SAHOO: Safeguarded Alignment for High-Order Optimization Objectives in Recursive Self-Improvement

Self-improvement training, where models learn from self-generated solutions, promises sustained capability gains but suffers from a pervasive failure mode: across multiple rounds, compounding reasoning errors cause accuracy to stall or degrade. We trace this drift to standard filtering criteria that retain solutions based solely on final answer correctness, which lets lucky guesses (correct answers with flawed reasoning) contaminate the training data. We propose Verified Self-Improvement (VSI), a framework that conditions data retention on step-level structural integrity rather than just the final output. VSI validates solutions by recomputing arithmetic steps via a computer-algebra library (sympy), checking intermediate consistency, and enforcing domain constraints. Evaluating VSI on GSM8K with Qwen3-4B-Thinking across 5 rounds of self-improvement against four baselines (no verification, outcome verification, majority voting, and VSI with DPO) shows that VSI rejects approximately 34% of correct-answer solutions, successfully isolating lucky guesses. This cleaner training signal drives sustained accuracy gains across all rounds (80.5% to 91.0%), whereas outcome verification plateaus and unverified training collapses. Finally, converting VSI checks into DPO preference pairs trains the model to distinguish sound reasoning from lucky answers, boosting reward accuracy from 46% to 63%. VSI offers a simple, reproducible recipe for robust self-improvement whenever automated reasoning checks are available.

With the rise of large language models, agentic systems are becoming increasingly autonomous, as they write code, call tools, revise prompts, and propose modifications to their own components. This autonomy raises a fundamental question for safe self-improvement: can an automated gate like a reward model or an automated reviewer reliably decide whether a proposed update should be accepted? The difficulty is structural. The gate must be a terminating, computable procedure, while the property we ultimately care about, namely whether an update is a genuine semantic improvement, depends on potentially unbounded program behavior. We formalize this mismatch by modeling agents as programs and defining improvement in terms of a semantic score that need not be computable. We study two evaluation abstractions. The first is a certifier that approves or rejects an update. The second is a search procedure that, given access to a proposal generator, attempts to output some improving candidate if one exists. We show that, in the fully general setting, no terminating procedure can simultaneously guarantee reliability (never accepting a non-improvement) and non-conservatism (never missing an available improvement): self-referential constructions can force any such gate or searcher to fail. Randomized gates with bounded error face the same limit. We then identify a standard decidable regime used in practice (bounded evaluation and a finite proposal budget) under which exhaustive simulation yields a terminating search procedure with the desired guarantees. We also show the limit of this recovery: a chain of accepted updates can raise the bounded score in every round while the true score falls. Our results draw a boundary between unbounded, universal settings where correctness guarantees collapse and bounded settings where they can be recovered by design.

AI systems increasingly participate in their own improvement: revising their outputs, adapting their own harnesses during deployment, training on data they generate, and, increasingly, conducting AI research itself. This literature is described under a vocabulary ("self-refine," "self-reward," "self-play," "self-evolve") that conflates fundamentally different ambitions. We survey 1,250 arXiv papers (2024-2026) along two axes: what the system improves -- its behavior in deployment, its policy through training, its evaluator, or the research process itself -- and the degree of loop closure (human-in-the-loop to fully closed). The taxonomy separates bounded self-refinement -- convergent, evaluable, and already industrial practice -- from open-ended recursive self-improvement (RSI), which remains bounded by grounding requirements, collapse dynamics, and compute constraints on every measured axis. Its distinctive feature is a dedicated category for self-evaluation: every improvement loop is a claim that some signal can substitute for human judgment. We survey the evaluator design space -- judges, process reward models, verifiers, rubrics, meta-evaluation -- order the signals into a verification hierarchy from formal verifiers (strongest) to intrinsic self-assessment (weakest), and observe that demonstrated self-improvement strength tracks this hierarchy, that its failure modes (self-confirming loops, model collapse, diversity collapse) follow from its violations, and that the "research direction-setting" bottleneck keeping humans in the loop sits at the top of that hierarchy. We connect the technical literature to the theory of RSI limits and to the safety and governance questions raised by frontier-lab accounts of closing the loop, and identify governance-grade measurement of self-improvement as the field's most underpopulated niche.