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Scalable LinUCB: Low-Rank Design Matrix Updates for Recommenders with Large Action Spaces

Evgenia Shustova, Marina Sheshukova, Sergey Samsonov, Evgeny Frolov

TL;DR

This paper tackles the scalability of LinUCB in recommender systems with large context dimensions and action spaces by maintaining a dynamical low-rank inverse of the regularized design matrix. It introduces PSI-LinUCB, which uses a Cholesky-style factorization $A_t=L_tL_t^T$ and rank-$r$ low-rank updates, with a projector-splitting integrator to cap the rank and ensure stable updates. The approach achieves $O(dr)$ average update cost per interaction and $O(dr)$ memory, with $O(dr)$ inference per action, and demonstrates substantial speedups (about $8.8\times$) and memory savings (about $15.9\times$) on large datasets like Amazon and MovieLens while preserving hit-rate performance. This work enables scalable, online contextual bandit recommendations in settings with high-dimensional features and extensive item collections, with potential applicability to other dynamical low-rank update schemes.

Abstract

Linear contextual bandits, especially LinUCB, are widely used in recommender systems. However, its training, inference, and memory costs grow with feature dimensionality and the size of the action space. The key bottleneck becomes the need to update, invert and store a design matrix that absorbs contextual information from interaction history. In this paper, we introduce Scalable LinUCB, the algorithm that enables fast and memory efficient operations with the inverse regularized design matrix. We achieve this through a dynamical low-rank parametrization of its inverse Cholesky-style factors. We derive numerically stable rank-1 and batched updates that maintain the inverse without directly forming the entire matrix. To control memory growth, we employ a projector-splitting integrator for dynamical low-rank approximation, yielding average per-step update cost $O(dr)$ and memory $O(dr)$ for approximation rank $r$. Inference complexity of the suggested algorithm is $O(dr)$ per action evaluation. Experiments on recommender system datasets demonstrate the effectiveness of our algorithm.

Scalable LinUCB: Low-Rank Design Matrix Updates for Recommenders with Large Action Spaces

TL;DR

This paper tackles the scalability of LinUCB in recommender systems with large context dimensions and action spaces by maintaining a dynamical low-rank inverse of the regularized design matrix. It introduces PSI-LinUCB, which uses a Cholesky-style factorization and rank- low-rank updates, with a projector-splitting integrator to cap the rank and ensure stable updates. The approach achieves average update cost per interaction and memory, with inference per action, and demonstrates substantial speedups (about ) and memory savings (about ) on large datasets like Amazon and MovieLens while preserving hit-rate performance. This work enables scalable, online contextual bandit recommendations in settings with high-dimensional features and extensive item collections, with potential applicability to other dynamical low-rank update schemes.

Abstract

Linear contextual bandits, especially LinUCB, are widely used in recommender systems. However, its training, inference, and memory costs grow with feature dimensionality and the size of the action space. The key bottleneck becomes the need to update, invert and store a design matrix that absorbs contextual information from interaction history. In this paper, we introduce Scalable LinUCB, the algorithm that enables fast and memory efficient operations with the inverse regularized design matrix. We achieve this through a dynamical low-rank parametrization of its inverse Cholesky-style factors. We derive numerically stable rank-1 and batched updates that maintain the inverse without directly forming the entire matrix. To control memory growth, we employ a projector-splitting integrator for dynamical low-rank approximation, yielding average per-step update cost and memory for approximation rank . Inference complexity of the suggested algorithm is per action evaluation. Experiments on recommender system datasets demonstrate the effectiveness of our algorithm.
Paper Structure (26 sections, 3 theorems, 52 equations, 7 figures, 3 tables, 5 algorithms)

This paper contains 26 sections, 3 theorems, 52 equations, 7 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Related works
  4. Scalable LinUCB with low-rank updates
  5. Rank-1 updates
  6. Batch updates
  7. Projector-Splitting Integrator
  8. Experimental Setup
  9. Baseline selection
  10. Training Protocol
  11. Feature Extraction
  12. Context Representation
  13. Results
  14. Scalability with context size
  15. Scalability with the number of arms
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\varepsilon > 0$, $L_0 = \varepsilon^{-1/2} I \in \mathbb{R}^{d \times d}$, and $U_0$, $V_0$ be empty matrices. Given a sequence of context vectors $\{x_t\}_{t \in \mathbb{N}}$ set Then the inverse root $L_{t+1}^{-1}$ can be expressed as where the matrices $U_{t+1} \in \mathbb{R}^{d \times (t+1)}$ and $V_{t+1} \in \mathbb{R}^{d \times (t+1)}$ are recursively updated with column-wise concate

Figures (7)

  • Figure 1: Performance comparison across different context sizes on Magazine Subscriptions dataset .
  • Figure 2: Algorithm scaling with number of arms on Beauty dataset.
  • Figure 3: Quality comparison by months: PSI vs LinUCB
  • Figure 4: Average Hit rate for different $m$ for CBSCFD.
  • Figure 5: Average Hit rate for different rank values for PSI-LinUCB.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof