Stochastic Semi-Gradient Descent for Learning Mean Field Games with Population-Aware Function Approximation

## Abstract

Abstract Mean field games (MFGs) model the interactions within a large-population multi-agent system using the population distribution.
Traditional learning methods for MFGs are based on fixed-point iteration (FPI), which calculates best responses and induced population distribution separately and sequentially.
However, FPI-type methods suffer from inefficiency and instability, due to oscillations caused by the forward-backward procedure.
This paper considers an online learning method for MFGs, where an agent updates its policy and population estimates simultaneously and fully asynchronously, resulting in a simple stochastic gradient descent (SGD) type method called SemiSGD.
Not only does SemiSGD exhibit numerical stability and efficiency, but it also provides a novel perspective by treating the value function and population distribution as a unified parameter.
We theoretically show that SemiSGD directs this unified parameter along a descent direction to the mean field equilibrium.
Motivated by this perspective, we develop a linear function approximation (LFA) for both the value function and the population distribution, resulting in the first population-aware LFA for MFGs on continuous state-action space.
Finite-time convergence and approximation error analysis are provided for SemiSGD equipped with population-aware LFA.

### 1 Introduction

Mean field games (MFGs) [, ] offer a tractable framework for modeling multi-agent systems with a large homogeneous population. In MFGs, the impact of other agents on a particular agent is encapsulated by a *population mass* (or *mean field*), providing a reliable approximation of actual interactions among agents when the number of agents is large.
Consequently, understanding the *population* is fundamental in MFGs, as learning these games entails considering both the agent’s policy and the population dynamics.

Prior work on learning MFGs has mainly focused on fixed-point iteration (FPI) methods and their variations, which *sequentially* calculate the best response w.r.t. a *fixed* population, and the induced population distribution (or measure) w.r.t. a *fixed* policy [, , , , ].
However, FPI-type methods face several limitations:
1) Calculating best responses (backward) and induced population distributions (forward) typically involves distinct iterative processes.
These processes are implemented separately and executed sequentially, hindering parallel computing and potentially increasing the overall *computational burden*.
2) FPI methods suffer from *instability*. Fixing a population measure far from the equilibrium measure can lead to a corresponding best response that is also far from the equilibrium policy, resulting in slow convergence. Moreover, drastic differences between best responses and induced population measures in consecutive iterations can cause oscillations in the learning process, a phenomenon commonly observed in practice (see, e.g., Figure 1(a) and []).

Figure: (a) Convergence performance.
Refer to caption: extracted/5793980/fig/rr/mse_new.png

This work delves into the rapidly growing field of online learning for MFGs, where an online agent interacts with the environment and gathers observations on a Markov chain to update its policy and population estimate. Notably, this Markov chain is *parameterized* by the policy and a population distribution, which collectively determine the transition kernel. Thus, the goal of finding a mean field equilibrium (MFE) can be translated into finding an *optimal parameter*, which corresponds to the fixed point of the Bellman operator and the transition operator. This perspective unlocks a large toolbox of stochastic parameter approximation methods for learning MFGs, such as simple stochastic gradient descent (SGD)-type methods. Specifically, we treat the policy and the population as a unified parameter and update their estimates *fully asynchronously* using the same samples, thereby eliminating the forward-backward process.

We term our method SemiSGD, as it uses semi-gradients in lieu of actual gradients.
Albeit simple, an SGD-type procedure raises several questions:
1) Simultaneously and asynchronously updating the behavior policy and the population estimate creates a *strong coupling* between the two, rendering the underlying Markov chain *non-stationary*.
Online FPI avoids this issue by fixing one estimate while updating the other.
How can we analyze this coupling and the non-stationary process to ensure the convergence of SemiSGD?
2) How does SemiSGD, following a non-stationary Markov chain, converge? Futhermore, how can it address the instability and oscillation issues of FPI-type methods?

For the first question, note that the Bellman and transition operators are *contractive*.
Thus, with sufficiently smooth dynamics, the non-stationarity is dominated by the contraction property.
Remarkably, we discover that SemiSGD indeed follows a *descent* direction towards the MFE (Lemma 1), provided that the dynamics are Lipschitz continuous with samll Lipschitz constants.
By following a descent direction, updating parameters incrementally, and dropping the forward-backward procedure, SemiSGD achieves *automatic stabilization* without additional mechanics (see Figure 1(a) for a numerical example and Figure 1(b) for an intuitive illustration).

Treating the policy and population as a unified parameter also illuminates the gap in learning MFGs on continuous state-action spaces: linear function approximation (LFA), a technique widely employed in learning value functions on continuous state-action spaces [, ], can be applied to the population measure as well.
As a framework for infinite-player games, common MFGs are often on *infinite or continuous* state-action spaces, such as autonomous driving [], flocking [] and crowd modelling [, ].
Equipped with population-aware LFA (PA-LFA), SemiSGD is capable of learning MFGs on continuous state-action spaces, fully online and model-free with strong theoretical guarantees.

Main results.
We highlight some contributions and novelties of this work.

- •
We propose SemiSGD, a simple online SGD-type method for learning MFGs. We innovatively treat the policy and the population as a unified parameter. Algorithmically, we update both simultaneously and asynchronously using the same online observations and learning rate, thus eliminating the forward-backward process typical of FPI methods. SemiSGD enjoys *automatic stabilization* without additional mechanisms such as fictitious play or online mirror descent (Figure 1(a)). Theoretically, we show that the unified parameter in SemiSGD follows a *descent direction* towards the MFE, whereas neither the policy nor population alone is guaranteed to do so (Lemma 1).
- •
We formulate a novel framework of linear MFGs, characterizing a class of MFGs with linear structure concerning the population measure.
Linear MFGs accommodates continuous state-action spaces and includes MFGs on finite state-action spaces as a special case.
We prove that linear MFGs enable linear parameterization of both the value function and the population measure.
- •
We extend the linear parameterization to develop a *population-aware linear function approximation (PA-LFA)* for general MFGs.
SemiSGD equipped with PA-LFA is the first method to apply LFA to the population measure in MFGs.
Notably, updates in SemiSGD with PA-LFA maintain the same operation complexity as in the finite state-action space case, highlighting the simplicity and efficiency of our method.
- •
Sample complexity and approximation error analysis are provided for SemiSGD with PA-LFA.
We novelly regard the learning process as a non-stationary Markov decision process parameterized by the unified parameter.
This perspective enables a straightforward SGD-type analysis, elegantly handling the strong coupling between the policy and population and offering insights into the learning dynamics of SemiSGD.
- •
We conduct various experiments to demonstrate the properties of our methods:
1) For continuous state-action spaces, PA-LFA is more efficient and accurate than discretization;
2) SemiSGD is inherently smoother and automatically stabilizes without additional mechanics;
3) By eliminating the forward-backward procedure, SemiSGD is significantly faster than other FPI-type methods.

**Table 1: Comparison of learning methods for MFGs**
| Reference | Algo.<br>type | Additional<br>mechanism | Oracle-<br>free | Function approx. | Sample<br>complex. |  |
| --- | --- | --- | --- | --- | --- | --- |
| Population | Policy |  |  |  |  |  |
| [, , ] | FPI | - | - | - | - | - |
| [, ] | FPI | FP & OMD | - | - | - | - |
| [, ] | FPI | regularization | - | - | - | - |
| [, ] | FPI | multi-timescale | ✓ | - | - | - |
| [] | FPI | FP | ✓ | - | - | $\widetilde{O}(\epsilon^{-2})$ |
| [] | FPI | two-timescale | ✓ | - | - | $O(\epsilon^{-4})$ |
| [] | FPI | two-timescale | - | - | ✓ | $\widetilde{O}(\epsilon^{-5})$ |
| [] | FPI | - | - | - | ✓ | $\widetilde{O}(\epsilon^{-2})$ |
| This paper | SGD | - | ✓ | ✓ | ✓ | $\widetilde{O}(\epsilon^{-2})$ |

Related work.
There is a growing body of literature on learning MFGs (summarized in Table. 1). Readers can refer to [, ] for a comprehensive review. To address the instability issue of FPI, researchers have proposed various methods to stabilize the learning process, broadly categorized into two classes: 1) regularization [, ]; 2) fictitious play (FP) [, , ]; and 3) online mirror descent (OMD) [, ]. A multi-timescale [, , , ] update rule is utilized for the population and value function in the forward-backward process. By fixing one element (or make its update much slower) while updating the other, these methods circumvent the strong coupling between the value function and population measure, and thus inherit the convergence properties of FPI-type methods. To deal with large state and action spaces and obtain strong theoretical guarantees, [, ] consider linear function approximation (LFA) [, , ] for the policy learning in the backward process of MFGs. The LFA scheme is not incorporated into the coupled forward-backward process, which renders the learning algorithm still a FPI-type method, and sample complexity results fail to capture the population estimate due to the requirement of population oracles in the forward process.

Notation.
A complete list of symbols and their meanings is provided in Table 3.
We denote the set of probability measures on a space $\mathcal{X}$ as $\mathcal{D}(\mathcal{X})$, and the set of signed measures as $\mathcal{M}(\mathcal{X})$.
When $\mathcal{X}$ is finite with $d$ elements, the probability simplex on it is denoted as $\Delta^{d}\subset\mathbb{R}^{d}$.
$\delta_{x}$ is the Dirac delta measure at $x$.
Without subscript, the inner product denotes the vector inner product $\left<x,y\right>=x^{T}y$.
$x\oplus y$ is the direct sum of elements in linear spaces, which reduces to their concatenation $(x;y)$ when $x$ and $y$ are vectors.
The norm without subscript denotes the $\ell_{2}$ norm for vectors, and $\ell_{2}$ operator norm for matrices.
For (vector-valued) functions on $\mathcal{S}$, the $L_{2}$ inner product is defined as $\left<f,g\right>_{L_{2}}=\int_{\mathcal{S}}f(s)g(s)\mathrm{d}s$,
and the $L_{1}$ norm is denoted as $\|f\|_{\mathrm{TV}}=\int_{\mathcal{S}}\|f(s)\|_{1}\mathrm{d}s$.

### 2 Stochastic semi-gradient descent for MFGs on finite state-action spaces

#### 2.1 Revisit online learning for MFGs on finite state-action spaces

We consider an infinite-horizon Markov decision process (MDP) denoted by $(\mathcal{S},\mathcal{A},r,P,\gamma)$, with the state space $\mathcal{S}$ and action space $\mathcal{A}$ being finite.
In MFGs, the reward function $r$ and transition kernel $P$ depend on the population distribution over the state space.
Specifically, for a given state-action pair $(s,a)\in\mathcal{S}\times\mathcal{A}$ and population distribution $\mu\in\mathcal{D}(\mathcal{S})$,
$r(s,a,\mu)$ and $P(s^{\prime}\,|\,s,a,\mu)$ denote the reward received and the probability of transitioning to state $s^{\prime}\in\mathcal{S}$. $\gamma$ is the discount factor.

An agent in an MFG aims to find a policy $\pi$, which maps a state to a distribution on the action space determining the agent’s actions,
that maximizes its expected cumulative discounted reward while interacting with the population.
We utilize a value-based approach to calculate policies and assume access to a policy operator $\Gamma_{\pi}$ that returns a policy based on an (action-)value function $Q\colon\mathcal{S}\times\mathcal{A}\to\mathbb{R}$.
We write $\pi_{Q}\coloneqq\Gamma_{\pi}(Q)$.
To define the learning goal, the mean field equilibrium (MFE), we first introduce the following two operators for any value functions $Q,Q^{\prime}\in\mathbb{R}^{|\mathcal{S}|\times|\mathcal{A}|}$ and population measures $M,M^{\prime}\in\mathcal{D}(\mathcal{S})$:

$$ $\displaystyle\mathcal{T}_{(Q,M)}Q^{\prime}(s,a)\coloneqq$ $\displaystyle\mathbb{E}_{\scriptscriptstyle(Q,M)}\left[r(s,a,M)+\gamma Q^{ \prime}(s^{\prime},a^{\prime})\right],\text{ with }a^{\prime}\!\sim\!\pi_{Q},s ^{\prime}\!\sim\!P(\cdot\,|\,s,a,M)$ (Bellman) $\displaystyle\mathcal{P}_{(Q,M)}M^{\prime}(s^{\prime})\coloneqq$ $\displaystyle\mathbb{E}_{\scriptscriptstyle(M^{\prime},Q)}\left[P(s^{\prime}\, |\,s,a,M)\right],\text{ with }s\!\sim\!M^{\prime},a\!\sim\!\pi_{Q}.$ (Transition) $$

###### Definition 1 (Mean field equilibrium) .

A value function-population distribution pair $(Q,M)$ is a *mean field equilibrium (MFE)* if it satisfies
$Q=\mathcal{T}_{(Q,M)}Q$ and $M=\mathcal{P}_{(Q,M)}M$.

A typical FPI-type method (approximately) calculates the fixed point of the Bellman operator given the current population distribution, i.e., the best response, and the fixed point of the transition operator given the current value function, i.e., the induced population distribution, alternately.
Under certain conditions, FPI converges to the MFE [].

This work focuses on online learning (model-free and stochastic) methods for MFGs.
In such methods, an agent maintains estimates about the value function and population measure, and uses online observations to update its estimates.
For the value function, temporal difference (TD) methods are widely used [].
Given an online observation tuple $O=(s,a,s^{\prime},a^{\prime})$ and step-size $\alpha$, on-policy TD (or SARSA []), updates the Q-value function as follows:

$$ $Q(s,a)\leftarrow Q(s,a)-\alpha\mathfrak{g}(Q;O),\ \ \text{with}\ \ \mathfrak{g }(Q;O)\coloneqq Q(s,a)-r(s,a,M)-\gamma Q(s^{\prime},a^{\prime}).$ (2) $$

For the population distribution, Monte Carlo (MC) sampling [] is a common choice.
Given a new state sample $s^{\prime}$ and using a general step-size $\alpha$, the population measure is updated as follows:

$$ $M\leftarrow M-\alpha\mathfrak{g}(M;O),\ \ \text{with}\ \ \mathfrak{g}(M;O) \coloneqq M-\delta_{s^{\prime}},$ (3) $$

where $\delta_{s^{\prime}}$ is the Dirac delta measure and is a one-hot vector for finite state spaces.

TD and MC updates are widely used in online learning methods for MFGs [, , ].
Fixing the policy and population measure (i.e., fixing the transition kernel), TD converges to the optimal value function, and MC converges to the induced population distribution.

###### Definition 1 (Mean field equilibrium) .

#### 2.2 Stochastic semi-gradient descent

Prior work on online learning for MFGs either uses two-layered loops with the policy and/or population measure fixed in the inner loop [, ],
or uses a two-timescale to update the policy and population measure in different rates [, ], essentially following a FPI-type forward-backward process.
In contrast, we propose simply combining Equations 2 and 3 in one pass, resulting in a simple SGD-type method that converges to the MFE efficiently.

Figure: Algorithm 1 Stochastic semi-gradient descent (SemiSGD)

To elaborate, $\mathfrak{g}$ in Equation 2 is referred as semi-gradients, as it is not a real gradient of any loss function, but provides a similar descent direction to the *stationary point*, i.e., the zero point of $\mathfrak{g}$.(^1^11With a slight abuse of notation, we use a single operator $\mathfrak{g}$ to return semi-gradients throughout the paper, for both the action-value function and population measure with or without LFA; it should be clear from the context and its arguments which one it refers to.)
Please refer to Section 4 for the derivation of the population measure semi-gradient in Equation 3.
Consequently, we propose updating the value function and population measure estimates simultaneously and asynchronously using semi-gradients in one pass.
We term our method SemiSGD and present it in Algorithm 1.
In Algorithm 1, we denote $(Q_{t};M_{t})$ as the concatenated representation of the value function and population measure, as they are both vectors for finite state-action spaces.
Similarly, $\mathfrak{g}$ represents the concatenation of the semi-gradients for the value function and population measure.

Algorithm 1 is an online single-loop method that completely removes the need for a forward-backward process.
Not only it updates the value function and population measure estimates in an SGD-like manner, the following lemma shows that the a descent direction is provided by the *mean-path* semi-gradient, defined as $\bar{\mathfrak{g}}_{(Q,M)}(\cdot)\coloneqq\mathbb{E}_{(Q,M)}\mathfrak{g}(\cdot
;O)$, where $O$ is the online observation tuple following the steady distribution induced by the policy and transition kernel determined by $Q$ and $M$.

###### Lemma 1 (Descent direction) .

Suppose $(Q_{*};M_{*})$ is an MFE and Assumptions 1, 2 and 3 hold.(^2^22Section 3 demonstrates linear MFGs encompass finite MFGs (Example 1). Thus, our analysis focuses on linear MFGs, with results subsequently reduced to finite MFGs in Appendix H. As a result, in this section we sometimes refer to the analysis for linear MFGs in later sections; in such cases, finite MFGs are considered as linear MFGs.)
For any value function $Q$ and population measure $M$, with $\Delta Q\coloneqq Q-Q_{*}$ and $\Delta M\coloneqq M-M_{*}$, we have

$$ $\displaystyle-\left<\Delta M,\bar{\mathfrak{g}}_{{(Q,M)}}(M)\right>\leq-\| \Delta M\|^{2}+\sigma L\|\Delta M\|(\|\Delta Q\|+\|\Delta M\|),$ (4) $\displaystyle-\left<\Delta Q,\bar{\mathfrak{g}}_{{(Q,M)}}(Q)\right>\leq-(1- \gamma)\lambda\|\Delta Q\|^{2}+(\sigma LH+L_{r})\|\Delta Q\|(\|\Delta Q\|+\| \Delta M\|),$ $$

where $\lambda\coloneqq\min_{s,a}M_{*}(s)\pi_{Q_{*}}(a\,|\,s)$, $H=O(\sqrt{|\mathcal{S}||\mathcal{A}|}R/(1-\gamma))$, and other constants are defined in Assumptions 1, 2 and 3.
Neither $-\bar{\mathfrak{g}}_{{(Q,M)}}(M)$ nor $-\bar{\mathfrak{g}}_{{(Q,M)}}(Q)$ is guaranteed to be a descent direction, no matter how small the Lipschitz constants are.
However, if $3\sigma LH+L_{r}\leq(1-\gamma)\lambda$, it holds that

$$ $-\left<{(Q;M)}-{(Q_{*};M_{*})},\bar{\mathfrak{g}}_{{(Q,M)}}({(Q;M)})\right> \leq-\frac{1}{2}(1-\gamma)\lambda\|{(Q;M)}-{(Q_{*};M_{*})}\|^{2}.$ (5) $$

A generalized version of Lemma 1 is proved in Section F.1 for linear MFGs.
We now discuss some key insights from Lemma 1.

Descent direction.
The concatenated semi-gradient points a descent direction for the concatenated estimate $(Q;M)$, while neither the value function nor population measure alone is guaranteed to follow a descent direction.
See Figure 9 for an illustrative example.
Remarkably, we will show that Algorithm 1 follows the direction of $-\bar{\mathfrak{g}}$ with higher-order noises (Theorem 1).
Therefore, Lemma 1 argues that SemiSGD generally follows a descent direction to the MFE.

Explaining oscillations in FPI.
Lemma 1 also gives an intuitive explanation for the instability in FPI-type methods.
When calculating the best response, if FPI fixes a population measure far from the equilibrium measure (large $\|M-M_{*}\|$), the semi-gradient of the value function may not give a descent direction, potentially driving the best response away from the optimal policy.
Similar oscillations can occur when FPI calculates the induced population distribution.

Incremental update by design.
To stabilize FPI, fictitious play (FP) and online mirror descent (OMD) incrementally update the policy or population measure [], rather than switching to entirely new estimates.
As an SGD-type method, SemiSGD naturally updates the estimates incrementally, providing another layer of stabilization, and thus does not require those additional mechanisms.

Learn MFG on a non-stationary Markov chain.
SemiSGD samples a non-stationary Markovian trajectory, as its transition kernel depends on the changing value function and population measure. Generally, rapidly changing Markov chains exhibit chattering []. Thus, Lemma 1 requires that the MDP components be Lipschitz continuous with sufficiently small Lipschitz constants. The proof (Section F.1) shows that this descent direction arises from the contraction properties of the Bellman and transition operators. When the MDP is smooth enough, non-stationarity is dominated by this contraction.
The contraction property of the operators translates to the attraction property of their fixed point, i.e., the MFE, which corresponds to a *stationary parameter* where the mean-field semi-gradient is zero (Proposition 2).
This fact again underscores that the landscape of an MFG is best described by the unified parameter, making it more natural to learn this parameter directly.

###### Lemma 1 (Descent direction) .

### 3 Linear mean field games

From this point, we remove the assumption that the state-action space is finite.
In Section 2, we have shown that it is beneficial to treat the value function and population measure as a unified parameter governing the MDP dynamics.
As a result, linear MDP [] and LFA for the value function can be naturally extended to address population measure on continuous state spaces, which is crucial in various applications such as autonomous driving [], flocking [] and crowd modelling [, ]. We now introduce linear MFGs, a class of MFGs with an underlying linear MDP and a linear structure on the transition kernel w.r.t. the population measure.

###### Definition 2 (Linear mean field games) .

An MDP $(\mathcal{S},\mathcal{A},P,r,\gamma)$ is a linear MDP [] with feature map $\phi\colon\mathcal{S}\times\mathcal{A}\to\mathbb{R}^{d_{1}}$ if there exists $d_{1}$ (signed) population-dependent measures ${\omega}_{M}=(\omega_{M}^{(i)})_{i=1}^{d_{1}}\in\mathcal{M}(\mathcal{S})^{d_{1}}$ and an unknown population-dependent vector ${\nu}_{M}\in\mathbb{R}^{d_{1}}$, such that for any state-action pair $(s,a)\in\mathcal{S}\times\mathcal{A}$ and population distribution $M\in\mathcal{D}(\mathcal{S})$, we have

$$ $P(s^{\prime}\,|\,s,a,M)=\left<\phi(s,a),{\omega}_{M}(s^{\prime})\right>,\qquad r (s,a,M)=\left<\phi(s,a),{\nu}_{M}\right>.$ (6) $$

A linear MFG further assumes a measure basis $\psi\in\mathcal{D}(\mathcal{S})^{d_{2}}$ such that for any population $M$, there exists an unknown matrix $\Omega_{M}\in\mathbb{R}^{d_{1}\times d_{2}}$ such that ${\omega}_{M}=\Omega_{M}\psi$, indicating that $P(s^{\prime}\,|\,s,a,M)=\left<\phi(s,a),\Omega_{M}\psi(s^{\prime})\right>$.
We require $\phi$ to be $L_{\infty}$ and $\psi$ to be $L_{\infty}$ and $L_{2}$ (thus its Gram matrix exists).
Without loss of generality, we assume $\sup_{s,a}\|\phi(s,a)\|_{2}\leq 1$ and $\sup_{s}\|\psi(s)\|_{1}\leq F$.

A linear MFG assumes the MDP components, the transition kernel and reward function, lie in a linear function space with known basis functions.
This indicates the linear structure of the value function and population measure.

