# Primal-Dual Spectral Representation for Off-policy Evaluation

arXiv: https://arxiv.org/abs/2410.17538 · HTML: https://mlsanity.com/paper/2410.17538/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 47 of its 68 references that are in the map.
- First posted on arXiv 23 Oct 2024.

## 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, 30 of 47)

- [QT-Opt: Scalable Deep Reinforcement Learning for Vision-Based Robotic Manipulation](https://mlsanity.com/paper/1806.10293/) (2018)
- [Provably Efficient Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/1907.05388/) (2019)
- [Doubly Robust Off-policy Value Evaluation for Reinforcement Learning](https://mlsanity.com/paper/1511.03722/) (2015)
- [Provable Guarantees for Self-Supervised Deep Learning with Spectral Contrastive Loss](https://mlsanity.com/paper/2106.04156/) (2021)
- [Contextual Decision Processes with Low Bellman Rank are PAC-Learnable](https://mlsanity.com/paper/1610.09512/) (2016)
- [Information-Theoretic Considerations in Batch Reinforcement Learning](https://mlsanity.com/paper/1905.00360/) (2019)
- [Data-Efficient Off-Policy Policy Evaluation for Reinforcement Learning](https://mlsanity.com/paper/1604.00923/) (2016)
- [Way Off-Policy Batch Deep Reinforcement Learning of Implicit Human Preferences in Dialog](https://mlsanity.com/paper/1907.00456/) (2019)
- [DualDICE: Behavior-Agnostic Estimation of Discounted Stationary Distribution Corrections](https://mlsanity.com/paper/1906.04733/) (2019)
- [Batch Policy Learning under Constraints](https://mlsanity.com/paper/1903.08738/) (2019)
- [FLAMBE: Structural Complexity and Representation Learning of Low Rank MDPs](https://mlsanity.com/paper/2006.10814/) (2020)
- [Breaking the Curse of Horizon: Infinite-Horizon Off-Policy Estimation](https://mlsanity.com/paper/1810.12429/) (2018)
- [Reinforcement Learning in Feature Space: Matrix Bandit, Kernels, and Regret Bound](https://mlsanity.com/paper/1905.10389/) (2019)
- [AlgaeDICE: Policy Gradient from Arbitrary Experience](https://mlsanity.com/paper/1912.02074/) (2019)
- [Top-K Off-Policy Correction for a REINFORCE Recommender System](https://mlsanity.com/paper/1812.02353/) (2018)
- [Pessimistic Model-based Offline Reinforcement Learning under Partial Coverage](https://mlsanity.com/paper/2107.06226/) (2021)
- [Minimax Weight and Q-Function Learning for Off-Policy Evaluation](https://mlsanity.com/paper/1910.12809/) (2019)
- [Representation Learning for Online and Offline RL in Low-rank MDPs](https://mlsanity.com/paper/2110.04652/) (2021)
- [Offline Reinforcement Learning with Realizability and Single-policy Concentrability](https://mlsanity.com/paper/2202.04634/) (2022)
- [GenDICE: Generalized Offline Estimation of Stationary Values](https://mlsanity.com/paper/2002.09072/) (2020)
- [Towards Optimal Off-Policy Evaluation for Reinforcement Learning with Marginalized Importance Sampling](https://mlsanity.com/paper/1906.03393/) (2019)
- [Double Reinforcement Learning for Efficient Off-Policy Evaluation in Markov Decision Processes](https://mlsanity.com/paper/1908.08526/) (2019)
- [Reinforcement Learning via Fenchel-Rockafellar Duality](https://mlsanity.com/paper/2001.01866/) (2020)
- [Benchmarks for Deep Off-Policy Evaluation](https://mlsanity.com/paper/2103.16596/) (2021)
- [Off-Policy Evaluation via the Regularized Lagrangian](https://mlsanity.com/paper/2007.03438/) (2020)
- [Offline Reinforcement Learning: Fundamental Barriers for Value Function Approximation](https://mlsanity.com/paper/2111.10919/) (2021)
- [CoinDICE: Off-Policy Confidence Interval Estimation](https://mlsanity.com/paper/2010.11652/) (2020)
- [Evaluating Reinforcement Learning Algorithms in Observational Health Settings](https://mlsanity.com/paper/1805.12298/) (2018)
- [Making Linear MDPs Practical via Contrastive Representation Learning](https://mlsanity.com/paper/2207.07150/) (2022)
- [Efficient Reinforcement Learning in Block MDPs: A Model-free Representation Learning Approach](https://mlsanity.com/paper/2202.00063/) (2022)

## Built on by (most cited first, 4 of 4)

- [Spectral Representation-based Reinforcement Learning](https://mlsanity.com/paper/2512.15036/) (2025)
- [Outcome-Aware Spectral Feature Learning for Instrumental Variable Regression](https://mlsanity.com/paper/2512.00919/) (2025)
- [Toward Scalable and Valid Conditional Independence Testing with Spectral Representations](https://mlsanity.com/paper/2512.19510/) (2025)
- [Spectral Ghost in Representation Learning: from Component Analysis to Self-Supervised Learning](https://mlsanity.com/paper/2601.20154/) (2026)

## Abstract (arXiv)

Off-policy evaluation (OPE) is one of the most fundamental problems in reinforcement learning (RL) to estimate the expected long-term payoff of a given target policy with only experiences from another behavior policy that is potentially unknown. The distribution correction estimation (DICE) family of estimators have advanced the state of the art in OPE by breaking the curse of horizon. However, the major bottleneck of applying DICE estimators lies in the difficulty of solving the saddle-point optimization involved, especially with neural network implementations. In this paper, we tackle this challenge by establishing a linear representation of value function and stationary distribution correction ratio, i.e., primal and dual variables in the DICE framework, using the spectral decomposition of the transition operator. Such primal-dual representation not only bypasses the non-convex non-concave optimization in vanilla DICE, therefore enabling an computational efficient algorithm, but also paves the way for more efficient utilization of historical data. We highlight that our algorithm, SpectralDICE, is the first to leverage the linear representation of primal-dual variables that is both computation and sample efficient, the performance of which is supported by a rigorous theoretical sample complexity guarantee and a thorough empirical evaluation on various benchmarks.