# Actively learning a Bayesian matrix fusion model with deep side information

Yangyang Yu & Jordan W. Suchow
School of Business
Stevens Institute of Technology
Hoboken, NJ 07030
<EMAIL_ADDRESS>

###### Abstract

High-dimensional deep neural network representations of images and concepts
can be aligned to predict human annotations of diverse stimuli. However, such
alignment requires the costly collection of behavioral responses, such that,
in practice, the deep-feature spaces are only ever sparsely sampled. Here, we
propose an active learning approach to adaptively sampling experimental
stimuli to efficiently learn a Bayesian matrix factorization model with deep
side information. We observe a significant efficiency gain over a passive
baseline. Furthermore, with a sequential batched sampling strategy, the
algorithm is applicable not only to small datasets collected from traditional
laboratory experiments but also to settings where large-scale crowdsourced
data collection is needed to accurately align the high-dimensional deep
feature representations derived from pre-trained networks.

_Keywords_ active learning, Bayesian modeling, cognitive science, multi-modal
learning

##  1 Introduction

In cognitive research, Bayesian probabilistic models typically serve two
principal roles: one as a hypothesis positing how individuals draw inferences
from their observations of the environment, and the other as a tool enabling
scientists to learn from observations of human behavior (Vul et al. (2014);
Griffiths et al. (2008); Mamassian et al. (2002)). Our work acts as an
intermediate approach that bridges these two uses of Bayesian models. We use
Bayesian Probabilistic Matrix Factorization (BPMF) with deep side information
to align a machine representation of entities to human behavioral responses to
those entities, such that the model serves as both a model of people’s mental
representations and as a predictive model of their behavior. This method
enables accurate inference of behavioral responses by generating low-rank
predictions of perceptual response matrices. It offers a viable model
structure to align machine vision systems with human visual perception. For
instance, it can integrate the bimodal information from facial imagery and
psychological attributes and yield predictions of people’s impressions of
human faces.

BPMF has proven effective in consolidating multi-source information and
predicting missing responses while constructing confidence intervals
(Salakhutdinov and Mnih (2008); Adams et al. (2010)). In traditional
laboratory experiments, it has been successfully applied to predict
individuals’ perceptual outcomes based on human interpretable features (Zhang
et al. (2020)).

When implementing BPMF in large-scale behavioral prediction tasks that involve
object features from multiple modalities, two significant challenges arise.
The first challenge is that high-quality prediction demands advanced machine-
generated features that outperform traditional human-defined ones. Machine
learning algorithms have the capacity to generate a vast number of new objects
based on previously perceived ones, enhancing diversity and realism while
reducing bias among stimuli. However, using deep learning algorithms such as
generative adversarial networks (GANs) (Kammoun et al. (2022)) to extract
high-dimensional, informative features from various objects increases
computational costs nearly linearly with the number of data features. This
poses a challenge to the integration of BPMF with deep learning methods,
making it practical only to sparsely sample from the deep-feature space.
Second, there is a lack of crucial information needed for making reliable
predictions due to the highly sparse response matrix. This issue becomes
especially problematic in human-subjects research, where data collection
operates under constrained budgets and resources, deals with a large
participant population, and may have an extensive list of questions. Budgetary
constraints may limit the number of questions asked, and lengthy experimental
instruments could lead to inaccurate responses due to participants resorting
to mental shortcuts (Krosnick (1991)). Hence, large-scale experiments can
often only collect responses from a small fraction of the total instrument
from each participant. To effectively implement BPMF with deep side
information on large-scale behavioral prediction tasks, it is therefore
crucial to develop a data sampling strategy that can effectively target the
most informative data points, enabling the achievement of satisfactory
predictive results despite these constraints.

Active learning is a data acquisition technique that can interactively
identify the most informative samples to efficiently create a training data
set. Although this training set can be compact, it possesses a powerful
predictive capacity. Active learning has been widely employed to tackle
problems associated with accuracy in sparse matrix completion (Elahi et al.
(2016); Chakraborty et al. (2013a)). One of its key strengths is the capacity
to accurately infer the complete response distribution from a limited
selection of samples, obviating the need to query the majority of the response
matrix.

Here, we introduce a method for active learning in a BPMF model, utilizing
uncertainty sampling (Sugiyama and Ridgeway (2006)) and k-Center Greedy
selection (Sener and Savarese (2017)). We demonstrate that this method more
efficiently learns a multimodal model compared to a passive baseline. We
further investigate the impact of varying MCMC approximation simulation chain
lengths on the active sampling performance to optimally integrate active
learning into the BPMF model framework. Estimating the posterior uncertainty
brings a cost associated with the number of posterior MCMC samples collected.
A trade-off exists between a slow yet precise estimate and a quick but less
accurate one. We investigate this trade-off by manipulating the number of MCMC
samples used for the posterior uncertainty estimation in model parameters and
measuring its effect on the overall algorithm performance under a fixed
computational budget.

