by Chen et al.
(2022) when considering the off-policy NAC.
With entropy regularization and a projection step, Cayci et al.
(2021) obtain a linear convergence for log-linear policies.
Same entropy regularization and a projection step are applied by Cayci et al.
(2022) for the neural NAC to improve the O (1 / √ k ) convergence rate of Wang et al.
(2020) to O (1 /k ) .
In contrast, we show that by using a simple geometrically increasing step size, fast linear convergence can be achieved for log-linear policies without any additional regularization nor a projection step.
We notice that Chen & Theja Maguluri (2022, Theorem 3.4) 4 also uses increasing step size and achieves linear convergence for log-linear policies without regularization.
The main differences between our result and Theorem 3.4 in Chen & Theja Maguluri (2022) are fourfold.
First, they rely on the contraction property of the generalized Bellman operator, while we consider the
PMD analysis approach.
So the proof techniques are completely different.
Second, their parameter update results in the off-policy multi-step temporal difference learning, whereas we require to solve a linear regression problem to minimize the function approximation error.
Third, their step size still depends on the iterates which is thus an adaptive step size and is proportional to the total number of iterations K , while ours is independent to the iterates nor to K .
Finally, their assumption on the modeling error requires an L ∞ supremum norm, i.e., (cid:107) Q s ( θ ( k ) )
− Φ w ( k ) (cid:63) (cid:107) ∞ ≤ (cid:15) bias for all states s of the state space, our convergence guarantee depends on the expected error (e.g., Assumption 2,or 8) which is a much weaker assumption.
After our submission, we are aware of the concurrent work of Alfano & Rebeschini (2022).
They only analyze the Q-NPG method and achieve similar linear convergence results as our Theorem 1.
In particular, their result in Theorem 4.7 has a better concentrability coefﬁcient compared to our Theorem 1.
However, their Assumption 4.6 assumes that the relative condition number upper bounds a time-varying ratio which depends on the iterates, while our Assumption 3 is independent to the iterates, as deﬁned in (24).
Furthermore, they only consider the case when the initial state distribution is the same as the target state distribution, while our analysis generalizes with any target state distribution, which is extensively discussed on the distribution mismatch coefﬁcients in Appendix H. See Table 1 a complete overview of NPG in the function approximation regime.
[Table]
[Table]
Fast linear convergence of other policy gradient methods.
Different to the PMD analysis approach, by leveraging a gradient dominance property (Polyak, 1963; Łojasiewicz, 1963), fast linear convergence results have also been established for the PG methods under different settings, such as the linear quadratic control problems (Fazel et al., 2018) and the exact PG method with softmax tabular policy and entropy regularization (Mei et al., 2020; Yuan et al., 2022).
Such gradient domination property is widely explored by Bhandari & Russo (2019) to identify more general structural MDP settings.
Linear convergence of PG can also be obtained through exact line search (Bhandari & Russo, 2021, Theorem 1 (a)) or by exploiting non-uniform smoothness (Mei et al., 2021).
Alternatively, by considering a general strongly-concave utility function of the state-action occupancy measure and by exploiting the hidden convexity of the problem, Zhang et al.
(2020) also achieve the linear convergence of a variational PG method.
When the object is relaxed to a general concave utility function, Zhang et al.
(2021) still achieve the linear convergence by leveraging the hidden convexity of the problem and by adding variance reduction to the PG method.
A. F UTURE W ORK
The main focus of this paper was the theoretical analysis of NPG method.
The results we have obtained open up several experimental questions related to parameter settings for NPG and Q-NPG.
We leave such questions as an important future work to further support our theoretical ﬁndings.
An interesting application from our work is to investigate the sample complexity of natural actorcritic with our PMD analysis.
Indeed, our paper obtains w ( k ) by a regression solver.
One can also use temporal difference (TD) learning (e.g., Cayci et al.
; Chen & Theja Maguluri (2022);
Telgarsky (2022)) with Markovian sampling to achieve similar O (1 /(cid:15) 2 ) sample complexity result.
The performance analysis of TD learning will be expressed for (cid:15) stat , which directly imply the total sample complexity results through our theorems.
One natural question is whether we can extend our analysis to the general policy classes.
Here we provide one possible way.
It can be extended by using a similar compatible function approximation framework.
Concretely, consider the parameterized policy
[Equation]
where f s,a ( θ ) is parameterized by θ ∈ R m and is differential.
As Agarwal et al.
(2021) mentioned, the gradient can be written as
[Equation]
where
[Equation]
The NPG update is equivalent to the following compatible function approximation framework
[Equation]
As Alfano & Rebeschini (2022, Remark 4.8) mentioned, if we assume that for all ( s, a ) ∈ S × A , function f ( θ ) satisﬁes
[Equation]
which is the case for the log-linear policies, then one can easily verify that the NPG update resulted in a new policy is also equivalent to the policy mirror descent update
[Equation]
where G ( k ) s ∈ R
|A|× m is a matrix with rows ( g s,a ( θ ( k ) ))
(cid:62) ∈ R 1 × m for a ∈ A .
Consequently, one can extend our work naturally in this general setting to derive linear convergence analysis for NPG.
Perhaps one can consider the exponential tilting , a generalization of Softmax to more general probability distributions.
Another interesting venue of investigation is to consider the generalized linear model instead of linear function approximation for the Q function and the advantage function.
One interesting open question is that is there a way to increase stepsize when the discount factor is unknown.
So far the PMD proof techniques used in Lan (2022); Xiao (2022) and ours require that the discount factor is known.
Perhaps the work of Li et al.
(2022a) can help to ﬁnd a way to increase stepsize when the discount factor is unknown.
Indeed, Li et al.
(2022a) consider the averaged
MDP setting.
So there is no discount factor.
They achieve linear convergence for NPG by increasing the stepsize with some regularization parameters.
It will be interesting to investigate if the way of increasing stepsize in Li et al.
(2022a) can be applied in our setting.
B S TANDARD R EINFORCEMENT L EARNING R ESULTS
In this section, we prove the standard reinforcement learning results used in our main paper, including the NPG updates written through the compatible function approximation (11) and the NPG updates formalized as policy mirror descent ((16) and (17)).
Then, we prove the performance difference lemma (Kakade & Langford, 2002), which is the ﬁrst key ingredient for our PMD analysis.
The three-point descent lemma (Lemma 11) is the second key ingredient for our PMD analysis.
Lemma 1 (NPG updates via compatible function approximation, Theorem 1 in Kakade (2001)) .
Consider the NPG updates (8)
[Equation]
and the updates using the compatible function approximation (11)
[Equation]
where w ( k ) (cid:63) ∈ argmin w ∈
R m L A (cid:0) w, θ ( k ) , ¯ d ( k ) (cid:1) .
If the parametrized policy is differentiable for all θ ∈ R m , then the two updates are equivalent up to a constant scaling (1 − γ ) of η k .
Proof.
Indeed, using the policy gradient (7) and the fact that (cid:80) a ∈A ∇ π s,a ( θ ) = 0 for all s ∈ S , as π ( θ ) is differentiable on θ and (cid:80) a ∈A π s,a = 1 , we have the policy gradient theorem (Sutton et al., 2000)
[Equation]
Furthermore
[Equation]
consider the optima w ( k ) (cid:63) .
By the ﬁrst-order optimality condition
[Equation]
we have
[Equation]
⇐⇒
[Equation]
⇐⇒
[Equation]
Thus
[Equation]
we have
[Equation]
which yields the update (8) up to a constant scaling (1 − γ ) of η k .
Lemma 2 (NPG updates as policy mirror descent) .
The closed form solution to (16) is given by
[Equation]
where (cid:12) is the element-wise product between vectors, and ¯Φ ( k )
s ∈ R
|A|× m is deﬁned in (17) , i.e.
[Equation]
Such policy update coincides the inexact NPG updates (32) of the log-linear policy, if θ ( k +1)
= θ ( k ) − η k w ( k ) with w ( k ) ≈ argmin w L A ( w, θ ( k ) , ˜ d ( k ) ) ; and coincides the inexact Q-NPG updates (18) of the log-linear policy, if θ ( k +1)
= θ ( k ) − η k w ( k ) with w ( k )
≈
[Equation]
w L Q ( w, θ ( k ) , ˜ d ( k ) )
[Equation]
Proof.
For shorthand, let g =
Φ s w ( k ) .
Thus, (16) ﬁts the format of Lemma 10 in Appendix I where q = π ( k ) s .
Consequently, the closed form solution is given by (99), that is
[Equation]
where the last equality is obtained as
[Equation]
with c s ∈ R some constant independent to a .
Similarly, by applying Lemma 10 with g = ¯Φ ( k ) s w ( k ) , the closed form solution to (39) is (40).
