Orthogonalized Policy Optimization:Policy Optimization as Orthogonal Projection in Hilbert Space
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908851666681856 |
|---|---|
| author | Zixian, Wang |
| author_facet | Zixian, Wang |
| contents | We propose Orthogonalized Policy Optimization (OPO), a principled framework for large language model alignment derived from optimization in the Hilbert function space L2(pi_k). Lifting policy updates from the probability simplex into L2(pi_k) transforms the nonlinear normalization constraint into a linear orthogonality condition <v, 1>_{pi_k} = 0 on the density fluctuation field v = pi/pi_k - 1. By the Hilbert projection theorem, the unique closed-form update is v_star = (omega_alpha - E[omega_alpha]) / mu, where the subtracted mean acts as a chemical potential enforcing probability conservation. This interpretation reveals advantage z-score normalization as a conservation-law projection rather than a variance-reduction heuristic.
OPO cleanly decouples sampling geometry, controlled by the escort exponent alpha, from optimization geometry, governed by the stiffness parameter mu, a separation not attainable under KL-based objectives. The same update can also be derived as a Euclidean mirror-descent step and as the linear-response law of near-equilibrium statistical mechanics, establishing its structural uniqueness within ratio geometry.
Structurally, OPO induces constant curvature, non-saturating linear gradient dynamics, and an intrinsic chi-square trust region. Experiments on MATH benchmarks show that the Hilbert projection formulation prevents gradient saturation typical of KL-constrained methods. By sustaining non-vanishing gradients in high-confidence regimes, OPO avoids premature plateaus and achieves stronger long-horizon training rewards and improved out-of-distribution generalization compared to clipping-based baselines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12415 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Orthogonalized Policy Optimization:Policy Optimization as Orthogonal Projection in Hilbert Space Zixian, Wang Machine Learning Artificial Intelligence We propose Orthogonalized Policy Optimization (OPO), a principled framework for large language model alignment derived from optimization in the Hilbert function space L2(pi_k). Lifting policy updates from the probability simplex into L2(pi_k) transforms the nonlinear normalization constraint into a linear orthogonality condition <v, 1>_{pi_k} = 0 on the density fluctuation field v = pi/pi_k - 1. By the Hilbert projection theorem, the unique closed-form update is v_star = (omega_alpha - E[omega_alpha]) / mu, where the subtracted mean acts as a chemical potential enforcing probability conservation. This interpretation reveals advantage z-score normalization as a conservation-law projection rather than a variance-reduction heuristic. OPO cleanly decouples sampling geometry, controlled by the escort exponent alpha, from optimization geometry, governed by the stiffness parameter mu, a separation not attainable under KL-based objectives. The same update can also be derived as a Euclidean mirror-descent step and as the linear-response law of near-equilibrium statistical mechanics, establishing its structural uniqueness within ratio geometry. Structurally, OPO induces constant curvature, non-saturating linear gradient dynamics, and an intrinsic chi-square trust region. Experiments on MATH benchmarks show that the Hilbert projection formulation prevents gradient saturation typical of KL-constrained methods. By sustaining non-vanishing gradients in high-confidence regimes, OPO avoids premature plateaus and achieves stronger long-horizon training rewards and improved out-of-distribution generalization compared to clipping-based baselines. |
| title | Orthogonalized Policy Optimization:Policy Optimization as Orthogonal Projection in Hilbert Space |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2601.12415 |