In this paper, we first explain the framework for actively learning the deep
Bayesian matrix factorization model. We then train the model on a large
behavioral dataset — the One Million Impressions dataset (Peterson et al.
(2022)) — and validate the Bayesian model learning efficiency as well as
performance improvement by applying a suitable active strategy to acquire a
more informative training pool. Finally, we demonstrate that our method
achieves promising predictive results by adaptively querying a minority of the
data for model training.

##  2 Related work

We discuss related work in three separate sections below, on the topics of
using Bayesian probabilistic matrix factorization (BPMF) for human behavioral
prediction, high-dimensional deep feature representations as the side
information of BPMF and active learning.

###  2.1 Bayesian Probabilistic Matrix Factorization

BPMF acts as a bridge between human cognition and mathematical inductive
inference. Based on assumptions regarding cognitive problems, it integrates
prior knowledge from multiple channels, guiding participants to update their
beliefs using mathematical forms. Moreover, it dictates how participants
update their beliefs in response to new data with mathematical forms. In the
context of visual perception modeling, BPMF uses a matrix structure to combine
2-way side information - visual objects and observers’ features, over priors.
Through Monte Carlo Markov Chain (MCMC) simulations, Bayesian posteriors are
computed and linked to perceptual inference outcomes (Mamassian et al. (2002);
Kersten et al. (2004)). Two components in the MCMC simulation chain are warm-
up steps and posterior samples. Warm-ups initialize model parameters so that
the Markov chain can efficiently sample from the neighborhood of substantial
posterior probability mass and travel in equilibrium. Posterior samples,
formed after warm-up, shape the posterior distribution and generate predicted
outcomes. A balance between the number of warm-ups and posterior samples is
needed to avoid significant fluctuations and overkill in MCMC simulation
performance (McElreath (2020)). It helps the BPMF achieve satisfactory
predictive results through MCMC.

###  2.2 Modeling Perception Using Deep Features

Most human perceptual inference tasks involve combining information from
multiple modalities, such as visual and linguistic information. Previous
research (Peterson et al. (2022); Zhang et al. (2018)) has demonstrated the
superiority of machine-generated deep features over conventional human-
interpretable attributes (e.g., color or size in images). This is primarily
due to 1) the comprehensive nature of high-dimensional deep object
representations, and 2) the ability to self-generate new objects, which is
crucial for predicting novel objects in the model system. Pretrained networks
from deep learning domains, such as StyleGAN2 (Karras et al. (2020)) for
images and Sentence-BERT (Reimers and Gurevych (2019)) for text, can generate
high-quality object feature sets for cognitive studies, thus reducing
considerable data-collection costs. Hence, within the BPMF paradigm, deep
features serve as a more effective choice than human-interpretable features
for generating predictions.

###  2.3 Active Learning

Taking inspiration from sparse matrix completion research (Settles (2009);
Chakraborty et al. (2013b)), we leverage active learning as a practical
approach to impute a full response distribution for each participant over
diverse objects, given a response matrix with limited entries. This approach
prioritizes data points for querying in the response matrix, allowing for a
faster alignment of machine inferences with human thinking with fewer queried
data points. Active learning accelerates this alignment by adaptively choosing
the most informative data samples, a critical approach in large-scale
cognitive experiments with constrained resources. In our model, two active
strategies are deployed to enhance BPMF performance with a limited training
set: 1) Uncertainty sampling (Chakraborty et al. (2013b); Sutherland et al.
(2013)), prioritizing data points with the highest predictive uncertainties in
each iteration as the most informative; 2) k-Center Greedy sampling (Sener and
Savarese (2017)), seeking k data points that maximize the mutual information
between them and the remaining unselected data pool in every iteration. This
is achieved by pre-setting k centers and identifying data points that minimize
the maximum distance of these points to the nearest centers.

##  3 Methods

Our approach improves the performance of perceptual learning on limited
training cases by implementing the active strategy to the Bayesian matrix
factorization with deep side features. The details of this approach and
notation are articulated in two sections below.

###  3.1 Deep Bayesian Probabilistic Matrix Factorization

First, in predicting first impressions, a pre-trained deep network is used to
create face and trait features. Utilizing indices $j$ for face images, $h$ for
traits, and $i$ for participants, Peterson et al. (Peterson et al. (2022))
proposed a method for extracting deep face features $f_{j}$ using StyleGAN2
generator. Meanwhile, we employ the Sentence-BERT model to create a deep trait
feature set $t_{h}$. This becomes in two latent spaces for the response matrix
$R$: a 512-dimensional image feature latent space $\mathit{\mathcal{F}}$ for
faces and a 300-dimensional linguistic feature space $\mathit{\mathcal{T}}$
for traits. These spaces are represented by conditioned, unit-variance,
multivariate normal latent variables: $\omega_{f_{j}}$ for the face space and
$\omega_{t_{h}}$ for the trait space. A parallel experiment involved lower-
dimensional dense deep features, conducted by further processing pre-trained
image and trait features through a multi-layer perceptron (MLP) model (Yu and
Suchow (2022)). The goal is to determine whether encapsulating deep features
to a lower dimensionality provides a sufficient representation and if it’s
necessary to include an intermediate dimension reduction operation before
inputting side features into the Bayesian model. If so, understanding the
appropriate degree of dimension reduction is key to ensuring adequate feature
representativeness while minimizing computational cost and simulation time.