###### Proposition 1 .

For a linear MFG, for any population distribution $M$ and policy $\pi$, we denote $Q^{\pi}_{M}$ as the action-value function and $\mu^{\pi}_{M}$ as the induced population measure w.r.t. the MDP determined by $M$ and $\pi$.
Then, there exist vector parameters $\theta\in\mathbb{R}^{d_{1}}$ and $\eta\in\mathbb{R}^{d_{2}}$ such that

$$ $Q^{\pi}_{M}(s,a)=\left<\phi(s,a),\theta\right>,\qquad\mu^{\pi}_{M}(s^{\prime}) =\left<\psi(s^{\prime}),\eta\right>,\qquad\forall(s,a,s^{\prime})\in\mathcal{S }\times\mathcal{A}\times\mathcal{S}.$ (7) $$

The proof of Proposition 1 is deferred to Appendix C.
Remarkably, requiring that the transition kernel being linear w.r.t. a measure basis is essential for the linear structure of the population measure.
In this paper, we reserve letters $\theta$ for the value function parameter and $\eta$ for the population measure parameter.
And we denote the *concatenated parameter* as $\xi=(\eta;\theta)$.
Additionally, we will use parameters ($\eta$, $\theta$, $\xi$) and their corresponding functions ($Q$, $M$, $(Q,M)$) interchangeably in our analysis.
For example, we say $\xi\in\mathbb{R}^{d_{1}+d_{2}}$ is a mean field equilibrium (MFE) if its corresponding value function $Q=\left<\phi,\theta\right>$ and popumeasure $M=\left<\psi,\eta\right>$ satisfy Definition 1.

###### Example 1 (Linear MDP plus population-independent transition kernel) .

MFGs with a linear MDP and population-independent transition kernel are a trivial example of linear MFGs, where $\Omega_{M}=I$ for all $M\in\mathcal{D}(\mathcal{S})$.

###### Example 2 (Finite state-action space) .

For finite MFGs, let the feature map $\phi$ return the one-hot vector of each state-action pair, i.e., $\phi(s,a)=e_{(s,a)}\in\mathbb{R}^{|\mathcal{S}||\mathcal{A}|}$,
and the measure basis $\psi$ return the Dirac delta measure at each state, i.e., $\psi(s^{\prime})=\delta_{s^{\prime}}\in\Delta^{|\mathcal{S}|}$.
Construct the matrix $\Omega_{M}\in\mathbb{R}^{|\mathcal{S}||\mathcal{A}|\times|\mathcal{S}|}$ such that $(\Omega_{M})_{(s,a),s^{\prime}}=P(s^{\prime}\,|\,s,a,M)$.
Similarly, construct the vector $\nu_{M}\in\mathbb{R}^{|\mathcal{S}||\mathcal{A}|}$ such that $(\nu_{M})_{(s,a)}=r(s,a,M)$.
Then, for any $(s,a,s^{\prime})\in\mathcal{S}\times\mathcal{A}\times\mathcal{S}$, we have

$$ $P(s^{\prime}\,|\,s,a,M)=e_{(s,a)}^{T}\Omega_{M}\delta_{s^{\prime}}=\left<e_{(s ,a)},\Omega_{M}\delta(s^{\prime})\right>,\quad r(s,a,M)=e_{(s,a)}^{T}\nu_{M}= \left<e_{(s,a)},\nu_{M}\right>.$ (8) $$

Example 2 implies that all the analysis for linear MFGs applies to finite MFGs (see also Appendix H).

###### Definition 2 (Linear mean field games) .

###### Proposition 1 .

###### Example 1 (Linear MDP plus population-independent transition kernel) .

###### Example 2 (Finite state-action space) .

### 4 SemiSGD with population-aware linear function approximation

Section 3 introduce the first linear parameterization of the population measure in MFGs, necessitating tailored stochastic update rule for the population measure estimate.
We now derive the semi-gradient for the population measure with LFA.
Let $M_{*}$ be the objective population measure and define the loss function $\mathcal{L}\coloneqq\frac{1}{2}\|M-M_{*}\|^{2}_{L_{2}}$. Then, the gradient of the loss is

$$ $\nabla_{\eta}\mathcal{L}=\left<\nabla_{\eta}M,M-M_{*}\right>_{L_{2}}=\left< \psi,\left<\psi,\eta\right>-M_{*}\right>_{L_{2}}.$ (9) $$

As the MFE population measure $M_{*}$ is unknown, we replace it with the empirical Delta distribution (bootstrapping), giving the semi-gradient

$$ $\mathfrak{g}(\eta;s^{\prime})=\left<\psi,\left<\psi,\eta\right>-\delta_{s^{ \prime}}\right>_{L_{2}}=\int_{\mathcal{S}}\psi(s)\psi(s)^{T}\eta\mathrm{d}s- \int_{\mathcal{S}}\psi(s)\delta_{s^{\prime}}(s)\mathrm{d}s\eqqcolon G_{\psi} \eta-\psi(s^{\prime}),$ (10) $$

where $G_{\psi}\coloneqq\int_{\mathcal{S}}\psi(s)\psi(s)^{T}\mathrm{d}s$ is the Gram matrix of measure basis $\psi$.

For finite MFGs, Equation 3 retains the updated population measure as a probability vector, which is not necessarily the case when using Equation 10 as the semi-gradient, due to the presence of general ${G}_{\psi}$ and $\psi(s^{\prime})$ that may not be an identity matrix or a probability vector.
Therefore, we need to apply a projection to the updated parameter to ensure that it remains a probability vector, giving our stochastic update rule:

$$ $\displaystyle\eta_{t+1}=\Pi_{\Delta}(\eta_{t}-\alpha_{t}\mathfrak{g}_{t}(\eta_ {t})),$ (11) $$

where $\mathfrak{g}_{t}(\eta)\coloneqq\mathfrak{g}(\eta;s_{t+1})$ and $\Pi_{\Delta}$ is the projection operator onto the probability simplex $\Delta^{d_{2}}$.

###### Remark 1 (Operation complexity) .

The simplex projection has a worst-case complexity of $O(d_{2}^{2})$ [].
In semi-gradient evaluation, we note that ${G}_{\psi}$ is a constant matrix, and thus we only need to evaluate $\psi$ at the next state $s^{\prime}$ and multiply parameter with a fixed matrix, with the matrix-vector multiplication having a complexity of $O(d_{2}^{2})$.
Therefore, the total worst-case operation complexity of the update rule Equation 11 is $O(d_{2}^{2})$.
Additionally, the simplex projection has an expected complexity of $O(d_{2})$ [].
And if ${G}_{\psi}$ has precomputed properties and/or special structures, such as being sparse with $O(d_{2})$ non-zero elements, the complexity of the semi-gradient evaluation is $O(d_{2})$, giving a total expected complexity of $O(d_{2})$, which is the same as the operation complexity of Equation 3 for finite MFGs with a state space of size $|\mathcal{S}|=O(d_{2})$.
In summary, Equation 11 is as simple and efficient as the population measure update for finite MFGs.

Similarly, we can get the TD update rule for the value function parameter $\theta$[]:

$$ $\mathfrak{g}(\theta;O)=\phi(s,a)\left(\left<\phi(s,a)-\gamma\phi(s^{\prime},a^ {\prime}),\theta\right>-r\right)\eqqcolon{G}_{\phi}(O)\theta-\varphi(O),$ (12) $$

where $O=(s,a,r,s^{\prime},a^{\prime})$, and ${G}_{\phi}(O)\coloneqq\phi(s,a)(\phi(s,a)-\gamma\phi(s^{\prime},a^{\prime}))^{T}$ and $\varphi(O)\coloneqq\phi(s,a)r$.
The update rule for the action-value function parameter is

$$ $\displaystyle\theta_{t+1}=\Pi_{D}(\theta_{t}-\alpha_{t}\mathfrak{g}_{t}(\theta _{t})),$ (13) $$

where we denote $\mathfrak{g}_{t}(\theta)\coloneqq\mathfrak{g}(\theta;O_{t})$, $O_{t}=(s_{t},a_{t},r_{t},s_{t+1},a_{t+1})$, and $\Pi_{D}$ is the projection operator onto the Euclidean ball $B_{D}^{d_{1}}\coloneqq\{\theta\in\mathbb{R}^{d_{1}}:\|\theta\|\leq D\}$.
Similar to the population measure update, the projection is commonly used with LFA to ensure that the value function parameter remains in a bounded region [, ], which is automatically satisfied in finite MFGs Appendix H.
Specifically, we need $D\geq\|\theta_{*}\|$, where $\theta_{*}$ is the MFE value function parameter.

Combining Equations 11 and 13 gives the update rule for the unified parameter:

$$ $\displaystyle\xi_{t+1}=\Pi(\xi_{t}-\alpha_{t}\mathfrak{g}_{t}(\xi_{t})) \coloneqq\Pi_{B^{d_{1}}_{D}\times\Delta^{d_{2}}}\left(\xi_{t}-\alpha_{t}\left( ({G}_{\phi}(O_{t})\oplus{G}_{\psi})\xi_{t}-(\varphi\oplus{\psi})(O_{t})\right) \right).$ (14) $$

Equation 14 updates the action-value function and population measure using the same online observations with the same learning rate.
Replacing Algorithm 1 in Algorithm 1 with Equation 14 gives SemiSGD with LFA.

The next section shows that SemiSGD converges to a zero point of the mean-path semi-gradient $\bar{\mathfrak{g}}_{\xi}(\xi)\coloneqq\mathbb{E}_{O\sim\mu_{\xi}}\mathfrak{g}(
\xi;O)$,
where $\mu_{\xi}$ is the observation distribution induced by the parameter $\xi$.
The next proposition states that this point is an MFE, thus validating the derivation in this section.

###### Proposition 2 (MFE as a stationary point) .

$\xi$ is an MFE if and only if $\bar{\mathfrak{g}}_{\xi}(\xi)=0$.

We prove an extended version of Proposition 2 in Appendix D, showing that the stationary point actually corresponds to a *projected* MFE for general (non-linear) MFGs.

###### Remark 1 (Operation complexity) .

###### Proposition 2 (MFE as a stationary point) .

### 5 Sample complexity analysis

This section provides a finite-sample analysis of the convergence of SemiSGD with LFA, thus covering finite MFGs as a special case.
We denote $P_{M}(s^{\prime}\,|\,s,a)\coloneqq P(s^{\prime}\,|\,s,a,M)$ and $r_{M}(s,a)\coloneqq r(s,a,M)$ for short.
For transition kernels, we define the total variation operator norm $\|P\|_{\mathrm{TV}}\coloneqq\sup_{p\in\mathcal{M}(\mathcal{S}\times\mathcal{A}
),\|p\|_{\mathrm{TV}\leq 1}}\|\int P(\cdot\,|\,s,a)p(\mathrm{d}s,\mathrm{d}a)
\|_{\mathrm{TV}}.$
We now state the assumptions for the analysis.

###### Assumption 1 (Lipschitz MDP) .

The transition kernel and reward function are Lipschitz continuous w.r.t. the population measure.
That is, there exists positive constants $L_{P}$ and $L_{r}$ such that for any state-action pair $(s,a)$ and population measures $M_{1},M_{2}$, we have

$$ $\|P_{M_{1}}-P_{M_{2}}\|_{\mathrm{TV}}\leq(L_{P}/\sqrt{d_{2}})\|M_{1}-M_{2}\|_{ \mathrm{TV}},\qquad\|r_{M_{1}}-r_{M_{2}}\|_{\infty}\leq(L_{r}/\sqrt{d_{2}})\|M _{1}-M_{2}\|_{\mathrm{TV}}.$ (15) $$

###### Assumption 2 (Lipschitz policy operator) .

There exists a constant $L_{\pi}$ such that for any state $s$ and value functions $Q_{1},Q_{2}$, we have $\|\Gamma_{\pi}(Q_{1})(\cdot\,|\,s)-\Gamma_{\pi}(Q_{2})(\cdot\,|\,s)\|_{\mathrm
{TV}}\leq L_{\pi}\|Q_{1}-Q_{2}\|_{\infty}.$

###### Assumption 3 (Uniform ergodicity) .

The MDP is uniformly ergodic for any parameter any value function $Q$ and population measure $M$. That is, there exists constants $m\geq 1$, $\rho\in(0,1)$, and $\mu_{(Q,M)}$, such that for any initial distribution $M_{0}\in\mathcal{D}(\mathcal{S})$, it holds that $\|\mu_{(Q,M)}-\mathcal{P}_{(Q,M)}^{t}M_{0}\|_{\mathrm{TV}}\leq m\rho^{t}.$

Assumptions 1 and 2 are regularity conditions ensuring learning dynamics’ smoothness, i.e., the updates are not too drastic between iterations.
Assumption 3 is a standard assumption in for online methods with Markovian sampling [, , , ], ensuring the environment is relatively easy to explore.
We can now bound the mean squared error of SemiSGD recursively, which directly gives several finite-time error bounds.

###### Theorem 1 (One-step progress) .

Let $\xi_{*}=(\theta_{*};\eta_{*})$ be an MFE parameter.
Let $\{\xi_{t}=(\theta_{t};\eta_{t})\}$ be a sequence of parameters generated by SemiSGD.
Then, under Assumptions 1, 2 and 3, there exists $w\in(0,1/2]$ such that if the Lipschitz constants are small enough ($3\sigma\max\{L_{P},L_{\pi}\}H+L_{r}\leq 2w$, see Lemma 9 in Section F.1),
we have

$$ $\mathbb{E}\!\left\|\xi_{t+1}\!\!-\!\xi_{*}\right\|^{2}\!\!\leq\!\!(1-\alpha_{t }w)\mathbb{E}\|\xi_{t}-\xi_{*}\|^{2}+H^{2}\cdot O(\alpha_{t}^{2}\log\alpha^{-1 }_{t})+\frac{\max\{L_{P},\!L_{\pi},\!L_{r}\}^{2}\!H^{4}}{w}\cdot O(\alpha_{t}^ {3}\log^{4}\alpha_{t}^{-1}),$ (16) $$

where $H\coloneqq(1+\gamma)D+R+2F$ can be seen as the scale of the problem.

###### Corollary 1 (Constant step-size) .

Given a constant step-size $\alpha_{t}\equiv\alpha_{0}$, we have

$$ $\mathbb{E}\|\xi_{T}-\xi_{*}\|^{2}\leq e^{-\alpha_{0}wT}\|\xi_{0}-\xi_{*}\|^{2} +w^{-1}H^{2}\cdot O(\alpha_{0}\log\alpha_{0}^{-1}).$ (17) $$

Let $\alpha_{0}=(\log T)/(wT)$. Then Corollary 1 states
$\mathbb{E}\|\xi_{T}-\xi_{*}\|^{2}=O\left((H^{2}\log^{2}T)/(w^{2}T)\right)$,
giving an $O(\epsilon^{-2}\log^{2}\epsilon^{-1})$ sample complexity for an $\epsilon$-MFE ($\mathbb{E}\|\xi_{T}-\xi_{*}\|\leq\epsilon$).
A linearly decaying step-size sequence improves the logarithmic factor, giving a convergence rate of $O\left((H^{2}\log T)/(w^{2}T)\right)$ (see Corollary 3).
Reducing to finite MFGs, the complexity becomes $\widetilde{O}\left(\frac{SAR^{2}}{\lambda^{2}(1-\gamma)^{4}T}\right)$ (see Corollary 4).

###### Remark 2 (Small Lipschitz constants) .

SemiSGD follows a non-stationary Markov chain with its kernel determined by the value function and population measure, which are updated on the go.
When the kernel is sensitive to the parameter update (large Lipschitz constants), the Markov chain is *rapidly changing* [], whose convergence is not well understood.
As a result, for theoretical analysis, we require the Lipschitz constants to be small (see Lemma 9), so the descent direction (attraction property of MFE) dominants the non-stationarity.
Such requirements are not implemented in the experiments.

###### Assumption 1 (Lipschitz MDP) .

###### Assumption 2 (Lipschitz policy operator) .

###### Assumption 3 (Uniform ergodicity) .

###### Theorem 1 (One-step progress) .

###### Corollary 1 (Constant step-size) .

###### Remark 2 (Small Lipschitz constants) .

### 6 Approximation error for non-linear MFGs

SemiSGD with PA-LFA applies to non-linear MFGs as long as the feature map $\phi$ and measure basis $\psi$ are specified.
However, for non-linear MFGs, it remains unclear:
1) What point does SemiSGD with PA-LFA converge to?
2) What factors characterize the approximation error caused by PA-LFA?

Actually, the zero point of the mean-path semi-gradient developed in Section 4 corresponds to a *projected* MFE
defined by $\left<\theta,\phi\right>=\Pi_{\phi}\mathcal{T}_{\xi}\left<\theta,\phi\right>$ and $\left<\eta,\psi\right>=\Pi_{\psi}\mathcal{P}_{\xi}\left<\eta,\psi\right>$,
where $\Pi_{\phi}$ and $\Pi_{\psi}$ are orthogonal projection operators onto the linear span of $\phi$ and $\psi$, respectively.
The formal definitions and proof of the statement are deferred to Appendix D.
For linear MFGs, as images of $\mathcal{T}$ and $\mathcal{P}$ are within the linear span of $\phi$ and $\psi$ (Appendix C), the projected MFE is the MFE itself (Proposition 2).

Let $\xi_{\diamond}=(\theta_{\diamond};\eta_{\diamond})$ be the convergence point of SemiSGD with PA-LFA (i.e., the stationary point w.r.t. the semi-gradient, or the projected MFE).
Let $(q_{*},\mu_{*})$ be the actual MFE.
We write $\mathcal{P}_{\diamond}\coloneqq\mathcal{P}_{\xi_{\diamond}}$ for short, and similarly for $\mathcal{P}_{*}$, $\mathcal{T}_{\diamond}$, and $\mathcal{T}_{*}$.
We have the following approximation error characterization.

###### Theorem 2 (Approximation error) .

Assumption 3 indicates the existence of $k\in\mathbb{N}$ such that

$$ $\displaystyle\epsilon_{q}\coloneqq\|q_{*}-\left<\phi,\theta_{\diamond}\right> \|_{\infty}\leq$ $\displaystyle\left(\gamma+\frac{\gamma\sigma RL_{\pi}}{1-\gamma}\right) \epsilon_{q}+\left(L_{r}+\frac{\gamma\sigma RL_{P}}{(1-\gamma)\sqrt{d_{2}}} \right)\epsilon_{\mu}+\left\|q_{*}-\Pi_{\phi}q_{*}\right\|_{\infty}$ (18) $\displaystyle\epsilon_{\mu}\coloneqq\|\mu_{*}-\left<\psi,\eta_{\diamond}\right >\|_{\mathrm{TV}}\leq$ $\displaystyle\left(\rho^{k}+\frac{kL_{P}}{\sqrt{d_{2}}}\right)\epsilon_{\mu}+ kL_{\pi}\epsilon_{q}+k\left\|\mathcal{P}_{*}-\Pi_{\psi}\mathcal{P}_{*}\right\| _{\mathrm{TV}}.$ (19) $$

If the Lipschitz constants are small enough: $2L_{\pi}(\gamma\sigma R+(1-\gamma)k)\leq(1-\gamma)^{2}$ and $2(L_{P}(\gamma\sigma R+(1-\gamma)k)+L_{r}(1-\gamma)\sqrt{d_{2}})\leq(1+\rho-2
\rho^{k})(1-\gamma)\sqrt{d_{2}}$, then

$$ $\epsilon_{q}+\epsilon_{\mu}\leq\frac{2}{1-\min\left\{\gamma,\rho\right\}}\left (\epsilon_{\phi}+k\epsilon_{\psi}\right),$ (20) $$

where $\epsilon_{\phi}\coloneqq\|q_{*}-\Pi_{\phi}q_{*}\|_{\infty}$ and $\epsilon_{\psi}\coloneqq\|\mathcal{P}_{*}-\Pi_{\psi}\mathcal{P}_{*}\|_{\mathrm
{TV}}$ are *inherent* approximation error induced by the projection onto the linear span of basis $\phi$ and $\psi$, which is independent of the algorithm.
Moreover, if $\|\mathcal{P}_{*}-\mathcal{P}_{*}^{\infty}\|_{\mathrm{TV}}=\rho$ (e.g., $\mathcal{P}_{*}$ induces a reversible Markov chain), $k=1$.

The approximation error in Theorem 2 scales with the inherent approximation error determined by the chosen basis, which is consistent with the approximation error of LFA for value function [].

###### Theorem 2 (Approximation error) .

### 7 Numerical experiments

We conduct experiments on two MFG examples in autonomous driving scenarios []: speed control on a ring road and routing game on a network. We equip FPI with entropy regularization (ER), fictitious play (FP), and online mirror descent (OMD). The following parameters are shared for all methods and two experiments: total number of samples $T=10^{5}$, with the number of outer loops being $K=200$ for FPI-type methods, constant step-size $\alpha=10^{-3}$, policy operator $\Gamma_{\pi}=\operatorname{softmax}$ with a large inverse temperature (near-greedy). The convergence performance include exploitability and mean squared error (MSE). We compare distributions with the reference MFE distribution, which is calculated by *model-based FPI with FP*. All results are averaged over $10$ independent runs with random initialization. $95\%$ confidence regions are plotted. Please refer to Appendix A for the full setup of the experiments. In the main body, we present only the results of the speed control problem due to page limit. The details of the other game (routing) can be found in Appendix A.

###### Speed control on a ring road.

We consider a speed control game on a ring road $\mathcal{S}\cong\mathbb{S}^{1}$. The cost function $r(s,a,\mu)=-\left(b(s)+\left(1-\mu(s)/\mu_{\mathrm{jam}}\right)/2-a/a_{\mathrm
{max}}\right)^{2}\Delta s/2$,
is a variant of the Lighthill-Whitham-Richards function and a bonus term $b$ that encodes the location preference, $\mu_{\mathrm{jam}}$ is the jam density, and $a_{\mathrm{max}}$ is the maximum speed.
ER is implemented with a low inverse temperature of $10^{4}$, while all other methods use an inverse temperature of $10^{9}$, guaranteeing the policies are near-greedy.
The MSE, exploitabity [], and learned population comparison are reported in Figure 1(a), Figure 2, and Figure 3, respectively.
The MSE comparison demonstrates that SemiSGD
1) is more efficient, with a much faster convergence,
2) is more stable without any stabilization techniques,
3) and is more accurate, with a lower MSE.
For the exploitability, SemiSGD seems to oscillate and be outperformed by ER. However, exploitability is an operator determined by the underlying MDP, which may not be smooth and does not directly reflecting the learning dynamics. The learned population distribution shows that FPI with ER is still far from the reference MFE.
We also conduct a comparison between LFA and discretization, which shows that LFA is more accurate than discretization for small dimensions (Table 2; please refer to Appendix A for the full report).

### 8 Conclusion and discussion

The proposed SemiSGD makes learning MFGs more accessible and practical in real-world applications, including but not limited to online autonomous vehicle driving, real-time pedestrian or crowd dynamics, internet packet routing, social opinion dynamics, and epidemiology. Our methodology is generalizable and can be extended to : (finite) population games, graphon games, policy optimization for MFGs, and evolutionary MDPs.

An open challenge is to relax or remove the small Lipschitz constants assumption in our theoretical analysis. As demonstrated in the experiments, SemiSGD is stable without requiring the Lipschitz constants to be small.
Bridging this gap requires a deeper understanding of the rapidly changing Markov chains (RCMCs) induced by SemiSGD.
It is also interesting to establish similar theoretical guarantees for *monotone* MFGs, a parallel condition to the smoothness assumption.

## Appendix

### Appendix A Additional experiments

#### A.1 General setup and remark

###### General setup.

We focus on two primary metrics: the mean squared error (MSE) of the population distribution and the exploitability of the policy.

###### Reference MFE and mean squared error.

We compare our results with the reference MFE distribution $M_{\mathrm{ref}}$, which is calculated by model-based FPI with FP.
The model-based FPI consists of a value iteration [] for the best responses and direct computation of the induced population measure using the transition operator.
Then, the (population measure) mean squared error (MSE) is calculated as

$$ $\operatorname{MSE}(M):=\left\|M-M_{\mathrm{ref}}\right\|_{2}^{2}=\sum_{s\in \mathcal{S}}\left(M(s)-M_{\mathrm{ref}}(s)\right)^{2}.$ (21) $$

Note that this is equivalent to the 1-Wasserstein distance between the two measures for finite state spaces.

