Despite recent progress, continual learning still does not match the performance of batch training. To avoid catastrophic forgetting, we need to build compact memory of essential past knowledge, but no clear solution has yet emerged, even for shallow neural networks with just one or two layers. In this paper, we present a new method to build compact memory for logistic regression. Our method is based on a result by Khan and Swaroop [2021] who show the existence of optimal memory for such models. We formulate the search for the optimal memory as Hessian-matching and propose a probabilistic PCA method to estimate them. Our approach can drastically improve accuracy compared to Experience Replay. For instance, on Split-ImageNet, we get 60% accuracy compared to 30% obtained by replay with memory-size equivalent to 0.3% of the data size. Increasing the memory size to 2% further boosts the accuracy to 74%, closing the gap to the batch accuracy of 77.6% on this task. Our work opens a new direction for building compact memory that can also be useful in the future for continual deep learning.

Bayesian neural networks (BNNs) estimate the posterior distribution of model parameters and utilize posterior samples for Bayesian Model Averaging (BMA) in prediction. However, despite the crucial role of flatness in the loss landscape in improving the generalization of neural networks, its impact on BMA has been largely overlooked. In this work, we explore how posterior flatness influences BMA generalization and empirically demonstrate that \emph{(1) most approximate Bayesian inference methods fail to yield a flat posterior} and \emph{(2) BMA predictions, without considering posterior flatness, are less effective at improving generalization}. To address this, we propose Flat Posterior-aware Bayesian Model Averaging (FP-BMA), a novel training objective that explicitly encourages flat posteriors in a principled Bayesian manner. We also introduce a Flat Posterior-aware Bayesian Transfer Learning scheme that enhances generalization in downstream tasks. Empirically, we show that FP-BMA successfully captures flat posteriors, improving generalization performance.

Convolutional deep sets is a neural network architecture that can model stationary stochastic processes. This architecture uses the kernel smoother and deep convolutional neural network to construct translation equivariant functional representations. However, the non-parametric nature of the kernel smoother can produce ambiguous representations when the number of data points is not given sufficiently. To address this issue, we introduce bayesian convolutional deep sets, which constructs random translation equivariant functional representations with a stationary prior. Furthermore, we present how to impose the task-dependent prior for each dataset because a wrongly imposed prior can result in an even worse representation than that of the kernel smoother. Empirically, we demonstrate that the proposed architecture alleviates the targeted issue in various experiments with time-series and image datasets.

Based on the Fourier duality between a stationary kernel and its spectral density, modeling the spectral density using a Gaussian mixture density enables one to construct a flexible kernel, known as a Spectral Mixture kernel, that can model any stationary kernel. However, despite its expressive power, training this kernel is typically difficult because scalability and overfitting issues often arise due to a large number of training parameters. To resolve these issues, we propose an approximate inference method for estimating the Spectral mixture kernel hyperparameters. Specifically, we approximate this kernel by using the finite random spectral points based on Random Fourier Feature and optimize the parameters for the distribution of spectral points by sampling-based variational inference. To improve this inference procedure, we analyze the training loss and propose two special methods: a sampling method of spectral points to reduce the error of the approximate kernel in training, and an approximate natural gradient to accelerate the convergence of parameter inference.

Attention computes the dependency between representations, and it encourages the model to focus on the important selective features. Attention-based models, such as Transformer and graph attention network (GAT), are widely utilized for sequential data and graph-structured data. This paper suggests a new interpretation and generalized structure of the attention in Transformer and GAT. For the attention in Transformer and GAT, we derive that the attention is a product of two parts: 1) the RBF kernel to measure the similarity of two instances and 2) the exponential of L2 norm to compute the importance of individual instances. From this decomposition, we generalize the attention in three ways. First, we propose implicit kernel attention with an implicit kernel function instead of manual kernel selection. Second, we generalize L2 norm as the Lp norm. Third, we extend our attention to structured multi-head attention. Our generalized attention shows better performance on classification, translation, and regression tasks.