Beyond the Best Guess: Improving LLM Solution Coverage with Evolution Strategies

Large Language Models (LLMs) are increasingly deployed in discovery domains such as math and science. The usual approach is to present the problem to the model and use its answer as the proposed solution. However, beyond this best guess, discovery can be enhanced by increasing test-time compute. In a process called pass@k, the model is allowed to explore the solution space and generate diverse candidate solutions. Unfortunately, the standard approach to post-training LLMs through Reinforcement Learning (RL) may limit pass@k: the model's output distribution narrows around high-reward outputs, causing the solution coverage to collapse. The alternative is to use Evolution Strategies (ES), a population-based, gradient-free post-training method that optimizes directly in weight space through random perturbations. As this paper shows, ES achieves consistently higher pass@k than RL and produces a broader output distribution with greater solution coverage. This coverage in turn makes it possible to achieve better results in e.g. standard math benchmarks. Thus, ES provides a better foundation for post-training in discovery problems and other domains where diverse solution coverage is critical.

When RLVR Shrinks the Reasoning Boundary: Diagnosing Pass@k Inversion

Reinforcement learning with verifiable rewards (RLVR) can improve one-sample accuracy while making a model worse under repeated sampling. We study this pass@k inversion: after training, the policy may solve fewer distinct problems than its base model at large $k$. The failure concentrates on boundary prompts, where the base model contains rare correct trajectories that are recoverable by sampling but too sparse to reliably appear in finite RLVR rollout groups. We argue that a two-mode account explains this as an absence-of-evidence failure: rare correct trajectories may disappear before RLVR samples and reinforces them often enough. The main contribution is this diagnostic and mechanistic framing. Per-Problem Base Anchoring (PBA) is a deliberately simple proof-of-concept: sharpen prompts with sufficient frozen-base correct evidence, and anchor risky prompts to the base distribution. Across three training seeds on Omni-MATH-Test, with MATH500 as a secondary high-coverage validation benchmark, PBA improves both \PassK{1} and high-budget coverage over matched GRPO. A 3000-prompt regime-controlled diagnostic study is consistent across seeds with the expected signature: ordinary GRPO loses base-solvable boundary prompts, while PBA preserves rare verifier-positive trajectories. We use mathematical verifiers as a controlled testbed for verifier-guided optimization; the same pass@k inversion risk applies to ECCV-relevant vision-language agents when repeated visual, spatial, or chart-reasoning attempts are checked by external tools or verifiers. Reasoning post-training should decide not only how strongly to optimize, but which prompts are safe to optimize.

On-Policy Self-Distillation with Sampled Demonstrations Reduces Output Diversity

On-policy self-distillation achieves strong pass@1 accuracy by using a single model as both teacher and student, with the teacher conditioned on a correct demonstration to provide dense token-level feedback. We show that this could come at a hidden cost: rollout diversity decreases and pass@k curves flatten (i.e., generating more rollouts fails to improve accuracy). We trace this to compounding biases in the design of self-distillation with sampled demonstrations. The teacher scores each student rollout while conditioned on a sampled correct rollout, channeling its feedback through the model's own biases. We theoretically analyze the optimal self-distillation policy and show that it tilts the base distribution by a pointwise conditional mutual information score between the student's rollout and the correct rollout used as context. Unlike the ideal optimal on-policy reinforcement learning (RL), which preserves probability ratios among equally correct rollouts, self-distillation can amplify existing probability gaps, concentrating mass on already-dominant modes. On a controlled graph path-finding task and science question-answering benchmarks, self-distilled models match or exceed RL on average performance but exhibit substantially lower functional and semantic diversity, failing on out-of-distribution settings that require diverse strategies.

Curriculum Reinforcement Learning Can Incentivize Reasoning Capacity in LLMs Beyond the Base Model

Reinforcement learning with verifiable rewards (RLVR) is widely viewed as a promising path toward continuously improving large language models. Recent works, however, suggest that mainstream RLVR often reallocates sampling probabilities among trajectories already present in the base model: it can improve sampling efficiency, reflected by higher pass@1 scores, but yields limited gains, and can even decrease pass@k scores when k is large, and therefore may fail to expand the base model's reasoning capacity boundary. In this paper, we present a boundary-aware Curriculum RL approach to move beyond the base model's reasoning capacity boundary. Our approach first uses pass@k sampling to locate the current reasoning capacity boundary, then applies targeted teacher guidance to examples near or beyond that boundary, and finally uses RL to consolidate the newly introduced reasoning patterns. Across Qwen, Llama, and DeepSeek base models, boundary-aware Curriculum RL improves both pass@1 scores and pass@256 scores, with pass@1 reflecting one-attempt performance and pass@256 serving as an empirical proxy for the reasoning capacity boundary. In our experiments, average pass@256 improves by 9.8 percentage points over the base models and by 10.3 percentage points over Vanilla RLVR. These results suggest that boundary-aware Curriculum RL can provide a scalable route for LLMs to continuously improve beyond the base model's empirical reasoning capacity boundary.