###### Exploitablity

Perrin et al. [] defines the exploitability of a policy as follows:

$$ $\operatorname{exploitability}(\pi)\coloneqq\max_{\pi^{\prime}}\mathbb{E}_{s \sim\mu_{\pi}}V(s;\pi^{\prime},\mu_{\pi})-\mathbb{E}_{s\sim\mu_{\pi}}V(s;\pi, \mu_{\pi}),$ (22) $$

where $V(s;\pi,\mu)$ is the value function determined by policy $\pi$ and population distribution $\mu$.
The exploitability results in our experiments are calculated using model-based value iteration.
We remark that although exploitability is a useful metric to evaluate the performance of the learned policy, it is an operator determined by the underlying MDP, which may not be smooth and does not directly reflect the learning dynamics.

#### A.2 Speed control on a ring road.

We consider a speed control game of on a ring road, i.e., the unit circle $\SS^{1}\cong[0,1)$.
At location $s\in\SS^{1}$, the representative vehicle selects a speed $a$, and then moves to the next location following transition $s^{\prime}=s+a\Delta t\pmod{1}$,
where $\Delta t$ is the time interval between two consecutive decisions.
Without loss of generality, we assume that the speed is bounded by $1$, i.e., the speed space is also $[0,1)$.
Then we discretize both the location space and the speed space using a granularity of $\Delta s=\Delta a=0.02$.
Thus, both our discretized state (location) space and action (speed) space can be represented by $\mathcal{S}=\mathcal{A}=\{0,0.02,\ldots,0.98\}\cong[50]$.
By the Courant-Friedrichs-Lewy condition, we choose the time interval to be $\Delta t=0.02\leq\Delta s/\max a$.
The objective of a vehicle is to maintain some desired speed while avoiding collisions with other vehicles. Thus, it needs to reduce the speed in areas with high population density.
A classic cost function for this goal is the Lighthill-Whitham-Richards function:

$$ $r^{(\mathrm{LWR})}(s,a,\mu)=-\frac{1}{2}\left(\left(1-\frac{\mu(s)}{\mu_{ \mathrm{jam}}}\right)-\frac{a}{a_{\mathrm{max}}}\right)^{2}\Delta s,$ (23) $$

where $\mu_{\mathrm{jam}}$ is the jam density, and $a_{\mathrm{max}}$ is the maximum speed.
However, in an infinite horizon game, this cost function induces a *trivial* MFNE, where the equilibrium policy and population are both constant across the state space.
Therefore, we introduce a stimulus term $b$ that varies across different locations:

$$ $r(s,a,\mu)=-\frac{1}{2}\left(b(s)+\frac{1}{2}\left(1-\frac{\mu(s)}{\mu_{ \mathrm{jam}}}\right)-\frac{a}{a_{\mathrm{max}}}\right)^{2}\Delta s,$ (24) $$

where the factor of one-half before the population distribution term is included to account for the presence of the new stimulus term.
This new cost function makes the MFNE more complex and corresponds to real-world situations where vehicles may have distinct desired speeds at different locations due to environmental variations.
Specifically, we choose the stimulus term as $b(s)=0.2(\sin(4\pi s)+1)$, and set $\mu_{\mathrm{jam}}=3/S$ and $a_{\mathrm{max}}=1$.
The discount factor is set as $\gamma=0.98$.

The convergence of model-based FPI with FP is shown in Figure 4, arguing the near-optimality of the reference MFE.

Figure: Figure 4: Exploitablity of model-based FPI with FP.
Refer to caption: rr/expl_opt.png

##### A.2.1 Comparison of PA-LFA and grid discretization

To demonstrate the effectiveness of PA-LFA, we compare it with grid discretization.
The reference MFE is calculated using a grid discretization with a granularity of $1/50$.
We only apply LFA to the population measure estimate, as this is our main focus.
The measure basis is chosen as

$$ $\psi_{i}(s)=\mathcal{N}(0)-\mathcal{N}(\tan((s-s_{i})\pi)),$ (25) $$

where $\mathcal{N}$ is the normal distribution, and $s_{i}$ is the center of the basis function.
We run SemiSGD $d_{2}$ states (grid discretization) and $d_{2}$ basis functions (PA-LFA) for $T=10^{4}$ steps.
Other parameters are the same as the general setup.
The final MSE (accuracy) comparison is reported in Table 2.
The MSE curves and learned population distributions are shown in Figure 6 and Figure 5, respectively.
As we can see, when $d_{2}$ is small, PA-LFA generalizes better than grid discretization, as the representation capacity of $\operatorname{span}\psi$ is larger than the grid discretization.
However, when $d_{2}$ is large, i.e., the discretization is fine enough, grid discretization performs better than PA-LFA.
This is because the PA-LFA approximation error is larger than the discretization error when the discretization is fine enough.

Figure: (a) $d_{2}=2$.
Refer to caption: rr/lfa_2.png

Figure: (a) $d_{2}=2$.
Refer to caption: rr/lfa_2_mse.png

#### A.3 Routing game on a network.

We consider a routing game on the Sioux Falls network,(^3^33The topology of the network is available at [https://github.com/bstabler/TransportationNetworks](https://github.com/bstabler/TransportationNetworks).) a graph with $24$ nodes and $74$ directed edges. We designate node $1$ as the starting point and node $20$ as the destination.
To construct an infinite-horizon game, we add a *restart* edge $e_{75}$ from the destination back to the starting point.
On each edge, a vehicle selects its next edge to travel to. We consider a deterministic environment, meaning that the vehicle will follow the chosen edge without any randomness.
Therefore, both the state space and the action space can be represented by the edge set, i.e., $\mathcal{S}=\mathcal{A}=\{e_{1},\ldots,e_{75}\}\cong[75]$, where $e_{75}$ is the restart edge.
It is worth noting that a vehicle can only select from the outgoing edges of its current location as its next edge.

The objective of a vehicle is to reach the destination as fast as possible. Due to congestion, a vehicle spends a longer time on an edge with higher population distribution. Specifically, the cost (time) on a non-restart edge is $r^{(\text{cong.})}(s,a,\mu)=-c_{1}\mu(s)^{2}\mathbbm{1}\{s\neq e_{75}\}$,
where $c_{1}$ is a cost constant. To drive the vehicle to the destination, we impose a reward at the restart edge: $r^{(\text{term.})}(s,a,\mu)=c_{2}\mathbbm{1}\{s=e_{75}\}$. Together, we get the cost function:

$$ $r(s,a,\mu)=\underbrace{-c_{1}\mu(s)^{2}\mathbbm{1}\{s\neq e_{75}\}}_{\text{ congestion cost}}+\underbrace{c_{2}\mathbbm{1}\{s=e_{75}\}}_{\text{terminal reward}}.$ (26) $$

We set $c_{1}=10^{5}$ and $c_{2}=10$.
The other algorithmic parameters are chosen as follows: the discount factor $\gamma=0.8$, the initial state is uniformly sampled, the initial value function is set as all-zero, the initial population is randomly generated.

Similarly, ER is implemented with a low inverse temperature of $10^{-3}$, while all other methods use an inverse temperature of $10^{2}$.
The performance comparison is reported in Figure 7.
Notably, all methods except FPI with ER have an oscillatory behavior in the exploitability around the same value.
This is due to the highly non-smooth nature of the underlying environment and the choice of near-greedy policies.
Applying ER to all methods results in similar performance, as shown in Figure 8.

Figure: (a) MSE.
Refer to caption: graph/err.png

Figure: (a) MSE under ER.
Refer to caption: graph/err_soft.png

### Appendix B Notation

Table 3 provides a summary of the symbol notation used in this paper.
We introduce some supplementary notation to assist analysis.

**Table 3: Notation**
| Notation | Definition |
| --- | --- |
| $\mathcal{D}(\mathcal{X}),\mathcal{M}(\mathcal{X})$ | Space of probability and signed measures on space $\mathcal{X}$ |
| $\Delta$ | Probability simplex |
| $\delta$ | Dirac delta measure |
| $a,\mathcal{A}$ | Action and action space |
| $s,\mathcal{S}$ | State and state space |
| $r,R$ | Reward function and its supremum norm bound |
| $\gamma$ | Discount factor |
| $P,\mathcal{P}$ | Transition kernel and operator |
| $\mathcal{T}$ | Bellman operator |
| $Q,q$ | Action-value function |
| $M,\mu$ | Population measure |
| $\pi,\Gamma_{\pi}$ | Policy and policy operator |
| $\phi$ | State-action feature map |
| $\psi$ | State measure basis |
| $F$ | Norm bound of $\psi$ |
| $d$ | Feature/measure space dimension |
| $\nu$ | Linear reward function parameter |
| $\omega,\Omega$ | Linear transition kernel parameter |
| $\eta$ | Population measure parameter |
| $\theta$ | Value function parameter |
| $\xi,\Xi$ | Concatenated parameter and its parameter space |
| $\Pi$ | Projection operator |
| $D$ | Value function projection bound |
| $H$ | Problem scale ($H=(1+\gamma)D+R+2F$) |
| $\mathfrak{g}$ | Semi-gradient |
| ${G}_{\phi},\varphi$ | Temporal difference operator |
| $G$ | Gram matrix |
| $L$ | Lipschitz constant |
| $w$ | Contraction constant |
| $\alpha$ | Step-size |
| $m,\rho,\sigma,k$ | Ergodicity constants |
| $\tau$ | Backtracking period |

###### Concatenation and direct sum.

We consider finite-dimensional Euclidean spaces as the parameter spaces. Thus, for any $x\in\mathbb{R}^{d_{1}}$ and $y\in\mathbb{R}^{d_{2}}$, we denote their concatenation as $(x;y)\in\mathbb{R}^{d_{1}+d_{2}}$.
We sometimes write it as the general direct sum between two vectors $x\oplus y\in\mathbb{R}^{d_{1}}\oplus\mathbb{R}^{d_{2}}\cong\mathbb{R}^{d_{1}+d
_{2}}$.
For matrices and operators, we have $(A\oplus B)(x\oplus y)=Ax\oplus By$.
This notation is especially useful for handling the unified parameter $\xi=\theta\oplus\eta$ (see, e.g., Lemma 7).
Additionally, we use $\Xi$ to denote the unified parameter space $\Xi\coloneqq\mathbb{R}^{d_{1}}\oplus\mathbb{R}^{d_{2}}$.

###### Unprojected parameters.

We denote the unprojected parameters as $\breve{\xi}$, $\breve{\theta}$, and $\breve{\eta}$. That is,

$$ $\xi_{t+1}=\Pi(\breve{\xi})=\Pi(\xi_{t}-\alpha_{t}\mathfrak{g}_{t}(\xi_{t})).$ (27) $$

For any parameter $\xi_{*}$ in the projected region, we have $\|\xi_{*}-\Pi(\breve{\xi})\|\leq\|\xi_{*}-\breve{\xi}\|$.

###### Steady distributions.

We denote $\mu_{\xi}\in\mathcal{D}(\mathcal{S})$ as the steady state distribution induced by parameter $\xi=(\theta;\eta)$, i.e., by policy $\Gamma_{\pi}(\theta)$ and transition kernel and reward function determined by population measure parameter $\eta$.
$\mu_{\xi}$ is the marginal distribution of the following two steady distributions:

$$ $\mu_{\xi}^{\dagger}(s,a)\coloneqq\mu_{\xi}(s)\pi_{\theta}(a\,|\,s),\quad\mu_{ \xi}^{\ddagger}(s,a,s^{\prime},a^{\prime})\coloneqq\mu_{\xi}(s)\pi_{\theta}(a \,|\,s)P(s^{\prime}\,|\,s,a,\eta)\pi_{\theta}(a^{\prime}\,|\,s^{\prime}).$ (28) $$

We write $\mathbb{E}_{\xi}$ as the expectation over the steady distribution induced by $\xi$; it should be clear from the context which steady distribution is used.

###### Semi-gradients and temporal difference operators.

Here we review the definition of semi-gradients. With a slight abuse of notation, we use a single operator $\mathfrak{g}$ to return semi-gradients for both the action-value function and population measure.
It should be clear from the argument of $\mathfrak{g}$ which parameter the semi-gradient is for.
Specifically, with a sample tuple $O=(s,a,r,s^{\prime},a^{\prime})$, we have

$$ $\mathfrak{g}(\xi;O)=\left(\mathfrak{g}(\theta;O);\mathfrak{g}(\eta;O)\right)= \left({G}_{\phi}(O)\theta-\varphi(O);{G}_{\psi}\eta-\psi(s^{\prime})\right),$ (29) $$

where ${G}_{\psi}$ is the gram matrix of the measure basis $\psi$, and ${G}_{\phi}$ and $\varphi$ are the temporal difference operators defined as

$$ ${G}_{\phi}(O)=\phi(s,a)(\phi(s,a)-\gamma\phi(s^{\prime},a^{\prime}))^{T},\quad \varphi(O)=\phi(s,a)r.$ (30) $$

Notably, ${G}_{\psi}$ is a constant matrix, and ${G}_{\phi}$ is a Gram-like matrix that depends on the sample tuple.
Therefore, we sometimes drop the subscript $\phi$ in ${G}_{\phi}$ and use other subscripts to indicate its dependence on the sample tuple. It should be clear that any $G$ with a subscript other than $\psi$ refers to ${G}_{\phi}$.
When the sample tuple is an online observation at time step $t$, i.e., $O_{t}=(s_{t},a_{t},r_{t},s_{t+1},a_{t+1})$, we use shorthand

$$ $\mathfrak{g}_{t}(\cdot)=\mathfrak{g}(\cdot;O_{t}),\quad G_{t}={G}_{\phi}(O_{t} ),\quad\varphi_{t}=\varphi(O_{t}),\quad\psi_{t}=\psi(O_{t}).$ (31) $$

Backtracking is an analysis technique introduced to tackle *rapidly changing Markov chains* [, , ]. It considers a virtual stationary Markov chain by backtracking a period $\tau$, fixing the parameter $\xi_{t-\tau}$, and then sampling the Markovian observations with the fixed parameter.
By the ergodicity of stationary Markov chains (Assumption 3), the virtual trajectory rapidly converges to the stationary distribution induced by $\xi_{t-\tau}$.
We denote $\widetilde{O}_{t}$ as the virtual observation tuple on this virtual trajectory at time $t$.
When we consider the semi-gradients on this virtual trajectory, we write out its dependence on $\widetilde{O}_{t}$ explicitly:

$$ $\mathfrak{g}_{t-\tau}(\cdot;\widetilde{O}_{t}),\quad G_{t-\tau}(\widetilde{O}_ {t}),\quad\varphi_{t-\tau}(\widetilde{O}_{t}),\quad\psi_{t-\tau}(\widetilde{O} _{t})$ (32) $$

with the subscript indicating the backtracking period $\tau$.

*Mean-path* semi-gradients are the expectation of semi-gradients over a steady distribution induced by a parameter:

$$ $\bar{\mathfrak{g}}_{\xi}=\mathbb{E}_{\xi}\mathfrak{g}(\xi;O),$ (33) $$

where the subscript $\xi$ indicates that the observation tuple $O$ follows the steady distribution induced by $\xi=(\theta;\eta)$.
More explicitly, the states follows the steady distribution corresponding to transition kernel $P(\cdot\,|\,\cdot,\cdot,\eta)$, and the actions follow policy $\Gamma_{\pi}(\theta)$, and the rewards are generated by $r(\cdot,\cdot,\eta)$.
Similarly, we have

$$ $\bar{G}_{\xi}=\mathbb{E}_{\xi}{G}_{\phi}(O),\quad\bar{\varphi}_{\xi}=\mathbb{E }_{\xi}\varphi(O),\quad\bar{\psi}_{\xi}=\mathbb{E}_{\xi}\psi(s^{\prime}).$ (34) $$

When the parameter has a subscript $\xi_{\circ}$, we also use shorthand

$$ $\bar{\mathfrak{g}}_{\circ}=\bar{\mathfrak{g}}_{\xi_{\circ}},\quad\bar{A}_{ \circ}=\bar{A}_{\xi_{\circ}},\quad\bar{b}_{\circ}=\bar{b}_{\xi_{\circ}},\quad \bar{\psi}_{\circ}=\bar{\psi}_{\xi_{\circ}}.$ (35) $$

For example, $\bar{\mathfrak{g}}_{t}=\bar{\mathfrak{g}}_{\xi_{t}}$ and $\bar{\mathfrak{g}}_{*}=\bar{\mathfrak{g}}_{\xi_{*}}$.

### Appendix C Proof of Proposition 1

###### Proof.

By the Bellman equation, we have

$$ $\displaystyle Q^{\pi}_{M}(s,a)=r(s,a,M)+\gamma\int_{\mathcal{S}\times\mathcal{ A}}Q^{\pi}_{M}(s^{\prime},a^{\prime})\pi(a^{\prime}\,|\,s^{\prime})P(s^{\prime }\,|\,s,a,M)\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}$ (36) $\displaystyle=$ $\displaystyle\left<\phi(s,a),{\nu}_{M}\right>+\gamma\int_{\mathcal{S}\times \mathcal{A}}Q^{\pi}_{M}(s^{\prime},a^{\prime})\pi(a^{\prime}\,|\,s^{\prime}) \left<\phi(s,a),\Omega_{M}\psi(s^{\prime})\right>\mathrm{d}s^{\prime}\mathrm{d }a^{\prime}$ $\displaystyle=$ $\displaystyle\left<\phi(s,a),\underbrace{{\nu}_{M}+\gamma\int_{\mathcal{S} \times\mathcal{A}}Q^{\pi}_{M}(s^{\prime},a^{\prime})\pi(a^{\prime}\,|\,s^{ \prime})\Omega_{M}\psi(s^{\prime})\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}}_{ \theta}\right>.$ $$

By the transition equation, we have

$$ $\displaystyle\mu^{\pi}_{M}$ $\displaystyle(s^{\prime})=\int_{\mathcal{S}\times\mathcal{A}}P(s^{\prime}\,|\, s,a,M)\pi(a\,|\,s)\mu^{\pi}_{M}(s)\mathrm{d}s\mathrm{d}a=\int_{\mathcal{S} \times\mathcal{A}}\left<\phi(s,a),\Omega_{M}\psi(s^{\prime})\right>\pi(a\,|\,s )\mu^{\pi}_{M}(s)\mathrm{d}s\mathrm{d}a$ (37) $\displaystyle=$ $\displaystyle\int_{\mathcal{S}\times\mathcal{A}}\left<\psi(s^{\prime}),\Omega_ {M}^{T}\phi(s,a)\right>\pi(a\,|\,s)\mu^{\pi}_{M}(s)\mathrm{d}s\mathrm{d}a= \left<\psi(s^{\prime}),\underbrace{\int_{\mathcal{S}\times\mathcal{A}}\Omega_{ M}^{T}\phi(s,a)\pi(a\,|\,s)\mu_{M}^{\pi}(s)\mathrm{d}s\mathrm{d}a}_{\eta} \right>.$ $$

∎

###### Proof.

### Appendix D Projected MFE and stationary point

We call a value function-population measure pair a *stationary point* if the mean-path semi-gradient evaluated at this point is zero.
Sections 5 and F show that SemiSGD converges to a stationary point.
Ideally, we want the stationary point to be an MFE (Definition 1).
Propositions 1 and C show that for linear MFGs, images of $\mathcal{T}$ and $\mathcal{P}$ are within the linear span of $\phi$ and $\psi$, indicating the linear structure of the MFE.
However, this does not hold for general (non-linear) MFGs, hinting the discrepancy between the stationary point and the MFE for non-linear MFGs.

We define the *projected MFE* using the Bellman operator and transition operator composed with the projection operators.

###### Definition 3 (Projected MFE) .

We say $\xi=(\theta;\eta)$ constitutes a projected MFE if

$$ $\left<\phi,\theta\right>=\Pi_{\phi}\mathcal{T}_{\xi}\left<\phi,\theta\right>, \quad\left<\psi,\eta\right>=\Pi_{\psi}\mathcal{P}_{\xi}\left<\psi,\eta\right>,$ (38) $$

where $\Pi_{\phi}$ and $\Pi_{\psi}$ are orthogonal projection operators onto the linear span of $\phi$ and $\psi$, respectively.

It should be noted that the projection operators are determined by the inner product structure of the function spaces.
They can be explicitly expressed as

$$ $\Pi_{\phi}=\phi^{T}(\langle\phi,\phi^{T}\rangle_{\circ})^{-1}\langle\phi,\cdot \rangle_{\circ},$ (39) $$

where $\phi$ is the function basis and $\left<\cdot,\cdot\right>_{\circ}$ is the chosen inner product.
Specifically, we choose the $L_{2}$ inner product on $\mathcal{M}(\mathcal{S})$, giving

$$ $\Pi_{\psi}=\ \psi^{T}{G}_{\psi}^{-1}\langle\psi,\cdot\rangle_{L_{2}}.$ (40) $$

For the projection acting on $\mathcal{T}_{\xi}$, we choose the inner product induced by the steady distribution $\mu_{\xi}$, i.e., $\langle f,g\rangle_{\mu_{\xi}}=\int f(o)g(o)\mu^{\ddagger}_{\xi}(\mathrm{d}o)$.
Then, we have

$$ $\Pi_{\phi}\mathcal{T}_{\xi}=\phi^{T}\widehat{G}_{\xi}^{-1}\langle\phi,\mathcal {T}_{\xi}\cdot\rangle_{\mu_{\xi}},$ (41) $$

where $\widehat{G}_{\xi}$ is the Gram matrix of $\phi$ w.r.t. the inner product $\langle\cdot,\cdot\rangle_{\mu_{\xi}}$.
Note that $\widehat{G}_{\xi}$ is different from the TD operator ${G}_{\phi}$ or $G_{\xi}$ defined in Appendix B, which is only Gram-like.

We are now ready to prove a generalized version of Proposition 2. Recall that for linear MFGs, the projected MFE is the MFE itself.

###### Proposition 3 (Projected MFE as a stationary point) .

$\xi$ is a projected MFE if and only if $\bar{\mathfrak{g}}_{\xi}(\xi)=0$.

