Generalized Agent Iteration: One Formal Framework for Iterative Policy Improvement and Recursive Self-Improvement
Organizations: Tianjin University · Shanxi University
Abstract
When we speak of recursive self-improvement (RSI), are we speaking of a phenomenon, a mechanism, or a prospect? Towards autonomous and evolving intelligence, RSI is being claimed at many scales, while no single framework that formally describes these emerging instances exists. Its counterpart in the classical realm, iterative policy improvement, is characterized by generalized policy iteration (GPI), a framework of broad applicability with well-understood theoretical properties, but only where the update principle and the evaluation base lie outside the agent. In this paper, we propose Generalized Agent Iteration (GAI), a formal framework that describes iterative policy improvement and RSI as two cases of a single learning paradigm. GAI defines the agent as a configuration of modifiable components within a system and models the learning process as a cycle of agent evaluation and agent improvement. Two pivotal dials then distinguish the instances: whether the improving mechanism is part of the agent and whether the standard it is measured against is grounded outside it. The former dial delineates the boundary between GPI and RSI, and the latter determines a system's polarity as anchored, goal drift, or fully self-referential. Moreover, we use these coordinates to place existing systems on the same two axes and make the defects of recursive self-improvement statable one condition at a time. We see this paper as a first step toward exploring a formal characterization of RSI that rests on the classical account, makes existing systems comparable, and provides a principled basis for analyzing and designing new ones.
Figures & tables
| Method | 0 Modification | 1 Improvement Mechanism | 2 Evaluation Basis | Guarantee |
| Policy / value iteration ( Sutton and Barto, 2018 ) | • policy • value | external, fixed (greedy; Bellman backup) | environment reward | converges to an optimal policy |
| DQN ( Mnih et al., 2013 ) | • deep Q-network (action values) • implicit greedy policy | external, fixed (temporal-difference update; greedy) | environment reward | no exact guarantee under function approximation |
| Reflexion / Self-Refine ( Shinn et al., 2023 ; Madaan et al., 2023 ) | • own outputs • reflection memory | external, fixed (retry loop; the model rewrites its own output) | task feedback / self-critique | empirical gains, bounded by the fixed loop |
| STOP ( Zelikman et al., 2023 ) | • own optimizer (for code generation) | internal (the optimizer rewrites itself) | validation examples | empirical gains |
| Gödel Agent / Darwin Gödel Machine ( Yin et al., 2024 ; Zhang et al., 2025 ) | • own code • policy and improvement mechanism • open-ended archive search (Darwin) | internal (the self-modification routine is modifiable) | external benchmark | each rewrite empirically validated |
| Red Queen Gödel Machine ( Iacob et al., 2026 ) | • the agent • its evaluator | internal (the agent and its evaluator co-evolve) | co-evolving evaluator; no fixed external basis | no guarantee against a fixed basis |
| Polarity | State of | Consequence | Representative systems |
| Anchored | external content, , so the agent cannot rewrite it | Improves toward a fixed external standard; a modifier satisfying the self-consistency conditions above tracks it. | Gödel machine ( Schmidhuber, 2007 ) ; STOP ( Zelikman et al., 2023 ) |
| Goal Drift | external content, but , so the agent may rewrite it | The standard itself can be rewritten, so self-improvement can drift away from the external goal. | Red Queen Gödel Machine ( Iacob et al., 2026 ) |
| Fully Self-Referential | no external content: empty, or depends on | No external signal reaches the self-improvement loop, and the critic equation reduces to internal self-consistency. | Socratic learning ( Schaul, 2024 ) (position paper) |
| System | Agent | Dial 1 [1pt] modifiable? | Dial 2 [1pt] Polarity | Classification |
| Policy/value iteration, PPO ( Sutton and Barto, 2018 ; Schulman et al., 2017 ) | no; is the fixed update principle (Bellman backup and greedy improvement; the clipped policy update for PPO) | Anchored; the evaluation base is the environment reward | (Anchored) GPI | |
| Fixed outer loops ( Gao et al., 2025 ) | (the optimized object) | no; the loop (proposal, training, selection) is fixed by its designers | Anchored; the evaluation base is the objective or validation metric set outside | (Anchored) GPI |
| Gödel machine ( Schmidhuber, 2003 ; Schmidhuber, 2007 ) | (solver and its rewriter) | yes; is the rewriting routine that edits the program when a proof succeeds | Anchored; the evaluation base is a fixed external utility, with a proof gate on rewrites | Anchored RSI (guarded) |
| Gödel Agent, Darwin Gödel Machine, STOP, SICA, Polaris, Hyperagents ( Yin et al., 2024 ; Zhang et al., 2025 ; Zelikman et al., 2023 ; Robeyns et al., 2025 ; Kakade et al., 2026 ; Zhang et al., 2026 ) | (the harness; model weights and benchmark outside) | yes; is the self-modification routine that edits prompts, memory, tools, or code | Anchored; the evaluation base is an external benchmark or validation set | Anchored RSI |
| Red Queen Gödel Machine ( Iacob et al., 2026 ) | (the agent and its evaluator, which co-evolve 1 1 1 In this reading the evaluator corresponds to the modification critic , which is an agent component whenever it is modifiable. ) | yes; rewrites both the agent and its evaluator | Goal Drift; the evaluation base is updated during the process, so no fixed external standard remains | Drifting RSI |
| Closed-system proposals ( Schaul, 2024 ) | all components, included | yes; may rewrite any component, the base included | Fully Self-Referential; the evaluation base is empty or self-referential, so no external signal remains | Fully Self-Referential RSI |
Self-Improving Agents
Self-improving agents use experience and feedback to update their models, memories, skills, or workflows for better future task performance.
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