Opal: An Operator Algebra View of RLHF
This work provides a foundational framework for standardizing and analyzing RLHF methods, which is incremental as it builds on existing techniques like DPO and RRHF.
The paper tackles the problem of unifying reinforcement learning from human feedback (RLHF) methods by introducing Opal, an operator algebra view that expresses objectives as ladders of primitives, and GKPO, a canonical schema for representation and conversion, demonstrating non-reducibility in cases like reference shift or non-additive gates.
We present Opal, an operator view of reinforcement learning from human feedback (RLHF). Objectives are expressed as ladders of two primitives on a base utility: additive penalties and multiplicative pairwise weights. We describe a simple reduction law with if-and-only-if conditions: such ladders collapse to a normal form on pairwise margins when the reference is fixed, penalties are additive, and weights are independent of intermediate margins. When these assumptions do not hold (reference shift, non-additive gates, score-dependent weights), small examples demonstrate non-reducibility. Building on this view, we introduce GKPO (Generalized Kernel Preference Object), a canonical schema in which many RLHF methods can be represented and, when reducible, mapped back from. GKPO provides a standard JSON serialization, canonicalization and hashing rules, and explicit flags with finite witnesses when assumptions fail. We illustrate these ideas with GKPO examples for DPO, RRHF, and ORPO, along with cross-method conversions (where assumptions permit) and minimal stress tests (SHIFT/GATE/SCORE) that highlight non-reducibility. A lightweight Python reference library accompanies the schema, implementing canonical hashing and adapters for DPO and RRHF.