Topology-Enhanced Alignment for Large Language Models: Trajectory Topology Loss and Topological Preference Optimization
Authors: Yurui Pan, Ke Xu, Bo Peng
Organizations: School of Computing and Intelligent Innovation, Fudan University · School of Economics and Management, Tongji University · College of Information Technology, Shanghai Ocean University
Alignment of large language models (LLMs) via SFT and RLHF/DPO typically ignores the global geometry of the representation space, relying instead on local token likelihoods or scalar scores. We view generation as tracing a semantic trajectory in hidden space and propose a topology-enhanced alignment framework that regularizes these trajectories using 0-dimensional persistent homology. First, for SFT, we introduce Trajectory Topology Loss (TTL). Treating prompt and gold-answer embeddings as a mixed point cloud, we use a 0D persistent homology algorithm to extract "prompt-answer bridges." TTL aligns the model's actual update direction with these topological bridges rather than arbitrary directions. Second, for DPO, we propose Topological Preference Optimization (TPO). TPO constructs topic-specific semantic preference vectors and aligns the improvement direction between rejected and chosen responses with these vectors in an intermediate hidden layer. We also introduce a dynamic weighting scheme to balance DPO and TPO losses. Evaluating on Qwen2.5-7B-Instruct using UltraChat and Anthropic HH-RLHF, our topology-enhanced objectives consistently outperform strong non-topological baselines (e.g., per-example, nearest-neighbor, random regularizers) on automatic preference metrics and LLM-judge evaluations, while maintaining or improving toxicity. Results show persistent homology and trajectory geometry offer a promising direction for controllable alignment.
Large language models are commonly aligned through supervised fine-tuning, yet little is known about how their internal representations evolve during this process. We study alignment dynamics using persistent homology by tracking the topology of activation spaces throughout fine-tuning. Across four transformer language models ranging from 1B to 7B parameters and three alignment objectives corresponding to helpful, harmless, and mixed training data, we find that the majority of topological reorganization occurs during the earliest stages of training. A dense checkpoint analysis reveals a transient peak in topological activity followed by rapid stabilization. We further show that different alignment objectives induce distinguishable topological trajectories, while instruction-tuned and pretrained models exhibit qualitatively different patterns of evolution. Our results suggest that persistent homology provides a complementary perspective on alignment, revealing representation-level changes that are not apparent from behavioral metrics alone.
Aligning large language models (LLMs) with human preferences is commonly done via reinforcement learning from human feedback (RLHF) with Proximal Policy Optimization (PPO) or, more simply, via Direct Preference Optimization (DPO). While DPO is stable and RL-free, it treats preferences as flat winner vs. loser signals and is sensitive to noisy or brittle preferences arising from fragile chains of thought. We propose TUR-DPO, a topology- and uncertainty-aware variant of DPO that rewards how answers are derived, not only what they say, by eliciting lightweight reasoning topologies and combining semantic faithfulness, utility, and topology quality into a calibrated uncertainty signal. A small learnable reward is factorized over these signals and incorporated into an uncertainty-weighted DPO objective that remains RL-free and relies only on a fixed or moving reference policy. Empirically, across open 7-8B models and benchmarks spanning mathematical reasoning, factual question answering, summarization, and helpful/harmless dialogue, TUR-DPO improves judge win-rates, faithfulness, and calibration relative to DPO while preserving training simplicity and avoiding online rollouts. We further observe consistent gains in multimodal and long-context settings, and show that TUR-DPO matches or exceeds PPO on reasoning-centric tasks while maintaining operational simplicity.
Abdulhady Abas Abdullah, Fatemeh Daneshfar, Seyedali Mirjalili +1
Test-time alignment methods offer a promising alternative to fine-tuning by steering the outputs of large language models (LLMs) at inference time with lightweight interventions on their internal representations. Recently, a prominent and effective approach, RE-Control (Kong et al., 2024), has proposed leveraging an external value function trained over the LLM's hidden states to guide generation via gradient-based editing. While effective, this method overlooks a key characteristic of alignment tasks, i.e. that they are typically formulated as learning from human preferences between candidate responses. To address this, in this paper we propose a novel preference-based training framework, Pref-CTRL, that uses a multi-objective value function to better reflect the structure of preference data. Our approach has outperformed RE-Control on two benchmark datasets and showed greater generalization on out-of-domain datasets. Our source code is available at https://github.com/UTS-nlPUG/pref-ctrl.