###### Proof.

For a parameter $\xi=(\theta;\eta)$, by the definition of mean-path semi-gradients, we have

$$ $\displaystyle\bar{\mathfrak{g}}_{\xi}(\theta)=$ $\displaystyle\mathbb{E}_{\xi}\left[\phi(s,a)\left(\phi^{T}(s,a)\theta-\gamma \phi^{T}(s^{\prime},a^{\prime})\theta-r(s,a,\eta)\right)\right],$ (42) $\displaystyle\bar{\mathfrak{g}}_{\xi}(\eta)=$ $\displaystyle\mathbb{E}_{\xi}\left[{G}_{\psi}\eta-\psi(s^{\prime})\right],$ (43) $$

where the observation tuple $(s,a,s^{\prime},a^{\prime})$ follows the steady distribution induced by $\xi$.
On the other hand, by the definition of the projection operators, we have

$$ $\left(\Pi_{\phi}\mathcal{T}_{\xi}-\operatorname{Id}\right)\left<\phi,\theta \right>=\phi^{T}\widehat{G}_{\xi}^{-1}\langle\phi,\mathcal{T}_{\xi}\left<\phi, \theta\right>\rangle_{\mu_{\xi}}-\phi^{T}\theta,$ (44) $$

where $\operatorname{Id}$ is the identity operator.
Suppose $\phi$ is linearly independent. Then, we get

$$ $\displaystyle\left(\Pi_{\phi}\mathcal{T}_{\xi}-\operatorname{Id}\right)\left< \phi,\theta\right>=0\iff$ $\displaystyle\widehat{G}_{\xi}^{-1}\langle\phi,\mathcal{T}_{\xi}\left<\phi, \theta\right>\rangle_{\mu_{\xi}}=\theta$ (45) $\displaystyle\iff$ $\displaystyle\langle\phi,\mathcal{T}_{\xi}\left<\phi,\theta\right>\rangle_{\mu _{\xi}}=\widehat{G}_{\xi}\theta$ (46) $\displaystyle\iff$ $\displaystyle\left<\phi,\mathbb{E}_{\xi}\left[r(\cdot,\cdot,\eta)+\gamma\phi^{ T}(s^{\prime},a^{\prime})\theta\right]\right>_{\mu_{\xi}}-\mathbb{E}_{\xi}[ \phi(s,a)\phi^{T}(s,a)]\theta=0$ (47) $\displaystyle\iff$ $\displaystyle\mathbb{E}_{\xi}\left[\phi(s,a)\left(r(s,a,\eta)+\gamma\phi^{T}(s ^{\prime},a^{\prime})\theta\right)\right]-\mathbb{E}_{\xi}[\phi(s,a)\phi^{T}(s ,a)\theta]=0$ (48) $\displaystyle\iff$ $\displaystyle\mathbb{E}_{\xi}\left[\phi(s,a)\left(\phi^{T}(s,a)\theta-\gamma \phi^{T}(s^{\prime},a^{\prime})\theta-r(s,a,\eta)\right)\right]=0$ (49) $\displaystyle\iff$ $\displaystyle\bar{\mathfrak{g}}_{\xi}(\theta)=0.$ (50) $$

Similarly, for the projected transition operator, we have

$$ $\displaystyle\left(\Pi_{\psi}\mathcal{P}_{\xi}-\operatorname{Id}\right)\left< \psi,\eta\right>=0\iff$ $\displaystyle\psi^{T}{G}_{\psi}^{-1}\langle\psi,\mathcal{P}_{\xi}\left<\psi, \eta\right>\rangle_{L_{2}}-\psi^{T}\eta=0$ (51) $\displaystyle\iff$ $\displaystyle\langle\psi,\mathcal{P}_{\xi}\left<\psi,\eta\right>\rangle_{L_{2} }={G}_{\psi}\eta$ (52) $\displaystyle\iff$ $\displaystyle\int_{\mathcal{S}^{2}\times\mathcal{A}}\psi(s^{\prime})P(s^{ \prime}\,|\,s,a,\eta)\pi_{\theta}(a\,|\,s)\psi^{T}(s)\eta\mathrm{d}s\mathrm{d} a\mathrm{d}s^{\prime}-{G}_{\psi}\eta=0$ (53) $\displaystyle\iff$ $\displaystyle\mathbb{E}_{\xi}[\psi(s^{\prime})]-{G}_{\phi}\eta=0$ (54) $\displaystyle\iff$ $\displaystyle\bar{\mathfrak{g}}_{\xi}(\eta)=0.$ (55) $$

Therefore, by Definition 3, $\xi$ is a projected MFE if and only if $\mathfrak{g}_{\xi}(\xi)=0$.
∎

###### Definition 3 (Projected MFE) .

###### Proposition 3 (Projected MFE as a stationary point) .

###### Proof.

### Appendix E Preliminary lemmas

We present some preliminary lemmas that are used throughout the analysis.

###### Lemma 2 (Norm relations) .

For any vectors $x,y$, we have

- •
$\|x\oplus y\|_{1}=\|x\|_{1}+\|y\|_{1},\quad\|x\oplus y\|_{1}^{2}\leq\|y\|_{1}^
{2}+\|x\|_{1}^{2}$.
- •
$\|x\oplus y\|_{2}\leq\|x\|_{2}+\|y\|_{2},\quad\|x\oplus y\|_{2}^{2}=\|y\|_{2}^
{2}+\|x\|_{2}^{2}$.
- •
$\|x\oplus y\|_{\infty}=\max\left\{\|x\|_{\infty},\|y\|_{\infty}\right\},\quad
\|x\oplus y\|_{\infty}^{2}\leq\|y\|_{\infty}^{2}+\|x\|_{\infty}^{2}$.
- •
$\|x\|_{1}\|y\|_{1}\leq\frac{1}{4}\|x\oplus y\|_{1}^{2}$.
- •
$\|x\|_{2}\|y\|_{2}\leq\frac{1}{2}\|x\oplus y\|_{2}^{2}$.
- •
$\|x\|_{\infty}\|y\|_{\infty}\leq\|x\oplus y\|_{\infty}^{2}$.

###### Proof.

All relations are basic facts of norms and can be easily verified.
∎

###### Lemma 3 (Gradient bounds) .

For any parameter $\xi=(\theta;\eta)$ and any observation tuple $O$, we have

$$ $\displaystyle\|\mathfrak{g}(\theta;O)\|\leq$ $\displaystyle(1+\gamma)\|\theta\|+R,$ (56) $\displaystyle\|\mathfrak{g}(\eta;O)\|\leq$ $\displaystyle F\|\eta\|+F.$ (57) $$

Moreover, suppose $\|\theta\|\leq D$ and $\|\eta\|_{1}=1$. Let $H\coloneqq(1+\gamma)D+R+2F$. Then, we have

$$ $\|\mathfrak{g}(\xi;O)\|\leq H.$ (58) $$

###### Proof.

By definition, we have

$$ $\displaystyle\|\mathfrak{g}(\theta;O)\|=$ $\displaystyle\|{G}_{\phi}(O)\theta-\varphi(O)\|\leq\|{G}_{\phi}(O)\|\|\theta\| +\|\varphi(O)\|$ (59) $\displaystyle\leq$ $\displaystyle\|\phi(s,a)\|\|\phi(s,a)-\gamma\phi(s^{\prime},a^{\prime})\|\| \theta\|+\|\phi(s,a)\||r(s,a,\eta)|$ (60) $\displaystyle\leq$ $\displaystyle(1+\gamma)\|\theta\|+R,$ (61) $$

where we use the fact that $\|\phi(s,a)\|\leq 1$.
Similarly, we have

$$ $\|\mathfrak{g}(\eta;O)\|\leq\|\mathfrak{g}(\eta;O)\|_{1}=\|{G}_{\psi}\eta-\psi (s^{\prime})\|_{1}\leq\|{G}_{\psi}\|_{\mathrm{op}}\|\eta\|_{1}+\|\psi(s^{ \prime})\|_{1}.$ (62) $$

The operator norm of ${G}_{\psi}$ satisfies

$$ $\|{G}_{\psi}\|_{\mathrm{op}}=\sup_{\|\eta\|_{1}=1}\|{G}_{\psi}\eta\|_{1}=\sup_ {\|\eta\|_{1}=1}\left\|\int_{\mathcal{S}}\psi(s)\psi^{T}(s)\eta\mathrm{d}s \right\|_{1}\leq\sup_{\|\eta\|_{1}=1}\int_{\mathcal{S}}\|\psi(s)\|_{1}\left< \psi,\eta\right>\mathrm{d}s\leq F,$ (63) $$

where the last inequality uses the norm bound of $\psi$ and the fact that $\left<\psi,\eta\right>$ is a probability measure.
Therefore, we get

$$ $\|\mathfrak{g}(\eta;O)\|\leq 2F.$ (64) $$

Then, Lemma 2 indicates that $\|\mathfrak{g}(\xi;O)\|\leq H$ given that $\|\theta\|\leq D$.
∎

To be more general, Assumptions 1 and 2 are stated in terms of the differences of population measures and value functions.
For the ease of presentation, we will develop our results in terms of the parameters, and state the more general results in terms of the differences of population measures and value functions without proof.
We need to first translate the Lipschitzness assumptions in terms of the parameters.
In the rest of the paper, we refer to the following lemma when we need to use the Lipschitzness assumptions in terms of the parameters.

###### Lemma 4 (Lipschitzness in parameters) .

Assumption 1 and Assumption 2 imply that for any two parameters $\xi_{1}=(\theta_{1};\eta_{1})$ and $\xi_{2}=(\theta_{2};\eta_{2})$, we have

$$ $\begin{gathered}\|P_{\eta_{1}}-P_{\eta_{2}}\|_{\mathrm{TV}}\leq L_{P}\|\eta_{1 }-\eta_{2}\|,\quad\|r_{\eta_{1}}-r_{\eta_{2}}\|_{\infty}\leq L_{r}\|\eta_{1}- \eta_{2}\|,\\ \|\pi_{\theta_{1}}(\cdot\,|\,s)-\pi_{\theta_{2}}(\cdot\,|\,s)\|_{\mathrm{TV}} \leq L_{\pi}\|\theta_{1}-\theta_{2}\|.\end{gathered}$ (65) $$

###### Proof.

The proof is straightforward noticing the fact that

$$ $\|\left<\psi,\eta\right>\|_{\mathrm{TV}}=\int_{\mathcal{S}}|\eta^{T}\psi(s)| \mathrm{d}s=\int_{\mathcal{S}}\|\eta\|_{1}\left|\frac{\eta^{T}}{\|\eta\|_{1}} \psi(s)\right|\mathrm{d}s\leq\|\eta\|_{1}\leq\sqrt{d_{2}}\|\eta\|,$ (66) $$

and

$$ $\|\left<\phi,\theta\right>\|_{\infty}=\|\theta^{T}\phi(s,a)\|_{\infty}\leq \left\|\|\theta\|\|\phi(s,a)\|\right\|_{\infty}\leq\|\theta\|.$ (67) $$

∎

###### Lemma 5 (Lipschitz steady distributions) .

For any two steady distributions $\mu^{\ddagger}_{\xi_{1}}$ and $\mu^{\ddagger}_{\xi_{2}}$ induced by parameters $\xi_{1}=(\theta_{1};\eta_{1})$ and $\xi_{2}=(\theta_{2};\eta_{2})$, we have

$$ $\|\mu^{\ddagger}_{\xi_{1}}-\mu^{\ddagger}_{\xi_{2}}\|_{\mathrm{TV}}\leq\sigma L \left(\|\theta_{1}-\theta_{2}\|+\|\eta_{1}-\eta_{2}\|\right),$ (68) $$

where $\sigma\coloneqq 2+\hat{n}+m\rho^{\hat{n}}/(1-\rho)$, $\hat{n}\coloneqq\left\lceil\log_{\rho}m^{-1}\right\rceil$, and
$L=\max\left\{L_{P},L_{\pi}\right\}$. Involved constants are defined in Assumptions 3, 1 and 2.
Since $\mu^{\dagger}_{\xi}$ and $\mu_{\xi}$ are marginal distributions of $\mu^{\ddagger}_{\xi}$, as a corollary, we have

$$ $\displaystyle\|\mu^{\dagger}_{\xi_{1}}-\mu^{\dagger}_{\xi_{2}}\|_{\mathrm{TV}}$ $\displaystyle\leq\sigma L\left(\|\theta_{1}-\theta_{2}\|+\|\eta_{1}-\eta_{2}\| \right),$ (69) $\displaystyle\|\mu_{\xi_{1}}-\mu_{\xi_{2}}\|_{\mathrm{TV}}$ $\displaystyle\leq\sigma L\left(\|\theta_{1}-\theta_{2}\|+\|\eta_{1}-\eta_{2}\| \right).$ (70) $$

###### Proof.

We first prove the last inequality in the lemma.
By Mitrophanov [, Corollary 3.1], we have

$$ $\|\mu_{\xi_{1}}-\mu_{\xi_{2}}\|_{\mathrm{TV}}\leq(\sigma-2)\|P_{\xi_{1}}-P_{ \xi_{2}}\|_{\mathrm{TV}},$ (71) $$

where $P_{\xi}$ represents the transition kernel determined by policy $\Gamma_{\pi}(\theta)$ and population measure $\eta$, and

$$ $\|P_{\xi}\|_{\mathrm{TV}}\coloneqq\sup_{\begin{subarray}{c}q\in\mathcal{M}( \mathcal{S})\\ \|q\|_{\mathrm{TV}}=1\end{subarray}}\left\|\int_{\mathcal{S}}q(s)P(\cdot\,|\,s ,\theta,\eta)\mathrm{d}s\right\|_{\mathrm{TV}}=\sup_{\begin{subarray}{c}q\in \mathcal{M}(\mathcal{S})\\ \|q\|_{\mathrm{TV}}=1\end{subarray}}\left\|\int_{\mathcal{S}\times\mathcal{A}} q(s)\pi_{\theta}(a\,|\,s)P(\cdot\,|\,s,a,\eta)\mathrm{d}s\mathrm{d}a\right\|_{ \mathrm{TV}}.$ (72) $$

By the triangle inequality, we have

$$ $\displaystyle\|\mu_{\xi_{1}}-\mu_{\xi_{2}}\|_{\mathrm{TV}}\leq\sigma\|P_{( \theta_{1};\eta_{1})}-P_{(\theta_{1};\eta_{2})}\|_{\mathrm{TV}}+\sigma\|P_{( \theta_{1};\eta_{2})}-P_{(\theta_{2};\eta_{2})}\|_{\mathrm{TV}}.$ (73) $$

For the first term, by Assumption 1, we have

$$ $\displaystyle\left\|P_{(\theta_{1};\eta_{1})}-P_{(\theta_{1};\eta_{2})}\right \|_{\mathrm{TV}}=$ $\displaystyle\sup_{\stackrel{{\scriptstyle q\in\mathcal{M}(\mathcal{S})}}{{\|q \|_{\mathrm{TV}}=1}}}\left\|\int_{\mathcal{S}\times\mathcal{A}}q(s)\pi_{\theta _{1}}(a\,|\,s)\left(P(\cdot\,|\,s,a,\eta_{1})-P(\cdot\,|\,s,a,\eta_{2})\right) \mathrm{d}s\mathrm{d}a\right\|_{\mathrm{TV}}$ (74) $\displaystyle\leq$ $\displaystyle\sup_{\stackrel{{\scriptstyle q\in\mathcal{M}(\mathcal{S}\times \mathcal{A})}}{{\|q\|_{\mathrm{TV}}=1}}}\left\|\int_{\mathcal{S}\times\mathcal {A}}q(s,a)\left(P(\cdot\,|\,s,a,\eta_{1})-P(\cdot\,|\,s,a,\eta_{2})\right) \mathrm{d}s\mathrm{d}a\right\|_{\mathrm{TV}}$ (75) $\displaystyle=$ $\displaystyle\left\|P_{\eta_{1}}-P_{\eta_{2}}\right\|_{\mathrm{TV}}$ (76) $\displaystyle\leq$ $\displaystyle L_{P}\|\eta_{1}-\eta_{2}\|.$ (77) $$

Similarly, for the second term in Equation 73, by Assumption 2, we have

$$ $\displaystyle\left\|P_{(\theta_{1};\eta_{2})}-P_{(\theta_{2};\eta_{2})}\right \|_{\mathrm{TV}}=$ $\displaystyle\sup_{\stackrel{{\scriptstyle q\in\mathcal{D}(\mathcal{S})}}{{\|q \|_{\mathrm{TV}}=1}}}\left\|\int_{\mathcal{S}\times\mathcal{A}}q(s)P(\cdot\,| \,s,a,\eta_{2})(\pi_{\theta_{1}}(a\,|\,s)-\pi_{\theta_{2}}(a\,|\,s))\mathrm{d} s\mathrm{d}a\right\|_{\mathrm{TV}}$ (78) $\displaystyle\leq$ $\displaystyle\sup_{\stackrel{{\scriptstyle q\in\mathcal{D}(\mathcal{S})}}{{\|q \|_{\mathrm{TV}}=1}}}\int_{\mathcal{S}^{2}\times\mathcal{A}}q(s)P(s^{\prime}\, |\,s,a,\eta_{2})|\pi_{\theta_{1}}(a\,|\,s)-\pi_{\theta_{2}}(a\,|\,s)|\mathrm{d }a\mathrm{d}s\mathrm{d}s^{\prime}$ (79) $\displaystyle=$ $\displaystyle\sup_{\stackrel{{\scriptstyle q\in\mathcal{D}(\mathcal{S})}}{{\|q \|_{\mathrm{TV}}=1}}}\int_{\mathcal{S}}q(s)\left\|\pi_{\theta_{1}}(\cdot\,|\,s )-\pi_{\theta_{2}}(\cdot\,|\,s)\right\|_{\mathrm{TV}}\mathrm{d}s$ (80) $\displaystyle\leq$ $\displaystyle\sup_{\stackrel{{\scriptstyle q\in\mathcal{D}(\mathcal{S})}}{{\|q \|_{\mathrm{TV}}=1}}}\int_{\mathcal{S}}q(s)\cdot L_{\pi}\|\theta_{1}-\theta_{2 }\|\mathrm{d}s$ (81) $\displaystyle=$ $\displaystyle L_{\pi}\|\theta_{1}-\theta_{2}\|.$ (82) $$

Let $L\coloneqq\max\left\{L_{P},L_{\pi}\right\}$. Plugging the above two inequalities into Equation 73 gives

$$ $\|\mu_{\xi_{1}}-\mu_{\xi_{2}}\|_{\mathrm{TV}}\leq\sigma L\left(\|\eta_{1}-\eta _{2}\|+\|\theta_{1}-\theta_{2}\|\right).$ (83) $$

Then, by the definition of $\mu^{\dagger}_{\xi}$, we have

$$ $\displaystyle\|\mu^{\dagger}_{\xi_{1}}-\mu^{\dagger}_{\xi_{2}}\|_{\mathrm{TV}}=$ $\displaystyle\int_{\mathcal{S}\times\mathcal{A}}\left|\mu_{\xi_{1}}(s)\pi_{ \theta_{1}}(a\,|\,s)-\mu_{\xi_{2}}(s)\pi_{\theta_{2}}(a\,|\,s)\right|\mathrm{d }s\mathrm{d}a$ (84) $\displaystyle\leq$ $\displaystyle\int_{\mathcal{S}\times\mathcal{A}}\left(\left|\mu_{\xi_{1}}(s)- \mu_{\xi_{2}}(s)\right|\pi_{\theta_{1}}(a\,|\,s)+\mu_{\xi_{2}}(s)\left|\pi_{ \theta_{1}}(a\,|\,s)-\pi_{\theta_{2}}(a\,|\,s)\right|\right)\mathrm{d}s\mathrm {d}a$ (85) $\displaystyle=$ $\displaystyle\|\mu_{\xi_{1}}-\mu_{\xi_{2}}\|_{\mathrm{TV}}+\int_{\mathcal{S}} \mu_{\xi_{2}}(s)\left\|\pi_{\theta_{1}}(\cdot\,|\,s)-\pi_{\theta_{2}}(\cdot\,| \,s)\right\|_{\mathrm{TV}}\mathrm{d}s$ (86) $\displaystyle\leq$ $\displaystyle(\sigma-2)L\left(\|\eta_{1}-\eta_{2}\|+\|\theta_{1}-\theta_{2}\| \right)+L_{\pi}\|\theta_{1}-\theta_{2}\|$ (87) $\displaystyle\leq$ $\displaystyle(\sigma-1)L\left(\|\eta_{1}-\eta_{2}\|+\|\theta_{1}-\theta_{2}\| \right).$ (88) $$

Similarly, by the definition of $\mu^{\ddagger}_{\xi}$, we have

$$ $\displaystyle\|\mu^{\ddagger}_{\xi_{1}}-\mu^{\ddagger}_{\xi_{2}}\|_{\mathrm{TV }}\leq$ $\displaystyle\|\mu^{\dagger}_{\xi_{1}}-\mu^{\dagger}_{\xi_{2}}\|_{\mathrm{TV}} +L_{P}\|\eta_{1}-\eta_{2}\|+L_{\pi}\|\theta_{1}-\theta_{2}\|$ (89) $\displaystyle\leq$ $\displaystyle\sigma L\left(\|\eta_{1}-\eta_{2}\|+\|\theta_{1}-\theta_{2}\| \right).$ (90) $$

∎

###### Corollary 2 .

Similarly, for steady distributions induced by general value functions and population measures $\mu_{1}\coloneqq\mu_{(Q_{1},M_{1})},\mu_{2}\coloneqq\mu_{(Q_{2},M_{2})}$, we have

$$ $\max\left\{\left\|\mu_{1}-\mu_{2}\right\|_{\mathrm{TV}},\|\mu_{1}^{\dagger}- \mu_{2}^{\dagger}\|_{\mathrm{TV}},\|\mu_{1}^{\ddagger}-\mu_{2}^{\ddagger}\|_{ \mathrm{TV}}\right\}\leq\sigma\left(\frac{L_{P}}{\sqrt{d_{2}}}\|M_{1}-M_{2}\|+ L_{\pi}\|Q_{1}-Q_{2}\|_{\infty}\right).$ (91) $$

###### Lemma 6 (Lipschitz temporal difference operators) .

For any two sets of mean-path temporal difference operators $\bar{G}_{\xi_{1}},\bar{\varphi}_{\xi_{1}},\bar{\psi}_{\xi_{1}}$ and $\bar{G}_{\xi_{2}},\bar{\varphi}_{\xi_{2}},\bar{\psi}_{\xi_{2}}$ determined by parameters $\xi_{1}=(\theta_{1};\eta_{1})$ and $\xi_{2}=(\theta_{2};\eta_{2})$, we have

$$ $\displaystyle\|\bar{G}_{\xi_{1}}-\bar{G}_{\xi_{2}}\|\leq$ $\displaystyle\sigma L(1+\gamma)\left(\|\theta_{1}-\theta_{2}\|+\|\eta_{1}-\eta _{2}\|\right),$ (92) $\displaystyle\|\bar{\varphi}_{\xi_{1}}-\bar{\varphi}_{\xi_{2}}\|\leq$ $\displaystyle\sigma LR\left(\|\theta_{1}-\theta_{2}\|+\|\eta_{1}-\eta_{2}\| \right)+L_{r}\|\eta_{1}-\eta_{2}\|$ (93) $\displaystyle\|\bar{\psi}_{\xi_{1}}-\bar{\psi}_{\xi_{2}}\|\leq$ $\displaystyle\sigma LF\left(\|\theta_{1}-\theta_{2}\|+\|\eta_{1}-\eta_{2}\| \right),$ (94) $$

where $\sigma$ and $L$ are defined in Lemma 5, and $L_{r}$ is defined in Assumption 1.

