Smooth Gate Functions for Soft Advantage Policy Optimization

A significant portion of recent research on Large Language Model (LLM) alignment focuses on developing new policy optimization methods based on Group Relative Policy Optimization (GRPO). Two prominent directions have emerged: (i) a shift toward sequence-level importance sampling weights that better align with the sequence-level rewards used in many tasks, and (ii) alternatives to the PPO-style clipping that aim to avoid the associated loss of training signal and entropy collapse. We introduce Soft Sequence Policy Optimization, an off-policy reinforcement learning objective that incorporates soft gating functions over token-level probability ratios within sequence-level importance weights. We provide theoretical motivation for SSPO and investigate practical modifications to improve optimization behavior. Empirically, we demonstrate that SSPO improves training stability and performance both in mathematical reasoning and coding tasks.

Group Relative Policy Optimization (GRPO) has emerged as a memory-efficient reinforcement fine-tuning (RFT) technique for reasoning-intensive tasks. How- ever, GRPO training dynamics on small language models (SLMs) remain poorly understood, limiting its reliable adoption and reproducibility in open and resource- constrained environments. In this work, we present a systematic study of GRPO fine-tuning for SLMs ranging from 1.5B to 7B parameters under a practical single- node 8xA100 compute budget. Our study spans multiple model families and reasoning domains, including mathematics, coding, and multiple-choice question answering (MCQ) in science. Across these settings, we analyze how group size affects policy convergence, training stability, and downstream benchmark per- formance. We further characterize tensor-level update dynamics during GRPO training and investigate whether the choice of LoRA target modules and layers can improve the performance of GRPO-tuned models. While our initial GRPO-tuned models outperform their base counterparts on approximately 80% of mathematical benchmark evaluations, they demonstrate limited capability on MCQ and code reasoning tasks. Guided by our mechanistic evaluations, we refined our LoRA and reward-shaping configurations to improve performance in latter domains. These findings provide practical guidance for GRPO training for SLMs.

Group Relative Policy Optimisation (GRPO) enhances large language models by estimating advantages across a group of sampled trajectories. However, mapping these trajectory-level advantages to policy updates requires aggregating token-level probabilities within each sequence. Relying on a fixed aggregation mechanism for this step fundamentally limits the algorithm's adaptability. Empirically, we observe a critical trade-off: certain fixed aggregations frequently suffer from training collapse, while others fail to yield satisfactory performance. To resolve this, we propose HölderPO, a generalised policy optimisation framework unifying token-level probability aggregation via the Hölder mean. By explicitly modulating the parameter $p$, our framework provides continuous control over the trade-off between gradient concentration and variance bounds. Theoretically, we prove that a larger $p$ concentrates the gradient to amplify sparse learning signals, whereas a smaller $p$ strictly bounds gradient variance. Because no static configuration can universally resolve this concentration-stability trade-off, we instantiate the framework with a dynamic annealing algorithm that progressively schedules $p$ across the training lifecycle. Extensive evaluations demonstrate superior stability and convergence over existing baselines. Specifically, our approach achieves a state-of-the-art average accuracy of $54.9\%$ across multiple mathematical benchmarks, yielding a substantial $7.2\%$ relative gain over standard GRPO and secures an exceptional $93.8\%$ success rate on ALFWorld.