Mixed Likelihood Variational Gaussian Processes

TL;DR

Gaussian processes enable well-calibrated uncertainty, but standard GP models struggle to ingest diverse auxiliary data. The authors introduce mixed likelihood variational GPs to jointly model multiple data types from a single latent function by maximizing an ELBO that sums over likelihoods: $\log p(\mathbf{y}|\mathbf{f}) = \sum_{t=1}^T \log p_t(\mathbf{y}^{(t)}|\mathbf{f}^{(t)})$ with a KL term for $q(\mathbf{u})$. They demonstrate practical gains in active learning and preference learning across visual, haptic, and robotic domains, using both synthetic and real-world data and a novel Likert-scale likelihood for confidence ratings. The results show that incorporating auxiliary information via mixed likelihoods improves data efficiency and predictive performance in human-in-the-loop experiments.

Abstract

Gaussian processes (GPs) are powerful models for human-in-the-loop experiments due to their flexibility and well-calibrated uncertainty. However, GPs modeling human responses typically ignore auxiliary information, including a priori domain expertise and non-task performance information like user confidence ratings. We propose mixed likelihood variational GPs to leverage auxiliary information, which combine multiple likelihoods in a single evidence lower bound to model multiple types of data. We demonstrate the benefits of mixing likelihoods in three real-world experiments with human participants. First, we use mixed likelihood training to impose prior knowledge constraints in GP classifiers, which accelerates active learning in a visual perception task where users are asked to identify geometric errors resulting from camera position errors in virtual reality. Second, we show that leveraging Likert scale confidence ratings by mixed likelihood training improves model fitting for haptic perception of surface roughness. Lastly, we show that Likert scale confidence ratings improve human preference learning in robot gait optimization. The modeling performance improvements found using our framework across this diverse set of applications illustrates the benefits of incorporating auxiliary information into active learning and preference learning by using mixed likelihoods to jointly model multiple inputs.

Figures (13)

• Figure 1: Illustrative depictions of perceived stereoscopic 3D distortions when render cameras are offset from the viewer's eyes. Left: Stereoscopic images rendered with cameras at the viewer's eyes have no 3D distortion. Center: Small camera offsets result in minimal perceived distortions, and participants cannot reliably identify any errors. Right: Large camera offsets result in obvious distortions and are easily recognized. Bernoulli Level Set Estimation: If participants are presented the left and middle options in randomized order and asked to select the distorted option, the probability of selecting the correct option will be close to $50\%$, i.e., at chance. On the other hand, when the left and right options are presented, the distorted option will be selected close to $100\%$ of the time. We aim to identify the space of camera placement configurations such that the distortion detection probability of a participant is below $75\%$, a threshold that is often considered a just-detectable difference from zero error.
• Figure 2: Left: A standard variational GP fit to Bernoulli observations. Right: A mixed likelihood GP trained on the same data with two constraints $f(0) = 0$ and $f(2) = 2$. The mixed likelihood-trained GP has near-zero uncertainty at the constraint locations. The true latent function is $1/2 \cdot x^2$.
• Figure 3: F1 scores (higher is better) of active learning for sublevel set estimation using two different acquisition functions (GlobalMI and EAVC). Domain knowledge for each problem is added either by mixing Bernoulli and Gaussian likelihoods (solid lines) or by adding Bernoulli pseudo data (dotted lines). For both acquisition functions, incorporating domain knowledge with mixed likelihoods led to better F1 scores than the pseudo data approach. Shaded areas show one standard errors over 100 different random seeds.
• Figure 4: Preference probabilities $\Pr(x_1 \succeq x_2)$ predicted by GPs trained on synthetic data. Left: The ground truth probabilities $\Phi(x_1 - x_2)$. Mid: A standard variational GP trained on preference observations only, which tends to be under confident at top left and bottom right corners. Right: A mixed likelihood GP trained on the same data but with additional synthetic Likert scale ratings on a scale of 0 to 2.
• Figure 5: The Brier scores ($\downarrow$) and F1 scores ($\uparrow$) of GPs trained on haptic data collected from five participants. The error bars show one standard error. Mixed likelihood GPs that include Likert scale confidence ratings generally achieve lower Brier scores and higher F1 scores.