###### Proof.

By definition, we have

$$ $\displaystyle\|\bar{G}_{\xi_{1}}-\bar{G}_{\xi_{2}}\|=$ $\displaystyle\left\|\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\phi(s,a)(\phi( s,a)-\gamma\phi(s^{\prime},a^{\prime}))^{T}\left(\mu^{\ddagger}_{\xi_{1}}(s,a, s^{\prime},a^{\prime})-\mu^{\ddagger}_{\xi_{2}}(s,a,s^{\prime},a^{\prime}) \right)\mathrm{d}s\mathrm{d}a\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}\right\|$ (95) $\displaystyle\leq$ $\displaystyle\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\left\|\phi(s,a)(\phi( s,a)-\gamma\phi(s^{\prime},a^{\prime}))^{T}\right\|\left(\mu^{\ddagger}_{\xi_{ 1}}(s,a,s^{\prime},a^{\prime})-\mu^{\ddagger}_{\xi_{2}}(s,a,s^{\prime},a^{ \prime})\right)\mathrm{d}s\mathrm{d}a\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}.$ (96) $$

Since $\|\phi(s,a)\|\leq 1$, we have $\left\|\phi(s,a)(\phi(s,a)-\gamma\phi(s^{\prime},a^{\prime}))^{T}\right\|\leq 1+\gamma$.
Then, by Lemma 5, we have

$$ $\|\bar{G}_{\xi_{1}}-\bar{G}_{\xi_{2}}\|\leq(1+\gamma)\sigma L\left(\|\theta_{1 }-\theta_{2}\|+\|\eta_{1}-\eta_{2}\|\right).$ (97) $$

Similarly, by definition, we have

$$ $\displaystyle\|\bar{\varphi}_{\xi_{1}}-\bar{\varphi}_{\xi_{2}}\|=$ $\displaystyle\left\|\int_{\mathcal{S}\times\mathcal{A}}\phi(s,a)r(s,a,\eta_{1} )\mu^{\dagger}_{\xi_{1}}(s,a)-\phi(s,a)r(s,a,\eta_{2})\mu^{\dagger}_{\xi_{2}}( s,a)\mathrm{d}s\mathrm{d}a\right\|$ (98) $\displaystyle\leq$ $\displaystyle\int_{\mathcal{S}\times\mathcal{A}}\left(r(s,a,\eta_{1})\left|\mu ^{\dagger}_{\xi_{1}}(s,a)-\mu^{\dagger}_{\xi_{2}}(s,a)\right|+\mu^{\dagger}_{ \xi_{2}}(s,a)\left|r(s,a,\eta_{1})-r(s,a,\eta_{2})\right|\right)\mathrm{d}s \mathrm{d}a$ (99) $\displaystyle\leq$ $\displaystyle R\|\mu^{\dagger}_{\xi_{1}}-\mu^{\dagger}_{\xi_{2}}\|_{\mathrm{TV }}+L_{r}\|\eta_{1}-\eta_{2}\|$ (100) $\displaystyle\leq$ $\displaystyle\sigma LR\left(\|\theta_{1}-\theta_{2}\|+\|\eta_{1}-\eta_{2}\| \right)+L_{r}\|\eta_{1}-\eta_{2}\|,$ (101) $$

and

$$ $\|\bar{\psi}_{\xi_{1}}-\bar{\psi}_{\xi_{2}}\|\leq F\|\mu_{\xi_{1}}-\mu_{\xi_{2 }}\|_{\mathrm{TV}}\leq\sigma LF\left(\|\theta_{1}-\theta_{2}\|+\|\eta_{1}-\eta _{2}\|\right).$ (102) $$

∎

###### Lemma 7 (Lipschitz semi-gradient) .

Given a fixed semi-gradient operator $\mathfrak{g}$ (which can be a mean-path semi-gradient), for any parameters $\xi_{1},\xi_{2}\in\Xi$, we have

$$ $\|\mathfrak{g}(\xi_{1})-\mathfrak{g}(\xi_{2})\|\leq H\|\xi_{1}-\xi_{2}\|.$ (103) $$

Let $\xi_{1}=(\theta_{1};\eta_{1})$ and $\xi_{2}=(\theta_{2};\eta_{2})$.
For any $\xi=(\theta;\eta)$ such that $\|\theta\|\leq G$, we have

$$ $\|\bar{\mathfrak{g}}_{\xi_{1}}(\xi)-\bar{\mathfrak{g}}_{\xi_{2}}(\xi)\|\leq \sigma H\max\left\{L_{P},L_{\pi}\right\}\left(\|\theta_{1}-\theta_{2}\|+\|\eta _{1}-\eta_{2}\|\right)+L_{r}\|\eta_{1}-\eta_{2}\|.$ (104) $$

###### Proof.

We first show that the semi-gradient operator is Lipschitz in its argument.
By definition, we have

$$ $\|\mathfrak{g}(\xi_{1};O)-\mathfrak{g}(\xi_{2};O)\|=\|\left({G}_{\phi}(O) \oplus{G}_{\psi}\right)(\xi_{1}-\xi_{2})\|\leq\|{G}_{\phi}(O)\oplus{G}_{\psi} \|_{\mathrm{op}}\|\xi_{1}-\xi_{2}\|.$ (105) $$

We get the result by noticing that

$$ $\|{G}_{\phi}(O)\oplus{G}_{\psi}\|_{\mathrm{op}}\leq\max\left\{\|{G}_{\phi}(O) \|_{\mathrm{op}},\|{G}_{\psi}\|_{\mathrm{op}}\right\}\leq\max\left\{1+\gamma,F \right\}\leq H.$ (106) $$

Next, we show that the mean-path semi-gradient operator is Lipschitz in the parameter.
Another representation of the mean-path semi-gradient is

$$ $\bar{\mathfrak{g}}_{\xi_{1}}=\left(\bar{G}_{\xi_{1}}\theta-\bar{\varphi}_{\xi_ {1}};{G}_{\psi}\eta-\bar{\psi}_{\xi_{1}}\right).$ (107) $$

Therefore, by Lemma 6, we have

$$ $\displaystyle\|\bar{\mathfrak{g}}_{\xi_{1}}(\xi)-\bar{\mathfrak{g}}_{\xi_{2}}( \xi)\|\leq$ $\displaystyle\left\|\bar{G}_{\xi_{1}}\theta-\bar{G}_{\xi_{2}}\theta-\bar{b}_{ \xi_{1}}+\bar{b}_{\xi_{2}}\right\|+\left\|{G}_{\psi}\eta_{1}-{G}_{\psi}\eta_{2 }-\bar{\psi}_{\xi_{1}}+\bar{\psi}_{\xi_{2}}\right\|$ (108) $\displaystyle\leq$ $\displaystyle D\left\|\bar{G}_{\xi_{1}}-\bar{G}_{\xi_{2}}\right\|+\left\|\bar{ b}_{\xi_{1}}-\bar{b}_{\xi_{2}}\right\|+\left\|\bar{\psi}_{\xi_{1}}-\bar{\psi}_ {\xi_{2}}\right\|$ $\displaystyle\leq$ $\displaystyle\sigma L((1+\gamma)D+R+F)\left(\|\theta_{1}-\theta_{2}\|+\|\eta_{ 1}-\eta_{2}\|\right)+L_{r}\|\eta_{1}-\eta_{2}\|.$ $$

∎

###### Lemma 2 (Norm relations) .

###### Proof.

###### Lemma 3 (Gradient bounds) .

###### Proof.

###### Lemma 4 (Lipschitzness in parameters) .

###### Proof.

###### Lemma 5 (Lipschitz steady distributions) .

###### Proof.

###### Corollary 2 .

###### Lemma 6 (Lipschitz temporal difference operators) .

###### Proof.

###### Lemma 7 (Lipschitz semi-gradient) .

###### Proof.

### Appendix F Sample complexity analysis

#### F.1 Key lemmas

In this section, we first decompose the mean square error into different terms, and then bound each term separately.

###### Lemma 8 (Error decomposition) .

Let $\xi_{*}$ be the optimal parameter. Recall that by Proposition 2, $\bar{\mathfrak{g}}_{*}(\xi_{*})=0$. We have

$$ $\displaystyle\mathbb{E}\|\xi_{t+1}-\xi_{*}\|^{2}\leq$ $\displaystyle\mathbb{E}\|\breve{\xi}_{t+1}-\xi_{*}\|^{2}$ (109) $\displaystyle=$ $\displaystyle\mathbb{E}\|\xi_{t}-\xi_{*}\|^{2}-2\alpha_{t}\mathbb{E}\left<\xi_ {t}-\xi_{*},{\mathfrak{g}}_{t}(\xi_{t})\right>+\alpha_{t}^{2}\|\mathfrak{g}_{t }(\xi_{t})\|^{2}$ (110) $\displaystyle=$ $\displaystyle\mathbb{E}\|\xi_{t}-\xi_{*}\|^{2}+\alpha_{t}^{2}\|\mathfrak{g}_{t }(\xi_{t})\|^{2}$ (111) $\displaystyle-2\alpha_{t}\mathbb{E}\left<\xi_{t}-\xi_{*},\bar{\mathfrak{g}}_{t }(\xi_{t})-\bar{\mathfrak{g}}_{*}(\xi_{*})\right>$ (descent) $\displaystyle-2\alpha_{t}\mathbb{E}\left<\xi_{t}-\xi_{*},\bar{\mathfrak{g}}_{t -\tau}(\xi_{t})-\bar{\mathfrak{g}}_{t}(\xi_{t})\right>$ (progress) $\displaystyle-2\alpha_{t}\mathbb{E}\left<\xi_{t}-\xi_{*},\mathfrak{g}_{t-\tau} (\xi_{t})-\bar{\mathfrak{g}}_{t-\tau}(\xi_{t})\right>$ (mix) $\displaystyle-2\alpha_{t}\mathbb{E}\left<\xi_{t}-\xi_{*},\mathfrak{g}_{t}(\xi_ {t})-{\mathfrak{g}}_{t-\tau}(\xi_{t})\right>.$ (backtrack) $$

###### Lemma 9 (Descent) .

Let $\xi_{*}=(\theta_{*};\eta_{*})$ be a projected MFE.
For any parameter $\xi=(\theta;\eta)\in B^{d_{1}}_{D}\times\Delta^{d_{2}}$, we denote $\Delta\theta=\theta-\theta_{*}$ and $\Delta\eta=\eta-\eta_{*}$.
We have

$$ $\displaystyle-\left<\eta-\eta_{*},\bar{\mathfrak{g}}_{\xi}(\eta)-\bar{ \mathfrak{g}}_{\xi_{*}}(\eta_{*})\right>$ $\displaystyle\leq-\lambda_{\min}({G}_{\psi})\|\Delta\eta\|^{2}+\sigma LF\| \Delta\eta\|\left(\|\Delta\theta\|+\|\Delta\eta\|\right),$ (112) $\displaystyle-\left<\theta-\theta_{*},\bar{\mathfrak{g}}_{\xi}(\theta)-\bar{ \mathfrak{g}}_{\xi_{*}}(\theta_{*})\right>$ $\displaystyle\leq-(1-\gamma)\lambda_{\min}(\hat{G}_{\xi_{*}})\|\Delta\theta\|^ {2}+\sigma LH\|\Delta\theta\|(\|\Delta\theta\|+\|\Delta\eta\|)+L_{r}\|\Delta \eta\|\|\Delta\theta\|.$ $$

That is, neither $-\bar{\mathfrak{g}}_{\xi}(\eta)$ nor $-\bar{\mathfrak{g}}_{\xi}(\theta)$ is
guaranteed to be a descent direction.
Let $w\coloneqq\frac{1}{2}\min\left\{\lambda_{\min}({G}_{\psi}),(1-\gamma)\lambda_{
\min}(\hat{G}_{\xi_{*}})\right\}$.
Suppose $3\sigma LH+L_{r}\leq 2w$. Then, we have

$$ $-\left<\xi-\xi_{*},\bar{\mathfrak{g}}_{\xi}(\xi)-\bar{\mathfrak{g}}_{\xi_{*}}( \xi_{*})\right>\leq-w\|\xi-\xi_{*}\|^{2}.$ (113) $$

###### Proof.

We first focus on the first two inequalities. We denote $\Delta\eta\coloneqq\eta-\eta_{*}$ and $\Delta\theta\coloneqq\theta-\theta_{*}$.
For the population measure parameter, by definition, we have

$$ $-\left<\Delta\eta,\bar{\mathfrak{g}}_{\xi}(\eta)-\bar{\mathfrak{g}}_{\xi_{*}}( \eta_{*})\right>=\left<\Delta\eta,-{G}_{\psi}\Delta\eta+\bar{\psi}_{\xi}-\bar{ \psi}_{\xi_{*}}\right>.$ (114) $$

Note that ${G}_{\psi}$ is a positive definite Gram matrix if $\psi$ is linearly independent.
Then, by Lemma 6, we have

$$ $\displaystyle-\left<\Delta\eta,\bar{\mathfrak{g}}_{\xi}(\eta)-\bar{\mathfrak{g }}_{\xi_{*}}(\eta_{*})\right>\leq$ $\displaystyle-\left<\Delta\eta,{G}_{\psi}\Delta\eta\right>+\left\|\bar{\psi}_{ \xi}-\bar{\psi}_{\xi_{*}}\right\|\|\Delta\eta\|$ (115) $\displaystyle\leq$ $\displaystyle-\lambda_{\min}({G}_{\psi})\|\Delta\eta\|^{2}+\sigma LF\|\Delta \eta\|(\|\Delta\eta\|+\|\Delta\theta\|).$ (116) $$

For the value function parameter, we have

$$ $\displaystyle-\left<\Delta\theta,\bar{\mathfrak{g}}_{\xi}(\theta)-\bar{ \mathfrak{g}}_{\xi_{*}}(\theta_{*})\right>=$ $\displaystyle\left<\Delta\theta,\bar{G}_{\xi_{*}}\theta_{*}-\bar{\varphi}_{\xi _{*}}-\bar{G}_{\xi}\theta+\bar{\varphi}_{\xi}\right>$ (117) $\displaystyle=$ $\displaystyle\left<\Delta\theta,-\bar{G}_{\xi_{*}}\Delta\theta+(\bar{G}_{\xi_{ *}}-\bar{G}_{\xi})\theta+\bar{\varphi}_{\xi}-\bar{\varphi}_{\xi_{*}}\right>$ (118) $\displaystyle\leq$ $\displaystyle-\left<\Delta\theta,\bar{G}_{\xi_{*}}\Delta\theta\right>+D\|\bar{ G}_{\xi_{*}}-\bar{G}_{\xi}\|\|\Delta\theta\|+\|\bar{\varphi}_{\xi}-\bar{ \varphi}_{\xi_{*}}\|\|\Delta\theta\|.$ (119) $$

By Lemma 6, we have

$$ $\displaystyle-\left<\Delta\theta,\bar{\mathfrak{g}}_{\xi}(\theta)-\bar{ \mathfrak{g}}_{\xi_{*}}(\theta_{*})\right>\leq$ $\displaystyle-\left<\Delta\theta,\bar{G}_{\xi_{*}}\Delta\theta\right>+\|\Delta \theta\|((1+\gamma)\sigma LD(\|\Delta\theta\|+\|\Delta\eta\|)$ (120) $\displaystyle+\sigma LR(\|\Delta\theta\|+\|\Delta\eta\|)+L_{r}\|\Delta\eta\|).$ (121) $$

For a matrix $G$, we define $w_{\min}(G)\coloneqq\min_{\|x\|=1}\left<x,Gx\right>$. For $\bar{G}_{\xi_{*}}$, we have

$$ $w_{\min}(\bar{G}_{\xi_{*}})=\sup_{\|\theta\|=1}\mathbb{E}_{\xi_{*}}\left[ \theta^{T}\phi(s,a)(\phi(s,a)-\gamma\phi(s^{\prime},a^{\prime}))^{T}\theta \right]\eqqcolon\sup_{\|\theta\|=1}\mathbb{E}_{\xi_{*}}\left[u^{2}-\gamma uu^{ \prime}\right],$ (122) $$

where $u\coloneqq\theta^{T}\phi(s,a)$ and $u^{\prime}\coloneqq\theta^{T}\phi(s^{\prime},a^{\prime})$.
We have

$$ $\mathbb{E}[uu^{\prime}]\leq\frac{1}{2}\left(\mathbb{E}[u^{2}]+\mathbb{E}[u^{ \prime 2}]\right)=\frac{1}{2}\left(\mathbb{E}[u^{2}]+\mathbb{E}[u^{2}]\right)= \mathbb{E}[u^{2}].$ (123) $$

Therefore,

$$ $w_{\min}(\bar{G}_{\xi_{*}})\geq(1-\gamma)\sup_{\|\theta\|=1}\mathbb{E}_{\xi_{* }}[u^{2}]=(1-\gamma)w_{\min}\left(\mathbb{E}_{\xi_{*}}[\phi\phi^{T}]\right)=(1 -\gamma)w_{\min}\left(\hat{G}_{\xi_{*}}\right)=(1-\gamma)\lambda_{\min}\left( \hat{G}_{\xi_{*}}\right),$ (124) $$

where $\hat{G}_{\xi_{*}}$ is the Gram matrix of the feature map $\phi$ under the steady distribution induced by $\xi_{\xi_{*}}$, and the last equality uses the property of normal matrices.
Plugging the above derivation into Equation 121 gives

$$ $-\left<\Delta\theta,\bar{\mathfrak{g}}_{\xi}(\theta)-\bar{\mathfrak{g}}_{\xi_{ *}}(\theta_{*})\right>\leq-(1-\gamma)\lambda_{\min}(\hat{G}_{\xi_{*}})\|\Delta \theta\|^{2}+\sigma L(H-2F)\|\Delta\theta\|(\|\Delta\theta\|+\|\Delta\eta\|)+L _{r}\|\Delta\eta\|\|\Delta\theta\|.$ (125) $$

It is clear that Equations 125 and 116 are controlled by both $\|\Delta\theta\|$ and $\|\Delta\eta\|$.
When $\|\Delta\eta\|\gg\|\Delta\theta\|$, Equation 125 suggests that $-\bar{\mathfrak{g}}_{\xi}(\theta)$ may not be a descent direction for $\|\Delta\theta\|$; when $\|\Delta\theta\|\gg\|\Delta\eta\|$, Equation 116 suggests that $-\bar{\mathfrak{g}}_{\xi}(\eta)$ may not be a descent direction for $\|\Delta\eta\|$.

However, combining Equations 125 and 116 gives

$$ $\displaystyle-\left<\Delta\xi,\bar{\mathfrak{g}}_{\xi}(\xi)-\bar{\mathfrak{g}} _{\xi_{*}}(\xi_{*})\right>\leq$ $\displaystyle-(1-\gamma)\lambda_{\min}(\hat{G}_{\xi_{*}})\|\Delta\theta\|^{2}- \lambda_{\min}({G}_{\psi})\|\Delta\eta\|^{2}+\sigma LF\|\Delta\eta\|^{2}$ (126) $\displaystyle+\sigma L(H-2F)\|\Delta\theta\|^{2}+(\sigma LF+\sigma L(H-2F)+L_{ r})\|\Delta\theta\|\|\Delta\eta\|.$ (127) $$

By Lemma 2, we have

$$ $\displaystyle-\left<\Delta\xi,\bar{\mathfrak{g}}_{\xi}(\xi)-\bar{\mathfrak{g}} _{\xi_{*}}(\xi_{*})\right>$ (128) $\displaystyle\leq$ $\displaystyle-\min\{(1-\gamma)\lambda_{\min}(\hat{G}_{\xi_{*}}),\lambda_{\min} ({G}_{\psi})\}\|\Delta\xi\|^{2}+\sigma LH\|\Delta\xi\|^{2}+\frac{1}{2}(\sigma LH +L_{r})\|\Delta\xi\|^{2}$ (129) $\displaystyle\leq$ $\displaystyle\left(-\min\{(1-\gamma)\lambda_{\min}(\hat{G}_{\xi_{*}}),\lambda_ {\min}({G}_{\psi})\}+\frac{1}{2}(3\sigma LH+L_{r})\right)\|\Delta\xi\|^{2}.$ (130) $$

Let $w\coloneqq\frac{1}{2}\min\left\{(1-\gamma)\lambda_{\min}(\hat{G}_{\xi_{*}}),
\lambda_{\min}({G}_{\psi})\right\}$.
Suppose $3\sigma LH+L_{r}\leq 2w$.
Then, we have

$$ $-\left<\Delta\xi,\bar{g}_{\xi}(\xi)-\bar{\mathfrak{g}}_{\xi_{*}}(\xi_{*}) \right>\leq-w\|\Delta\xi\|^{2}.$ (131) $$

The above inequality suggests that $-\bar{\mathfrak{g}}_{\xi}(\xi)$ is a descent direction for $\|\Delta\xi\|$ if the Lipschitz contants are small enough.
∎

Figure: Figure 9: An illustrative example for Lemma 1, where $-\bar{\mathfrak{g}}_{\xi}(\xi)$ gives a descent direction, while $-\bar{\mathfrak{g}}_{\xi}(\theta)$ does not.

###### Lemma 10 (Progress) .

For any $\xi=(\theta;\eta)$ in $B_{D}^{d_{1}}\times\Delta^{d_{2}}$, and for any time step $t$ and period $\tau$, we have

$$ $\|\bar{\mathfrak{g}}_{t}(\xi)-\bar{\mathfrak{g}}_{t-\tau}(\xi)\|\leq\alpha_{t- \tau}\tau C_{\mathrm{prog}},$ (132) $$

where $C_{\mathrm{prog}}\coloneqq\sigma\max\{L_{\pi},L_{P}\}H^{2}+2L_{r}F$.

###### Proof.

By Lemma 7, we have

$$ $\|\bar{\mathfrak{g}}_{t}(\xi)-\bar{\mathfrak{g}}_{t-\tau}(\xi)\|\leq\sigma LH \left(\|\theta_{t}-\theta_{t-\tau}\|+\|\eta_{t}-\eta_{t-\tau}\|\right)+L_{r}\| \eta_{t}-\eta_{t-\tau}\|.$ (133) $$

According to the update rule of $\theta$ and $\eta$, we have

$$ $\displaystyle\|\theta_{t}-\theta_{t-1}\|\leq$ $\displaystyle\|\breve{\theta}_{t}-\theta_{t-1}\|=\|\alpha_{t-1}\mathfrak{g}_{t -1}(\theta_{t-1})\|$ (134) $\displaystyle\|\eta_{t}-\eta_{t-1}\|\leq$ $\displaystyle\|\breve{\eta}_{t}-\eta_{t-1}\|=\|\alpha_{t-1}\mathfrak{g}_{t-1}( \eta_{t-1})\|.$ $$

Since $\|\theta_{t-1}\|\leq D$ and $\|\eta_{t-1}\|\leq\|\eta_{t-1}\|_{1}=1$, by Lemma 3, we have

$$ $\|\theta_{t}-\theta_{t-1}\|+\|\eta_{t}-\eta_{t-1}\|\leq\alpha_{t-1}\left((1+ \gamma)D+R\right)+\alpha_{t-1}\cdot 2F=\alpha_{t-1}H.$ (135) $$

By the triangle inequality, we get

$$ $\displaystyle\|\bar{\mathfrak{g}}_{t}(\xi)-\bar{\mathfrak{g}}_{t-\tau}(\xi)\|\leq$ $\displaystyle\sigma LH\sum_{l=0}^{\tau-1}\left(\|\theta_{t-l}-\theta_{t-l-1}\| +\|\eta_{t-l}-\eta_{t-l-1}\|\right)+L_{r}\sum_{l=0}^{\tau-1}\|\eta_{t-l}-\eta_ {t-l-1}\|$ (136) $\displaystyle\leq$ $\displaystyle\alpha_{t-\tau}\tau\sigma LH^{2}+2\alpha_{t-\tau}\tau L_{r}F$ (137) $\displaystyle\eqqcolon$ $\displaystyle\alpha_{t-\tau}\tau C_{\mathrm{prog}},$ (138) $$

where we require that the step size $\alpha_{t}$ is non-increasing, and $C_{\mathrm{prog}}\coloneqq\sigma\max\{L_{\pi},L_{P}\}H^{2}+2L_{r}F$.
∎

###### Lemma 11 (Mix) .

Let $\mathcal{F}_{t-\tau}$ be the filtration generated by the history up to time $t-\tau$. For any $\xi=(\theta;\eta)\in B_{D}^{d_{1}}\times\Delta^{d_{2}}$ that is independent of $\mathfrak{g}_{t-\tau}$ and $\bar{\mathfrak{g}}_{t-\tau}$ conditioned on $\mathcal{F}_{t-\tau}$, and for any time step $t$ and period $\tau$, we have

$$ $\left\|\mathbb{E}\left[\mathfrak{g}_{t-\tau}(\xi;\widetilde{O}_{t})-\bar{ \mathfrak{g}}_{t-\tau}(\xi)\,\middle|\,\mathcal{F}_{t-\tau}\right]\right\|\leq m \rho^{\tau}H.$ (139) $$

###### Proof.

We denote $P_{t-\tau}$ as the observation distribution on the virtual trajectory by fixing the transition kernel $P_{\xi_{t-\tau}}$ at time $t-\tau$. We have

$$ $\displaystyle\left\|\mathbb{E}\left[\mathfrak{g}_{t-\tau}(\xi,\widetilde{O}_{t })-\bar{\mathfrak{g}}_{t-\tau}(\xi)\,\middle|\,\mathcal{F}_{t-\tau}\right]\right\|$ (140) $\displaystyle=$ $\displaystyle\left\|\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\left(\phi(s,a) \left(r(s,a)+\gamma\phi^{T}(s^{\prime},a^{\prime})\theta-\phi^{T}(s,a)\theta \right);\hat{\psi}(s^{\prime})\right)\left(P_{t-\tau}(\widetilde{O}_{t}=o\,|\, \mathcal{F}_{t-\tau})-\mu_{t-\tau}^{\ddagger}(o)\right)\mathrm{d}o\right\|$ $\displaystyle\leq$ $\displaystyle((1+\gamma)D+R+F)\left\|P_{t-\tau}-\mu_{\xi_{t-\tau}}^{\ddagger} \right\|_{\mathrm{TV}}.$ $$

Since $P_{t-\tau}$ and $\mu_{\xi_{t-\tau}}^{\ddagger}$ share the same policy $\pi_{\theta_{t-\tau}}$ and transition kernel $P_{\xi_{t-\tau}}$, we have

$$ $\left\|P_{t-\tau}-\mu_{\xi_{t-\tau}}^{\ddagger}\right\|_{\mathrm{TV}}=\left\|P _{t-\tau}(s_{t}=\cdot\,|\,\mathcal{F}_{t-\tau})-\mu_{\xi_{t-\tau}}\right\|_{ \mathrm{TV}}\leq m\rho^{\tau}H,$ (141) $$

where the last inequality uses Assumption 3.
∎

###### Lemma 12 (Bakctrack) .

Let $\mathcal{F}_{t-\tau}$ be the filtration generated by the history up to time $t-\tau$. For any $\xi=(\theta;\eta)\in B_{D}^{d_{1}}\times\Delta^{d_{2}}$ that is independent of $\mathfrak{g}_{t}$ and $\mathfrak{g}_{t-\tau}$ conditioned on $\mathcal{F}_{t-\tau}$, and for any time step $t$ and period $\tau$, we have

$$ $\left\|\mathbb{E}\left[\mathfrak{g}_{t}(\xi,{O}_{t})-\mathfrak{g}_{t-\tau}(\xi ,\widetilde{O}_{t})\,\middle|\,\mathcal{F}_{t-\tau}\right]\right\|\leq\alpha_{ t-\tau}\tau C_{\mathrm{back}}(\tau),$ (142) $$

where $C_{\mathrm{back}}(\tau)\coloneqq(\tau+1)LH^{2}+2L_{r}F$.