As for the closed form updates of the policy for NPG (32) and Q-NPG (18) with the parameter updates θ ( k +1)
= θ ( k ) − η k w ( k ) , it is straightforward to verify that it coincides (37) and (38) given the speciﬁc structure of the log-linear policy (6), which concludes the proof.
Lemma 3 (Performance difference lemma (Kakade & Langford, 2002)) .
For any policy π, π (cid:48) ∈ ∆( A ) S and ρ ∈ ∆( S ) ,
[Equation]
where Q s ( π ) is the shorthand for [ Q s,a ( π )] a ∈A ∈ R |A| for any policy π .
Proof.
From Lemma 2 in Agarwal et al.
(2021), we have
[Equation]
where A s ( π ) is the shorthand for [ A s,a ( π )] a ∈A ∈ R |A| for any policy π .
To show (42), it sufﬁces to show
[Equation]
for all s ∈ S and π, π
[Equation]
∈ ∆( A ) S .
Let 1 n denote a vector in R n with coordinates equal to 1 element-wisely.
Indeed, using the deﬁnition of the advantage function
[Equation]
we have
[Equation]
from which we conclude the proof.
C A LGORITHMS C. NPG AND Q-NPG A LGORITHM
Algorithm 1 combined with the sampling procedure (Algorithm 4) and the averaged SGD procedure, called NPG-SGD (Algorithm 5), provide the sample-based NPG methods.
Algorithm 1: Natural policy gradient
Input: Initial state-action distribution ν , policy π (0) , discounted factor γ ∈ [0 , 1) , step size η 0 > 0 for NPG update, step size α > 0 for NPG-SGD update, number of iterations T for
NPG-SGD 1 for k = 0 to K − 1 do
Compute w ( k ) of (32) by NPG-SGD , i.e., Algorithm 5 with inputs ( T, ν, π ( k ) , γ, α )
Update θ ( k +1)
=
θ ( k ) − η k w ( k ) and η k
Output: π ( K )
Similarly, Algorithm 2 combined with the sampling procedure (Algorithm 3) and the averaged SGD procedure, called Q-NPG-SGD (Algorithm 6), provide the sample-based Q-NPG methods.
C. S AMPLING P ROCEDURES
In practice, we cannot compute the true minimizer w ( k ) (cid:63) of the regression problem in either (32) or (18), since computing the expectation L A or L Q requires averaging over all state-action pairs ( s, a ) ∼ ˜ d ( k ) and averaging over all trajectories ( s 0 , a 0 , c 0 , s 1 , · · · ) to compute the values of Q ( k ) s,a
Output: π θ ( K ) and A ( k ) s,a .
So instead, we provide a sampler which is able to obtain unbiased estimates of Q s,a ( θ ) (or
A s,a ( θ ) ) with ( s, a ) ∼ ˜
d θ ( ν ) for any π ( θ ) .
To solve (18), we sample ( s, a ) ∼ ˜ d ( k ) and (cid:98) Q ( k ) s,a by a standard rollout, formalized in Algorithm 3.
This sampling procedure is commonly used, for example in Agarwal et al.
(2021, Algorithm 1).
estimate
Q probability γ :
Sample
Sample a θ
Q s a θ )
Q s h a h t
Otherwise with probability (1 − γ ) :
(cid:46) Accept (cid:98) Q θ )
Output: ( s h , a h )
(cid:98) Q s h ,a h ( θ )
It is straightforward to verify that ( s h , a h ) and
(cid:98) Q s h ,a h ( θ ) obtained in Algorithm 3 are unbiased for any π ( θ ) .
The expected length of the trajectory is 11 − γ .
We provide its proof here for completeness.
Lemma 4.
Consider the output ( s h , a h ) and (cid:98) Q s h ,a h ( θ ) of Algorithm 3.
It follows that
[Equation]
Proof.
The expected length ( h + 1) of sampling ( s, a ) is
[Equation]
The probability of the state-action pair ( s, a ) being sampled by Algorithm 3 is
[Equation]
Now we verify that (cid:98) Q s h ,a h ( θ ) obtained from Algorithm 3 is an unbiased estimate of Q s h ,a h ( θ ) .
Indeed, from Algorithm 3, we have
[Equation]
where ( H + 1) is the length of the horizon executed between lines 13 and 19 in Algorithm 3 for calculating (cid:98) Q s h ,a h ( θ ) .
To simplify notation, we consider the estimate of (cid:98) Q s,a for any ( s, a ) ∈ S ×
A following the same procedure starting from line 10 in Algorithm 3.
Taking expectation, we have
[Equation]
The desired result is obtained by setting s = s h and a = a h .
Similar to Algorithm 3, to solve (32), we sample ( s, a ) ∼ ˜ d ( k ) by the same procedure and estimate (cid:98) A ( k ) s,a with a slight modiﬁcation, namely Algorithm 4 (also see Agarwal et al., 2021, Algorithm 3).
Notice that the sampling procedure for estimating Q s,a ( θ ) in Algorithm 3 is simpler than that for estimating A s,a ( θ ) in Algorithm 4, since Algorithm 4 requires an additional estimation of V s ( θ ) and thus doubles the number of samples to estimate A s,a ( θ ) .
As in Lemma 4, we verify in the following lemma that the output ( s h , a h ) is sampled from the distribution
˜ d θ and (cid:98)
A s h ,a h ( θ ) in Algorithm 4 is an unbiased estimator of A s h ,a h ( θ ) for all policy π ( θ ) .
Lemma
5. Consider the output ( s h , a h ) and (cid:98)
A s h ,a h ( θ ) of Algorithm 4.
It follows that
[Equation]
1 Initialize the the
X =2 while do
With probability γ :
Sample s h +1 ∼ P ( · | s h , a h )
Sample a h +1 ∼ π s h +1 ( θ )
h ← h +
Otherwise with probability (1 − γ ) :
X =(cid:46)
Accept ( s h , a h ) 9 X =10 Set
the estimate (cid:98) Q s h ,a h ( θ ) =
c ( s h , a h ) (cid:46) Start to estimate (cid:98) Q s h ,a h ( θ ) 11 t =
h 12 while X = 1 do
With probability γ :
Sample s t +1 ∼ P ( · | s t , a t )
Sample a t +1 ∼ π s t
+1 ( θ ) (cid:98) Q s h ,a h ( θ ) ← (cid:98) Q s h ,a h ( θ ) + c ( s t +1 , a t +1 ) t ← t +
Otherwise with probability (1 − γ ) :
X =(cid:46)
Accept (cid:98) Q s h ,a h ( θ ) 20 X =21 Set the estimate (cid:98) V s h ( θ ) =(
cid:46)
Start to estimate (
cid:98) V s h ( θ ) 22 t =
h 23 while X = 1 do
Sample a t ∼ π s t ( θ ) (cid:98)
V s h ( θ )
← (cid:98) V s h ( θ )
+ c ( s t , a t )
With probability γ :
Sample s t +1 ∼ P ( · | s t , a t ) t ← t +
Otherwise with probability (1 − γ ) :
X =(cid:46)
Accept (cid:98) V s h ( θ )
Output: ( s h , a h ) and (cid:98)
A s h ,a h ( θ ) = (cid:98) Q s h ,a h ( θ ) − (cid:98) V s h ( θ )
Proof.
Since the procedure of sampling ( s h , a h ) in Algorithm 4 is identical to the one in Algorithm 3, from Lemma 4, the ﬁrst two results are veriﬁed.
It remains to show that (cid:98)
A s h ,a h ( θ ) is unbiased.
The estimation of (cid:98) A s h ,a h ( θ ) is decomposed into the estimations of (cid:98) Q s h ,a h ( θ ) and (cid:98) V s h ( θ ) .
The procedure of estimating (cid:98) Q s h ,a h ( θ ) is also identical to the one in Algorithm 3.
Thus, from Lemma 4, we have
[Equation]
By following the similar arguments of Lemma 4, one can verify that
[Equation]
Combine the above two equalities and obtain that
[Equation]
C. SGD P ROCEDURES FOR S OLVING THE R EGRESSION P ROBLEMS OF NPG AND Q-NPG
Once we obtain the sampled ( s, a ) and (cid:98) A s,a ( θ ( k ) ) from Algorithm 4, we can apply the averaged
SGD algorithm as in Bach & Moulines (2013) to solve the regression problem (32) of NPG for every iteration k .
Here we suppress the superscript ( k ) .
For any parameter θ ∈ R m , recall the compatible function approximation L A in (
[Equation]
With the output ( s, a ) ∼ ˜
d θ and (cid:98)
A s,a ( θ ) from Algorithm 4 (here we suppress the subscript h )
[Equation]
we compute the stochastic gradient estimator of the function L A in (32) by
[Equation]
Next, we show that (44) is an unbiased gradient estimator of the loss function L A .