Second, under the BPMF setting, the latent variables represent the
computational coefficients for the participants’ impression ratings $R_{jh}$.
The coefficients are estimates based on two priors of
$\omega_{f_{j}},\omega_{t_{h}}$ following spherical Gaussian distributions
(Gurkan and Suchow (2022)). In Formulas 1 and 2, the two-sided features are
fused together after being converted into the same dimensionality in the
latent space, and then the responses $\hat{R}_{jh}$ are predicted through
simulation.

$\begin{split}p(\omega_{f_{j}}|\sigma)=\text{Normal}(\omega_{f_{j}}|\sigma^{-1}\mathbf{I});\quad
p(\omega_{t_{h}}|\sigma)=\text{Normal}(\omega_{t_{h}}|\theta^{-1}\mathbf{I}),\end{split}$
(1)

where $\sigma$ and $\theta$ are independent and gamma distributed.

$\begin{split}&F_{j}=f_{j}\times\omega_{f_{j}}^{T};\quad\quad\quad\;\;\quad
T_{h}=t_{h}\times\omega_{t_{h}}^{T};\\\ &R^{*}_{jh}=F_{j}\times
T_{h}^{T}+\varepsilon_{jh};\quad\hat{R}_{jh}=\text{sigmoid}(R^{*}_{jh})\times
100,\end{split}$ (2)

where $f_{j}$ and $t_{h}$ are row vectors and $R^{*}_{jh}\in(-\infty,\infty)$
is projected into predicted ratings as a continuous value
$\hat{R}_{jh}\in(0,100)$.

Third, we further analyze the impact of the simulation process on our deep
BPMF model’s performance. By running parallel simulation chains of equal
length but varying proportions of warm-ups to posterior samples, we identify
the optimal mean for impression inference. Furthermore, we employ sequential
Markov chain Monte Carlo (MCMC) (Yang and Dunson (2013)) as the approximation
method for modeling BPMF across the whole dataset. This choice is guided by
the MCMC method family’s suitability for large hierarchical model computation
(Banerjee et al. (2003)), and its effectiveness in our case for integrating
over thousands of parameters associated with deep features. We observe that
aggregating parameters from the last ten Bayesian posteriors rather than all
posteriors stabilizes model predictions, reducing sampling randomness.
Compared to traditional MCMC, sequential MCMC more readily extracts the last
few posteriors in each iteration, offering reduced memory usage and faster
computation when using Numpyro in Python as the development platform.

###  3.2 Actively Learning The Matrix Factorization Model

By implementing active learning strategies in the training data sampling
process of BPMF, we aim to use as few filled matrix entries as possible in
training, predict all the rest in the response matrix, and converge the
Bayesian model in fewer iterations. We uniformly choose a very small initial
training pool $\boldsymbol{S^{0}}$ with size $L$ (5, 8, etc.), selected
randomly from the whole rated data pool $\boldsymbol{S}$, which has size $N$.
Notice that only the initial pool is accessible to us. The feature pair
$(f_{jn},t_{hn})$ of an arbitrary data entry $\mathcal{C}_{n}$ in
$\boldsymbol{S}$ is shortened as $\mathcal{U}_{n},\,n\in\\{1,..N\\}$, and its
rating as $R_{in}$. The feature pair of a data entry $\mathcal{C}_{l}$ in the
initial pool $\boldsymbol{S^{0}}$ is denoted as $\mathcal{U}_{l}$ with
observable rating as $R_{il}$, $l\in\\{1,...,L\\}$. Besides the initial pool,
we assume $Q$ opportunities as the budget for asking an oracle for information
about an extra $p$ data points and a learning algorithm $A_{\boldsymbol{S}}$
to guide us in choosing the appropriate points at each time.
$A_{\boldsymbol{S}}$ generates an updated set of parameters
$\\{\omega_{f_{j}}^{*},\omega_{t_{h}}^{*}\\}$ based on
$\boldsymbol{{S^{0}}^{*}}=\boldsymbol{S^{0}}\cup\boldsymbol{S^{p}}$. The
choice of $\boldsymbol{S^{p}}$ for the oracle to label is a subset of
$\boldsymbol{S}$, which minimizes the future expected learning loss. Formula 3
shows the future expected loss after $q$ times label querying.