###### Proof.

By definition< we have

$$ $\displaystyle\left\|\mathbb{E}\left[\mathfrak{g}_{t}(\xi,{O}_{t})-\mathfrak{g} _{t-\tau}(\xi,\widetilde{O}_{t})\,\middle|\,\mathcal{F}_{t-\tau}\right]\right\|$ (143) $\displaystyle=$ $\displaystyle\Bigl{\|}\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\left({G}_{ \phi}(o)\oplus{G}_{\psi}\right)\xi\cdot\left(P_{t}(O_{t}=o\,|\,\mathcal{F}_{t- \tau})-P_{t-\tau}(\widetilde{O}_{t}=o\,|\,\mathcal{F}_{t-\tau})\right)\mathrm{ d}o$ (144) $\displaystyle-\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\left(\varphi_{\eta_{ t}}\oplus\psi\right)(o)P_{t}(O_{t}=o\,|\,\mathcal{F}_{t-\tau})-\left(\varphi_{ \eta_{t-\tau}}\oplus\psi\right)(o)P_{t-\tau}(\widetilde{O}_{t}=o\,|\,\mathcal{ F}_{t-\tau})\mathrm{d}o\Bigr{\|}$ (145) $\displaystyle\leq$ $\displaystyle((1+\gamma)D+F)\|P_{t}-P_{t-\tau}\|_{\mathrm{TV}}$ (146) $\displaystyle+\left\|\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\left(\varphi_ {\eta_{t}}\oplus\psi\right)(o)\left(P_{t}(O_{t}=o\,|\,\mathcal{F}_{t-\tau})-P_ {t-\tau}(\widetilde{O}_{t}=o\,|\,\mathcal{F}_{t-\tau})\right)\mathrm{d}o\right \|_{\mathrm{TV}}$ (147) $\displaystyle+\left\|\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\left(\varphi_ {\eta_{t}}-\varphi_{\eta_{t-\tau}}\right)(o)P_{t-\tau}(\widetilde{O}_{t}=o\,| \,\mathcal{F}_{t-\tau})\mathrm{d}o\right\|_{\mathrm{TV}}$ (148) $\displaystyle\leq$ $\displaystyle((1+\gamma)D+F+R+F)\|P_{t}-P_{t-\tau}\|_{\mathrm{TV}}+L_{r}\|\eta _{t}-\eta_{t-\tau}\|$ (149) $$

where $P_{t}$ and $P_{t-\tau}$ are the distributions of observation at time step $t$ on the actual trajectory and the virtual trajectory, respectively.
By the proof of Lemma 10, we have $\|\eta_{t}-\eta_{t-\tau}\|\leq\alpha_{t-\tau}\tau\cdot 2F$.
Therefore, we have

$$ $\left\|\mathbb{E}\left[\mathfrak{g}_{t}(\xi,{O}_{t})-\mathfrak{g}_{t-\tau}(\xi ,\widetilde{O}_{t})\,\middle|\,\mathcal{F}_{t-\tau}\right]\right\|\leq H\|P_{t }-P_{t-\tau}\|_{\mathrm{TV}}+2\alpha_{t-\tau}\tau L_{r}F.$ (150) $$

Let $\Xi$ be the set of all parameters.
We first expand $P_{t}$ with conditional probabilities:

$$ $\displaystyle P_{t}(O_{t}=(s,a,s^{\prime},a^{\prime})\,|\,\mathcal{F}_{t-\tau})$ (151) $\displaystyle=$ $\displaystyle\int_{\Xi^{2}}P_{t}(s_{t}=s\,|\,\xi_{t-\tau},s_{t-\tau})P_{t}(\xi _{t-1}=\xi\,|\,\xi_{t-\tau},s_{t-\tau},s_{t}=s)\pi_{\theta}(a\,|\,s)$ (152) $\displaystyle\qquad P(s^{\prime}\,|\,s,a,\eta^{\prime})P_{t}(\xi_{t}=\xi^{ \prime}\,|\,\xi_{t-\tau},s_{t-\tau},\xi_{t-1}=\xi,s_{t}=s,a_{t}=a)\pi_{\theta^ {\prime}}(a^{\prime}\,|\,s^{\prime})\mathrm{d}\xi\mathrm{d}\xi^{\prime},$ (153) $$

where $\xi=(\theta;\eta)$ and $\xi^{\prime}=(\theta^{\prime};\eta^{\prime})$.
We then decompose $P_{t-\tau}$ into a similar form:

$$ $\displaystyle P_{t-\tau}(\widetilde{O}_{t}=(s,a,s^{\prime},a^{\prime})\,|\, \mathcal{F}_{t-\tau})$ (154) $\displaystyle=$ $\displaystyle P_{t-\tau}(\widetilde{s}_{t}=s\,|\,\xi_{t-\tau},s_{t-\tau})\pi_{ \theta_{t-\tau}}(a\,|\,s)P(s^{\prime}\,|\,s,a,\eta_{t-\tau})\pi_{\theta_{t- \tau}}(a^{\prime}\,|\,s^{\prime})$ (155) $\displaystyle=$ $\displaystyle P_{t-\tau}(\widetilde{s}_{t}=s\,|\,\xi_{t-\tau},s_{t-\tau})\pi_{ \theta_{t-\tau}}(a\,|\,s)P(s^{\prime}\,|\,s,a,\eta_{t-\tau})\pi_{\theta_{t- \tau}}(a^{\prime}\,|\,s^{\prime})$ (156) $\displaystyle\cdot\int_{\Xi^{2}}P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t-\tau},s_{t-\tau },s_{t}=s)P_{t}(\xi_{t}=\xi^{\prime}\,|\,\xi_{t-\tau},s_{t-\tau},\xi_{t-1}=\xi ,s_{t}=s,a_{t}=a)\mathrm{d}\xi\mathrm{d}\xi^{\prime}.$ (157) $$

Therefore, we can decompose the distribution difference into four parts:

$$ $\displaystyle P_{t}-P_{t-\tau}$ (158) $\displaystyle=$ $\displaystyle\int_{\Xi^{2}}\left(P_{t}(s_{t}=s\,|\,\mathcal{F}_{t-\tau})-P_{t- \tau}(\widetilde{s}_{t}=s\,|\,\mathcal{F}_{t-\tau})\right)P_{t}(\xi_{t-1}=\xi \,|\,\xi_{t-\tau},s_{t-\tau},s_{t}=s)$ (159) $\displaystyle\qquad\pi_{\theta}(a\,|\,s)P(s^{\prime}\,|\,s,a,\eta^{\prime})P_{ t}(\xi_{t}=\xi^{\prime}\,|\,\xi_{t-\tau},s_{t-\tau},\xi_{t-1}=\xi,s_{t}=s,a_{t }=a)\pi_{\theta^{\prime}}(a^{\prime}\,|\,s^{\prime})\mathrm{d}\xi\mathrm{d}\xi ^{\prime}$ (S 1 ) $\displaystyle+\int_{\Xi^{2}}P_{t-\tau}(\widetilde{s}_{t}=s\,|\,\mathcal{F}_{t- \tau})P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t-\tau},s_{t-\tau},s_{t}=s)\left(\pi_{ \theta}(a\,|\,s)-\pi_{\theta_{t-\tau}}(a\,|\,s)\right)$ (160) $\displaystyle\qquad P(s^{\prime}\,|\,s,a,\eta^{\prime})P_{t}(\xi_{t}=\xi^{ \prime}\,|\,\xi_{t-\tau},s_{t-\tau},\xi_{t-1}=\xi,s_{t}=s,a_{t}=a)\pi_{\theta^ {\prime}}(a^{\prime}\,|\,s^{\prime})\mathrm{d}\xi\mathrm{d}\xi^{\prime}$ (S 2 ) $\displaystyle+\int_{\Xi^{2}}P_{t-\tau}(\widetilde{s}_{t}=s\,|\,\mathcal{F}_{t- \tau})P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t-\tau},s_{t-\tau},s_{t}=s)\pi_{\theta_{t- \tau}}(a\,|\,s)\left(P(s^{\prime}\,|\,s,a,\eta^{\prime})-P(s^{\prime}\,|\,s,a, \eta_{t-\tau})\right)$ (161) $\displaystyle\qquad P_{t}(\xi_{t}=\xi^{\prime}\,|\,\xi_{t-\tau},s_{t-\tau},\xi _{t-1}=\xi,s_{t}=s,a_{t}=a)\pi_{\theta^{\prime}}(a^{\prime}\,|\,s^{\prime}) \mathrm{d}\xi\mathrm{d}\xi^{\prime}$ (S 3 ) $\displaystyle+\int_{\Xi^{2}}P_{t-\tau}(\widetilde{s}_{t}=s\,|\,\mathcal{F}_{t- \tau})P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t-\tau},s_{t-\tau},s_{t}=s)\pi_{\theta_{t- \tau}}(a\,|\,s)P(s^{\prime}\,|\,s,a,\eta_{t-\tau})$ (162) $\displaystyle\qquad P_{t}(\xi_{t}=\xi^{\prime}\,|\,\xi_{t-\tau},s_{t-\tau},\xi _{t-1}=\xi,s_{t}=s,a_{t}=a)\left(\pi_{\theta^{\prime}}(a^{\prime}\,|\,s^{ \prime})-\pi_{\theta_{t-\tau}}(a^{\prime}\,|\,s^{\prime})\right)\mathrm{d}\xi \mathrm{d}\xi^{\prime}.$ (S 4 ) $$

We now bound each part separately.
By integrating out the later part, S1 becomes the marginal difference of the state distribution:

$$ $\displaystyle\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\mathrm{S}_{1}\mathrm{ d}s\mathrm{d}a\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}=$ $\displaystyle\int_{\mathcal{S}}\left(P_{t}(s_{t}=s\,|\,\mathcal{F}_{t-\tau})-P _{t-\tau}(\widetilde{s}_{t}=s\,|\,\mathcal{F}_{t-\tau})\right)$ (163) $\displaystyle\cdot\Biggl{(}\int_{\mathcal{S}\times\mathcal{A}^{2}\times\Xi^{2} }P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t-\tau},s_{t-\tau},s_{t}=s)\pi_{\theta}(a\,|\,s) P(s^{\prime}\,|\,s,a,\eta^{\prime})$ (164) $\displaystyle P_{t}(\xi_{t}=\xi^{\prime}\,|\,\xi_{t-\tau},s_{t-\tau},\xi_{t-1} =\xi,s_{t}=s,a_{t}=a)\pi_{\theta^{\prime}}(a^{\prime}\,|\,s^{\prime})\mathrm{d }\xi\mathrm{d}\xi^{\prime}\mathrm{d}a\mathrm{d}s^{\prime}\mathrm{d}a^{\prime} \Biggr{)}\mathrm{d}s$ (165) $\displaystyle=$ $\displaystyle\int_{\mathcal{S}}\left(P_{t}(s_{t}=s\,|\,\mathcal{F}_{t-\tau})-P _{t-\tau}(\widetilde{s}_{t}=s\,|\,\mathcal{F}_{t-\tau})\right)\mathrm{d}s$ (166) $\displaystyle\leq$ $\displaystyle\left\|P_{t}(s_{t}=\cdot\,|\,\mathcal{F}_{t-\tau})-P_{t-\tau}( \widetilde{s}_{t}=\cdot\,|\,\mathcal{F}_{t-\tau})\right\|_{\mathrm{TV}}$ (167) $\displaystyle=$ $\displaystyle\left\|P_{t-1}(s_{t}=\cdot\,|\,\mathcal{F}_{t-\tau})-P_{t-\tau}( \widetilde{s}_{t}=\cdot\,|\,\mathcal{F}_{t-\tau})\right\|_{\mathrm{TV}}.$ (168) $$

∎

By Jensen’s inequality, we have

$$ $\displaystyle\left\|P_{t-1}(s_{t}=\cdot\,|\,\mathcal{F}_{t-\tau})-P_{t-\tau}( \widetilde{s}_{t}=\cdot\,|\,\mathcal{F}_{t-\tau})\right\|_{\mathrm{TV}}$ (169) $\displaystyle=$ $\displaystyle\left\|\int_{\mathcal{S}\times\mathcal{A}^{2}}P_{t-1}(O_{t-1}=(s, a,\cdot,a^{\prime})\,|\,\mathcal{F}_{t-\tau})-P_{t-\tau}(\widetilde{O}_{t-1}=( s,a,\cdot,a^{\prime})\,|\,\mathcal{F}_{t-\tau})\mathrm{d}s\mathrm{d}a\mathrm{d }a^{\prime}\right\|_{\mathrm{TV}}$ (170) $\displaystyle\leq$ $\displaystyle\int_{\mathcal{S}\times\mathcal{A}^{2}}\left\|P_{t-1}(O_{t-1}=(s, a,\cdot,a^{\prime})\,|\,\mathcal{F}_{t-\tau})-P_{t-\tau}(\widetilde{O}_{t-1}=( s,a,\cdot,a^{\prime})\,|\,\mathcal{F}_{t-\tau})\right\|_{\mathrm{TV}}\mathrm{d }s\mathrm{d}a\mathrm{d}a^{\prime}$ (171) $\displaystyle=$ $\displaystyle\left\|P_{t-1}(O_{t}=\cdot\,|\,\mathcal{F}_{t-\tau})-P_{t-\tau}( \widetilde{O}=\cdot\,|\,\mathcal{F}_{t-\tau})\right\|_{\mathrm{TV}}.$ (172) $$

That is, S1 is recursively bounded by $\|P_{t-1}-P_{t-\tau}\|_{\mathrm{TV}}$.

For S2, we have

$$ $\displaystyle\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\mathrm{S}_{2}\mathrm{ d}s\mathrm{d}a\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}$ (173) $\displaystyle=$ $\displaystyle\int_{\mathcal{S}\times\mathcal{A}\times\Xi}P_{t-\tau}(\widetilde {s}_{t}=s\,|\,\mathcal{F}_{t-\tau})P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t-\tau},s_{t- \tau},s_{t}=s)\left(\pi_{\theta}(a\,|\,s)-\pi_{\theta_{t-\tau}}(a\,|\,s)\right)$ (174) $\displaystyle\cdot\Biggl{(}\int_{\mathcal{S}\times\mathcal{A}\times\Xi}P(s^{ \prime}\,|\,s,a,\eta^{\prime})P_{t}(\xi_{t}=\xi^{\prime}\,|\,\xi_{t-\tau},s_{t -\tau},\xi_{t-1}=\xi,s_{t}=s,a_{t}=a)\pi_{\theta^{\prime}}(a^{\prime}\,|\,s^{ \prime})\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}\mathrm{d}\xi^{\prime}\Biggr{) }\mathrm{d}s\mathrm{d}a\mathrm{d}\xi$ (175) $\displaystyle=$ $\displaystyle\int_{\mathcal{S}\times\mathcal{A}\times\Xi}P_{t-\tau}(\widetilde {s}_{t}=s\,|\,\mathcal{F}_{t-\tau})P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t-\tau},s_{t- \tau},s_{t}=s)\left(\pi_{\theta}(a\,|\,s)-\pi_{\theta_{t-\tau}}(a\,|\,s)\right )\mathrm{d}s\mathrm{d}a\mathrm{d}\xi$ (176) $\displaystyle\leq$ $\displaystyle\int_{\mathcal{S}\times\Xi}P_{t-\tau}(\widetilde{s}_{t}=s\,|\, \mathcal{F}_{t-\tau})P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t-\tau},s_{t-\tau},s_{t}=s) \left\|\pi_{\theta}(\cdot\,|\,s)-\pi_{\theta_{t-\tau}}(\cdot\,|\,s)\right\|_{ \mathrm{TV}}\mathrm{d}s\mathrm{d}\xi.$ (177) $$

By Assumption 2 and Lemma 10, for any $\xi\in\Xi$ such that $P_{t}(\xi_{t-1}=\xi\,|\,\mathcal{F}_{t-\tau})\neq 0$ (which is equivalent to $P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t-\tau},s_{t-\tau},s_{t}=s)\neq 0$ as all policies are ergodic), we have

$$ $\|\pi_{\theta}(\cdot\,|\,s)-\pi_{\theta_{t-\tau}}(\cdot\,|\,s)\|_{\mathrm{TV}} \leq\alpha_{t-\tau}(\tau-1)L_{\pi}((1+\gamma)G+R).$ (178) $$

Therefore, we get

$$ $\displaystyle\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\mathrm{S}_{2}\mathrm{ d}s\mathrm{d}a\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}\leq$ $\displaystyle\alpha_{t-\tau}(\tau-1)L_{\pi}((1+\gamma)G+R)\int_{\mathcal{S} \times\Xi}P_{t-\tau}(\widetilde{s}_{t}=s\,|\,\mathcal{F}_{t-\tau})P_{t}(\xi_{t -1}=\xi\,|\,\xi_{t-\tau},s_{t-\tau},s_{t}=s)\mathrm{d}s\mathrm{d}\xi$ (179) $\displaystyle=$ $\displaystyle\alpha_{t-\tau}(\tau-1)L_{\pi}((1+\gamma)G+R).$ (180) $$

For S3, we have

$$ $\displaystyle\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\mathrm{S}_{3}\mathrm{ d}s\mathrm{d}a\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}$ (181) $\displaystyle=$ $\displaystyle\int_{\mathcal{S}^{2}\times\mathcal{A}\times\Xi^{2}}P_{t-\tau}( \widetilde{s}_{t}=s\,|\,\mathcal{F}_{t-\tau})P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t- \tau},s_{t-\tau},s_{t}=s)\pi_{\theta_{t-\tau}}(a\,|\,s)$ (182) $\displaystyle\cdot\left(P(s^{\prime}\,|\,s,a,\eta^{\prime})-P(s^{\prime}\,|\,s ,a,\eta_{t-\tau})\right)P_{t}(\xi_{t}=\xi^{\prime}\,|\,\xi_{t-\tau},s_{t-\tau} ,\xi_{t-1}=\xi,s_{t}=s,a_{t}=a)$ (183) $\displaystyle\cdot\left(\int_{\mathcal{A}}\pi_{\theta^{\prime}}(a^{\prime}\,| \,s^{\prime})\mathrm{d}a^{\prime}\right)\mathrm{d}s\mathrm{d}a\mathrm{d}s^{ \prime}\xi\mathrm{d}\xi^{\prime}$ (184) $\displaystyle\leq$ $\displaystyle\int_{\mathcal{S}\times\mathcal{A}\times\Xi^{2}}P_{t-\tau}( \widetilde{s}_{t}=s\,|\,\mathcal{F}_{t-\tau})P_{t}(\xi_{t-1}=\xi\,|\,\xi_{t- \tau},s_{t-\tau},s_{t}=s)\pi_{\theta_{t-\tau}}(a\,|\,s)$ (185) $\displaystyle\cdot\left\|P(\cdot\,|\,s,a,\eta^{\prime})-P(\cdot\,|\,s,a,\eta_{ t-\tau})\right\|_{\mathrm{TV}}P_{t}(\xi_{t}=\xi^{\prime}\,|\,\xi_{t-\tau},s_{t -\tau},\xi_{t-1}=\xi,s_{t}=s,a_{t}=a)\mathrm{d}s\mathrm{d}a\mathrm{d}\xi \mathrm{d}\xi^{\prime}.$ (186) $$

Similar to the bound for S2, by Assumption 1 and Lemma 10, for any $\xi^{\prime}\in\Xi$ such that $P_{t}(\xi_{t}=\xi^{\prime}\,|\,\mathcal{F}_{t-\tau})\neq 0$, we have

$$ $\left\|P(\cdot\,|\,s,a,\eta^{\prime})-P(\cdot\,|\,s,a,\eta_{t-\tau})\right\|_{ \mathrm{TV}}\leq 2\alpha_{t-\tau}\tau L_{P}.$ (187) $$

Therefore, we get

$$ $\displaystyle\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\mathrm{S}_{3}\mathrm{ d}s\mathrm{d}a\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}\leq$ $\displaystyle\alpha_{t-\tau}\tau L_{P}H\int_{\mathcal{S}\times\mathcal{A} \times\Xi^{2}}P_{t-\tau}(\widetilde{s}_{t}=s\,|\,\mathcal{F}_{t-\tau})P_{t}( \xi_{t-1}=\xi\,|\,\xi_{t-\tau},s_{t-\tau},s_{t}=s)$ (188) $\displaystyle\pi_{\theta_{t-\tau}}(a\,|\,s)P_{t}(\xi_{t}=\xi^{\prime}\,|\,\xi_ {t-\tau},s_{t-\tau},\xi_{t-1}=\xi,s_{t}=s,a_{t}=a)\mathrm{d}s\mathrm{d}a \mathrm{d}\xi\mathrm{d}\xi^{\prime}$ (189) $\displaystyle=$ $\displaystyle 2\alpha_{t-\tau}\tau L_{P}.$ (190) $$

Similar to the bound for S2, for S4, we have

$$ $\int_{\mathcal{S}^{2}\times\mathcal{A}^{2}}\mathrm{S}_{4}\mathrm{d}s\mathrm{d} a\mathrm{d}s^{\prime}\mathrm{d}a^{\prime}\leq\alpha_{t-\tau}\tau L_{\pi}((1+ \gamma)G+R).$ (191) $$

Plugging back the above bounds for S1, S2, S3, and S4, we get

$$ $\displaystyle\left\|P_{t}-P_{t-\tau}\right\|_{\mathrm{TV}}\leq$ $\displaystyle\left\|P_{t-1}-P_{t-\tau}\right\|_{\mathrm{TV}}+2\tau\alpha_{t- \tau}(L_{\pi}((1+\gamma)D+R)+L_{P}F)$ (192) . (193) $$

Recursively, we get

$$ $\|P_{t}-P_{t-\tau}\|_{\mathrm{TV}}\leq 2LH\sum_{l=1}^{\tau}\left(l\alpha_{t-l} \right)\leq\alpha_{t-\tau}\tau(\tau+1)LH,$ (194) $$

where we require the step size to be non-increasing.
Plugging the above bound back to Equation 150 gives the desired result.