Lemma 6.
Consider the output ( s, a ) and (cid:98)
A s,a ( θ ) of Algorithm 4 and the stochastic gradient (44) .
It follows that
[Equation]
where the expectation is with respect to the randomness in the sequence of the sampled s 0 , a 0 , · · · , s t , a t from Algorithm 4.
Proof.
The total expectation of the stochastic gradient is given by
[Equation]
45) where the second line is obtained by ( s, a )
∼ ˜
d θ from Lemma 5.
From Lemma 5, we have
[Equation] [Equation]
Combining the above two equalities yield
[Equation]
as desired.
Since (44) is unbiased shown in Lemma 6, we can use it for the averaged SGD algorithm to minimize
L A , called NPG-SGD in Algorithm 5 (also see Agarwal et al., 2021, Algorithm 4).
Similar to Algorithm 5, once we obtain the sampled ( s, a ) and (cid:98) Q s,a ( θ ) from Algorithm 3, we can apply the averaged SGD algorithm to solve (18) of Q-NPG.
Recall the compatible function approximation L Q in (18)
[Equation]
With the output ( s, a ) ∼ ˜ d θ and (cid:98) Q s,a ( θ ) from Algorithm 3, we compute the stochastic gradient estimator of the function L Q in (18) by
[Equation]
(47) and use it for the averaged SGD algorithm to minimize L Q , called Q-NPG-SGD in Algorithm 6 (also see Agarwal et al., 2021, Algorithm 2).
Compared to (44), the cost of computing (47) is |A| times cheaper than that of computing (47).
Indeed, to compute (47), we only need one single action for φ s,a , while to compute (44), one needs to go through all the actions to compute ¯ φ s,a ( θ ) .
Thus, the computational cost of Q-NPG-SGD is |A| times cheaper than that of NPG-SGD .
Input: Number of iterations T , step size α > 0 , initialization w 0 ∈ R m , initial state-action measure ν , policy π ( θ ) , discounted factor γ ∈ [0 , 1) 1 for t = 0 to T − 1 do
Call Algorithm 3 with the inputs ( ν, π ( θ ) , γ ) to sample ( s, a ) ∼ ˜
d θ and (cid:98) Q s,a ( θ )
Update w t +1 =
w t − α (cid:98) ∇ w L Q ( w, θ, ˜ d θ ) by using (47)
Output: w out = 1 T (cid:80) Tt =
1 w t
The estimator (cid:98) ∇ w L Q ( w, θ, ˜ d θ ) is also unbiased following the similar argument of the proof of
Lemma 6.
We formalize this in the following and omit the proof.
Lemma 7. Consider the output ( s, a ) and (cid:98) Q s,a ( θ ) of Algorithm 3 and the stochastic gradient (47) .
It follows that
[Equation]
where the expectation is with respect to the randomness in the sequence of the sampled s 0 , a 0 , · · · , s t , a t from Algorithm 3.
D P ROOF OF S ECTION
Throughout this section and the next, we use the shorthand V ( k ) ρ for V ρ ( θ ( k ) ) and similarly, Q ( k ) s,a for
Q s,a (
We also introduce a weighted KL divergence given by
[Equation]
If we choose the uniform initial policy, i.e., π (0) s,a = 1 / |A| for all ( s, a ) ∈ S × A (or θ (0) = 0 ), then
D ∗ 0 ≤ log |A| , (48) for all ρ ∈ ∆( S ) and for any π ∗ ∈ ∆( A ) S .
We ﬁrst provide the one step analysis of the Q-NPG update, which will be helpful for proving
Theorem 1 and 3 and the sublinear convergence result of Theorem 2 further in Appendix D.3.
D. T HE O NE S TEP Q-NPG L EMMA
The following one step analysis of Q-NPG is based on the mirror descent approach of Xiao (2022).
Lemma 8 (One step Q-NPG lemma) .
Fix a state distribution ρ ; an initial state-action distribution ν ; an arbitrary comparator policy π ∗ .
Let w ( k ) (cid:63) ∈ argmin w L Q ( w, θ ( k ) , ˜ d ( k ) ) denote the exact minimizer.
Consider the w ( k ) and π ( k ) given in (18) and (16) respectively.
We have that
[Equation]
Proof.
In the context of the PMD method (16)
[Equation]
we apply the three-point descent lemma (Lemma 11) with C = ∆( A ) , f is the linear function η k (cid:10) Φ s w ( k ) , · (cid:11) and h : ∆( A ) → R is the negative entropy with h ( p )
= (cid:80) a ∈A p a log p a .
Thus, h is of Legendre type with rint dom h ∩ C = rint ∆( A ) (cid:54) = ∅ and D h ( · , · ) is the KL divergence D ( · , · ) .
From Lemma 11, we obtain that for any p ∈ ∆( A ) , we have
[Equation]
Rearranging terms and dividing both sides by η k , we get
[Equation]
Letting p =
π ( k )
[Equation]
yields
[Equation]
Letting p =
π ∗ s and subtract and add π ( k ) s within the inner product term in (50) yields
[Equation]
Note that we dropped the nonnegative term 1 η k D ( π ( k +1) s , π ( k ) s ) on the left hand side to the inequality.
Taking expectation with respect to the distribution d ∗ , we have
[Equation]
For the ﬁrst expectation in (52)
[Equation]
we have
[Equation]
where the last equality is due to the performance difference lemma (42) in Lemma 3 and the two inequalities above are obtained by the negative sign of (cid:68) Φ s w ( k ) , π ( k +
[Equation]
s − π ( k
[Equation]
s (cid:69) shown in (51) and by using the following inequality
[Equation]
The second term of (53) can be decomposed into four terms.
That is,
[Equation]
= 1 + 2 + 3 +
[Equation]
(54) where 1 , 2 , 3 and 4 are deﬁned in (49).
For the second expectation in (52), by applying again the performance difference lemma (42), we have
[Equation]
Similarly, we decompose the second term of (55) into four terms.
That is,
[Equation]
where a , b , c and d are deﬁned in (49)
[Equation]
Plugging (53) with the decomposition (54) and (55) with the decomposition (56) into (52) concludes the proof.
Consequently, the convergence analysis of Q-NPG (Theorem 1, 3 and Theorem 2 further in Appendix D.3) will be obtained by upper bounding the absolute values of 1 , 2 , 3 , 4 , a , b , c , d in (49) with different set of assumptions (assumptions in Theorem 1 or assumptions in Theorem 3) and with different step size scheme (geometrically increasing step size for Theorem 1 and 3 or constant step size for Theorem 2).
D. P ROOF OF T HEOREM
Here we provide the general Q-NPG convergence analysis with any initial policy, which is in contrast to Theorem 1 with the uniform initial policy.
Theorem 5.
Fix a state distribution ρ , an state-action distribution ν and a comparator policy π ∗ .
We consider the Q-NPG method (18) with the step sizes satisfying η 0 ≥ 1 − γγ D ∗ 0 and η k +1 ≥ 1 γ η k .
Suppose that Assumptions 1, 2, 3 and 4 all hold.
Then we have for all k ≥ 0 ,
E
[Equation]
Remark.
A deviation from the setting of Theorem 1 is that here we require the step sizes satisfying η 0 ≥ 1 − γγ D ∗ 0 instead of η 0 ≥ 1 − γγ log |A| .
This is because when using uniform initial policy in
Theorem 1, from (48), we know that η 0 ≥ 1 − γγ log |A| is a sufﬁcient condition for η 0 ≥ 1 − γγ D ∗ 0 .
Consequently, to prove Theorem 1, it sufﬁces to prove the more general Theorem 5.
Proof.
From (49) in Lemma 8, we will upper bound | 1 | and | 3 | by the statistical error assumption (19) and upper bound | 2 | and | 4 | by using the transfer error assumption (22).
Indeed, to upper bound | 1 | , by Cauchy-Schwartz’s inequality, we have
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table] [Equation] [Table] [Equation]
[Table]
[Equation] [Table] [Equation] [Table]
[Equation] [Table] [Equation] [Table]
[Equation] [Table] [Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
where the second inequality is obtained by Cauchy-Schwartz’s inequality
[Equation]
and the third inequality is obtained by the following inequality
[Equation]
Then, by using Assumption 3 with the deﬁnition of κ ν , (57) is upper bounded by
[Equation]
where we use the shorthand
[Equation] [Equation]
Besides
[Equation]
by the ﬁrst-order optimality conditions for the optima w ( k ) (cid:63)
[Equation]
we have
[Equation]
for all
[Equation]
Therefore
[Equation]
for all w ∈ R m ,
[Equation]
[Equation] [Equation]
Deﬁne
[Equation]
Note that from (19), we have
[Equation]
[Equation]
Plugging (62) into (59), we have
[Equation]
Similar to (57), we get the same upper bound for | 3 | by just replacing π ( k +1)
s,a into π ( k ) s,a .
That is,
[Equation]
|
[Equation] [Equation]
stat .
[Equation]
[Equation]
To upper bound | 2 | and | 4 |
[Equation]
we introduce the following term
[Equation]
Note that from (22), we have
[Equation]
By Cauchy-Schwartz’s inequality
[Equation]
we have
[Equation]
Similar to (67)
[Equation]
we get the same upper bound for | 4 | by just replacing π ( k +1)
s,a into π ( k ) s,a .
That is,
[Equation] [Equation]
Next, we will upper bound the absolute values of a , b , c and d of (49) separately by using again the statistical error (19) and by using the transfer error assumption (22).
Indeed, to upper bound | a |
[Equation]
by Cauchy-Schwartz’s inequality
[Equation]
we have
[Equation]
a
[Equation] [Equation]
(cid:115) |A
[Equation]
From the deﬁnition of κ ν , we further obtain
[Equation]
Similar to (69)
[Equation]
we get the same upper bound for | c | by just replacing π ( k ) s,a into π ∗ s,a .
That is,
[Equation]
To upper bound | b |
[Equation]
by Cauchy-Schwartz’s inequality
[Equation]
we have
[Equation]
Similar to (71), we get the same upper bound for | d | by just replacing π ( k ) s,a into π ∗ s,a .
That is,
[Equation]
Plugging all the upper bounds (64) of | 1 | , (67) of | 2 | , (65) of | 3 | , (68) of | 4 | , (69) of | a | , (71) of | b | , (70) of | c | and (72) of | d | into (49) yields
[Equation]
(73) where δ k def = V ( k ) ρ − V ρ
( π ∗ ) .
Dividing both sides by ϑ ρ and rearranging terms, we get δ k +
+
D ∗ k +
[Equation]
If the step sizes satisfy η k +1 ( ϑ ρ − 1) ≥ η k ϑ ρ , which is implied by η k +1 ≥ η k /γ and (20), then
[Equation]
Finally
[Equation]
by choosing η 0 ≥ 1 − γγ D ∗ 0 and using the fact that
[Equation]
we obtain
[Equation]
Taking the total expectation with respect to the randomness in the sequence of the iterates w (0) , · · ·
[Equation]
w ( k − 1)
[Equation]
we have
[Equation]
where the second inequality is obtained by Jensen’s inequality.
This concludes the proof
[Equation]
D. P ROOF OF T HEOREM
Similar to Theorem 5, to prove Theorem 2, it sufﬁces to prove the following sublinear convergence of Q-NPG with any initial policy.
Theorem 6.
Fix a state distribution ρ , an state-action distribution ν and an optimal policy π ∗ .
We consider the Q-NPG method (18) with any constant step size η k = η > 0 .
Suppose that
Assumptions 1, 2, 3 and 4 all hold.
Then we have for all k ≥ 0 ,
[Equation]
Remark.
A deviation from the setting of Theorem 1 is that here we require π ∗ to be an optimal policy 5 .
Compared to Theorem 20 in Agarwal et al.
(2021), our convergence rate is also sublinear, but with an improved convergence rate of O (1 /k ) , as opposed to O (1 / √ k ) .
Moreover, they use a diminishing step size of order O (1 / √ k ) while our constant step size is unconstrained.
Proof.
By (73) and using a constant step size η , we have
[Equation]
Taking the total expectation with respect to the randomness in the sequence of the iterates w (0) , · · · , w ( k − 1) , summing up from 0 to k − 1 and rearranging terms
[Equation]
we have
[Equation] [Equation]
where we use the following inequalities
[Equation]
Finally
[Equation]
dropping the positive term E
[ δ k ] on the left hand side as π ∗ is the optimal policy and dividing both side by k yields
[Equation]
D. P ROOF OF T HEOREM
Similar to Theorem 5, to prove Theorem 3, it sufﬁces to prove the following Q-NPG result with any initial policy.
Theorem 7.
Fix a state distribution ρ , an state-action distribution ν and a comparator policy π ∗ .
We consider the Q-NPG method (18) with the step sizes satisfying η 0 ≥ 1 − γ γ D ∗ 0 and η k +1 ≥ 1 γ η k .
Suppose that Assumptions 1, 5 and 6 hold.
Then we have for all k ≥ 0 ,
[Equation]
Proof.
Similar to the proof of Theorem 1, by Lemma 8, we upper bound the absolute values of 1 , 2 , 3 , 4 , a , b , c , d introduced in (49), separately, with the set of assumptions in Theorem 3.
In comparison with the proof of Theorem 1, we will also upper bound | 1 | , | 3 | , | a | and | c | by the statistical error assumption (19) as in the proof of Theorem 1.
However, we will upper bound | 2 | , | 4 | , | b | and | d | by using the approximation error assumption (27) instead of the transfer error assumption (22).
To upper bound | 1 | , by Cauchy-Schwartz’s inequality, we get
[Equation]
Similar to | 1 | , by using Assumption 6 and Cauchy-Schwartz’s inequality, and by simply replacing π ( k +1) into π ( k ) or π ∗ and replacing d ( k +1) into d ∗ , we obtain the same upper bound of | 3 | , | a | and | c | , that is
[Equation]
Next, we deﬁne
[Equation]
By Assumption 5, we know that
[Equation]
To upper bound | 2 | , by Cauchy-Schwartz’s inequality, we have | 2 | ≤ (cid:88) s ∈S (cid:88) a ∈A
[Equation]
≤
[Equation]
(cid:88) (
[Equation]
Similar to | 2 | , by using Assumption 5 and Cauchy-Schwartz’s inequality, and by simply replacing π ( k +1) into π ( k ) or π ∗ and replacing d ( k +1) into d ∗ , we obtain the same upper bound for | 4 | , | b | and | d | , that is
[Equation]
Consequently, plugging all these upper bounds into (49) leads to the following recurrent inequality
[Equation]
By using the same increasing step size as in Theorem 5 and following the same arguments in the proof of Theorem 5 after (73), we obtain the ﬁnal performance bound with the linear convergence rate
[Equation]
D. P ROOF OF C OROLLARY
In this section, we aim to prove Corollary 1 presented in Section 4.3.
Corollary 2 (Corollary 1) .
Consider the setting of Theorem 3.
Suppose that the sample-based
Q-NPG Algorithm 2 is run for K iterations, with T gradient steps of Q-NPG-SGD (Algorithm 6) per iteration.
Furthermore, suppose that for all ( s, a ) ∈ S × A , we have (cid:107) φ s,a (cid:107) ≤ B with B > 0 , and we choose the step size α = 12 B 2 and the initialization w 0 = 0 for Q-NPG-SGD .
If for all θ ∈ R m , the covariance matrix of the feature map followed by the initial state-action distribution ν satisﬁes
[Equation]
where I m ∈ R m × m is the identity matrix and µ > 0 , then
[Equation]
In order to better understand our proof, we ﬁrst identify an issue appeared in the sample complexity analysis of Q-NPG in Agarwal et al.
(2021, Corollay 26).
Agarwal et al.
(2021) adopts the optimization results of Shalev-Shwartz & Ben-David (2014, Theorem 14.8) where the stochastic gradient (cid:98) ∇ L Q ( w, θ, ˜ d θ ) in (47) needs to be bounded 6 .
However, although they consider a projection step for the iterate w t and assume that the feature map φ s,a is bounded, (cid:98) ∇ L Q ( w, θ, ˜ d θ ) is still not guaranteed to be bounded.
Indeed, recall the stochastic gradient of the function L Q in (47)
[Equation]
They incorrectly use the argument that w, φ s,a and (cid:98) Q s,a ( θ ) are bounded to imply that (cid:13)(cid:13) (cid:13)(cid:98) ∇ w L Q ( w, θ, ˜ d θ ) (cid:13)(cid:13) (cid:13) is bounded.
In fact, (cid:98) Q s,a ( θ ) can be unbounded even though E (cid:104) (cid:98) Q s,a ( θ ) (cid:105) =
Q s,a ( θ ) ∈ (cid:104) 0 , 11 − γ (cid:105) is bounded.
To see this, we can rewrite (cid:98) Q s,a ( θ ) from (43) as (cid:98) Q s,a ( θ ) =
H (cid:88) t = c ( s t , a t ) , with ( s 0 , a 0 ) =
( s, a ) ∼ ˜
d θ and H is the length of the sampled trajectory for estimating Q s,a ( θ ) in
Algorithm 3.
From Algorithm 3 and from the proof of Lemma 4, we know that the probability of
H =
k + 1 is that
[Equation]
So, with exponentially decreasing low probability, H can be unbounded.
Consequently, | (cid:98) Q s,a ( θ ) | upper bounded by H is not guaranteed to be bounded.