$\begin{split}&\underset{\boldsymbol{S^{p}_{q}}}{\text{min}}\,E_{(\mathcal{C}_{1},...,\mathcal{C}_{N})\sim\boldsymbol{S}}[\mathcal{L}(\mathcal{C}_{n};A_{\boldsymbol{S^{0}}\cup...\boldsymbol{S^{p}_{q}}})]\end{split}$
(3)

Compared to the classic definition of active learning, our active strategy has
two significant differences: 1) We set $A_{\boldsymbol{S}}$ to query a batch
of $p$ data points in each round instead of a single data point because
previous works (Sener and Savarese (2017); Zhang et al. (2020)) illustrate
that with a large dataset and high-dimensional deep features, the performance
improvement made by one data point in each round is negligible. 2) As active
learning has mainly been shown to be effective for tasks with discrete
prediction targets (Settles (2009); Yona et al. (2022)), we further expand its
capability of handling continuous prediction cases by reformulating the
learning metrics and loss functions. Here, we explore two types of batched
active strategies to construct and minimize the loss function for continuous
targets and compare their performance. One is pure uncertainty-based sampling,
and the other takes the trade-off of sampling diversity and uncertainty of
sampling into account.

####  3.2.1 Uncertainty Sampling

One classic approach of uncertainty selection is targeting samples with the
least confidence in predictions in each active iteration. Sugiyama and
Ridgeway (Sugiyama and Ridgeway (2006)) justified selecting data points with
the maximum standard deviations of predictive distribution,
$\sigma_{{\omega_{\mathcal{U}_{l}}^{*}}}$ (shorten as $\sigma$), as the most
uncertain samples in the context of Gaussian linear regression. Here, we
further extend this strategy to Bayesian matrix factorization because of two
critical similarities: 1) like their work, we have a continuous prediction
target, and 2) the prediction distribution is also a univariate Gaussian
distribution, $\rho(R^{*})$.

In this paper, we validate the performance of uncertainty-based active
Bayesian matrix factorization on the impression prediction task. Algorithm 1
and Formula 4 (in Appendices B) present how this strategy works iteratively to
select the $p$ most uncertain samples as a batch to add to the training pool.

Algorithm 1 Batched Uncertainty Sample Selection for Deep Bayesian Matrix
Factorization

The set of randomly selected $d$ ratings as initial pool $\boldsymbol{S^{0}}$;
The rating $R_{il}$ in $\boldsymbol{S^{0}}$ is given by a participant $i$ for
a face image in terms of a certain trait. And budget $Q>0$.

Predicted ratings for all entries of the response matrix, $\hat{R}$

Extract deep features face images $f_{jl}$ and trait $t_{hl}$

repeat $Q$ times, initialize $q=1$;

Update parameters ${\omega_{f_{j}}^{*}}_{q},{\omega_{t_{h}}^{*}}_{q}$ via MCMC

Compute predicted $\hat{R}_{q}$, $\sigma_{q}$ for the response matrix

Query $\boldsymbol{S^{p}_{q}}$ with
$\text{argmax}\sum_{z=1}^{p}\;{\sigma_{z}}_{q}$111${\sigma_{z}}_{q}$ denotes
the predicted standard deviation for the $z$th queried data point in a batch
$p$ samples during the $q$th iteration of the uncertainty strategy

$\boldsymbol{S^{0}_{q}}\leftarrow\boldsymbol{S^{p}_{q}}\cup\boldsymbol{S^{0}_{q-1}}$

q = q + 1

until $q=Q$

return $\hat{R}_{Q}$, $\boldsymbol{S}\setminus\boldsymbol{S^{0}_{Q}}$

####  3.2.2 k-Center Greedy Sampling

k-Center Greedy sampling is a pool-based active sampling strategy. Sener et
al. (Sener and Savarese (2017)) regard active learning loss for
classifications using CNN given a batch of $p$ samples,
$E_{(\mathcal{C}_{1},...,\mathcal{C}_{N})\sim\boldsymbol{S}}(\mathcal{L}(\mathcal{C}_{n};A_{\boldsymbol{S^{p}}}))$,
as containing three components: generalization loss, training loss, and core-
set loss. $p$ denotes the number of central points in the unexplored data pool
and also equates to the count of points chosen during each adaptive sampling
cycle.

In our scenario, which involves predicting continuous targets, the first two
losses are primarily managed by the Bayesian model, while the active learning
strategy concentrates on minimizing core-set loss. Core-set loss is defined as
the distance between average empirical loss over the points with known ratings
and that over the entire dataset, including entries with unknown ratings.
Sener et. al shows its upper bound as
$\mathcal{O}(d_{\boldsymbol{S^{0}}\cup\boldsymbol{S^{p}}})+\mathcal{O}(\sqrt{\frac{1}{N}})$.
Consequently, the optimization goal of loss can be converted to minimize the
coverage radius from $p$ centers
($d_{{\boldsymbol{S^{0}}\cup\boldsymbol{S^{p}}}}$), as illustrated in Figure 4
(in Appendices A and C). The minimization of
$d_{{\boldsymbol{S^{0}}\cup\boldsymbol{S^{p}}}}$ is further deduced as
$\underset{\boldsymbol{S^{p}}}{\text{min}}\,\underset{\mathcal{C}_{n}\in\boldsymbol{S}\setminus(\boldsymbol{S^{0}}\cup\boldsymbol{S^{p}})}{\text{max}}\,\underset{\mathcal{C}_{l}\in\boldsymbol{S^{0}}\cup\boldsymbol{S^{p}}}{\text{min}}\,\bigtriangleup(\mathcal{U}_{n},\mathcal{U}_{l})$
when $p$ data points are queried.This formula can be computed based on the
distances of feature pairs $\mathcal{U}_{n}=(f_{jn},t_{hn})$ and
$\mathcal{U}_{l}=(f_{jl},t_{hl})$ in a two-dimensional coordinate system. The
$\underset{\boldsymbol{S^{p}}}{\text{min}}\,{\boldsymbol{S^{0}}\cup\boldsymbol{S^{p}}}$
is iteratively recomputed in Algorithm 2 to search for new data points to be
queried.

