A Corrected Expected Improvement Acquisition Function Under Noisy Observations

Sequential maximization of expected improvement (EI) is one of the most widely used policies in Bayesian optimization because of its simplicity and ability to handle noisy observations. In particular, the improvement function often uses the best posterior mean as the best incumbent in noisy settings. However, the uncertainty associated with the incumbent solution is often neglected in many analytic EI-type methods: a closed-form acquisition function is derived in the noise-free setting, but then applied to the setting with noisy observations. To address this limitation, we propose a modification of EI that corrects its closed-form expression by incorporating the covariance information provided by the Gaussian Process (GP) model. This acquisition function specializes to the classical noise-free result, and we argue should replace that formula in Bayesian optimization software packages, tutorials, and textbooks. This enhanced acquisition provides good generality for noisy and noiseless settings. We show that our method achieves a sublinear convergence rate on the cumulative regret bound under heteroscedastic observation noise. Our empirical results demonstrate that our proposed acquisition function can outperform EI in the presence of noisy observations on benchmark functions for black-box optimization, as well as on parameter search for neural network model compression.

Pointwise visual field estimation from optical coherence tomography in glaucoma: a structure-function analysis using deep learning

Background/Aims: Standard Automated Perimetry (SAP) is the gold standard to monitor visual field (VF) loss in glaucoma management, but is prone to intra-subject variability. We developed and validated a deep learning (DL) regression model that estimates pointwise and overall VF loss from unsegmented optical coherence tomography (OCT) scans. Methods: Eight DL regression models were trained with various retinal imaging modalities: circumpapillary OCT at 3.5mm, 4.1mm, 4.7mm diameter, and scanning laser ophthalmoscopy (SLO) en face images to estimate mean deviation (MD) and 52 threshold values. This retrospective study used data from patients who underwent a complete glaucoma examination, including a reliable Humphrey Field Analyzer (HFA) 24-2 SITA Standard VF exam and a SPECTRALIS OCT scan using the Glaucoma Module Premium Edition. Results: A total of 1378 matched OCT-VF pairs of 496 patients (863 eyes) were included for training and evaluation of the DL models. Average sample MD was -7.53dB (from -33.8dB to +2.0dB). For 52 VF threshold values estimation, the circumpapillary OCT scan with the largest radius (4.7mm) achieved the best performance among all individual models (Pearson r=0.77, 95% CI=[0.72-0.82]). For MD, prediction averaging of OCT-trained models (3.5mm, 4.1mm, 4.7mm) resulted in a Pearson r of 0.78 [0.73-0.83] on the validation set and comparable performance on the test set (Pearson r=0.79 [0.75-0.82]). Conclusion: DL on unsegmented OCT scans accurately predicts pointwise and mean deviation of 24-2 VF in glaucoma patients. Automated VF from unsegmented OCT could be a solution for patients unable to produce reliable perimetry results.