Proof sketch.
Instead, we adopt the optimization results of Bach & Moulines (2013, Theorem 1) (see also Theorem 12), which does not require the boundedness of the stochastic gradient.
However, in our following proof, we can verify that E (cid:104) (cid:98) Q s,a ( θ ) 2 (cid:105) is bounded even though (cid:98) Q s,a ( θ ) is unbounded.
As to verify the condition (vi) in Theorem 12 in our proof, i.e., the covariance of the stochastic gradient at the optimum is upper bounded by the covariance of the feature map up to a ﬁnite constant, we use a conditional expectation argument to separate the correlated random variables ( s, a ) ∼ ˜
d θ and (cid:98) Q s,a ( θ ) appeared in the stochastic gradient.
Non-singularity of the covariance matrix Σ ν .
As for the condition (31), it is shown in Cayci et al.
(2021, Proposition 3) that with ν chosen as uniform distribution over S × A and φ s,a ∼ N (0 , I m ) sampled as Gaussian random features, (31) is guaranteed with high probability.
More generally, with m (cid:28) |S||A| , it is easy to ﬁnd m linearly independent φ s,a among all |S||A| features such that the covariance matrix Σ ν has full rank.
Proof.
From Theorem 3, it remains to upper bound the statistical error √ (cid:15) stat produced from the
Q-NPG-SGD procedure (Algorithm 6) for each iteration k .
We suppress the superscript ( k ) .
Let w out be the output of T steps Q-NPG-SGD with the constant step size 12 B 2 and the initialization w 0 = 0 , and let w (cid:63) ∈ argmin w L Q ( w, θ, ˜ d θ ) be the exact minimizer.
To upper bound (cid:15) stat from (19), we aim to apply the standard analysis for the averaged SGD, i.e., Theorem 12.
Now we verify all the assumptions in order for Q-NPG-SGD .
First, (i) is veriﬁed by considering the Euclidean space H =
R m .
The observations (cid:16) φ s,a , (cid:98) Q s,a ( θ ) φ s,a (cid:17) ∈ R m × R m are independent and identically distributed, sampled from Algorithm 3.
Thus, (ii) is veriﬁed with x n = φ s,a ∈ R m and z n = (cid:98) Q s,a ( θ ) φ s,a ∈ R m .
As the feature map (cid:107) φ s,a (cid:107) ≤ B , we have E (cid:104) (cid:107) φ s,a (cid:107) 2 (cid:105) ﬁnite.
From (31), we know that the covariance
E (cid:2) φ s,a φ (cid:62) s,a (cid:3) is invertible.
To verify (iii), it remains to verify that E (cid:20)(cid:13)(cid:13)(cid:13) (cid:98) Q s,a ( θ ) φ s,a (cid:13)(cid:13)(cid:13)(cid:21) is ﬁnite.
Indeed, by using (cid:107) φ s,a (cid:107) ≤ B , we have
E (cid:20)(cid:13)(cid:13)(cid:13) (cid:98) Q s,a ( θ ) φ s,a (cid:13)(cid:13)(cid:13)(cid:21) ≤ B 2 E (cid:104) (cid:98) Q s,a ( θ ) 2 (cid:105) .
Thus, it remains to show E (cid:20)(cid:16) (cid:98) Q s,a ( θ ) (cid:17) 2 (cid:21) ﬁnite for (iii).
From (43), we rewrite (cid:98) Q s,a ( θ ) as (cid:98) Q s,a ( θ ) =
H (cid:88) t = c ( s t , a t ) , with ( s 0 , a 0 ) =
( s, a ) ∼ ˜
d θ and H is the length of the trajectory for estimating Q s,a ( θ ) .
Thus, (iii) is veriﬁed as the variance of (cid:98) Q s,a ( θ ) is upper bounded by
[Equation]
where the ﬁrst inequality is obtained as | c ( s t , a t ) | ∈ [0 , 1] for all ( s t , a t ) ∈ S × A .
Next, we introduce the residual
[Equation]
From Lemma 7, we know that
[Equation]
So, we have that
[Equation]
where the last equality is obtained as w (cid:63) is the exact minimizer of the loss function L Q .
Thus, (iv) is veriﬁed with that f is 12 L Q , ξ n is ξ and θ is w in our context.
From Q-NPG-SGD update 47, we have (v) veriﬁed with step size α/ 2 in our context.
Finally, for (vi), from the boundedness of the feature map (cid:107) φ s
[Equation]
a (cid:107) ≤ B
[Equation]
we take R =
B such that
E
[Equation]
It remains to ﬁnd σ > 0 such that
[Equation]
We rewrite the covariance of ξ as
[Equation]
Thus, it sufﬁces to ﬁnd σ > 0 such that
[Equation]
for all ( s, a ) ∈ S × A to verify (vi).
Besides, we know that
[Equation]
We also know that | Q s,a ( θ ) | ≤ 11 − γ and (cid:107) φ s,a (cid:107) ≤ B .
Now we need to bound (cid:107) w (cid:63) (cid:107) .
Again, since w (cid:63) is the exact minimizer, we have ∇ w L Q ( w (cid:63) , θ, ˜ d θ ) = 0 .
That is
[Equation]
, which implies
[Equation]
By the boundness of the feature map (cid:107) φ s,a (cid:107) ≤ B and the Q-function | Q s,a ( θ ) | ≤ 11 − γ , and the condition (31), we have the minimizer w (cid:63) bounded by
[Equation]
By using the upper bounds of E (cid:20)(cid:16) (cid:98) Q s,a ( θ ) (cid:17) 2 | s, a (cid:21) , | Q s,a ( θ ) | , (cid:107) w (cid:63) (cid:107) and (cid:107) φ s,a (cid:107) , the left hand side of (76) can be upper bounded by
[Equation]
(cid:33) ≤ (1 − γ )
[Equation]
(cid:18)
B
[Equation]
µ (
[Equation]
(cid:19)
[Equation]
Thus, in order to satisfy (76), we choose
[Equation]
Now all the conditions (i) - (vi) in Theorem 12 are veriﬁed.
With step size α = 12 B 2 , the initialization w 0 = 0 and T steps of Q-NPG-SGD updates (47), we have
[Equation]
Consequently, Assumption 1 is veriﬁed by
[Equation]
The proof is completed by replacing the above upper bound of √ (cid:15) stat in the results of Theorem 3.
E P ROOF OF S ECTIONE. T HE O NE S TEP NPG L EMMA
To prove Theorem 4 and the sublinear convergence result of Theorem 9 further in Appendix F, we start from providing the one step analysis of the NPG update.
Lemma 9 (One step NPG lemma) .
Fix a state distribution ρ ; an initial state-action distribution ν ; an arbitrary comparator policy π ∗ .
At the k -th iteration, let w ( k ) (cid:63) ∈ argmin w L A ( w, θ ( k ) , ˜ d ( k ) ) denote the exact minimizer.
Consider the w ( k ) and π ( k ) NPG iterates given in (32) and (17) respectively.
Note
[Equation]
If Assumptions
[Equation]
8 and 9 hold for all k ≥
[Equation]
then we have that
[Equation]
Proof.
From the three-point descent lemma (Lemma 11) and (17)
[Equation]
we obtain that for any p ∈ ∆( A )
[Equation]
we have
[Equation]
Rearranging terms and dividing both sides by η k , we get
[Equation]
Letting p =
π ( k ) s and knowing that
[Equation]
for all k ≥ 0 , which is due to (12), we have
[Equation]
Letting p =
π ∗ s yields
[Equation]
Note that we dropped the nonnegative term 1 η k D (
[Equation]
, π
[Equation]
s ) on the left hand side to the inequality.
Taking expectation with respect to the distribution d ∗ , we have
[Equation]
For the ﬁrst expectation in (81)
[Equation]
we have
[Equation]
where the last line is obtained by the performance difference lemma (41)
[Equation]
and we use the shorthand ¯ φ ( k ) s,a as ¯ φ s,a ( θ ( k ) )
[Equation]
The second term of (82) can be lower bounded.
To do it, we ﬁrst decompose it into two terms.
That is,
[Equation]
We will upper bound the absolute values of the above two terms | 1 | and | 2 | separately.
More precisely, similar to the proof of Theorem 3, we will upper bound the ﬁrst term | 1 | by the statistical error assumption (33) and upper bound the second term | 2 | by using the approximation error assumption (34).
To upper bound 1 , we ﬁrst deﬁne the following covariance matrix of the centered feature map
[Equation]
Here we use the superscript ( k ) for Σ ( k ) ˜
d ( k ) to distinguish the covariance matrix of the feature map Σ ˜
d ( k ) deﬁned in (60) in the proof of Theorem 1, as the centered feature map ¯ φ ( k ) s,a depends on the iterates θ ( k ) .