Algorithm 2 k-Center Greedy Sample Selection for Deep Bayesian Matrix
Factorization

The set of randomly selected $d$ ratings as initial pool $\boldsymbol{S^{0}}$;
the rating $R_{il}$ in $\boldsymbol{S^{0}}$ is given by a participant $i$ for
a face image in terms of a certain trait. And budget $Q>0$.

Predicted ratings for all the entries of the response matrix, $\hat{R}$

Extract deep features face images $f_{jl}$ and trait $t_{hl}$

repeat $Q$ times, initialize $q=1$;

Update parameters ${\omega_{f_{j}}^{*}}_{q},{\omega_{t_{h}}^{*}}_{q}$ via MCMC

Set $p$ learning centers

Choose $p$ centers
$({\mathcal{C}_{1}}_{q},...,{\mathcal{C}_{p}}_{q})=S^{p}_{q}$
222${\boldsymbol{S}_{h}^{p}}_{q}$ denotes the $k$th queried data point in a
batch $p$ in the $q$th iteration of k-Center Greedy active strategy.by

$\boldsymbol{S^{p}_{q}}=\underset{{\mathcal{C}_{n}}_{z}\in
S\setminus{S_{z}^{p}}_{q}}{\text{argmax}}\;\text{min}_{{\mathcal{C}_{l}}_{z}\in{S_{z}^{p}}_{q}}\;\bigtriangleup({\mathcal{U}_{n}}_{z},{\mathcal{U}_{l}}_{z})$,

where set
$\boldsymbol{S^{p}_{q}}\subseteq\boldsymbol{S}\setminus\boldsymbol{S^{0}_{q-1}}$

$\boldsymbol{S^{0}_{q}}\leftarrow\boldsymbol{S^{0}_{q-1}}\cup\boldsymbol{S^{p}_{q}}$

q = q + 1

until $q=Q$

return $\hat{R}_{Q}$, $\boldsymbol{S}\setminus\boldsymbol{S^{0}_{Q}}$

##  4 Dataset

The One Million Impressions dataset (Peterson et al. (2022)) is a large-scale
dataset containing people’s perceptual judgments based on their first
impressions of human faces. This dataset contains over 1 million ratings of 34
traits for 1,000 machine-generated face images. Each face image is rated by 30
unique individuals for each trait. Given that each image only receives ratings
from a limited group of participants, the response matrix is relatively
sparse. This dataset is ideal for our approach as it allows for the
integration of deep image and trait attributes as side information in BPMF.

##  5 Experiments and Results

We conduct three experiment sets to explore the applicability of implementing
the active learning strategy in the training data sampling process for deep
Bayesian matrix factorization model. This approach comprises active learning,
deep input features, and BPMF. Each experiment tweaks one component. The
active learning method’s efficacy and efficiency are examined by the Root Mean
Square Error (RMSE). The test RMSE, computed on the dataset excluding the
queried training samples, is the most critical metric, reflecting the model’s
predictive ability for unknown data. The models’ evaluation metrics are
averaged over three experimental repetitions.

Figure 1: Test RMSEs over sample number with different batch sizes for
uncertainty vs. k-Center Greedy active strategies with $95\%$ confidence
intervals.333To smooth out fluctuations in RMSEs, each dot in Figures 3 and 2
is averaged from 200 samples of the adjacent epochs.

###  5.1 Choices of Strategy and Batch Size

The first experiment seeks to identify the optimal batch size for an active
learning task on a large scale. We employ two active sampling strategies,
uncertainty and k-Center Greedy, with deep Bayesian matrix factorization
models to predict first-impression ratings. These strategies adaptively query
batch samples, with batch size varying per experimental setting. Five active
models with distinct strategy and batch size combinations are run
concurrently. The initial training pool, equivalent in size to the chosen
batch size, is randomly drawn from the unknown data pool. The baseline is the
same deep Bayesian matrix factorization model with passive sampling that
randomly selects increasing numbers of samples across iterations, without
adaptive querying. Pretrained StyleGAN2 image features and Sentence-BERT trait
features serve as the deep BPMF’s input side information.

Figure 2: Test RMSEs over sample number for the BPMF with batch-10 uncertainty
active strategy under different choices of input features dimensions.

