Active Learning for Non-Parametric Choice Models

TL;DR

This work tackles actively learning non-parametric choice models, where identifiability can fail offline. It introduces a DAG representation that encodes all information that can be inferred from data and proves that exact probabilities suffice to reconstruct the DAG, while noisy estimates can cause error amplification. To address this, the authors develop AlgDAG, an active-learning algorithm that uses a set-cover-based inclusion-exclusion approach to estimate DAG edges efficiently, achieving polynomial sample complexity under a random-model assumption. Empirical results on synthetic and Sushi datasets show that AlgDAG more accurately recovers frequent preferences and outperforms non-active learning baselines, highlighting the practical value of active data collection for complex consumer-choice models.

Abstract

We study the problem of actively learning a non-parametric choice model based on consumers' decisions. We present a negative result showing that such choice models may not be identifiable. To overcome the identifiability problem, we introduce a directed acyclic graph (DAG) representation of the choice model. This representation provably encodes all the information about the choice model which can be inferred from the available data, in the sense that it permits computing all choice probabilities. We establish that given exact choice probabilities for a collection of item sets, one can reconstruct the DAG. However, attempting to extend this methodology to estimate the DAG from noisy choice frequency data obtained during an active learning process leads to inaccuracies. To address this challenge, we present an inclusion-exclusion approach that effectively manages error propagation across DAG levels, leading to a more accurate estimate of the DAG. Utilizing this technique, our algorithm estimates the DAG representation of an underlying non-parametric choice model. The algorithm operates efficiently (in polynomial time) when the set of frequent rankings is drawn uniformly at random. It learns the distribution over the most popular items among frequent preference types by actively and repeatedly offering assortments of items and observing the chosen item. We demonstrate that our algorithm more effectively recovers a set of frequent preferences on both synthetic and publicly available datasets on consumers' preferences, compared to corresponding non-active learning estimation algorithms. These findings underscore the value of our algorithm and the broader applicability of active-learning approaches in modeling consumer behavior.

Figures (4)

• Figure 1: The DAG corresponding to $(\Pi,p)$ with $p((1,2,3,4,5)) = p((1,2,3,5,4)) = p((1,2,4,3,5)) = p((2,3,4,1,5)) = p((2,4,1,3,5)) = 0.2$.
• Figure 2: The DAG corresponding to the indistinguishable pairs of rankings from Theorem \ref{['thm:impossible']}.
• Figure 3: The DAG corresponding to Example \ref{['example:naive_2']}. The rankings corresponding to the blue nodes interfere when trying to estimate the probability of the ranking corresponding to the purple node and edge. That is, the blue nodes represent the prefixes $A' \subsetneq A = \{1, 2, 3, 4, 5, 6\}$ in Lemma \ref{['lem:inter_2']}.
• Figure 4: L1 error in choice probabilities vs. market share of the set. In both figures, $n=16$, $n_0=8$, $\epsilon=0.01$, and $\kappa=0.01$. The only difference is that in the left figure, $\rho=0.01$, while in the right figure, $\rho=0.05$.

Theorems & Definitions (28)

• Definition 1: Indistinguishability
• Theorem 1: Impossibility Result
• Definition 2: Prefixes, Edges, and Their Probabilities
• Proposition 1
• Definition 3: DAG Representation of $(\Pi,\bf{p})$
• Definition 4: Truncated DAG
• Proposition 2: Computing Choice Probabilities from a DAG
• Lemma 1
• proof
• Theorem 2: Constructing the Exact DAG using Exact Choice Probabilities