By Cauchy-Schwartz’s inequality, we have
[Equation]
By further using the concentrability assumption
[Equation]
we have
[Equation]
where (85) uses that w ( k ) (cid:63) is a minimizer of L A and w ( k ) (cid:63) is feasible (see the same arguments of (62) in the proof of Theorem 1)
[Equation]
For the second term | 2 | in (83)
[Equation]
by Cauchy-Schwartz’s inequality, we have
[Equation]
Plugging (86) and (87) into (82) yields
[Equation]
[Equation]
Now for the second expectation in (81)
[Equation]
by using the performance difference lemma (41) in Lemma
[Equation]
we have
[Equation]
The second term of (89) can be lower bounded.
We ﬁrst decompose it into two terms.
That is,
[Equation]
Now we will upper bound the absolute values of the above two terms | a | and | b | separately.
For the ﬁrst one | a | , by Cauchy-Schwartz’s inequality, we have
[Equation]
For the second term | b | in (90), by Cauchy-Schwartz’s inequality, we have
[Equation]
Thus, we lower bound (90) by
[Equation]
Substituting (88) and (93) into (81), dividing both side by 1 − γ and rearranging terms, we get ϑ ρ ( δ k +
[Equation]
− δ
[Equation]
δ
[Equation]
≤
D
[Equation]
γ
[Equation]
∗
[Equation]
+
[Equation]
1 − γ (cid:18)(cid:113) (cid:15) ( k ) stat + (cid:113) (cid:15) ( k ) approx (cid:19)
[Equation]
E. P ROOF OF T HEOREM
To prove Theorem 4, we prove the following general result with any initial policy by knowing that
D ∗ 0 ≤ log |A| when using the uniform initial policy in (48).
Theorem 8.
Fix a state distribution ρ , a state-action distribution ν , and a comparator policy π ∗ .
We consider the NPG method (32) with the step sizes satisfying η 0 ≥ 1 − γγ D ∗ 0 and η k +1 ≥ 1 γ η k .
Suppose that Assumptions 7, 8 and 9 hold.
Then we have for all k ≥ 0 ,
[Equation]
Proof.
From (79) in Lemma 9, by using the same increasing step size as in Theorem 5, i.e. η 0 ≥ 1 − γγ D ∗ 0 and η k +1 ≥ η k /γ , and following the same arguments in the proof of Theorem 5 after (73), we obtain the ﬁnal performance bound with the linear convergence rate
[Equation]
F S UBLINEAR C ONVERGENCE OF NPG WITH C ONSTANT STEP SIZE
In this section, we provide the sublinear convergence of NPG with arbitrary constant step size.
Theorem 9.
Fix a state distribution ρ , an state-action distribution ν and an optimal policy π ∗ .
We consider the NPG method (32) with any constant step size η k = η > 0 .
Suppose that Assumptions 7, 8 and 9 hold.
Then we have for all k ≥ 0 ,
[Equation]
Compared to Theorem 2, again here we require π ∗ to be an optimal policy for the same reason as indicated in Footnote 5.
Furthermore our sublinear convergence guarantees for both Q-NPG and
NPG are the same.
Compared to Theorem 29 in Agarwal et al.
(2021), the main differences are also similar to those for Q-NPG as summarized right after Theorem 2: our convergence rate improves from O (1 / √ k ) to O (1 /k ) ; they use a diminishing step size of order O (1 / √ k ) while we can take any constant step size we want.
Despite the difference of using ˜ d ( k ) instead of ¯ d ( k ) for the compatible function approximation
L A ( w ( k ) , θ ( k ) , ˜ d ( k ) ) , notice that same sublinear convergence rate O (1 /k ) is established by Liu et al.
(2020) for NPG with constant step size, while their step size is bounded by the inverse of a smoothness constant and they further require that the feature map is bounded and the Fisher information matrix (9) is strictly lower bounded for all parameters θ ∈ R m (see this condition later in (94)).
With such additional conditions, we are able to provide a ˜ O (
[Equation]
) sample complexity result of NPG in
Appendix G.
Proof.
From (79) in Lemma 9 with the constant step size, we have
[Equation]
[Table]
Taking the total expectation with respect to the randomness in the sequence of the iterates w (0) , · · · , w ( k − 1) yields
[Equation]
By summing up from 0 to k − 1 , we get ϑ
[Equation]
] +
[Equation]
Finally, dropping the positive term E [ δ k ] on the left hand side as π ∗ is the optimal policy and dividing both side by k yields
[Equation]
G S AMPLE C OMPLEXITY OF NPG
Combined with a regression solver, NPG-SGD in Algorithm 5, which uses a slight modiﬁcation of Q-NPG-SGD for the unbiased gradient estimates of L A , we consider a sampled-based NPG
Algorithm 1 proposed in Appendix C and show its sample complexity result in the following corollary.
Corollary 3.
Consider the setting of Theorem 8.
Suppose that the sample-based NPG Algorithm 1 is run for K iterations, with T gradient steps of NPG-SGD (Algorithm 5) per iteration.
Furthermore, suppose that for all ( s, a ) ∈ S × A , we have (cid:107) φ s,a (cid:107) ≤ B with B > 0 , and we choose the step size α = 18 B 2 and the initialization w 0 = 0 for NPG-SGD .
If for all θ ∈ R m , the covariance matrix of the centered feature map induced by the policy π ( θ ) and the initial state-action distribution ν satisﬁes
[Equation] [Equation]
where I m ∈ R m × m is the identity matrix and µ > 0 , then
[Equation]
Now we compare our Corollary 3 with Corollary 33 in Agarwal et al.
(2021), which is their corresponding sample complexity results for NPG.
The main differences between Corollary 3 and Corollary 33 in Agarwal et al.
(2021) are similar to those for Q-NPG as summarized right after Corollary 1:
Their sample complexity is O (cid:0)
[Equation] [Equation]
[Equation]
6 (cid:1) while ours is ˜ O (cid:0)
[Equation]
[Equation]
; they consider a projection step for the iterates and incorrectly bound the stochastic gradient due to a similar error indicated in
Footnote 3 (and see Appendix G.1 for more details), while we assume Fisher-non-degeneracy (94).
Compared to Corollary 1, the sample complexities for both Q-NPG and NPG are the same.
The assumption (94) on the Fisher information matrix is much stronger than (31), as (31) is independent to the iterates.
However, despite the difference of using ν instead of ρ , the Fisher-non-degeneracy (94) is commonly used in the optimization literature (Byrd et al., 2016;
Gower et al., 2016; Wang et al., 2017) and in the RL literature (Liu et al., 2020; Ding et al., 2022; Yuan et al., 2022).
It characterizes that the Fisher information matrix behaves well as a preconditioner in the NPG update (8).
Indeed, (94) is directly assumed to be positive deﬁnite in the pioneering NPG work (Kakade, 2001) and in the follow-up works on natural actor-critic algorithms (Peters & Schaal, 2008; Bhatnagar et al., 2009).
It is satisﬁed by a wide families of policies, including the Gaussian policy (Duan et al., 2016; Papini et al., 2018;
Huang et al., 2020) and certain neural policy with log-linear policy as a special case.
We refer to Liu et al.
(2020, Section B.2) and Ding et al.
(2022, Section 8) for more discussions on the
Fisher-non-degenerate setting.
To prove Corollary 3, our approach is inspired from the proof of the sample complexity analysis of Liu et al.
(2020, Theorem 4.9).
That is, we require the Fisher-non-degeneracy (94) and apply
Theorem 12 to the minimization of function L A ( w, θ, ˜ d θ ) without relying on the boundedness of the stochastic gradient.
A proof sketch is provided in Appendix G.1.
Compared to their result, they obtain worse O (cid:0)
[Table]
sample complexity for NPG due to a slower O (1 /k ) convergence rate.
G. P ROOF OF C OROLLARY
There is a similar remark for the proof of Corollary 3 to the one right before the proof of Corollaryin Appendix D.5.
We notice that there is the same error occurred for the proof of NPG sample complexity analysis in Agarwal et al.
Recall the stochastic gradient of L A in (44)
[Equation]
It turns out that (cid:98) ∇ w L A ( w, θ, ˜ d θ ) is unbounded, since the estimate (cid:98)
A s,a ( θ ) of A s,a ( θ ) can be unbounded due to the unbounded length of the trajectory sampled in the sampling procedure, Algorithm 4.
Thus, Agarwal et al.
(2021) incorrectly verify (cid:98) ∇ L A ( w, θ, ˜ d θ ) bounded by claiming that (cid:98) A s,a ( θ ) is bounded by 21 − γ .
Proof sketch.
Despite the difference of using either ˜ d θ or ¯ d θ in the loss function L A , we use the same assumptions of Liu et al.
