# Sublinear Least-Squares Value Iteration via Locality Sensitive Hashing

arXiv: https://arxiv.org/abs/2105.08285 · HTML: https://mlsanity.com/paper/2105.08285/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 94 references that are in the map.
- First posted on arXiv 18 May 2021.

## Builds on (most cited first, 30 of 50)

- [Is Q-learning Provably Efficient?](https://mlsanity.com/paper/1807.03765/) (2018)
- [Deep Reinforcement Learning for Dialogue Generation](https://mlsanity.com/paper/1606.01541/) (2016)
- [Provably Efficient Exploration in Policy Optimization](https://mlsanity.com/paper/1912.05830/) (2019)
- [Reinforcement Learning in Feature Space: Matrix Bandit, Kernels, and Regret Bound](https://mlsanity.com/paper/1905.10389/) (2019)
- [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)
- [Almost Optimal Model-Free Reinforcement Learning via Reference-Advantage Decomposition](https://mlsanity.com/paper/2004.10019/) (2020)
- [Provably Efficient Reinforcement Learning with Linear Function Approximation Under Adaptivity Constraints](https://mlsanity.com/paper/2101.02195/) (2021)
- [Asymmetric LSH (ALSH) for Sublinear Time Maximum Inner Product Search (MIPS)](https://mlsanity.com/paper/1405.5869/) (2014)
- [Practical and Optimal LSH for Angular Distance](https://mlsanity.com/paper/1509.02897/) (2015)
- [Solving Linear Programs in the Current Matrix Multiplication Time](https://mlsanity.com/paper/1810.07896/) (2018)
- [On Reward-Free Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/2006.11274/) (2020)
- [Provably Efficient Q-Learning with Low Switching Cost](https://mlsanity.com/paper/1905.12849/) (2019)
- [Training (Overparametrized) Neural Networks in Near-Linear Time](https://mlsanity.com/paper/2006.11648/) (2020)
- [An Improved Cutting Plane Method for Convex Optimization, Convex-Concave Games and its Applications](https://mlsanity.com/paper/2004.04250/) (2020)
- [Solving Empirical Risk Minimization in the Current Matrix Multiplication Time](https://mlsanity.com/paper/1905.04447/) (2019)
- [Approximate Nearest Neighbor Search in High Dimensions](https://mlsanity.com/paper/1806.09823/) (2018)
- [SLIDE : In Defense of Smart Algorithms over Hardware Acceleration for Large-Scale Deep Learning Systems](https://mlsanity.com/paper/1903.03129/) (2019)
- [Optimal Data-Dependent Hashing for Approximate Near Neighbors](https://mlsanity.com/paper/1501.01062/) (2015)
- [Faster Dynamic Matrix Inverse for Faster LPs](https://mlsanity.com/paper/2004.07470/) (2020)
- [A Deterministic Linear Program Solver in Current Matrix Multiplication Time](https://mlsanity.com/paper/1910.11957/) (2019)
- [A Provably Efficient Algorithm for Linear Markov Decision Process with Low Switching Cost](https://mlsanity.com/paper/2101.00494/) (2021)
- [Hashing-Based-Estimators for Kernel Density in High Dimensions](https://mlsanity.com/paper/1808.10530/) (2018)
- [Quantization based Fast Inner Product Search](https://mlsanity.com/paper/1509.01469/) (2015)
- [A Nearly-Linear Time Algorithm for Linear Programs with Small Treewidth: A Multiscale Representation of Robust Central Path](https://mlsanity.com/paper/2011.05365/) (2020)
- [Kernel Density Estimation through Density Constrained Near Neighbor Search](https://mlsanity.com/paper/2011.06997/) (2020)
- [On Symmetric and Asymmetric LSHs for Inner Product Search](https://mlsanity.com/paper/1410.5518/) (2014)
- [Solving Tall Dense Linear Programs in Nearly Linear Time](https://mlsanity.com/paper/2002.02304/) (2020)
- [A Framework for Adversarially Robust Streaming Algorithms](https://mlsanity.com/paper/2003.14265/) (2020)
- [Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors](https://mlsanity.com/paper/1608.03580/) (2016)

## Built on by (most cited first, 15 of 15)

- [Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time](https://mlsanity.com/paper/2310.17157/) (2023)
- [Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time](https://mlsanity.com/paper/2302.11068/) (2023)
- [Sketching for First Order Method: Efficient Algorithm for Low-Bandwidth Channel and Vulnerability](https://mlsanity.com/paper/2210.08371/) (2022)
- [A Sublinear Adversarial Training Algorithm](https://mlsanity.com/paper/2208.05395/) (2022)
- [Solving Attention Kernel Regression Problem via Pre-conditioner](https://mlsanity.com/paper/2308.14304/) (2023)
- [Breaking the Linear Iteration Cost Barrier for Some Well-known Conditional Gradient Methods Using MaxIP Data-structures](https://mlsanity.com/paper/2111.15139/) (2021)
- [Dynamic Tensor Product Regression](https://mlsanity.com/paper/2210.03961/) (2022)
- [Adaptive and Dynamic Multi-Resolution Hashing for Pairwise Summations](https://mlsanity.com/paper/2212.11408/) (2022)
- [Toward Highly Efficient and Private Submodular Maximization via Matrix-Based Acceleration](https://mlsanity.com/paper/2305.08367/) (2023)
- [Efficient Alternating Minimization with Applications to Weighted Low Rank Approximation](https://mlsanity.com/paper/2306.04169/) (2023)
- [Sublinear Time Algorithm for Online Weighted Bipartite Matching](https://mlsanity.com/paper/2208.03367/) (2022)
- [Dynamic Maintenance of Kernel Density Estimation Data Structure: From Practice to Theory](https://mlsanity.com/paper/2208.03915/) (2022)
- [Online Adaptive Mahalanobis Distance Estimation](https://mlsanity.com/paper/2309.01030/) (2023)
- [Fast Heavy Inner Product Identification Between Weights and Inputs in Neural Network Training](https://mlsanity.com/paper/2311.11429/) (2023)
- [Efficient, Low-Regret, Online Reinforcement Learning for Linear MDPs](https://mlsanity.com/paper/2411.10906/) (2024)

## Abstract (arXiv)

We present the first provable Least-Squares Value Iteration (LSVI) algorithms that have runtime complexity sublinear in the number of actions. We formulate the value function estimation procedure in value iteration as an approximate maximum inner product search problem and propose a locality sensitive hashing (LSH) [Indyk and Motwani STOC'98, Andoni and Razenshteyn STOC'15, Andoni, Laarhoven, Razenshteyn and Waingarten SODA'17] type data structure to solve this problem with sublinear time complexity. Moreover, we build the connections between the theory of approximate maximum inner product search and the regret analysis of reinforcement learning. We prove that, with our choice of approximation factor, our Sublinear LSVI algorithms maintain the same regret as the original LSVI algorithms while reducing the runtime complexity to sublinear in the number of actions. To the best of our knowledge, this is the first work that combines LSH with reinforcement learning resulting in provable improvements. We hope that our novel way of combining data-structures and iterative algorithm will open the door for further study into cost reduction in optimization.