Approximate Nash Equilibrium via Inexact ADMM

• The paper presents a distributed inexact-ADMM method to compute ε-approximate Nash equilibria in convex and strongly monotone games.
• It leverages consensus constraints and proximal updates, ensuring convergence with an O(1/k) residual decay under limited information exchange.
• The algorithm’s tuning of penalty parameters balances convergence speed and stability, with empirical validation in networked scenarios like wireless congestion control.

An approximate Nash equilibrium seeking algorithm is a computational procedure designed to identify an action profile or strategy set for multiple agents in a noncooperative game, such that no agent can achieve more than a specified ε improvement in their cost or utility by deviating unilaterally. Rigorous development of such algorithms is central to multi-agent learning, distributed optimization, and equilibrium computation for games characterized by convexity, continuity, monotonicity, and possibly large-scale communication graphs. Among the foundational approaches, distributed inexact-ADMM (Alternating Direction Method of Multipliers) provides a principled framework for convergence-guaranteed iterative computation of approximate Nash equilibria under restricted information and network constraints (Salehisadaghiani et al., 2016).

1. Precise Problem Formulation

Consider a game $\mathcal{N}=\{1,\ldots,N\}$ of $N$ players, each selecting $x_i\in X_i\subset\mathbb{R}$ (convex, compact), with joint action vector $x=(x_i,x_{-i})\in X:=\prod_j X_j$, and cost function $J_i(x_i,x_{-i})$ for player $i$. The Nash equilibrium $x^*$ is characterized by

$$J_i(x_i^*,x_{-i}^*) \leq J_i(x_i,x_{-i}^*) \quad \forall x_i\in X_i,\ \forall i,$$

which is equivalently reformulated as a variational inequality (VI) involving the pseudo-gradient mapping

$F(x):=(\nabla_1 J_1(x),\ldots,\nabla_N J_N(x))^T.$

A solution $x^*$ to

$(F(x^*))^T(x-x^*) \geq 0 \quad \forall x\in X$

yields the Nash equilibrium. To enable distributed computation, each agent maintains a local copy $x^i$ and consensus constraints are imposed via a communication graph $G=(\mathcal{N},\mathcal{E})$.

An $\epsilon$-approximate Nash equilibrium is $x^\epsilon$ such that: $$\max_i\, \big[J_i(x^\epsilon_i, x^\epsilon_{-i}) - J_i(x_i^*, x^\epsilon_{-i})\big] \leq \epsilon.$$

2. Algorithmic Framework: Inexact-ADMM Approach

The distributed NE seeking problem under consensus constraints is framed as

$$\min_{ \{x^i\} } \sum_{i=1}^N \big[J_i(x_i^i, x_{-i}^i) + \mathbb{I}_{X_i}(x_i^i)\big] \ \quad\text{s.t.}\quad x^i-x^j=0,\ \forall(i,j)\in\mathcal{E},$$

with $\mathbb{I}_{X_i}$ the indicator function for $X_i$.

The edge-based augmented Lagrangian is: $$\mathcal{L}_\rho(\{x^i\}, \{\lambda^{ij}\}) = \sum_{i=1}^N [J_i(x_i^i, x_{-i}^i) + \mathbb{I}_{X_i}(x_i^i)] + \sum_{(i,j)\in\mathcal{E}} [\left(\lambda^{ij}\right)^T(x^i-x^j) + \tfrac{\rho}{2}\|x^i-x^j\|^2],$$ where $\lambda^{ij}$ are dual variables and $\rho>0$ is the penalty parameter.

ADMM update steps per player $i$, per iteration $k$:

1. Primal update (x-step):

$$x_i^{k+1} = \arg\min_{x_i\in X_i} \Big[J_i(x_i, x_{-i}^k) + (\lambda_i^k)^T(x_i-z_i^k) + \tfrac{\rho}{2}\|x_i-z_i^k\|^2 \Big]$$

2. Consensus update (z-step):

$$z_i^{k+1} = \frac{1}{|\mathcal{N}_i|}\sum_{j\in\mathcal{N}_i} \left( x_j^{k+1} + \frac{1}{\rho}\lambda_j^k \right)$$

3. Dual update (λ-step):

$$\lambda_i^{k+1} = \lambda_i^k + \rho (x_i^{k+1} - z_i^{k+1})$$

Communication per step requires player $i$ to receive $(x_j^{k+1}, \lambda_j^k)$ from each neighbor $j$.