(2020), i.e., the Fisher-non-degeneracy (94) and the boundedness of the feature map, and verify all the conditions of Theorem 12 without relying on the boundedness of the stochastic gradient.
In particular, similar to the proof of Corollary 1, we verify that E (cid:2) (cid:98) A s,a ( θ ) 2 (cid:3) is bounded even though (cid:98) A s,a ( θ ) is unbounded.
To verify the condition (vi) in Theorem 12 in our proof, we use the same conditional expectation argument as in the proof of Corollary 1 to separate the correlated random variables (cid:98)
A s,a ( θ ) and ¯ φ s,a ( θ ) with ( s, a ) ∼ ˜
d θ appeared in the stochastic gradient.
Thanks to this argument, we ﬁx a ﬂaw in the previous proof of Liu et al.
(2020, Proposition G.1) 7 .
Proof.
Similar to the proof of Corollary 1, we suppress the subscript k .
First, the centered feature map is bounded by (cid:13)(cid:13)
¯ φ s,a ( θ ) (cid:13)(cid:13) ≤ 2 B .
In order to apply Theorem 12, it remains to upper bound
[Equation] [Equation]
] , which is incorrect since s, and ¯ φ s,
) are correlated random variables.
To ﬁx it, we use the following conditional expectation computation argument
[Equation]
and bound the term E (cid:20)(cid:16)
[Table]
E
[Equation]
¯ φ s,a ( θ ) (cid:107) 2 (cid:3) and (cid:107) w (cid:63) (cid:107) with w (cid:63) ∈ argmin w L A ( w, θ, ˜ d θ ) , and ﬁnd σ > 0 such that
E
[Equation] [Equation]
holds for all ( s, a ) ∈ S
× A and θ ∈ R m .
Similar to the proof of Corollary 1, the closed form solution of w (cid:63) can be written as
[Equation]
From (94), we have
[Equation]
Now we need to upper bound E (cid:20)(cid:16) (cid:98)
A s,a ( θ ) (cid:17) 2 | s, a (cid:21) from (95).
Indeed, by using (cid:98) A s,a ( θ ) = (cid:98) Q s,a ( θ ) − (cid:98) V s ( θ ) , we have
[Equation]
(97) where the last line is obtained, as E (cid:20)(cid:16)(cid:98) V s,a ( θ ) (cid:17) 2 | s, a (cid:21) shares the same upper bound (74) of
E (cid:20)(cid:16) (cid:98) Q s,a ( θ ) (cid:17) 2 | s, a (cid:21) by using the similar argument.
From (97) and ¯ φ s,a ( θ ) ≤ 2 B , we verify E (cid:20)(cid:13)(cid:13)(cid:13) (cid:98)
A s,a ( θ )¯ φ s,a ( θ ) (cid:13)(cid:13)(cid:13)(cid:21) bounded as well.
By using the upper bounds of E (cid:20)(cid:16) (cid:98) A s,a ( θ ) (cid:17) 2 | s, a (cid:21) , (cid:107) w (cid:63) (cid:107) ,
| A s,a ( θ ) | ≤ 21 − γ and (cid:13)(cid:13) ¯ φ s,a ( θ ) (cid:13)(cid:13) ≤ 2 B , the left hand side of (95) is upper bounded by
E
[Equation]
−
[Equation]
(cid:62)
[Equation]
φ
[Equation]
a
[Equation]
θ ) (cid:17) 2 |
[Equation]
(cid:21) ≤
[Equation]
≤
[Equation]
(cid:19)
[Equation]
Thus, we choose
[Equation]
Now all the conditions (i) - (vi) in Theorem 12 are veriﬁed.
The reminder of the proof follows that of
Corollary 1.
H D ISCUSSION ON THE D ISTRIBUTION M ISMATCH C OEFFICIENTS , THE C ONCENTRABILITY C OEFFICIENTS AND THE I NCREASING S TEP S IZE
We have already mentioned in the comparison with Agarwal et al.
(2021) right after Theorem 1 that, although we have linear convergence rates, the magnitude of our error ﬂoor is worse (larger) by a factor of ϑ ρ (cid:112) C
ρ ( ϑ ρ √ C ν for Theorem 3 and 4), due to the concentrability C ρ and the distribution mismatch coefﬁcients ϑ ρ used in our proof.
Such difference comes from different nature of the proof techniques.
Here the distribution mismatch coefﬁcients ϑ ρ and the concentrability coefﬁcients C ρ and C ν are potentially large in our convergence theories.
We give extensive discussions on them, respectively.
We also provide further noteworthy observations about the connection between the use of the increasing step size and the policy iteration.
Distribution mismatch coefﬁcients ϑ ρ .
Our distribution mismatch coefﬁcient ϑ ρ in (20) is the same as the one in Xiao (2022).
It contains both an upper bound and a lower bound.
The linear convergence rate in our theories is 1 − 1 ϑ ρ > 0 .
Thus, the smaller ϑ ρ is, the faster the resulting linear convergence rate.
The best linear convergence rate is achieved when ϑ ρ achieves its lower bound.
Here our analysis is general that it includes all the distribution mismatch coefﬁcient ϑ ρ induced by any target state distribution ρ .
Our results generalizes and sometimes also improves with respect to prior results.
A very pessimistic and trivial upper bound on ϑ ρ is
[Equation]
However, if the target state distribution ρ ∈ ∆( S ) does not have full support, i.e., ρ s = 0 for some s ∈ S , then ϑ ρ might be inﬁnite from this upper bound.
Xiao (2022) just assumes that ϑ ρ is ﬁnite.
We further propose a solution to this particular issue.
Indeed, if ρ does not have full support, consider π ∗ as an optimal policy.
We can always convert the convergence guarantees for some state distribution ρ (cid:48) ∈ ∆( S ) with full support, i.e., ρ (cid:48) s > 0 for all s ∈ S as follows:
[Equation]
Then we only need convergence guarantees of V ρ (cid:48)
( π ( k ) )
− V ρ (cid:48) ( π ∗ ) for arbitrary ρ (cid:48) obtained from all our convergence analysis above.
In this case, the linear convergence rate depends on
[Equation]
Equation (20) provides the lower bound 11 − γ for ϑ ρ .
Such lower bound can be achieved when the target state distribution ρ satisﬁes that ρ = d π ∗ ( ρ ) where π ∗ is an optimal policy.
The advantage of this case is that, not only it implies the best linear convergence rate, more importantly, the fast linear convergence rate is known to be γ .
So we know the convergence rate explicitly without any estimation, even though the optimal policy or the policy iterates are unknown before training.
Hence, we know when to stop running the algorithm.
Lan (2022) only considers the case when ρ = d π ∗ ( ρ ) and we are able to recover the same linear convergence rate γ in their result.
Furthermore, the convergence performance V ρ ( π ( k ) ) − V ρ
( π ∗ ) depends on the target state distribution ρ .
If the optimal policy π ∗ is independent to the target state distribution ρ which is usually the case in RL problems, then we are always allowed to ﬁx ρ = d π ∗ ( ρ ) for the analysis without knowing ρ and π ∗ and derive this best linear convergence performance with rate γ , because we use the initial state-action distribution ν in training which is independent to ρ .
Finally, from (20), if d ( k ) converges to d ∗ , then ϑ k converges to 1.
This might imply superlinear convergence results as Section 4.3 in Xiao (2022).
In this case, the notion of the distribution mismatch coefﬁcients ϑ ρ no longer exists for the superlinear convergence analysis.
In other words, it is no longer concerned.
Concentrability coefﬁcients C ν .
The issue of having (potentially large) concentrability coefﬁcients is unavoidable in all the fast linear convergence analysis of the inexact NPG that we are aware of, including even the tabular setting (e.g., Lan (2022) and Xiao (2022)) and the log-linear policy setting (Cayci et al. (2021), Chen & Theja Maguluri (2022) and ours).
First, in the fast linear convergence analysis of inexact NPG, the concentrability coefﬁcients appear from the errors, including the statistical error and the approximation error.
Thus, one way to avoid having the concentrability coefﬁcients appear is to consider the exact NPG in the tabular setting (See Theorem 10 in Xiao (2022)).
Because the tabular setting makes no approximation error and the exact NPG makes no statistical error.
We consider the inexact NPG with the log-linear policy.
Consequently, we have the concentrability coefﬁcients multiplied by both the statistical error (cid:15) stat and the approximation error ( (cid:15) bias in Assumption 2 or (cid:15) approx in Assumption 5 and 8).
To remove the concentrability coefﬁcients, one has to make strong assumptions on the errors with the L ∞ supremum norm.
In the tabular setting, Lan (2022) and Xiao (2022) assume that (cid:107) (cid:98) Q ( π ) −
Q ( π ) (cid:107) ∞ ≤ (cid:15) stat .