Figure 3 presents adaptive sampling across 34,000 samples, where the
uncertainty active strategy surpasses both the k-Center Greedy strategy and
the passive baseline model. Among the uncertainty sampling batch sizes, a
batch size of 8 emerges as the optimal choice, demonstrating the fastest and
sharpest decrease in test RMSE, and experiencing minimal fluctuations and the
narrowest confidence interval as the training pool size increases. Its
advantage is particularly notable when the training pool has fewer than 10,000
samples, representing roughly $0.1\%$ of the entire dataset. In this case, the
test RMSE drops sharply from 32.1 to 27.8, approximately 1.0 lower than the
passive learning’s test RMSE. The uncertainty strategy with a batch size of 10
achieves the second most efficient convergence of the six models but exhibits
less stability in test RMSE during the intermediate training pool size in a
range of [15,000, 25,000]. The batch size 5 model encounters severe
fluctuations when the training pool grows larger than 20,000 and sometimes has
noticeably higher predicted RMSEs and a wider confidence interval than the
baseline model, indicating it is the worst model in the uncertainty strategy
group. On the contrary, regardless of training set size, neither batch-8 nor
batch-10 k-Center Greedy performs better than the baseline model. After the
training pool reaches 28,000 samples, models with batched uncertainty active
sampling gradually converge with the baseline model at a test RMSE of about
25.8, indicating the training pool has reached a size of containing extensive
information. Thus, even random selection works as effectively as the active
strategy due to the Bayesian model’s estimation and control for future
predictive uncertainty. Consequently, we confirm that the batched uncertainty
active strategy for deep Bayesian matrix factorization can efficiently
decrease predictive RMSE when training data size is limited. Furthermore, the
choice of batch size influences the model’s prediction stability.

###  5.2 Efficacies of Input Feature Dimensionalities

The second set of experiments probes the influence of different deep-feature
dimensionalities on deep Bayesian matrix factorization models with uncertainty
active sampling. We investigate the trade-off between feature dimensionality
and predictive performance and determine the optimal set of feature vectors.
We use the original pre-trained deep features (512-dimensions for images, 300
for traits) as inputs for the benchmark model (Model 0). Models 1–4
incorporate dense feature layers, from the bimodal MLP model (Yu and Suchow
(2022)), with reduced dimensionalities of 100, 80, 32, and 16 before fusion.

Figure 2 shows that while Model 0 consistently retains a leading position in
terms of test RMSE, Models 1 and 2, using 100- and 80-dimensional dense
features respectively, achieve comparable test RMSEs when the training pool
size is small to moderate (up to 30,000 samples). However, when the sample
size reaches 30,000, both Models 1 and 2 exhibit significant fluctuations in
RMSE, suggesting their inability to capture sufficient informative samples for
future iterations. Models 3 and 4, using 32- and 16-dimensional features, fail
to target informative data points conductive for learning performance
irrespective of training pool size. The models’ performance indicates that
retaining approximately $33\%$ of trait features and $20\%$ of face features’
dimensionality enables the active strategy to sample informative points when
the training set is sparse. However, the ability to query effectively degrades
when the training set is more enriched. We infer that the loss of subtle
feature information due to the nonlinear feature transformation of dimension
reduction is the primary reason for this. The consistently high test RMSEs for
Models 3 and 4, with the lowest input dimensions, support this inference. As
the feature dimensionality decreases, the information loss worsens from
negligible to critical.

MCMC chain length |  Learning Type |  Running time (minutes) |  Test RMSE
---|---|---|---
220 |  Active |  32.4633 |  27.024
280 |  Active |  36.1294 |  26.1506
64,000 |  Passive |  29.3237 |  27.4982
80,000 |  Passive |  35.8137 |  27.7274
88,000 |  Passive |  39.5863 |  27.1459
Table 1: Active learning vs. passive learning training time and test RMSE in
the third experiment.

###  5.3 Impact of Simulation Chain Length

(a) Test RMSEs for different model options over the lengths of the simulation
chain with 350 training samples

(b) Test RMSEs for the BPMF using batch-2 active strategy with different
simulation chain lengths over training sample numbers.

Figure 3: Result summary of the third experiment. 444To smooth out
fluctuations in RMSE values, the dots in Figures 4 are averaged from 5
adjacent epochs.

In the third set of experiments, we explored three configurations for the
Markov Chain Monte Carlo (MCMC) simulation chains: (1) increasing both warm-
ups and posterior samples in a 3:5 ratio, in increments of 8 samples, (2)
expanding the posterior samples by steps of 5 with no warm-ups, and (3)
raising the warm-ups in steps of 5 with a constant 5-step posterior samples.
After identifying the optimal configuration, we applied a batch-2 uncertainty
active sampling strategy to the BPMF model. Through the comparison between
active versus passive sampling on BPMF, we aimed to evaluate whether
increasing the MCMC simulation chain’s length can compensate for the
efficiency advantage gained by active learning. For this, we compared the
prediction efficiency of the deep BPMF with active versus passive sampling as
the simulation chain lengthened. Given computational limitations, these
experiments utilized a 1000-sample subset of the One Million Impressions
dataset. This subset consisted of ten arbitrary responses from each
combination of ten faces and ten traits, selected randomly beforehand. To
control the outcome unpredictability caused by insufficient training data and
to distinguish performance differences across experimental settings, we set
the training pool to 350 samples and assessed model performance by comparing
the predictive Root Mean Squared Error (RMSE) on the remaining 650 test data
points. Given the considerable reduction in the total size of the dataset, the
batch size for the active strategy was decreased to two.