3. Convergence Guarantees and Analysis

Assuming:

• Nonempty, compact, convex $X_i$ for all $i$.
• $J_i$ is $C^1$ in $x_i$, convex in $x_i$, joint continuity.
• $F(x)$ is $L$-Lipschitz and $\sigma$-strongly monotone.
• $G$ is connected.

Main convergence properties:

• For penalty $\rho>\rho_{\min} := \frac{L^2}{2\sigma} - c_{\min}$ (where $c_{\min}$ is the smallest eigenvalue of the Laplacian of $G$), the iterates converge to $x^*$, with residuals

$$r^k := \max_i \|x_i^k - z_i^k\|,\quad s^k := \rho\|z^k - z^{k-1}\|,$$

satisfying $r^k+s^k = O(1/k)$.

Proof is via:

• Proximal-ADMM firm nonexpansiveness.
• Lyapunov function combining primal and dual errors:

$$V_k = \|x^k - x^*\|^2 + \|\lambda^k - \lambda^*\|^2,$$

with $$V_{k+1} \le V_k - \alpha(\|x^{k+1}-x^k\|^2 + \|\lambda^{k+1}-\lambda^k\|^2)$$ for $\alpha>0$.

• Telescoping argument leads to $x^k \to x^*$, $\lambda^k \to \lambda^*$.

4. Approximation Error and Parameter Tuning

$\epsilon$-approximate Nash equilibrium: $x^\epsilon$ such that

$$\max_i [J_i(x^\epsilon_i, x^\epsilon_{-i}) - J_i(x_i^*, x^\epsilon_{-i})] \le \epsilon.$$

After $k$ steps,

$$\|x^k-z^k\| + \|z^k-z^{k-1}\| \le \frac{C}{\rho k},$$

implies, by the Lipschitz continuity of $J_i$, suboptimality $O(1/\rho k)$. To achieve $\epsilon$-accuracy, select $k \ge C/(\rho\epsilon)$.

Penalty selection:

• $\rho$ must fulfill: $\rho \geq \rho_{\min} = \frac{L^2}{2\sigma} - c$ (ensuring strong convexity of penalized subproblems).
• Practically, choose $\rho \approx O(L)$ to trade-off convergence speed and numerical stability.

5. Practical Implementation and Complexity

Each iteration comprises local minimization and averaging over neighbors:

• Communication per step: each node sends one vector $x_j$, one dual vector $\lambda_j$; total messages per iteration $\approx 2|E|$.

Empirical results (e.g., ad-hoc wireless network congestion control with $N=16$ nodes):

• Convergence of $x_i^k$ to $x_i^*$ observed within 200 iterations.
• The ADMM-based method reaches $10^{-3}$ accuracy in approximately 50 iterations, compared to 400 for a best-response gradient scheme.
• Residual decays as $O(1/k)$, consistent with theoretical prediction.

6. Structural, Spectral, and Communication Considerations

Convergence speed and approximation quality depend on:

• Graph connectivity and Laplacian spectrum.
• Degree of coupling in cost functions (condition number affects $\rho_{\min}$).
• Local computation resources (solving convex minimizations per update).

A strong monotonicity of the pseudo-gradient and convexity of cost ensures global convergence under the specified penalties, and communication graph properties critically influence step-size and rate bounds.

7. Significance and Extensions

The distributed inexact-ADMM algorithm exemplifies a scalable, provably convergent method for approximate Nash equilibrium seeking in multi-agent convex games with limited information exchange. The O(1/k) residual decay and tunable accuracy via penalty parameters provide guarantees suitable for large-scale networks and real-time scenarios. This methodology has influenced subsequent work on consensus-based splitting, operator-theoretic distributed algorithms, and robust game-theoretic computation (Salehisadaghiani et al., 2016).

In summary, approximate Nash equilibrium seeking via inexact-ADMM leverages local linearizations, consensus averaging, and primal-dual residual control to yield distributed convergence at quantifiable rates, under standard convexity, monotonicity, and graph-connectivity assumptions.