The cons of such strong assumption requires high sample complexity and is already explained in Appendix A.1.
In the log-linear policy setting, Chen & Theja Maguluri (2022) assume that (cid:107) Q s ( θ ( k ) ) − Φ w ( k ) (cid:63) (cid:107) ∞ ≤ (cid:15) bias for the approximation error, which is a very strong assumption in the function approximation regime.
Due to the supremum norm, (cid:15) bias is unlikely to be small, especially for large action spaces.
Under this strong assumption, Lan (2022), Xiao (2022) and Chen & Theja Maguluri (2022) are able to eliminate the concentrability coefﬁcients.
To avoid assuming such strong assumptions, Cayci et al.
and our paper consider the expected L 2 errors in the log-linear policy setting, which are much weaker assumptions, especially much more reasonable for the approximation error (cid:15) bias compared to the one in Chen & Theja Maguluri (2022).
The tradeoff is that, the concentrability coefﬁcients can not be eliminated in this case both in Cayci et al.
(2021) and our results.
Furthermore, as mentioned right after Theorem 4, under the expected error assumptions (Assumption 7 and 8), our concentrability coefﬁcient C ν is better presented than the one in Assumption 2 in Cayci et al.
(2021) in the sense that it is independent to the policies throughout the iterations thanks to the use of ˜ d ( k ) instead of ¯ d ( k ) (which is mentioned in Remark 1 as well) and is controllable to be ﬁnite by ν , while the one in Cayci et al.
(2021) depends on the iterates, thus is unknown and is not guaranteed to be ﬁnite.
Finally, like the distribution mismatch coefﬁcient, the upper bound of C ν in (30) is very pessimistic.
By the deﬁnition of C ν in (28), one can expect that C ν is closed to 1 , when π ( k ) and π ( k +1) converge to π ∗ with π ∗ the optimal policy.
So our concentrability coefﬁcient C ν is the “best” one among all concentrability coefﬁcients in the sense that, it takes the weakest assumptions on errors compared to Lan (2022), Xiao (2022) and Chen & Theja Maguluri (2022), it does not impose any restrictions on the MDP dynamics compared to
Cayci et al.
and it can be controlled to be ﬁnite by ν when other concentrability coefﬁcients are inﬁnite (Scherrer, 2014).
It is still an open question whether we can obtain fast linear convergence results of the inexact NPG in the log-linear policy setting, with small error ﬂoor and a much improved concentrability coefﬁcient, e.g., as the same magnitude as the one in Agarwal et al.
Increasing step size and the connection with policy iteration.
As for the use of the increasing step size, intuitively, the reason is that both NPG in (17) and Q-NPG in (16) behave more and more like policy iteration.
For instance, when η k → ∞ and we replace the linear approximation Φ s w ( k ) by Q s ( θ ( k ) ) , (16) becomes
[Equation]
which is exactly the classical Policy Iteration method (e.g., Puterman, 1994; Bertsekas, 2012).
We refer to Xiao (2022, Section 4.4) for more discussion on the connection with policy iteration.
I S TANDARD O PTIMIZATION R ESULTS
In this section, we present the standard optimization results from Beck (2017);
Xiao (2022); Bach &
Moulines (2013) used in our proofs.
First, we present the closed form update of mirror descent with KL divergence on the simplex.
We provide its proof for the completeness.
Lemma 10 (Mirror descent on the simplex, Example 9.10 in Beck (2017)) .
Let g ∈ R n which will often be a gradient and let η > 0 .
For p, q in the unit n -simplex ∆ n , the mirror descent step with respect to the KL divergence
[Equation]
is given by
[Equation]
where (cid:12) is the element-wise product between vectors.
Proof.
The Lagrangian of (98) is given by
[Equation]
where µ ∈ R and λ ∈ R n with non-negative coordinates are the Lagrangian multipliers.
Thus the
Karush–Kuhn–Tucker conditions are given by
[Equation]
where the division p/q is element-wise.
Isolating p in the top equation gives
[Equation]
Using the second constraint 1 (cid:62) n p = 1 gives that
[Equation]
Consequently, by plugging the above term into p , we have that
[Equation]
It remains to determine λ.
If q i =
then p i = 0 and thus λ i > 0 .
Conversely, if q i > 0 then p i >and thus λ i = 0 .
In either of these cases, we have that the solution is given by (99).
Now we present the three-point descent lemma on proximal optimization with Bregman divergences, which is another key ingredient for our PMD analysis.
Following Xiao (2022, Lemma 6), we adopt a slight variation of Lemma 3.2 in Chen & Teboulle (1993).
First, we need some technical conditions.
Deﬁnition 10 (Legendre function, Section 26 in Rockafellar (1970)) .
We say a function h is of
Legendre type or a Legendre function if the following properties are satisﬁed:
(i) h is strictly convex in the relative interior of dom h , denoted as rint dom h .
(ii) h is essentially smooth, i.e., h is differentiable in rint dom h and, for any boundary point x b of rint dom h , lim x → x b (cid:107)∇ h ( x ) (cid:107) → ∞ where x ∈ rint dom h .
Deﬁnition 11 (Bregman divergence (Bregman, 1967; Censor & Zenios, 1997)) .
Let h : dom
h
→ R be a Legendre function and assume that rint dom h is nonempty.
The Bregman divergence D h ( · , · ) : dom h × rint dom h → [0 , ∞ ) generated by h is a distance-like function deﬁned as
[Equation]
Under the above conditions, we have the following result.
We also provide its proof for selfcontainment.
(Xiao (2022) does not provide a formal proof.)
Lemma 11 (Three-point decent lemma, Lemma 6 in Xiao (2022)) .
Suppose that C ⊂ R m is a closed convex set, f : C → R is a proper, closed 8 convex function, D h ( · , · ) is the Bregman divergence generated by a function h of Lengendre type and rint dom h ∩ C (cid:54) = ∅ .
For any x ∈ rint dom h , let x + ∈ arg min u ∈ dom
h ∩C { f
[Equation]
u )
[Equation]
D h
( u
[Equation]
Then x + ∈ rint dom h ∩ C and for any u ∈ dom h ∩ C ,
[Equation]
Proof.
First, we prove that for any a, b ∈ rint dom h and c ∈ dom h , the following identity holds:
[Equation]
Indeed, using the deﬁnition of D h in (100), we have
[Equation]
Subtracting (102) and (103) from (104) yields (101).
Next, since h is of Legendre type, we have x + ∈ rint dom h ∩ C .
Otherwise, x + is a boundary point of dom h .
From the deﬁnition of Legendre function, (cid:107)∇ h ( x + ) (cid:107) = ∞ which is not possible, as x + is also the minimum point of f ( u ) + D h ( u, x ) .
By the ﬁrst-order optimality condition, we have
[Equation]
where g + ∈ ∂f ( x + ) is the subdifferential of f at x + .
From the deﬁnition of D h , the above inequality is equivalent to
[Equation]
Besides, plugging c = u, a = x + and b = x into (101), we obtain
[Equation]
Rearranging terms and adding f ( u ) on both sides, we have
[Equation]
which concludes the proof.
The last inequality is obtained by the convexity of f and g + ∈ ∂f ( x + ) .
Finally, we use the following linear regression analysis for the proof of our sample complexity results, i.e., Corollary 1 and 3.
Theorem 12 (Theorem 1 in Bach & Moulines (2013)) .
Consider the following assumptions:
(i) H is a m -dimensional Euclidean space.
(ii)
The observations ( x n , z n ) ∈ H
×
H are independent and identically distributed.
(iii) E (cid:104) (cid:107) x n (cid:107) 2 (cid:105) and E (cid:104) (cid:107)
z n (cid:107) 2 (cid:105) are ﬁnite.
The covariance E (cid:2) x n x (cid:62) n (cid:3) is assumed invertible.
(iv) The global minimum of f ( θ ) =
12 E (cid:104) (cid:104) θ, x n (cid:105) 2 − 2 (cid:104) θ, z n (cid:105) (cid:105) is attained at a certain θ ∗ ∈ H .
Let ξ n = z n − (cid:104) θ ∗ , x n (cid:105) x n denote the residual.
We have E [ ξ n ] = 0 .
(v) Consider the stochastic gradient recursion deﬁned as
[Equation]
started from θ 0 ∈ H and also consider the averaged iterates θ out = 1 n +1 (cid:80) nk =
θ k .
(vi) There exists R > 0 and σ > 0 such that E (cid:2)
ξ n ξ (cid:62) n (cid:3) ≤ σ 2 E (cid:2) x
n x (cid:62) n (cid:3) and E (cid:104) (cid:107) x n (cid:107) 2 x n x (cid:62) n (cid:105)
[Equation]
When η = 14 R
[Equation]
we have
[Equation]