# TensorPlan and the Few Actions Lower Bound for Planning in MDPs under Linear Realizability of Optimal Value Functions

arXiv: https://arxiv.org/abs/2110.02195 · HTML: https://mlsanity.com/paper/2110.02195/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 23 references that are in the map.
- First posted on arXiv 5 Oct 2021.

## Builds on (most cited first, 7 of 7)

- [Bilinear Classes: A Structural Framework for Provable Generalization in RL](https://mlsanity.com/paper/2103.10897/) (2021)
- [Is a Good Representation Sufficient for Sample Efficient Reinforcement Learning?](https://mlsanity.com/paper/1910.03016/) (2019)
- [Learning with Good Feature Representations in Bandits and in RL with a Generative Model](https://mlsanity.com/paper/1911.07676/) (2019)
- [Exponential Lower Bounds for Planning in MDPs With Linearly-Realizable Optimal Action-Value Functions](https://mlsanity.com/paper/2010.01374/) (2020)
- [An Exponential Lower Bound for Linearly-Realizable MDPs with Constant Suboptimality Gap](https://mlsanity.com/paper/2103.12690/) (2021)
- [Improved and Generalized Upper Bounds on the Complexity of Policy Iteration](https://mlsanity.com/paper/1306.0386/) (2013)
- [On Query-efficient Planning in MDPs under Linear Realizability of the Optimal State-value Function](https://mlsanity.com/paper/2102.02049/) (2021)

## Built on by (most cited first, 7 of 7)

- [Computationally Efficient PAC RL in POMDPs with Latent Determinism and Conditional Embeddings](https://mlsanity.com/paper/2206.12081/) (2022)
- [A Few Expert Queries Suffices for Sample-Efficient RL with Resets and Linear Value Approximation](https://mlsanity.com/paper/2207.08342/) (2022)
- [Stabilizing Q-learning with Linear Architectures for Provably Efficient Learning](https://mlsanity.com/paper/2206.00796/) (2022)
- [Exponential Hardness of Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/2302.12940/) (2023)
- [Computational-Statistical Gaps in Reinforcement Learning](https://mlsanity.com/paper/2202.05444/) (2022)
- [Linear Reinforcement Learning with Ball Structure Action Space](https://mlsanity.com/paper/2211.07419/) (2022)
- [Misspecified Q-Learning with Sparse Linear Function Approximation: Tight Bounds on Approximation Error](https://mlsanity.com/paper/2407.13622/) (2024)

## Abstract (arXiv)

We consider the minimax query complexity of online planning with a generative model in fixed-horizon Markov decision processes (MDPs) with linear function approximation. Following recent works, we consider broad classes of problems where either (i) the optimal value function v^star or (ii) the optimal action-value function q^star lie in the linear span of some features; or (iii) both v^star and q^star lie in the linear span when restricted to the states reachable from the starting state. Recently, Weisz et al. (2021b) showed that under (ii) the minimax query complexity of any planning algorithm is at least exponential in the horizon H or in the feature dimension d when the size A of the action set can be chosen to be exponential in min(d,H). On the other hand, for the setting (i), Weisz et al. (2021a) introduced TensorPlan, a planner whose query cost is polynomial in all relevant quantities when the number of actions is fixed. Among other things, these two works left open the question whether polynomial query complexity is possible when A is subexponential in min(d,H). In this paper we answer this question in the negative: we show that an exponentially large lower bound holds when A=Omega(min(d1/4,H1/2)), under either (i), (ii) or (iii). In particular, this implies a perhaps surprising exponential separation of query complexity compared to the work of Du et al. (2021) who prove a polynomial upper bound when (iii) holds for all states. Furthermore, we show that the upper bound of TensorPlan can be extended to hold under (iii) and, for MDPs with deterministic transitions and stochastic rewards, also under (ii).