Discrete choice modeling

Overview

Given the attributes of an offered set (assortment) of products and of a customer, predict the probability that each product is chosen 2207.12877 §1. The shared core is the random utility maximization (RUM) idea: each product and each customer has unobserved attributes, every product gets a utility from them, and the chosen product is the one with the highest utility. The choice probability is the chance that a product's utility is the maximum 2207.12877 §2.1. Variants differ in how they model the utility function and the unobserved attributes. The variants below all target this same quantity.

Variants

Parametric RUM models (MNL, nested logit, latent-class MNL)

These fix the utility function and the distribution of the unobserved parts in advance. The paper says the fit then depends on the modeller's choices 2207.12877 §1 §2.1. Multinomial logit (MNL) is the basic member. Latent-class MNL allows customer segments 2207.12877 §5.2 and is the stronger of the two parametric baselines in the paper's tables 2207.12877 tab. 1. No card yet for these; they appear only as benchmarks in the RUMnet card below.

RUMnet: a learned RUM approximator

RUMnet keeps the RUM structure but replaces the random fields with sample averages. It draws K samples of the unobserved product attributes and K samples of the unobserved customer attributes, each produced by a feed-forward network. A utility network scores every product for each of the K^2 sample pairs, a softmax turns those scores into probabilities, and the K^2 results are averaged 2207.12877 §2.2 §2.3. It is trained by minimising the negative log-likelihood of the observed choices 2207.12877 §4.1. The paper shows that the set of RUMnet models is dense among RUM models, so any RUM model can be approximated as closely as wanted given enough samples and network size 2207.12877 §3. Card: Representing Random Utility Choice Models with Neural Networks.

Neural special cases: TasteNet and DeepMNL

The paper treats these as special cases of RUMnet: with K = 0, or with a linear utility 2207.12877 §2.4. On Expedia they come next to RUMnet on loss (2.069 and 2.092) and accuracy (0.412 and 0.413) 2207.12877 tab. 1. No card yet.

Model-free classifiers: random forest and vanilla neural network

These do not impose the RUM structure 2207.12877 §1. The paper tunes the random forest over tree counts {50, 100, 200, 400} and depths {5, 10, 20} on validation loss 2207.12877 §6.2. The random forest has the lowest Swissmetro loss in the paper but the highest Expedia loss, and the paper ties its Expedia overfitting to the larger input dimension 2207.12877 tab. 1 §6.3 §6.4. Both are covered by the RUMnet card; neither has its own card.

When to use which

The paper's results are author-reported, on two datasets (Swissmetro with about 10K observations and three alternatives; Expedia with about 398K search queries and up to 39 alternatives) 2207.12877 §6.2 app. E. Treat the rows below as the conditions under which the paper's evidence applies.

Implementation sketch

Core of the RUMnet forward pass, one choice event. Each step is cited; the paper's Gumbel-noise motivation for the softmax is not needed in code 2207.12877 §2.2.

# Average softmax choice probabilities over K x K sample pairs  [§2.2] [§2.3]
# Input:  products (kappa, d_x), customer (d_z,), K product and customer nets
# Output: choice probabilities (kappa,), summing to 1

def rumnet_probs(products, customer, eps_nets, nu_nets, utility_net):
    # Start from zero probability for every alternative  [§2.3]
    total = zeros(products.shape[0])
    for eps_net in eps_nets:
        # Sampled unobserved product attributes  [§2.2]
        eps_sample = eps_net(products)
        for nu_net in nu_nets:
            # Sampled unobserved customer attributes  [§2.2]
            nu_sample = nu_net(customer)
            # Concatenate observed and sampled attributes per alternative  [§2.3]
            h = concat(products, eps_sample,
                       repeat(customer, products.shape[0]),
                       repeat(nu_sample, products.shape[0]))
            # Score each alternative under this sample pair  [§2.3]
            utilities = utility_net(h)
            # Softmax over alternatives replaces the argmax  [§2.2]
            total = total + softmax(utilities)
    # Average over the K^2 sample pairs  [§2.3]
    return total / (len(eps_nets) * len(nu_nets))

# Training objective: mean negative log-likelihood of observed choices  [eq. 2]
# Input:  chosen indices (T,), per-event probabilities (T, kappa)
# Output: scalar loss

def choice_nll(chosen, probs):
    # Probability the model gave to each observed choice  [eq. 2]
    chosen_probs = probs[arange(T), chosen]
    # Average negative log  [eq. 2]
    return -mean(log(chosen_probs))

Contradictions

• Significance of the RUMnet vs DeepMNL gap. The paper says the difference is statistically significant on both datasets 2207.12877 §6.3. Its footnote gives paired t-test p-values of 5.4e-2 on Swissmetro and 9.33e-10 on Expedia 2207.12877 §6.3. The Swissmetro value is above the common 0.05 threshold. This page reports both as the paper gives them.
• Training schedule: paper versus repository. The paper's table gives 1,000 epochs with patience 200 for Swissmetro and 100 epochs for Expedia 2207.12877 app. E. The authors' repository configs use different epoch counts on both datasets, and a different patience on Swissmetro. Which schedule produced the reported numbers is not stated.
• Activation: theory versus experiments. The theory uses ReLU networks 2207.12877 §4.1. The experiments use ELU 2207.12877 §6.2 app. E. The approximation result is stated for ReLU, so the results on ELU are not covered by the same theorem.

Papers

• Representing Random Utility Choice Models with Neural Networks: RUMnet, a neural approximator of RUM choice models, with a universal-approximation result and benchmarks on Swissmetro and Expedia.