# A Representation Optimization Dichotomy, Lie-Algebraic Policy Optimization

arXiv: https://arxiv.org/abs/2603.25525 · HTML: https://mlsanity.com/paper/2603.25525/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 15 of its 37 references that are in the map.
- First posted on arXiv 26 Mar 2026.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 15 of 15)

- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning](https://mlsanity.com/paper/2006.16908/) (2020)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [Scalable trust-region method for deep reinforcement learning using Kronecker-factored approximation](https://mlsanity.com/paper/1708.05144/) (2017)
- [Full-Capacity Unitary Recurrent Neural Networks](https://mlsanity.com/paper/1611.00035/) (2016)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Finite-Time Analysis of Natural Actor-Critic for POMDPs](https://mlsanity.com/paper/2202.09753/) (2022)
- [Optimization on Manifolds via Graph Gaussian Processes](https://mlsanity.com/paper/2210.10962/) (2022)

## Abstract (arXiv)

Structured reinforcement learning and stochastic optimization often involve parameters evolving on matrix Lie groups such as rotations and rigid-body transformations. We establish a representation-optimization dichotomy for Lie-algebra-parameterized Gaussian policy objectives in the Lie Group MDP class: the gradient Lipschitz constant L(R), governing step size, convergence, and sample complexity of first-order methods, depends only on the algebraic type of g, uniformly over all objectives, independent of reward or transition structure. Specifically, L = O(1) for compact g (e.g., so(n), su(n)), and L = Theta(exp(2R)) for g = gl(n), with O(exp(2R)) for all algebras with a hyperbolic element. A key lower bound shows this exponential growth cannot be canceled by interaction between the exponential map and the objective, making the dichotomy intrinsic to the algebra.This yields an algorithmic consequence: for compact algebras, radius-independent smoothness enables O(1/sqrt(T)) convergence using an O(n^2 J) Lie-algebraic projection step instead of O(d_g^3) Fisher inversion. A Kantorovich alignment bound alpha >= 2 kappa / (kappa + 1) provides a computable condition under which this projection approximates natural gradient updates. Experiments on SO(3)^J and SE(3) confirm the theory: constant smoothness for compact algebras, polynomial growth for SE(3), and alignment across condition regimes. The projection step achieves 1.1-1.7x speedup over Cholesky-based Fisher inversion, with increasing gains at larger scales.