###### Lemma 8 (Error decomposition) .

###### Lemma 9 (Descent) .

###### Proof.

###### Lemma 10 (Progress) .

###### Proof.

###### Lemma 11 (Mix) .

###### Proof.

###### Lemma 12 (Bakctrack) .

###### Proof.

#### F.2 Proof of Theorem 1

###### Theorem 1 .

Let $\xi_{*}=(\theta_{*};\eta_{*})$ be a (projected) MFE.
Let $\{\xi_{t}=(\theta_{t};\eta_{t})\}$ be a sequence of parameters generated by Algorithm 1.
Then, under Assumptions 1, 2 and 3, if $3\sigma LH+L_{r}\leq 2w$, we have

$$ $\mathbb{E}\left\|\xi_{t+1}-\xi_{*}\right\|^{2}\leq(1-\alpha_{t}w)\mathbb{E}\| \xi_{t}-\xi_{*}\|^{2}+\alpha_{t}^{2}H^{2}+O\left(\frac{\alpha_{t}^{3}\log^{4} \alpha_{t}^{-1}L^{2}H^{4}}{w}\right).$ (195) $$

###### Proof.

We denote $\Delta\xi_{t}=\xi_{t}-\xi_{*}$.
We first plug Lemmas 3 and 1 into Lemma 8 to get

$$ $\displaystyle\mathbb{E}\|\Delta\xi_{t+1}\|^{2}\leq$ $\displaystyle(1-2\alpha_{t}w)\mathbb{E}\|\Delta\xi_{t}\|^{2}+\alpha_{t}^{2}H^{2}$ (196) $\displaystyle+2\alpha_{t}\mathbb{E}\left<\Delta\xi,\bar{\mathfrak{g}}_{t}(\xi_ {t})-\bar{\mathfrak{g}}_{t-\tau}(\xi_{t})\right>$ (197) $\displaystyle+2\alpha_{t}\mathbb{E}\left<\Delta\xi,\bar{\mathfrak{g}}_{t-\tau} (\xi_{t})-{\mathfrak{g}}_{t-\tau}(\xi_{t})\right>$ (198) $\displaystyle+2\alpha_{t}\mathbb{E}\left<\Delta\xi,{\mathfrak{g}}_{t-\tau}(\xi _{t})-{\mathfrak{g}}_{t}(\xi_{t})\right>.$ (199) $$

For Equation 197, by Young’s inequality, for any $\beta>0$, we have

$$ $2\mathbb{E}\left<\Delta\xi_{t},\bar{\mathfrak{g}}_{t}(\xi_{t})-\bar{\mathfrak{ g}}_{t-\tau}(\xi_{t})\right>\leq\beta\mathbb{E}\left\|\Delta\xi_{t}\right\|^{2 }+\frac{1}{\beta}\mathbb{E}\left\|\bar{\mathfrak{g}}_{t-\tau}(\xi_{t})-\bar{ \mathfrak{g}}_{t}(\xi_{t})\right\|^{2}.$ (200) $$

By Lemma 10, we get

$$ $2\mathbb{E}\left<\Delta\xi_{t},\bar{\mathfrak{g}}_{t}(\xi_{t})-\bar{\mathfrak{ g}}_{t-\tau}(\xi_{t})\right>\leq\beta\mathbb{E}\left\|\Delta\xi_{t}\right\|^{2 }+\beta^{-1}\alpha^{2}_{t-\tau}\tau^{2}C^{2}_{\mathrm{prog}}.$ (201) $$

For Equation 198, note that conditioned on $\mathcal{F}_{t-\tau}$, $\bar{\mathfrak{g}}_{t-\tau}$ is determined and $\mathfrak{g}_{t-\tau}$ is only dependent on the virtual Markovian trajectory. Thus, $\xi_{t}$ is independent of $\mathfrak{g}_{t-\tau}$ and $\bar{\mathfrak{g}}_{t-\tau}$ conditioned on $\mathcal{F}_{t-\tau}$. Therefore, by Lemma 11 and Young’s inequality, we have

$$ $\displaystyle 2\mathbb{E}\left<\Delta\xi_{t},\bar{\mathfrak{g}}_{t-\tau}(\xi_{ t})-\mathfrak{g}_{t-\tau}(\xi_{t})\right>=$ $\displaystyle 2\mathbb{E}\left[\left<\mathbb{E}\left[\Delta\xi_{t}\,|\, \mathcal{F}_{t-\tau}\right],\mathbb{E}\left[\mathfrak{g}_{t-\tau}(\xi_{t})- \bar{\mathfrak{g}}_{t-\tau}(\xi_{t})\,|\,\mathcal{F}_{t-\tau}\right]\right>\right]$ (202) $\displaystyle\leq$ $\displaystyle 2\mathbb{E}\left[\mathbb{E}\left[\left\|\Delta\xi_{t}\right\|\,| \,\mathcal{F}_{t-\tau}\right]\cdot\left\|\mathbb{E}\left[\mathfrak{g}_{t-\tau} (\xi_{t})-\bar{\mathfrak{g}}_{t-\tau}(\xi_{t})\,|\,\mathcal{F}_{t-\tau}\right] \right\|\right]$ (203) $\displaystyle\leq$ $\displaystyle 2m\rho^{\tau}H\mathbb{E}\left\|\Delta\xi_{t}\right\|$ (204) $\displaystyle\leq$ $\displaystyle\beta\mathbb{E}\left\|\Delta\xi_{t}\right\|^{2}+\beta^{-1}m^{2} \rho^{2\tau}H^{2}.$ (205) $$

For Equation 199, we cannot directly apply Lemma 12 as $\xi_{t}$ and $\mathfrak{g}_{t}$ are dependent conditioned on $\mathcal{F}_{t-\tau}$. To proceed, we employ the following decomposition:

$$ $\displaystyle\mathbb{E}\left<\Delta\xi_{t},\mathfrak{g}_{t-\tau}(\xi_{t})- \mathfrak{g}_{t}(\xi_{t})\right>=$ $\displaystyle\underbrace{\mathbb{E}\left<\Delta\xi_{t},\left(\mathfrak{g}_{t- \tau}(\xi_{t})-\mathfrak{g}_{t-\tau}(\xi_{t-\tau})\right)-\left(\mathfrak{g}_{ t}(\xi_{t})-\mathfrak{g}_{t}(\xi_{t-\tau})\right)\right>}_{H_{1}}$ (206) $\displaystyle+\underbrace{\mathbb{E}\left<\xi_{t}-\xi_{t-\tau},\mathfrak{g}_{t -\tau}(\xi_{t-\tau})-\mathfrak{g}_{t}(\xi_{t-\tau})\right>}_{H_{2}}+ \underbrace{\mathbb{E}\left<\Delta\xi_{t-\tau},\mathfrak{g}_{t-\tau}(\xi_{t- \tau})-\mathfrak{g}_{t}(\xi_{t-\tau})\right>}_{H_{3}}.$ (207) $$

For $H_{1}$, by the Lipschitzness of the semi-gradient (Lemma 7) and Lemma 10, we have

$$ $H_{1}\leq 2\alpha_{t-\tau}\tau H^{2}\cdot\mathbb{E}\left\|\Delta\xi_{t}\right\|.$ (208) $$

For $H_{2}$, by Lemmas 3 and 10, we get

$$ $H_{2}\leq 2H\cdot\alpha_{t-\tau}\tau H.$ (209) $$

For $H_{3}$, $\xi_{t-\tau}$ is independent of $\mathfrak{g}_{t}$ and $\mathfrak{g}_{t-\tau}$ conditioned on $\mathcal{F}_{t-\tau}$.
Similar to the bound of Equation 205, by Lemma 12, we have

$$ $\displaystyle H_{3}=$ $\displaystyle\mathbb{E}\left[\left<\mathbb{E}\left[\Delta\xi_{t-\tau}\,\middle |\,\mathcal{F}_{t-\tau}\right],\mathbb{E}\left[\mathfrak{g}_{t-\tau}(\xi_{t- \tau})-\mathfrak{g}_{t}(\xi_{t-\tau})\,\middle|\,\mathcal{F}_{t-\tau}\right] \right>\right]$ (210) $\displaystyle\leq$ $\displaystyle\mathbb{E}\left[\mathbb{E}\left[\|\Delta\xi_{t-\tau}\|\,\middle| \,\mathcal{F}_{t-\tau}\right]\cdot\left\|\mathbb{E}\left[\mathfrak{g}_{t}(\xi_ {t-\tau})-\mathfrak{g}_{t-\tau}(\xi_{t-\tau})\,\middle|\,\mathcal{F}_{t-\tau} \right]\right\|\right]$ (211) $\displaystyle\leq$ $\displaystyle\alpha_{t-\tau}\tau C_{\mathrm{back}}\mathbb{E}\left\|\Delta\xi_{ t-\tau}\right\|.$ (212) $$

Then, applying Lemma 10 and Young’s inequality gives

$$ $2H_{3}\leq\alpha_{t-\tau}\tau C_{\mathrm{back}}\left(\alpha_{t-\tau}\tau H+ \mathbb{E}\|\Delta\xi_{t}\|^{2}\right)\leq 2\alpha^{2}_{t-\tau}\tau^{2}HC_{ \mathrm{back}}+\beta\mathbb{E}\|\Delta\xi_{t}\|^{2}+\beta^{-1}\alpha^{2}_{t- \tau}\tau^{2}C^{2}_{\mathrm{back}}.$ (213) $$

Plugging $H_{1}$, $H_{2}$, and $H_{3}$ back into the decomposition and applying Young’s inequality, we get

$$ $\displaystyle 2\mathbb{E}\left<\Delta\xi_{t},\mathfrak{g}_{t-\tau}(\xi_{t})- \mathfrak{g}_{t}(\xi_{t})\right>\leq$ $\displaystyle 2\beta\mathbb{E}\left\|\Delta\xi_{t}\right\|^{2}+4\alpha_{t-\tau }\tau H^{2}+\alpha^{2}_{t-\tau}\tau^{2}\left(2HC_{\mathrm{back}}+\beta^{-1} \left(C_{\mathrm{back}}^{2}+4H^{4}\right)\right).$ (214) $$

Finally, plugging the bounds of Equations 201, 205 and 214 back into Equations 197, 198 and 199 gives

$$ $\displaystyle\mathbb{E}\|\Delta\xi_{t+1}\|^{2}\leq$ $\displaystyle(1-2\alpha_{t}w+4\alpha_{t}\beta)\mathbb{E}\|\Delta\xi_{t}\|^{2}+ \alpha_{t}^{2}H^{2}+4\alpha_{t}\alpha_{t-\tau}\tau H^{2}$ (215) $\displaystyle+\alpha_{t}\left(\beta^{-1}\alpha_{t-\tau}^{2}\tau^{2}C_{\mathrm{ prog}}^{2}+\beta^{-1}m^{2}\rho^{2\tau}H^{2}+\alpha_{t-\tau}^{2}\tau^{2}(2HC_{ \mathrm{back}}+\beta^{-1}(C^{2}_{\mathrm{back}}+4H^{4}))\right).$ (216) $$

Now we choose $\tau=\left\lceil\frac{\log\alpha_{t}^{-1}}{\log\rho^{-1}}\right\rceil$. Then, $\rho^{\tau}\leq\alpha_{t}$.
We require the step-size to be non-increasing, and not too samll such that

$$ $\sum_{t=1}^{\infty}\alpha_{t}=+\infty\quad\text{and}\quad\log\alpha_{t}^{-1}=o (T).$ (217) $$

The first condition indicates that $\limsup_{t\to\infty}\alpha_{t/2}/\alpha_{t}\coloneqq C_{\alpha,1}<\infty$.
And there exists $C_{\alpha,2}>0$ such that $\alpha_{t/2}/\alpha_{t}\leq C_{\alpha,2}\cdot\limsup_{t\to\infty}\alpha_{t/2}/
\alpha_{t}$ for any time step $t$.
The second condition indicates the existence of $C_{\alpha,2}>0$ such that $\alpha_{t-\tau}\leq C_{\alpha,3}\alpha_{t/2}$ for any $t$.
In conclusion, we have

$$ $\alpha_{t-\tau}\leq C_{\alpha,3}\alpha_{t/2}\leq C_{\alpha,3}C_{\alpha,2}C_{ \alpha,1}\alpha_{t}\eqqcolon C_{\alpha}\alpha_{t}.$ (218) $$

Then, we choose $\beta=w/4$. Together, we get

$$ $\displaystyle\mathbb{E}\|\Delta\xi_{t+1}\|^{2}\leq$ $\displaystyle(1-\alpha_{t}w)\mathbb{E}\|\Delta\xi_{t}\|^{2}+\alpha^{2}_{t}(H^{ 2}+4\tau C_{\alpha}H^{2})$ (219) $\displaystyle+\alpha^{3}_{t}\left(\frac{4}{w}\left(\tau^{2}C^{2}_{\alpha}(C^{2 }_{\mathrm{prog}}+C_{\mathrm{back}}^{2}+4H^{4})+m^{2}H^{2}\right)+2\tau^{2}C_{ \alpha}^{2}C_{\mathrm{back}}H\right)$ (220) $\displaystyle\eqqcolon$ $\displaystyle(1-\alpha_{t}w)\mathbb{E}\|\Delta\xi_{t}\|^{2}+\alpha^{2}_{t} \cdot C_{2}(\alpha_{t})+\alpha^{3}_{t}\cdot C_{3}(\alpha_{t}),$ (221) $$

where we use $C_{2}$ and $C_{3}$ to encapsulate the terms in the right-hand side of Equation 219.
Plugging back the definitions of $C_{\mathrm{prog}}$ and $C_{\mathrm{back}}$ gives the final result

$$ $\mathbb{E}\|\Delta\xi_{t+1}\|^{2}=(1-\alpha_{t}w)\mathbb{E}\|\Delta\xi_{t}\|^{ 2}+H^{2}\cdot O(\alpha^{2}_{t}\log\alpha_{t}^{-1})+\frac{\max\left\{L_{\pi},L_ {P},L_{r}\right\}^{2}H^{4}}{w}\cdot O(\alpha^{3}_{t}\log^{4}\alpha_{t}^{-1}),$ (222) $$

where the asymptotic equivalence holds as $\alpha_{t}\to 0$.
∎

###### Theorem 1 .

###### Proof.

#### F.3 Proof of Corollaries

###### Corollary 1 .

For a constant step-size $\alpha_{t}\equiv\alpha_{0}$, we have

$$ $\mathbb{E}\|\xi_{t}-\xi_{*}\|^{2}\leq e^{-\alpha_{0}wt}\|\xi_{0}-\xi_{*}\|^{2} +\frac{\alpha_{0}H^{2}}{w}+{O}\left(\frac{L^{2}H^{4}\alpha_{0}^{2}}{w^{2}} \right).$ (223) $$

###### Proof.

By Theorem 1, we have

$$ $\displaystyle\mathbb{E}\|\xi_{T}-\xi_{*}\|^{2}\leq$ $\displaystyle(1-\alpha_{0}w)^{T}\|\xi_{0}-\xi_{*}\|^{2}+\sum_{t=0}^{T}(1- \alpha_{0}w)^{T-t}(\alpha_{0}^{2}C_{2}+\alpha_{0}^{3}C_{3})$ (224) $\displaystyle\leq$ $\displaystyle e^{-\alpha_{0}wT}\|\xi_{0}-\xi_{*}\|^{2}+\frac{\alpha_{0}}{w}(C_ {2}+\alpha_{0}C_{3})$ (225) $\displaystyle\leq$ $\displaystyle e^{-\alpha_{0}wT}\|\xi_{0}-\xi_{*}\|^{2}+w^{-1}H^{2}\cdot O( \alpha_{0}\log\alpha_{0}^{-1}).$ (226) $$

∎

###### Corollary 3 (Linearly decaying step-size) .

We define a linearly decaying step-size sequence $\alpha_{t}=4/(w(t+1))$,
and the convex combination $\tilde{\xi}_{T}\coloneqq\frac{2}{T(T+1)}\sum_{t=0}^{T}t\xi_{t}$.
Then, we have

$$ $\mathbb{E}\|\tilde{\xi}_{T}-\xi_{*}\|^{2}\leq O\left(\frac{H^{2}\log T}{w^{2}T }\right).$ (227) $$

###### Proof.

Rearranging the result of Theorem 1 gives

$$ $\frac{1}{2}\mathbb{E}\|\Delta\xi_{t}\|^{2}\leq\left(\frac{1}{\alpha_{t}w}- \frac{1}{2}\right)\mathbb{E}\|\Delta\xi_{t}\|^{2}-\frac{1}{\alpha_{t}w}\mathbb {E}\|\Delta\xi_{t+1}\|^{2}+\frac{\alpha_{t}}{w}C_{2}+\frac{\alpha^{2}_{t}}{w}C _{3}.$ (228) $$

Substituting $\alpha_{t}=4/(w(t+1))$ and multiplying by $t$ gives

$$ $t\mathbb{E}\|\Delta\xi_{t}\|^{2}\leq\frac{(t-1)t}{2}\mathbb{E}\|\Delta\xi_{t} \|^{2}-\frac{t(t+1)}{2}\mathbb{E}\|\Delta\xi_{t+1}\|^{2}+\frac{8t}{w^{2}(t+1)} C_{2}+\frac{32t}{w^{3}(t+1)^{2}}C_{3}.$ (229) $$

By Jensen’s inequality, we have

$$ $\mathbb{E}\|\tilde{\xi}_{T}-\xi_{*}\|^{2}\leq\frac{1}{T(T+1)}\sum_{t=0}^{T}t \mathbb{E}\|\Delta\xi_{t}\|^{2}.$ (230) $$

Combining Equations 229 and 230 gives

$$ $\mathbb{E}\|\tilde{\xi}_{T}-\xi_{*}\|^{2}\leq\frac{4}{w^{2}T(T+1)}\left(C_{2} \sum_{t=0}^{T}\frac{t}{t+1}+\frac{4C_{3}}{w}\sum_{t=0}^{T}\frac{t}{(t+1)^{2}} \right)=O\left(\frac{H^{2}\log T}{w^{2}T}\right).$ (231) $$

∎

###### Corollary 1 .

###### Proof.

###### Corollary 3 (Linearly decaying step-size) .

###### Proof.

### Appendix G Approximation error analysis

This section aims to characterize the approximation error of SemiSGD for general (non-linear) MFGs.
Recall that SemiSGD converges to the projected MFE (Appendix D).
Let $\xi_{\diamond}=(\theta_{\diamond};\eta_{\diamond})$ be the convergence point of SemiSGD with PA-LFA.
Let $(q_{*},\mu_{*})$ be the actual MFE.
The approximation error is defined as

$$ $\epsilon_{q}\coloneqq\|q_{*}-\left<\phi,\theta_{\diamond}\right>\|_{\infty}, \quad\epsilon_{\mu}\coloneqq\|\mu_{*}-\left<\psi,\eta_{\diamond}\right>\|_{ \mathrm{TV}}.$ (232) $$

Additionally, we define the inherent error of the chosen basis as

$$ $\epsilon_{\phi}\coloneqq\|q_{*}-\Pi_{\phi}q_{*}\|_{\infty},\quad\epsilon_{\psi }\coloneqq\|\mathcal{P}_{*}-\Pi_{\psi}\mathcal{P}_{*}\|_{\mathrm{TV}}.$ (233) $$

The next two lemmas bound the approximation errors in Equation 232 separately, showing how they are correlated.
Theorem 2 is a direct corollary of Lemmas 13 and 14 under the small Lipschitz constants assumption.

###### Lemma 13 (Value function approximation error) .

$$ $\epsilon_{q}\leq\left(\gamma+\frac{\gamma\sigma RL_{\pi}}{1-\gamma}\right) \epsilon_{q}+\left(L_{r}+\frac{\gamma\sigma RL_{P}}{(1-\gamma)\sqrt{d_{2}}} \right)\epsilon_{\mu}+\epsilon_{\phi}.$ (234) $$

###### Proof.

We first have the decomposition:

$$ $\epsilon_{q}\leq\|\left<\phi,\theta_{\diamond}\right>-\Pi_{\phi}q_{*}\|_{ \infty}+\|\Pi_{\phi}q_{*}-q_{*}\|_{\infty},$ (235) $$

where $\Pi_{\phi}$ is the orthogonal projection operator onto the linear span of basis $\phi$ w.r.t. the inner product induced by the $\xi_{\diamond}$.
Since $\xi_{\diamond}$ is a projected MFE, by Definition 3, we get

$$ $\left\|\left<\phi,\theta_{\diamond}\right>-\Pi_{\phi}q_{*}\right\|=\left\|\Pi_ {\phi}\mathcal{T}_{\diamond}\left<\phi,\theta_{\diamond}\right>-\Pi_{\phi}q_{* }\right\|\leq\left\|\mathcal{T}_{\diamond}\left<\phi,\theta_{\diamond}\right>- q_{*}\right\|,$ (236) $$

where $\mathcal{T}_{\diamond}\coloneqq\mathcal{T}_{\xi_{\diamond}}$ and the inequality uses the fact that $\Pi_{\phi}$ is a non-expansive operator.
Since $q_{*}$ is an equilibrium value function, we have

$$ $\|\mathcal{T}_{\diamond}\left<\phi,\theta_{\diamond}\right>-q_{*}\|_{\infty}= \|\mathcal{T}_{\diamond}\left<\phi,\theta_{\diamond}\right>-\mathcal{T}_{*}q_{ *}\|_{\infty}\leq\|\mathcal{T}_{\diamond}\left(\left<\phi,\theta_{\diamond} \right>-q_{*}\right)\|_{\infty}+\left\|\left(\mathcal{T}_{\diamond}-\mathcal{T }_{*}\right)q_{*}\right\|_{\infty}.$ (237) $$

For the first term in Equation 237, the Bellman operator’s definition gives

$$ $\left\|\mathcal{T}_{\diamond}\left(\left<\phi,\theta_{\diamond}\right>-q_{*} \right)\right\|_{\infty}\leq\gamma\left\|\left<\phi,\theta_{\diamond}\right>-q _{*}\right\|_{\infty}=\gamma\epsilon_{q}.$ (238) $$

Similar to the Lipschitzness of TD operators (Lemma 6),
the second term in Equation 237 can be bounded
as follows:

$$ $\displaystyle\left\|\left(\mathcal{T}_{\diamond}-\mathcal{T}_{*}\right)q_{*}(s ,a)\right\|_{\infty}$ (239) $\displaystyle=$ $\displaystyle\left\|r(s,a,\langle\psi,\eta_{\diamond}\rangle)-r(s,a,\mu_{*})+ \int_{\mathcal{S}\times\mathcal{A}}\gamma q_{*}(s^{\prime},a^{\prime})\left( \mu^{\ddagger}_{\xi_{\diamond}}(s,a,s^{\prime},a^{\prime})-\mu^{\ddagger}_{*}( s,a,s^{\prime},a^{\prime})\right)\mathrm{d}s^{\prime}\mathrm{d}a^{\prime} \right\|_{\infty}$ (240) $\displaystyle\leq$ $\displaystyle L_{r}\|\left<\psi,\eta_{\diamond}\right>-\mu_{*}\|+\gamma\|q_{*} \|_{\infty}\|\mu_{\xi_{\diamond}}^{\ddagger}-\mu_{*}^{\ddagger}\|_{\mathrm{TV}}$ (241) $\displaystyle\leq$ $\displaystyle L_{r}\epsilon_{\mu}+\gamma\|q_{*}\|_{\infty}\cdot\sigma\left( \frac{L_{P}}{\sqrt{d_{2}}}\epsilon_{\mu}+L_{\pi}\epsilon_{q}\right),$ (242) $$

where Equation 241 uses Assumption 1 and Equation 242 uses Corollary 2.
Since $q_{*}$ is the best response to $\mu_{*}$, we have

$$ $\|q_{*}\|_{\infty}=\left\|\mathbb{E}_{(q_{*},\mu_{*})}\sum_{t=0}^{\infty} \gamma^{t}r(s_{t},a_{t},\mu_{*})\right\|_{\infty}\leq\sum_{t=0}^{\infty}\gamma ^{t}R=\frac{R}{1-\gamma}.$ (243) $$