We first executed three passive-sampling BPMF models, each with different
simulation chain options, and compared their test RMSE trends over time
(Figure 4(a)). Our observations revealed that the MCMC chains from Option 1
offered the highest reduction efficiency in predictive RMSE as the simulation
steps increased. Options 2 and 3 consistently yielded higher test RMSEs than
Option 1, demonstrating that a simulation chain with proportional increases in
warm-ups and posteriors yields the most powerful predictions for human
impression ratings. Moreover, we applied the optimal simulation chain (Option
1) to the BPMF model with a batched uncertainty active sampling strategy. Its
results in Figure 4(a) showed a significantly better performance with fewer
simulation steps than all passive learning models, except during the initial
stages ($<$200 steps). By the time 220 simulation steps were reached, the
active model’s predictive RMSE dropped to 27.024, already 0.05 lower than the
best-performing passive model with much longer simulation chains. When the
simulation chain length extended to 1,800 steps, the active model achieved the
best test RMSE of 24.515, 2.533 lower than the passive model with the same
number of steps. After that point, the active model’s test RMSE increased
slightly but remained the lowest among all settings. As too few simulation
steps lead to incorrect predictions in practice, we demonstrate that deep
Bayesian matrix factorization with active strategy retains a sustained
advantage over passive sampling BPMF on model performance. The training time
analysis Table 1 further illustrates that the advantage of active learning
cannot be offset by extensively extending the simulation chain in terms of
both training time and predictive performance. With a reasonably short
simulation chain length of 220, the active model’s RMSE decreases to a level
that the passive models can’t reach. This training time analysis was run on a
single NVIDIA GeForce RTX 4090 GPU with 24 GB memory.

Upon querying 350 samples, the Bayesian model with active strategy achieves
optimal performance with a 1,800-step simulation chain. We investigate whether
this simulation chain length can maintain the performance advantage throughout
the adaptive sampling process. Figure 4(b) displays test RMSEs across varying
training pool sizes for different simulation chain lengths. The test RMSE for
the 1,800-step model remains among the lowest as the training set expands from
50 to 350 samples, confirming that the optimal performance achieved with 350
training samples is not an exceptional case for the model with 1,800-step
simulation during the active sampling process.

##  6 Conclusion and Discussion

In this paper, we propose and evaluate a method for actively learning a model
that fuses deep neural network representations using Bayesian probabilistic
matrix factorization to produce a predictive model of human first impressions.
The active strategy identifies the most informative stimulus-attribute pairs
to sample, considering the uncertainty in the posterior of the model
parameters. Empirical experiments demonstrate its superior performance
compared to passive learning. This advantage is crucial in real-world
crowdsourced cognitive studies with limited experimental budgets.
Additionally, our methodology exhibits broad applicability across numerous
domains, such as in the development of online recommendations for social
media, e-commerce, or behavioral prediction systems. With minimal data
collected from a small segment of existing customers or users, our approach
efficiently generates high-quality behavioral forecasts for new users or
objects, thereby offering a solution to the prevalent cold-start problem in
recommendation systems.

Four aspects related to the three main components of deep Bayesian matrix
factorization using an active learning strategy jointly impact performance and
computational efficiency on large behavioral datasets: the sampling strategy
and batch size in active learning, the dimensionality of the deep feature
space, and the lengths of the simulation chain in BPMF. Our first experiment
with human impression data demonstrated that the uncertainty sampling strategy
significantly outperforms the k-Center Greedy strategy. The uncertainty
strategy selects samples based on predicted standard error computed from the
posterior distribution over the latent feature space, while the k-Center
strategy targets samples covering the diversity of the original pretrained
deep features. With a small training pool, Bayesian inference is still capable
of learning from the least confident data retrieved by the previous sampling
iteration and extracts valuable information to efficiently reduce model
predictive error in the next iteration, benefiting the uncertainty strategy.
In contrast, the k-Center Greedy sampling struggles to choose cluster centers
from input features, which reflect the centers of similar groups in the
unknown pool. The choices of 8 or 10 centers fail to capture feature
diversity, causing the k-Center Greedy strategy to underperform against
passive learning. Choosing an appropriate batch size can enhance learning
efficiency for active sampling. However, this decision is dependent on the
size of the unknown data pool and the strategy mechanism. Although the unknown
pool size is usually beyond our control, the first experiment showed that an
active strategy accessing the Bayesian latent space, instead of initial input
features, makes it easier to determine a feasible batch size.

Our second experiment tested whether reducing the dimensionality of deep
features can optimize computational resources. The findings suggested that
while reducing feature dimensionality can economize computational resources,
it may impair model performance. This emphasized the importance of high-
dimensional pretrained deep features over low-dimensional, human-interpretable
features. We observed that reducing deep feature dimensionality by over $60\%$
was tolerable for a small training set. Consequently, the required degree of
dimension reduction and prospective training set size need careful
investigation when optimizing computational resources.

