Alternating Descent-Ascent Algorithm

• The alternating descent-ascent algorithm is an iterative method for saddle-point optimization, alternating updates to minimize and maximize the objective in two-player zero-sum games.
• By using the most recent updates, the method achieves improved numerical stability and an O(1/T) ergodic convergence rate, outperforming simultaneous update schemes in constrained settings.
• Its practical applications span large-scale game solving, variational formulations in online optimization, and evolving extensions to nonconvex, distributed, and manifold-based problems.

An alternating descent-ascent algorithm is an iterative first-order method for min-max (saddle-point) optimization, typically arising in two-player zero-sum games and convex-concave saddle problems. It is characterized by the alternation of descent updates in the minimization variable and ascent updates in the maximization variable, with each player using the most recent iterate of the other, rather than both updating simultaneously. This structure has gained prominence in variational formulations of learning games, online optimization, and large-scale game solving. Alternating protocols are empirically observed to exhibit stronger numerical stability, improved convergence rates, and finite regret properties compared to their simultaneous counterparts, especially in constrained and bilinear settings.

1. Problem Domain and Algorithmic Description

The prototypical setting is a finite two-player zero-sum bilinear game: given $A \in \mathbb{R}^{n \times m}$, the strategy sets $\mathcal{X} \subseteq \mathbb{R}^n$ and $\mathcal{Y} \subseteq \mathbb{R}^m$ (e.g., probability simplices), and objective $f(x, y) = y^\top A x$,

$$\min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x, y)$$

An alternating gradient descent-ascent (AltGDA) algorithm with step-size $\eta > 0$ proceeds, starting from $(x^0, y^0) \in \mathcal{X} \times \mathcal{Y}$: $$\begin{aligned} x^{t+1} &= \Pi_{\mathcal{X}} \left( x^t - \eta A^\top y^t \right) \ y^{t+1} &= \Pi_{\mathcal{Y}} \left( y^t + \eta A x^{t+1} \right) \end{aligned}$$ where $\Pi_{\mathcal{X}}, \Pi_{\mathcal{Y}}$ denote projections onto the respective domains. The output is often the ergodic average $$(\bar{x}, \bar{y}) = (\frac{1}{T}\sum_{t=0}^{T-1} x^t,\, \frac{1}{T}\sum_{t=0}^{T-1} y^t)$$.

This framework extends to general convex-concave and smooth saddle point problems, as well as to nonconvex or constrained strategies. Lyapunov and potential frameworks, duality gap analyses, and energy inequalities underpin the theoretical paper of AltGDA and its variants (Nan et al., 4 Oct 2025).

2. Theoretical Convergence and Performance Guarantees

The central theoretical findings concern the convergence rate of the ergodic duality gap and its dependence on problem structure.

For two-player zero-sum bilinear games with interior Nash equilibrium:

• Let $L = \|A\|_2$, and assume $(x^*, y^*) \in \mathcal{X} \times \mathcal{Y}$ with all $x^*_i, y^*_j > 0$.
• If $\eta \le 1 / (L \min_i x^*_i, L \min_j y^*_j)$, then after $T$ iterations,

$$\text{DualityGap}(\bar{x}, \bar{y}) \le \frac{9 + 4 \eta L}{\eta T}$$

• This delivers an $O(1/T)$ ergodic convergence rate, the first such guarantee for AltGDA in the constrained simplex/matrix game setting (Nan et al., 4 Oct 2025).

When the Nash equilibrium is not interior, AltGDA achieves local $O(1/T)$ convergence if initialized in a neighborhood of the equilibrium, with the bound incorporating support-related constants.

Simultaneous GDA (SimGDA), where $(x^{t+1}, y^{t+1})$ are updated using $(x^t, y^t)$, is limited to an $O(1/\sqrt{T})$ rate under the same constraints, even with optimal step-size, demonstrating the substantial theoretical improvement due to alternation.

The key structural claim is summarized in the following table:

Method	Ergodic Duality Gap	Conditions	Step-size Selection
AltGDA	$O(1/T)$	Interior NE; $\eta \lesssim 1/L$	Constant, $O(1/L)$
SimGDA	$O(1/\sqrt{T})$	Same	Any
AltGDA	$O(1/T)$ (local)	Non-interior NE, local	$\eta \le 1/(2L)$

The performance improvement is quantitatively attributed to exploitations of two-step Lyapunov inequalities, which allow strict contraction and cross-term cancellation not present in the simultaneous regime. For games without interior NE, the convergence is local, with an additional $\delta^2/(\eta T)$ term (Nan et al., 4 Oct 2025).

3. Analytical and Proof Principles

The foundational analysis relies on a series of potential and energy functions. For bilinear $f(x, y) = y^\top A x$, the two-step descent is expressed via functions such as: $$\phi_t(x, y) = \frac{1}{2}\|x^t - x\|^2 + \frac{1}{2}\|y^t - y\|^2 + \eta (y^t)^\top A x$$ and

$$\mathcal{E}(x, y) = \|x - x^*\|^2 + \|y - y^*\|^2 - \eta y^\top A x$$

Bounding the ergodic duality gap is reduced to telescoping these quantities and controlling "extra" non-negative cross terms that arise due to alternating updates.

In the unconstrained or interior NE setting, the potential decay is globally valid. In non-interior cases, local arguments ensure iterates remain inside a stability region, and the accumulated error is controlled.

4. Performance Estimation Programming (PEP) and Rate Optimality

A performance estimation programming (PEP) framework provides dimension-free worst-case gap bounds through a finite-dimensional semidefinite program. Within this setup:

• AltGDA achieves $P_T^* \approx c/T$ with nearly constant $\eta \approx 1.3$, confirming the $O(1/T)$ rate.
• SimGDA does not numerically reach $O(1/T)$ within this PEP even for optimized step-sizes (Nan et al., 4 Oct 2025).

PEP certification underscores that alternation realizes a qualitative rate advantage at the level of worst-case instances, not just in specific problem classes.

5. Intuition, Practical Impact, and Related Domains

The dominance of AltGDA over SimGDA stems from the use of the most recent primal iterates in the dual update, which systematically breaks the stale-gradient symmetry of simultaneous updating. Alternating schemes control the impact of boundary projections and exploit fresh-coordinate information, leading to faster decay of the duality gap.

This insight generalizes to broader settings, including:

• Large-scale equilibrium computation in matrix games, security games, and economic models.
• Online learning and online convex optimization, where alternating updates yield bounded regret and recurrent iterates for fixed step-sizes, as shown by volume preservation and Poincaré recurrence theorems (Bailey et al., 2019).

AltGDA is also readily parallelizable and computationally efficient, requiring only matrix-vector multiplications and projections.

6. Generalizations and Future Directions

While the $O(1/T)$ ergodic convergence for AltGDA is tight in the bilinear game setting, ongoing research extends the alternating concept to nonconvex optimization, variational inequalities, and manifold-based problems. Potential further work may involve:

• Characterizing global non-ergodic last-iterate convergence,
• Extending alternation principles to asynchronous and distributed settings,
• Designing adaptive step-size or momentum variants within the alternating framework.

The provable improvement of alternation over simultaneous updates in constrained bilinear games marks a definitive theoretical advance, providing a new foundation for algorithmic design in zero-sum and saddle-point optimization (Nan et al., 4 Oct 2025).