Subgame Perfect Proximal Point Algorithm

• SPPPA is a first-order method for nonsmooth convex optimization that uses a prox-oracle and history-dependent SOCP planning to achieve subgame perfection.
• The algorithm dynamically re-optimizes its continuation strategy at each iteration, ensuring improved worst-case performance beyond classical minimax guarantees.
• A limited-memory variant reduces computational cost by focusing on recent iterates, enabling efficient updates via Gram matrices and interior-point methods.

The Subgame Perfect Proximal Point Algorithm (SPPPA) is a first-order method designed for nonsmooth convex optimization utilizing a proximal operator oracle. SPPPA achieves the game-theoretic notion of subgame perfection, a property that requires not only minimax-optimality over the original problem class but also optimality on every subclass revealed through the observed information at each step. At every iteration, SPPPA dynamically re-optimizes its continuation strategy by solving a small, history-dependent second-order cone program (SOCP), which yields performance bounds that are guaranteed never to be worse and are typically strictly better than minimax-optimal guarantees for the original class (Grimmer et al., 17 Nov 2025).

1. Problem Setting and Oracle Model

The target is unconstrained convex minimization: $\min_{x\in\mathbb{R}^d} f(x)$ where $f:\mathbb{R}^d\to(-\infty,+\infty]$ is closed, proper, convex, and attains its minimum at some $y_\star$. The algorithm is initialized at $x_0$, and progress is evaluated by normalized suboptimality: $\frac{f(y_N)-f_\star}{\tfrac12\|x_0-y_\star\|^2}$ The model assumes access to a “prox-oracle of accuracy one”: for any query $x$, the oracle returns

$$y = \mathrm{prox}_{L,f}(x) = \arg\min_u \bigl\{f(u) + \tfrac{L}{2}\|u-x\|^2\bigr\},\quad g = L(x-y) \in \partial f(y),\quad f(y)$$

where $\{L_0, L_1, ..., L_{N-1}\}$ is a predefined sequence of positive proximal parameters.

Algorithms in this class, denoted $\mathcal{A}$, are deterministic and at each step query the prox-oracle, then select their next iterate within the affine span generated by past subgradients: $$x_n \in x_0 + \mathrm{span}\,\{g_0, \ldots, g_{n-1}\}$$ with this selection depending solely on previously observed history $\{(y_i, f_i, g_i, x_i)\}_{i=0}^{n-1}$.

2. The Subgame Perfect Proximal Point Algorithm: Structure and Update

SPPPA generalizes the Optimized Proximal Point Algorithm (OPPA) by dynamically re-optimizing its proof of progress at each iteration. While OPPA proceeds by a closed-form recurrence, SPPPA, at each iteration $n$, observes the current history

$$\mathcal{H}_{n-1} = \{(y_i, f_i, g_i, x_i, z_{i+1}, \tau_i)\}_{i=0}^{n-1}$$

and constructs matrices and vectors encoding this history:

• Select the index $m$ with minimal observed value $f_m = \min\{f_0,...,f_{n-1}\}$
• Form $Z = [z_1-x_0\ \cdots\ z_n-x_0]$, $G = [g_0\ \cdots\ g_{n-1}]$ in $\mathbb{R}^{d \times n}$
• Vectors: $\tau = (\tau_0,\dots,\tau_{n-1})^\top$, $f = (f_0,\dots,f_{n-1})^\top$
• Auxiliary vectors: $$a_i = \tfrac12\|z_{i+1}-x_0\|^2 + \tau_i(f_i - f_m),\quad b_i = (f_i - \langle g_i, y_i - x_0 \rangle) - f_m$$

The next step is determined by solving the following SOCP in non-negative variables $\mu, \lambda_\star \in \mathbb{R}^n$: $$\text{maximize }\, \langle\tau, \mu\rangle + \mathbf{1}^\top \lambda_\star$$

$$\text{subject to }\, \tfrac12\|Z\mu - G\lambda_\star\|^2 \le \langle\mu,a \rangle + \langle \lambda_\star, b \rangle$$

Denote optima by $(\mu^\star, \lambda_\star^\star)$, then

$$z' = x_0 + Z \mu^\star - G \lambda_\star^\star,\quad \tau' = \langle \tau, \mu^\star \rangle + \mathbf{1}^\top \lambda_\star^\star$$

With $(z',\tau')$, update as

$$\tau_n = \tau' + \frac{1}{L_n}\bigl( 1 + \sqrt{1 + 2L_n \tau'} \bigr)$$

$$x_n = \frac{\tau'}{\tau_n} y_m + \frac{\tau_n-\tau'}{\tau_n} z'$$

$$y_n = \mathrm{prox}_{L_n, f}(x_n),\quad g_n = L_n(x_n - y_n),\quad z_{n+1} = z' - (\tau_n - \tau') g_n$$

3. Subgame-Perfect Guarantee and Theoretical Properties

SPPPA exactly realizes the subgame-perfect criterion: at any iteration $n$ and for every realized history $\mathcal{H}_{n-1}$, the algorithm’s actions attain the smallest possible maximal worst-case normalized suboptimality for the continuation problem. This property is captured by

$$\min_{A \in \mathcal{A}^{\mathcal{H}_{n-1}}} \max_{(f, x_0) \text{ consistent with } \mathcal{H}_{n-1}} \frac{ f(y_N^A) - f_\star }{ \frac12 \|x_0 - y_\star\|^2 } = \frac{1}{\tau_{n, N}}$$

where $\tau_{n, N}$ is the terminal progress parameter, computed from the current “seed” $\tau_{n,n-1} = \tau'$ and propagated via the OPPA recurrence. For the initial step, this reduces to the standard minimax guarantee, and SPPPA only improves upon this at future steps as history is revealed.

Through maintaining an appropriate inductive potential, $$H_n = \tau_n(f_\star - f_n) - \tfrac12\|z_{n+1} - y_\star\|^2 + \tfrac12\|x_0 - y_\star\|^2$$, the algorithm establishes at each step a matching upper bound. For lower bounds, a convex “hard” function is constructed for each possible observed history, using a zero-chain construction and the SOCP dual certificate ($\xi, w$), to guarantee that no algorithm in $\mathcal{A}$ can improve on the bound delivered by SPPPA for the continuation subgame (Grimmer et al., 17 Nov 2025).

4. Computational Complexity and Practical Implementation

The per-iteration computational cost of SPPPA is governed by the size of the SOCP. At step $n$, the SOCP features $2n$ variables (length-$n$ nonnegative vectors $\mu, \lambda_\star$) and a single quadratic constraint. Notably, the problem dimension $d$ does not impact the SOCP's computational burden, as all matrix operations can be performed using the $n \times n$ Gram matrices $Z^\top Z, Z^\top G, G^\top G$ and vectors $a, b$, yielding total per-iteration time $O(n^3)$ with interior-point methods.

A limited-memory variant, which stores only the most recent $k$ iterates, reduces the SOCP and per-step cost to $O(k^3)$, maintaining independence from both $d$ and $N$. Gram matrices and vector quantities are updated at each step in $O(n^2)$ time. The algorithm maintains only a short prefix of the full search history—specifically $(y_i, g_i, z_{i+1}, \tau_i)$—and calls a standard SOCP solver with a single rotated-cone constraint of size $n$.

5. Special Cases and Connections to Related Algorithms

SPPPA generalizes several classical methods. In the special case where the sequence of proximal parameters is constant ($L_n \equiv L$), each prox-step coincides with a gradient-step on the Moreau envelope of $f$ with step size $1/L$. Thus, SPPPA reduces exactly to the Subgame Perfect Gradient Method (SPGM), recovering its planning subproblem and corresponding dynamic improvements over classical Nesterov’s methods.

Compared to the Optimized Proximal Point Algorithm (OPPA), SPPPA maintains the same minimax-optimality guarantee for the overall class at initialization, but, crucially, it never allows the worst-case bound to become worse and typically yields strictly improved guarantees on subclasses revealed by observed history—for example, when the locally realized convex function exhibits stronger convexity or smoother behavior than assumed by the global parameterization.

6. Summary and Significance in Convex Optimization

SPPPA introduces a paradigm by which first-order methods equipped with a prox-oracle can dynamically plan their actions to guarantee, for each possible subgame determined by cumulative oracle responses, the sharpest worst-case optimality bound theoretically attainable. This approach is enabled by the tractable SOCP-based planning subproblem, which depends only on the history length, guaranteeing scalability even in high-dimensional settings.

The algorithm establishes a stronger standard of optimality—subgame perfection—than classical minimax formulations, and it offers a new direction for the design of robust algorithms in convex optimization, especially in settings where revealed information can be leveraged for improved adaptation over time (Grimmer et al., 17 Nov 2025).