The impact of the MCMC simulation chain on the deep Bayesian matrix
factorization model with active strategy is investigated from multiple
perspectives in the third set of experiments. The outcomes indicate that the
batched active sampling strategy empowers the Bayesian model with a shorter
simulation chain, outperforming the passive models in predicting human first
impressions. This underscores our assertion that an appropriate active
strategy and batch size can enhance the performance of the deep BPMF,
particularly when constrained by data query budgets and computational
resources.

Despite the promising performance of deep Bayesian matrix factorization
combined with the uncertainty active strategy in predicting people’s
impressions, two aspects of our model require further attention: 1) Our
experiment utilized deep image and trait features as bimodal side information
to predict impression ratings through Bayesian factorization, generating a 2D
response matrix. Considering that Bayesian factorization can also handle 3D
tensors (Xiong et al. (2010)), we are interested in exploring the
generalizability of our approach to datasets with more complex structures and
diverse features. For instance, in predicting people’s impressions, a third
information type, such as participants’ features, could be included alongside
image and trait attributes. 2) We expanded two classic active strategies
initially designed for classifying discrete targets to predict continuous
rating values. Our study confirmed the effectiveness of the uncertainty
strategy and examined potential issues in the k-Center Greedy strategy. To
further enhance learning efficiency, the integration of the Bayesian model
with more advanced adaptive sampling strategies is worth thorough
investigation.

##  7 Appendices

##  Appendix A Queried data samples in k-Center greedy active learning
strategy

The following graph Figure 4 depicts $p$ centers and their associated coverage
radii $d_{{\boldsymbol{S^{0}}\cup\boldsymbol{S^{p}}}}$ during one batch
selection of k-Center Greedy active sampling. This representation is utilized
to illustrate one essential component of the upper bound of Core-set loss.

Figure 4: Queried data points and $d_{{S^{0}\cup S^{p}}}$ in one batch.

##  Appendix B Batched sample selection formula of uncertainty active learning
strategy

$\begin{split}&\boldsymbol{{S^{p}}^{*}}=\underset{\mathcal{C}_{n}\in\boldsymbol{S}\setminus\boldsymbol{S^{p}}}{\text{argmax}}\,\sum_{h=1}^{p}\;\sigma_{{\omega_{\mathcal{U}_{l}}^{*}}_{q}}(R^{*}|\rho(R^{*}))\end{split}$
(4)

##  Appendix C Extension of the mathematical details to formulate K-center
Greedy active learning strategy

The subsequent Formulas 5, 6, and 7 illustrate further deductions of the batch
selection criteria, informed by the knowledge that the upper bound of the
Core-set loss primarily consists of the component
$\mathcal{O}(d_{\boldsymbol{S}^{0}\cup\boldsymbol{S}^{p}})$. The task of
finding the upper bound for Formula 5 is equivalent to the k-Center problem
presented in Formula 6, which is described as the min-max facility location
problem in Wolf’s work (Wolf (2011)).

$\begin{split}&\frac{1}{N}{\textstyle\sum_{n\in\boldsymbol{S}}\mathcal{L}(\mathcal{C}_{n};A_{\boldsymbol{S}^{0}\cup\boldsymbol{S}^{p}})}-\frac{1}{L}{\textstyle\sum_{l\in\boldsymbol{S}^{p}}\mathcal{L}(\mathcal{C}_{l};A_{\boldsymbol{S}^{0}\cup\boldsymbol{S}^{p}})}\\\
&\leq\mathcal{O}(d_{\boldsymbol{S}^{0}\cup\boldsymbol{S}^{p}})+\mathcal{O}(\sqrt{\frac{1}{N}})\end{split}$
(5)

$\begin{split}&\underset{\boldsymbol{S}^{p}}{\text{min}}\,d_{\boldsymbol{S}^{0}\cup\boldsymbol{S}^{p}}\simeq\underset{\boldsymbol{S}^{p}}{\text{min}}\,\underset{\mathcal{C}_{n}\in
S\setminus(\boldsymbol{S}^{0}\cup\boldsymbol{S}^{p})}{\text{max}}\,\underset{\mathcal{C}_{l}\in\boldsymbol{S}^{0}\cup\boldsymbol{S}^{p}}{\text{min}}\,\bigtriangleup(\mathcal{U}_{n},\mathcal{U}_{l})\end{split}$
(6)

$\begin{split}&\boldsymbol{{S^{p}}^{*}}=\sum_{l=1}^{p}\;\underset{\mathcal{C}_{l}\in\boldsymbol{S}\setminus\boldsymbol{S}^{p}}{\text{argmax}}\;\text{min}_{\mathcal{C}_{L}\in\boldsymbol{S}^{p}}\;\bigtriangleup(\mathcal{U}_{l},\mathcal{U}_{L})\end{split}$
(7)

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