Plugging Equation 243 back into Equation 242 gives

$$ $\left\|\left(\mathcal{T}_{\diamond}-\mathcal{T}_{*}\right)q_{*}\right\|_{ \infty}\leq L_{r}\epsilon_{\mu}+\frac{\gamma\sigma R}{1-\gamma}\left(\frac{L_{ P}}{\sqrt{d_{2}}}\epsilon_{\mu}+L_{\pi}\epsilon_{q}\right).$ (244) $$

Plugging Equations 238 and 244 back into Equation 237 gives

$$ $\left\|\mathcal{T}_{\diamond}\left<\phi,\theta_{\diamond}\right>-q_{*}\right\| _{\infty}\leq\left(\gamma+\frac{\gamma\sigma RL_{\pi}}{1-\gamma}\right) \epsilon_{q}+\left(L_{r}+\frac{\gamma\sigma RL_{P}}{(1-\gamma)\sqrt{d_{2}}} \right)\epsilon_{\mu}.$ (245) $$

Plugging Equation 245 back into Equation 236 and then Equation 235 gives the desired result.
∎

###### Lemma 14 (Population measure approximation error) .

###### Proof.

For uniformly ergodic MDPs, $\mathcal{P}_{*}^{\infty}\coloneqq\lim_{t\to\infty}\mathcal{P}_{*}^{t}$ exists, which maps any distribution to $\mu_{*}$.
The uniform ergodicity is equivalent to strong ergodicity [], which implies following relation about the geometric convergence rate []:

$$ $\rho(\mathcal{P}_{*}-\mathcal{P}_{*}^{\infty})\leq\rho<1,$ (246) $$

where $\rho(\mathcal{P})$ returns the spectral radius of $\mathcal{P}$.
Without loss of generality, we assume $\rho>\rho(\mathcal{P}_{*}-\mathcal{P}_{*}^{\infty})$.
Then, by Isaacson and Luecke [, Corollary 3.9], for any $\rho>\rho(\mathcal{P}_{*}-\mathcal{P}_{*}^{\infty})$, there exists $k\in\mathbb{N}$ such that

$$ $\|\mathcal{P}_{*}^{k}-\mathcal{P}_{*}^{\infty}\|_{\mathrm{TV}}\leq\rho^{k},$ (247) $$

where the norm is the operator norm induced by the total variation norm.
Now we apply the decomposition:

$$ $\epsilon_{\mu}\leq\underbrace{\left\|\left<\psi,\eta_{\diamond}\right>- \mathcal{P}_{*}^{k}\left<\psi,\eta_{\diamond}\right>\right\|_{\mathrm{TV}}}_{E _{1}}+\underbrace{\left\|\mathcal{P}_{*}^{k}\left<\psi,\eta_{\diamond}\right>- \mu_{*}\right\|_{\mathrm{TV}}}_{E_{2}}.$ (248) $$

Since $\xi_{\diamond}$ is a projected MFE, by Definition 3, we have

$$ $E_{1}=\left\|\left(\left(\Pi_{\psi}\mathcal{P}_{\diamond}\right)^{k}-\mathcal{ P}_{*}^{k}\right)\left<\psi,\eta_{\diamond}\right>\right\|_{\mathrm{TV}}\leq \left\|\left(\Pi_{\psi}\mathcal{P}_{\diamond}\right)^{k}-\mathcal{P}_{*}^{k} \right\|_{\mathrm{TV}}.$ (249) $$

A further decomposition gives

$$ $E_{1}\leq\left\|\Pi_{\psi}\mathcal{P}_{\diamond}\left(\left(\Pi_{\psi}\mathcal {P}_{\diamond}\right)^{k-1}-\mathcal{P}_{*}^{k-1}\right)+\Pi_{\psi}\left( \mathcal{P}_{\diamond}-\mathcal{P}_{*}\right)\mathcal{P}_{*}^{k-1}+\left(\Pi_{ \psi}-\operatorname{Id}\right)\mathcal{P}_{*}^{k}\right\|_{\mathrm{TV}}.$ (250) $$

Note that both $\Pi_{\psi}$ and $\mathcal{P}$ are non-expansive operators.
By the sub-multiplicativity of operator norms, we have

$$ $E_{1}\leq\left\|\left(\Pi_{\psi}\mathcal{P}_{\diamond}\right)^{k-1}-\mathcal{P }_{*}^{k-1}\right\|_{\mathrm{TV}}+\underbrace{\|\mathcal{P}_{\diamond}- \mathcal{P}_{*}\|_{\mathrm{TV}}}_{E_{3}}+\underbrace{\left\|(\Pi_{\psi}- \operatorname{Id})\mathcal{P}_{*}\right\|_{\mathrm{TV}}}_{\epsilon_{\psi}}.$ (251) $$

We denote $\epsilon_{\psi}\coloneqq\|\mathcal{P}_{*}-\Pi_{\psi}\mathcal{P}_{*}\|_{\mathrm
{TV}}$; $\epsilon_{\psi}$ is the inherent approximation error induced by $\psi$.
For $E_{3}$, by definition, we have

$$ $\displaystyle E_{3}=$ $\displaystyle\sup_{\|\mu\|_{\mathrm{TV}}\leq 1}\left\|\int_{\mathcal{S}\times \mathcal{A}}\left(P(\cdot\,|\,s,a,\left<\psi,\eta_{\diamond}\right>)\pi_{ \theta_{\diamond}}(a\,|\,s)-P(\cdot\,|\,s,a,\mu_{*})\pi_{q_{*}}(a\,|\,s)\right )\mu(s)\mathrm{d}s\mathrm{d}a\right\|_{\mathrm{TV}}$ (252) $\displaystyle\leq$ $\displaystyle L_{\pi}\epsilon_{q}+\frac{L_{P}}{\sqrt{d_{2}}}\epsilon_{\mu}.$ (253) $$

Plugging the above bound of $E_{3}$ back into Equation 251 gives the following recursion:

$$ $\displaystyle E_{1}\leq$ $\displaystyle\left\|\left(\Pi_{\psi}\mathcal{P}_{\diamond}\right)^{k-1}- \mathcal{P}_{*}^{k-1}\right\|_{\mathrm{TV}}+L_{\pi}\epsilon_{q}+\frac{L_{P}}{ \sqrt{d_{2}}}\epsilon_{\mu}+\epsilon_{\psi}$ (254) $\displaystyle\leq$ $\displaystyle k\left(L_{\pi}\epsilon_{q}+\frac{L_{P}}{\sqrt{d_{2}}}\epsilon_{ \mu}+\epsilon_{\psi}\right).$ (255) $$

For $E_{2}$, since $\mu_{*}$ is the equilibrium population measure, and $\mathcal{P}_{*}^{\infty}$ maps any distribution to $\mu_{*}$, we have

$$ $\displaystyle E_{2}=$ $\displaystyle\left\|\mathcal{P}_{*}^{k}\left<\psi,\eta_{\diamond}\right>-\mu_{ *}+\mu_{*}-\mu_{*}\right\|_{\mathrm{TV}}$ (256) $\displaystyle=$ $\displaystyle\left\|\mathcal{P}_{*}^{k}\left<\psi,\eta_{\diamond}\right>- \mathcal{P}_{*}^{k}\mu_{*}+\mathcal{P}_{*}^{\infty}\left<\psi,\eta_{\diamond} \right>-\mathcal{P}_{*}^{\infty}\mu_{*}\right\|_{\mathrm{TV}}$ (257) $\displaystyle=$ $\displaystyle\left\|\left(\mathcal{P}_{*}^{k}-\mathcal{P}_{*}^{\infty}\right) \left(\left<\psi,\eta_{\diamond}\right>-\mu_{*}\right)\right\|_{\mathrm{TV}}$ (258) $\displaystyle\leq$ $\displaystyle\left\|\mathcal{P}_{*}^{k}-\mathcal{P}_{*}^{\infty}\right\|_{ \mathrm{TV}}\epsilon_{\mu}$ (259) $\displaystyle\leq$ $\displaystyle\rho^{k}\epsilon_{\mu}.$ (260) $$

Plugging Equations 255 and 260 back into Equation 248 gives

$$ $\epsilon_{\mu}\leq\left(\rho^{k}+\frac{kL_{P}}{\sqrt{d_{2}}}\right)\epsilon_{ \mu}+kL_{\pi}\epsilon_{q}+k\epsilon_{\psi}.$ (261) $$

Furthermore, if $\|\mathcal{P}_{*}-\mathcal{P}_{*}^{\infty}\|_{\mathrm{TV}}\leq\rho$, for example, $\mathcal{P}_{*}$ corresponds to a reversible Markov chain, then $k=1$.
∎

###### Lemma 13 (Value function approximation error) .

###### Proof.

###### Lemma 14 (Population measure approximation error) .

###### Proof.

### Appendix H Mean field games with finite state-action space

Recall that SemiSGD with PA-LFA (Algorithm 1) reduces to tabular SemiSGD (Algorithm 1) for finite state-action spaces, when the feature map and measure map are chosen as

$$ $\phi(s,a)=e_{(s,a)}\in\mathbb{R}^{|\mathcal{S}||\mathcal{A}|},\quad\psi(s^{ \prime})=e_{s^{\prime}}\in\Delta^{|\mathcal{S}|}.$ (262) $$

Then, $Q=\theta\in\mathbb{R}^{|\mathcal{S}||\mathcal{A}|}$ and $M=\eta\in\Delta^{|\mathcal{S}|}$ are the parameters themselves.

#### H.1 Implicit regularization

We first show that tabular SemiSGD does not need the projection step for regularizing the parameters (see Equation 14).
That is, tabular SemiSGD enjoys implicit regularization.
For $\|M\|$, we first have $\|\psi(s^{\prime})\|_{1}\leq 1\eqqcolon F$.
Recall the stochastic update rule of $M$:

$$ $M_{t+1}=M_{t}-\alpha\left(M_{t}-e_{s_{t+1}}\right)=(1-\alpha)M_{t}+\alpha e_{s _{t+1}}.$ (263) $$

Suppose $M_{t}\in\Delta^{|\mathcal{S}|}$ and the step-size is smaller than one. Them $M_{t+1}\geq 0$. Futhermore, we have

$$ $\|M_{t+1}\|_{1}=\sum_{s\in\mathcal{S}}|(1-\alpha)M_{t}(s)+\alpha e_{s_{t+1}}(s )|=(1-\alpha)\sum_{s\in\mathcal{S}}M_{t}(s)+\alpha\sum_{s\in\mathcal{S}}e_{s_{ t+1}}(s)=(1-\alpha)+\alpha=1,$ (264) $$

indicating that $M_{t+1}\in\Delta^{|\mathcal{S}|}$, without any projection.

For $\|Q\|$, notice that the true action-value function induced by any policy $\pi$ is bounded by
$\|q_{\pi}\|_{\infty}\leq\sum_{t=0}^{\infty}\gamma^{t}R=R/(1-\gamma)=:D_{\infty}$
Suppose current estimated value function satisfies that $\|Q_{t}\|_{\infty}\leq D_{\infty}$, then we have

$$ $\displaystyle\left|Q_{t+1}(s,a)\right|$ $\displaystyle=\left|Q_{t}(s_{t},a_{t})+\alpha\left(r_{t}+\gamma Q_{t}\left(s_{ t+1},a_{t+1}\right)-Q_{t}(s_{t},a_{t})\right)\right|$ (265) $\displaystyle=\left|(1-\alpha)Q_{t}(s_{t},a_{t})+\alpha\gamma Q_{t}\left(s_{t+ 1},a_{t+1}\right)+\alpha r_{t}\right|$ $\displaystyle\leq(1-\alpha)D_{\infty}+\alpha\gamma D_{\infty}+\alpha R$ $\displaystyle=(1-\alpha+\alpha\gamma)\frac{R}{1-\gamma}+\alpha R$ $\displaystyle=\frac{R}{1-\gamma}=D_{\infty}.$ $$

Therefore, if the bound holds for the initial estimated value function, it holds for all sequential estimated value functions.
Then, the following $\ell_{2}$ norm bound holds for all value functions:

$$ $\|Q\|_{2}\leq\sqrt{|\mathcal{S}||\mathcal{A}|}\|Q\|_{\infty}\leq\frac{\sqrt{| \mathcal{S}||\mathcal{A}|}R}{1-\gamma}=:D$ $$

Consequently,

$$ $H=\frac{((1+\gamma)\sqrt{|\mathcal{S}||\mathcal{A}|}+1-\gamma)R}{1-\gamma}+2=O \left(\frac{\sqrt{|\mathcal{S}||\mathcal{A}|}R}{1-\gamma}\right).$ (266) $$

#### H.2 Convergence rate

We now figure out the scale of the descent parameter $w$ for finite MFGs.
First, for tabular SemiSGD, ${G}_{\psi}=I$. According to Lemma 9, $\lambda_{\min}({G}_{\psi})=1$.
Second, $\hat{G}_{*}=\operatorname{diag}(\mu^{\dagger}_{*}(s,a))$, where $\mu^{\dagger}_{*}$ is the steady state-action distribution induced by $\xi_{*}$.
Thus, $\lambda_{\min}(\hat{G}_{*})=\min_{s,a}\mu^{\dagger}_{*}(s,a)\leq\frac{1}{|
\mathcal{S}||\mathcal{A}|}$.
We define $\lambda\coloneqq\min_{s,a}\mu_{*}^{\dagger}(s,a)>0$, the probability of visiting the least probable state-action pair in the MFE.
Then, we have

$$ $w=\frac{1}{2}\min\{(1-\gamma)\lambda_{\min}(\hat{G}_{\xi_{*}}),\lambda_{\min}( {G}_{\psi})\}\geq\frac{1}{2}(1-\gamma)\lambda.$ (267) $$

We are now ready to state the sample complexity of tabular SemiSGD.

###### Corollary 4 .

With either a constant step size $\alpha_{t}\equiv\alpha_{0}=\log T/(wT)$ or a linearly decaying step size $\alpha_{t}=1/(w(1+t))$, there exists a convex combination $\widetilde{\xi}_{T}$ of the iterates $\{\xi_{t}\}_{t=1}^{T}$ such that

$$ $\mathbb{E}\left\|\widetilde{\xi}_{T}-\xi_{*}\right\|^{2}=\widetilde{O}\left( \frac{|\mathcal{S}||\mathcal{A}|R^{2}}{\lambda^{2}(1-\gamma)^{4}T}\right),$ (268) $$

where $\widetilde{O}$ suppresses logarithmic factors.

Notably, tabular SemiSGD has the same sample complexity as TD Learning methods [, ].

###### Corollary 4 .

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## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 5
presentation: 3
questions: My main concerns and questions are outlined in Weaknesses.
Additionally, I have the following question:
* I am not sure how Assumption 5 is relatively weaker than Assumption 4. Could you provide a more intuitive explanation of Assumption 5?
soundness: 2
strengths: * The problem is well-motivated. Eliminating the forward-backward process of learning algorithms for mean-field games is significantly important.
* The proposed algorithm has a strong convergence guarantee with a rate.
summary: This paper primarily concentrates on the development of a stochastic gradient method for learning in mean-field games.
The proposed algorithm significantly reduces the computational cost associated with calculating the population distribution.
Moreover, the author demonstrates that under certain standard assumptions, the proposed algorithm converges to a stationary point.
Additionally, the algorithm's performance is validated experimentally in various mean-field games.
weaknesses: My primary concern is that the definition of the mean-field equilibrium in this study differs from that in existing studies.
Specifically, this study considers the Bellman operator, as opposed to the commonly used Bellman optimality operator.
Hence, I believe that there is no guarantee for the agent to maximize the expected cumulative discounted reward in the mean field equilibria.
In the context of learning in multi-agent systems, it seems more natural to consider maximizing the expected cumulative discounted reward.
Consequently, I'm wondering why the current definition of equilibrium was considered in this study.
Is it possible to extend the provided theoretical results to the reward-maximizing setting?
If not, I don't think we can fairly compare this study with existing studies on mean-field games.

Additionally, I believe that a more detailed explanation of the numerical experiments is necessary.
For example, I'm curious why exploitability was reported for the flocking game, despite the fact that reward maximization is not the main focus of this study.

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: Q1: Theorem 2: Please provide an example satisfying the assumptions of this theorem.

Q2: Section 7: Can you please explain what is FPI+MD? Is it the same as the online mirror descent of (Perolat et al., 2021)? If not, please compare with this algorithm, which is known to empirically converge much faster than fictitious play-type iterations.
soundness: 3
strengths: The paper is relatively clear and the algorithm is new, to the best of my knowledge. The convergence analysis includes sample complexity. The experiments cover different examples and contain a few baselines.
summary: In this paper, a numerical method for mean field games is proposed. Mean field games games approximate Nash equilibria for games with many players. Here, the proposed method relies on stochastic semi-gradient updates. Theoretical convergence is proved and numerical illustrations are also given.
weaknesses: I understand that empirical convergence is observed beyond the assumptions, but still: it is not clear to me how to check these assumptions in practice.
In the numerical examples, the baselines could be explained more clearly.

---

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: - Uniqueness of solutions seems to not be required. How it is possible that we obtain convergence in the presence of non-unique MFE? Does the theory imply convergence to multiple MFE, or is uniqueness implicitly assumed?
- What is the reasoning behind replacing $M_*$ (the non-unique? unknown MFE) by bootstrap estimates under the current policy (before Eq. (3))? Why can the current policy produce estimates for the mean field $M_*$ of the MFE?
- The methodology does not directly optimize exploitability, what is the difficulty in instead using "true" gradients on the unified parameters to minimize exploitability?
- Can you extend similar techniques to non-time-stationary MFGs?
soundness: 3
strengths: - The paper is well written and relatively clear despite the subject matter. 
- The analysis is novel, interesting and extensive, e.g., the combination of linear approximations with MFGs and producing finite-time error bounds, or the convergence results without (directly) assuming contractivity or monotonicity.
- Empirically, the approach appears to be able to outperform existing approaches in some of the demonstrated problems in terms of exploitability. Further, the method is practical, as it is applicable in an online manner to unknown models. 
- The introduction of synchronous updates as opposed to forward-backward computations seems somewhat significant to the study of MFG learning algorithms, as it allows for improved sample complexity and better empirical results.
summary: The manuscript goes beyond existing fixed-point iteration (FPI) methods for solving mean field games, by treating the mean field and agent policy jointly instead of performing forward and backward updates. Here, FPI-type methods are understood as those computing the full forward and backward equations for mean field and optimal policies. The advantage is allowing asynchronous updates to both, which can lead to improved results. An analysis of the resulting semi-gradient descent method is performed under linear models and varying sets of assumptions. Empirically, the designed algorithms are further demonstrated on a variety of problems.
weaknesses: - The approach seems to be limited to stationary mean field games, i.e. ones with time-stationary mean fields. This does not match with the compared / referenced literature, of which most are for non-stationary cases. 
- For the theoretical results, there remain limits in terms of significance, as also discussed by the authors. The significance of theoretical results is limited due to the requirement of strong assumptions such as linear models and regularized solutions.
- Some minor issues (incomplete / TeX errors) in references.
- Some points remained unclear to me, see questions below.

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confidence: 3
contribution: 3
initial_rating: 5
presentation: 3
questions: 1. In Assumption 5, $w$ is introduced to denote the upper bound of $\bar{w}$. However, it does not appears in Theorem 2. So what's the reason to mention such an upper bound?
soundness: 2
strengths: It seems interesting to me the idea that considers the policy and population parameters as a unified parameter to optimize.

The assumptions are clearly stated, and some explanations are provided. 

Numerical experiments are provided.

The comparison in Table 2 provides a good summary.
summary: This paper studies learning equilibrium policy in mean-field game. The authors propose semiSGD, a SGD style online algorithm, where they treat the policy and population as a unified parameter. For linear MFGs, under small Lipschitz factor case, they establish convergence analysis. Besides, when the Lipschitz factor of the problem is relative large, they show the algorithm converges to the neighborhood of Nash, whose radius scales with the Lipschitz factors.
weaknesses: 1. **About assumptions**: I think the authors should provide some justification about why it is reasonable to consider Assumption 2. In the more standard MFG setting, the equilibrium policy is defined to be the policy that no agent can deviate and increase its value, which correspond to the setting $\Gamma_\pi(Q)(\cdot|s) := \arg\max_a Q(s,a)$. However, Assumption 2 can not cover this setting with finite $L_\pi$, because of the discountinuity of the argmax operator.

    As implied by Assumption 2, the paper seems to consider a "smooth" version of standard Nash equilibrium. It is unclear that why this objective is of interests to study, and also what's the relationship between it and the standard Nash equilibrium.

2. **Comparison with Assumptions in previous work**: this paper consider contraction style assumptions to establish the convergence. Although in remark 2, the authors provide some discussion to justify the assumptions, it is still unclear to me how they compare with previous works. For example, does the contractivity assumption in [1] can be recovered as a special case by choosing some specific parameters in Assumption 2 and 4? Or maybe comparing with [2], which ensures contractivity by introducing the regularization, would Assumption 2 or 4 be satisfied by introducing large enough regularization?

3. **When Theorem 2 is meaningful?** Theorem 2 suggests the convergence with bounded bias. I think it is necessary to have some discussion about the magnitude of the bias term comparing with another algorithm randomly compute $\xi_T$. For instance, can you provide some examples when the bias term can be (much) lower than $\|\xi^*\|$, which suggests the algorithm is better than a random guess by directly assigning $\xi_T = 0$.

    It is also unclear how large $\bar{w}$ is. Given that $\bar{w}$ is a problem-dependent constant, can you explain when $\bar{w}$ can be large or small? That would be helpful to understand when Theorem 2 is meaningful and when it is vacuous.

[1] Guo et al., Learning mean-field games.

[2] Yardim et al., Policy mirror ascent for efficient and independent learning in mean field games

Meta-review: This paper treats the problem of learning in mean-field games (MFGs) and, in particular, analyzes a stochastic approximation method dubbed SemiSGD (stochastic semi-gradient descent) in which agents simultaneously update their policy and population estimates. The authors provide a finite-time analysis for SemiSGD, and they establish its convergence to regularized mean-field equilibria (or neighobrhoods of mean-field equilibria).

The reviewers were mostly positive regarding the paper, and found it relatively well-written and topical. Two main concerns that were raised concern the following points:
- The authors' convergence analysis in Theorem 1 concerns the game's regularized MFE, with the degree of regularization being sufficient to ensure contraction (and, a fortiori, a unique regularized MFE). This reduces the relevance of this result (the authors stated in their rebuttal that they were transparent about this, but the writing is not sufficiently clear in this regard).
- It was also unclear how the authors' assumptions can be verified in practice - especially the delicate conditions involving $w$ and the eigenvalues of the various operators in Appendix H (which, incidentally, make the paper very hard to read at that point).

The authors provided very extensive replies to the reviewers' other questions, but the above concerns remained. While I remain concerned that these points impose considerable limitations on the authors' results (which are not adequately discussed in the paper), I believe the paper's merits outweigh its flaws, so I recommend acceptance.

Decision: Accept (Poster)