Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adversarial Online Collaborative Filtering

Stephen Pasteris, Fabio Vitale, Mark Herbster, Claudio Gentile, Andre' Panisson

TL;DR

This work designs and analyzes an algorithm that works under biclustering assumptions on the user-item preference matrix, and shows that this algorithm exhibits an optimal regret guarantee, while being fully adaptive, and proposes a more robust version of this algorithm which operates with general matrices.

Abstract

We investigate the problem of online collaborative filtering under no-repetition constraints, whereby users need to be served content in an online fashion and a given user cannot be recommended the same content item more than once. We start by designing and analyzing an algorithm that works under biclustering assumptions on the user-item preference matrix, and show that this algorithm exhibits an optimal regret guarantee, while being fully adaptive, in that it is oblivious to any prior knowledge about the sequence of users, the universe of items, as well as the biclustering parameters of the preference matrix. We then propose a more robust version of this algorithm which operates with general matrices. Also this algorithm is parameter free, and we prove regret guarantees that scale with the amount by which the preference matrix deviates from a biclustered structure. To our knowledge, these are the first results on online collaborative filtering that hold at this level of generality and adaptivity under no-repetition constraints. Finally, we complement our theoretical findings with simple experiments on real-world datasets aimed at both validating the theory and empirically comparing to standard baselines. This comparison shows the competitive advantage of our approach over these baselines.

Adversarial Online Collaborative Filtering

TL;DR

This work designs and analyzes an algorithm that works under biclustering assumptions on the user-item preference matrix, and shows that this algorithm exhibits an optimal regret guarantee, while being fully adaptive, and proposes a more robust version of this algorithm which operates with general matrices.

Abstract

We investigate the problem of online collaborative filtering under no-repetition constraints, whereby users need to be served content in an online fashion and a given user cannot be recommended the same content item more than once. We start by designing and analyzing an algorithm that works under biclustering assumptions on the user-item preference matrix, and show that this algorithm exhibits an optimal regret guarantee, while being fully adaptive, in that it is oblivious to any prior knowledge about the sequence of users, the universe of items, as well as the biclustering parameters of the preference matrix. We then propose a more robust version of this algorithm which operates with general matrices. Also this algorithm is parameter free, and we prove regret guarantees that scale with the amount by which the preference matrix deviates from a biclustered structure. To our knowledge, these are the first results on online collaborative filtering that hold at this level of generality and adaptivity under no-repetition constraints. Finally, we complement our theoretical findings with simple experiments on real-world datasets aimed at both validating the theory and empirically comparing to standard baselines. This comparison shows the competitive advantage of our approach over these baselines.
Paper Structure (17 sections, 3 theorems, 30 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 17 sections, 3 theorems, 30 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries, Learning Tasks, and Overview of Results
  3. The Perturbation-Free Case
  4. The One-Time Recommendation Algorithm Orca
  5. Analysis
  6. The Adversarial Perturbation Case
  7. The One-Time Recommendation Algorithm Orca$^*$
  8. Analysis
  9. Experiments
  10. Conclusions and Limitations
  11. Further Related Work
  12. Proofs
  13. Proof of Theorem \ref{['thm:noisefree']}
  14. Proof of Theorem \ref{['thm:noisefree_lowerbound']}
  15. Proof of Theorem \ref{['t:orcastar']}
  16. ...and 2 more sections

Key Result

theorem 1

Let Orca be run on a $(C,D)$-biclustered matrix $\boldsymbol{L}$ of size $M\times N$ with an arbitrary sequence of users $i_1,\ldots, i_T$ and an arbitrary monotonically increasing sequence of item sets $\mathcal{I}_{1},\ldots, \mathcal{I}_{T}\subseteq[N]$. Then the expected regret of Orca is upper the expectation being over the internal randomization of the algorithm. Orca is parameter-free in t

Figures (3)

  • Figure 1: Left: The behavior of Orca (with a static inventory) at trial $t$ when $\ell:=\ell_{i_{t}}<\ell^*$. The four boxed diagrams on top represent the possible cases. In all cases, the yellow circle represents the set $\mathcal{U}_{\ell}$, the red circle contains the set of so far unrecommended items for user $i_t$, and the blue circle represents set $\mathcal{P}_{\ell}$. The figure illustrates the movement of $i_t$ across levels. In the upper-left diagram the condition in Line \ref{['tp1']} is true. In all other diagrams the condition in Line \ref{['tp2']} is true. Note that in the upper-left diagram $j_t$ is removed from the blue set $\mathcal{P}_{\ell}$. The item $j_t$ is always removed from the red set of items not yet recommended to user $i_t$. When $\ell=\ell^*$ the upper-left diagram still applies. Right: An example run of Orca-UC with user set $\{1,2,3\}$, item set $\{a,b,c\}$, and a static inventory.
  • Figure 2: An example of the construction of the forest structure of the sets in Orca-UC with $\Lambda=8$. The set corresponding to a node is a strict subset of that corresponding to its parent (if it exists) whilst the sets corresponding to nodes on different root-to-leaf paths are disjoint. For all $\ell'\in[\Lambda]$ the node numbered $\ell'$ corresponds to the set $\mathcal{U}_{\ell'}$ and the $\ell'$-th diagram depicts its construction (the blue node). The blue nodes in the 9-th diagram correspond to the sets $\mathcal{U}_{2}\setminus\mathcal{U}_{5}$ , $\mathcal{U}_{6}\setminus\mathcal{U}_{8}$ and $\mathcal{U}_{4}\setminus\mathcal{U}_{6}$ which are all non-empty.
  • Figure 3: Recommendation curves for subsets of MovieLens with varying number of items (from left to right: $N=50$, $N=100$, $N=200$ items). The curves are averages across 30 repetitions. The numbers in braces give the area under each curve (the higher the better). The displayed curves for Wrmf are the best performers across the number of latent factors.

Theorems & Definitions (9)

  • definition 1
  • theorem 1
  • proof
  • theorem 2
  • theorem 3
  • proof
  • proof
  • proof
  • proof