A Practical Adaptive Subgame Perfect Gradient Method

We present a performant gradient method for smooth convex optimization, drawing inspiration from several recent advances in the field. Our algorithm, the Adaptive Subgame Perfect Gradient Method (ASPGM) is based on the notion of subgame perfection, attaining a dynamic strengthening of minimax optimality. At each iteration, ASPGM makes a momentum-type update, optimized dynamically based on a (limited) memory/bundle of past first-order information. ASPGM is linesearch-free, parameter-free, and adaptive due to its use of recently developed auto-conditioning, restarting, and preconditioning ideas. We show that ASPGM is competitive with state-of-the-art L-BFGS methods on a wide range of smooth convex problems. Unlike quasi-Newton methods, however, our core algorithm underlying ASPGM has strong, subgame perfect, non-asymptotic guarantees, providing certificates of solution quality, resulting in simple stopping criteria and restarting conditions.

💡 Research Summary

The paper introduces the Adaptive Subgame Perfect Gradient Method (ASPGM), a novel first‑order algorithm for smooth convex optimization that unifies several recent advances: adaptive estimation of smoothness and strong convexity parameters, memory‑based subgame‑perfect updates, and online preconditioning. The authors begin by highlighting the longstanding trade‑off between strong theoretical guarantees and practical speed. Classical quasi‑Newton methods such as L‑BFGS achieve impressive empirical performance but lack non‑asymptotic convergence guarantees, while accelerated gradient schemes like the Optimized Gradient Method (OGM) are minimax‑optimal in theory but require prior knowledge of the Lipschitz constant L and often rely on line‑searches. Recent work on subgame‑perfect gradient methods (SPGM) showed how to exploit a limited bundle of past gradient information to compute an optimal two‑step update via a low‑dimensional second‑order cone program, yet SPGM still depends on an exact L.

To overcome these limitations, the authors first develop a backtracking‑free core algorithm called BSPGM (Backtracking‑free Subgame Perfect Gradient Method). BSPGM retains the subgame‑perfect property by solving the same SOCP at each iteration, but replaces the fixed L with an automatically estimated smoothness constant (\hat L). This estimate is updated using an “auto‑conditioning” rule that guarantees the cocoercivity inequality (Q_{i,j}(\hat L)\ge 0) for all stored pairs ((x_i,g_i)). When the current (\hat L) is too small, a null‑step is performed: the algorithm records the new information without moving the iterate, thereby preserving the potential function (H_n\ge 0) that underlies the convergence proof. Consequently, BSPGM is completely line‑search‑free and parameter‑free while still providing a non‑asymptotic convergence guarantee of (O(1/k^2)) for smooth convex functions.

Building on BSPGM, ASPGM incorporates two additional mechanisms. First, it estimates a local strong‑convexity parameter (\hat\mu) from the same bundle of data. When (\hat\mu>0) is detected, a robust restarting condition is triggered, resetting the momentum terms and enabling linear convergence with rate (O\big((1-\sqrt{\hat\mu/\hat L})^k\big)). Second, ASPGM applies an online preconditioner (B), typically a diagonal matrix updated via the Online Gradient Scaling technique. This preconditioner effectively rescales the problem geometry, reducing the condition number (L_B/\mu_B) and further accelerating convergence. The combined algorithm thus adapts to unknown smoothness, exploits strong convexity when present, and benefits from a data‑driven preconditioner—all without any line‑search or user‑specified hyper‑parameters.

The theoretical contributions are presented in a unified inductive framework. The authors first recall the potential‑function view of OGM, where a sequence (H_n) is maintained via specific choices of momentum coefficients (\tau_n). They then show that SPGM can be interpreted as a dynamic optimization of this induction, selecting the coefficients that minimize the worst‑case bound given the observed bundle. BSPGM inherits this structure, but the induction now includes the adaptive smoothness estimate and null‑step handling, preserving (H_n\ge0) regardless of estimation errors. ASPGM adds a strong‑convexity term to the potential function, and the restart rule guarantees that the potential never increases across restarts. Detailed proofs (deferred to the appendix) establish the subgame‑perfect property of BSPGM and the non‑asymptotic convergence rates for both the smooth and strongly convex regimes.

Empirically, the authors evaluate BSPGM and ASPGM on a diverse set of benchmark problems: logistic regression on large sparse datasets, L2‑regularized least squares, portfolio optimization, and small‑scale neural network training. They compare against state‑of‑the‑art adaptive first‑order methods (e.g., AdaGrad, Adam variants), OBL (Optimal Backtracking Linesearch), and L‑BFGS. Results are reported both in terms of oracle calls (gradient evaluations) and wall‑clock time. Across all experiments, ASPGM matches or exceeds L‑BFGS in wall‑clock performance while offering explicit convergence certificates and automatic stopping criteria. In the purely smooth setting, ASPGM’s (O(1/k^2)) decay matches OGM but without requiring L. When strong convexity is present, ASPGM’s restart mechanism yields a clear linear regime, outperforming both OBL and adaptive gradient methods. The authors also demonstrate that the preconditioner effectively reduces the effective condition number, leading to fewer iterations.

In conclusion, the paper delivers a practically useful algorithm that bridges the gap between theory and practice in smooth convex optimization. By integrating adaptive smoothness/strong‑convexity estimation, subgame‑perfect memory utilization, and online preconditioning, ASPGM provides strong non‑asymptotic guarantees, parameter‑free operation, and competitive empirical performance. The work opens several avenues for future research, including extensions to non‑convex settings, larger memory strategies, and distributed implementations.