Quantification using Permutation-Invariant Networks based on Histograms

TL;DR

Quantification targets estimating class prevalences in bags rather than individual labels, framing the task as symmetric learning where bags are labeled by prevalence. The authors introduce HistNetQ, a permutation-invariant architecture that encodes bag-level distributions with differentiable histograms and supports direct optimization of quantification losses. Across LeQua 2022 tasks and Fashion-MNIST, HistNetQ achieves state-of-the-art or competitive performance against set-based DNNs and traditional quantification methods, including EMQ, and remains effective under data-generation schemes like APP. The work demonstrates benefits of learning from aggregate labels, enabling tailored loss functions and broader applicability to problems like LLP and multi-instance learning.

Abstract

Quantification, also known as class prevalence estimation, is the supervised learning task in which a model is trained to predict the prevalence of each class in a given bag of examples. This paper investigates the application of deep neural networks to tasks of quantification in scenarios where it is possible to apply a symmetric supervised approach that eliminates the need for classification as an intermediary step, directly addressing the quantification problem. Additionally, it discusses existing permutation-invariant layers designed for set processing and assesses their suitability for quantification. In light of our analysis, we propose HistNetQ, a novel neural architecture that relies on a permutation-invariant representation based on histograms that is specially suited for quantification problems. Our experiments carried out in the only quantification competition held to date, show that HistNetQ outperforms other deep neural architectures devised for set processing, as well as the state-of-the-art quantification methods. Furthermore, HistNetQ offers two significant advantages over traditional quantification methods: i) it does not require the labels of the training examples but only the prevalence values of a collection of training bags, making it applicable to new scenarios; and ii) it is able to optimize any custom quantification-oriented loss function.

Figures (9)

• Figure 1: In this example, we observe the distributions of positive cases (green) and negative cases (blue) within the training dataset $D$. Additionally, we can see the mixture distribution (magenta) that provides the best approximation of the test bag distribution (black).
• Figure 2: Distribution of errors produced by EMQ-BCTS and "Mixer" heuristic in terms of Absolute Error (AE) and Relative AE (RAE) as evaluated in LeQua datasets T1A (top row) and T1B (bottom row) (see more details in Section \ref{['sec:experiments']}). EMQ-BCTS was trained and optimized using, respectively, the training and validation sets, and evaluated in the corresponding test bags, while for Mixer we run Montecarlo simulations generating bags out of the training examples of each task.
• Figure 3: Learnable histogram layer with hard binning and learnable bin centers and widths. The individual components are common operations used in DL frameworks that we use to compute Equation \ref{['eq:binning']}.
• Figure 4: An example of the common architecture used for DNNs methods. The feature extraction layer and the layer sizes correspond to a computer vision problem (Fashion-MNIST dataset). DenseFE and denseQ are sequences of fully-connected layers used in the feature extraction module and in the quantification module, respectively.
• Figure 5: Error distribution (measured in terms of RAE on a logarithmic scale) binned by the amount of prior probability shift ($|p_{D}-p_{B}|$) between the training set and each test bag. The green bars represent the distribution of bags per bin.

Theorems & Definitions (2)

• Lemma 4.1
• proof