We study the optimisation problem associated with Gaussian process regression using squared loss. The most common approach to this problem is to apply an exact solver, such as conjugate gradient descent, either directly, or to a reduced-order version of the problem. Recently, driven by successes in deep learning, stochastic gradient descent has gained traction as an alternative. In this paper, we show that when done right$\unicode{x2014}$by which we mean using specific insights from the optimisation and kernel communities$\unicode{x2014}$this approach is highly effective. We thus introduce a particular stochastic dual gradient descent algorithm, that may be implemented with a few lines of code using any deep learning framework. We explain our design decisions by illustrating their advantage against alternatives with ablation studies and show that the new method is highly competitive. Our evaluations on standard regression benchmarks and a Bayesian optimisation task set our approach apart from preconditioned conjugate gradients, variational Gaussian process approximations, and a previous version of stochastic gradient descent for Gaussian processes. On a molecular binding affinity prediction task, our method places Gaussian process regression on par in terms of performance with state-of-the-art graph neural networks.
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Numerical solution of discrete PDEs corresponding to saddle point problems is highly relevant to physical systems such as Stokes flow. However, scaling up numerical solvers for such systems is often met with challenges in efficiency and convergence. Multigrid is an approach with excellent applicability to elliptic problems such as the Stokes equations, and can be a solution to such challenges of scalability and efficiency. The degree of success of such methods, however, is highly contingent on the design of key components of a multigrid scheme, including the hierarchy of discretizations, and the relaxation scheme used. Additionally, in many practical cases, it may be more effective to use a multigrid scheme as a preconditioner to an iterative Krylov subspace solver, as opposed to striving for maximum efficacy of the relaxation scheme in all foreseeable settings. In this paper, we propose an efficient symmetric multigrid preconditioner for the Stokes Equations on a staggered finite-difference discretization. Our contribution is focused on crafting a preconditioner that (a) is symmetric indefinite, matching the property of the Stokes system itself, (b) is appropriate for preconditioning the SQMR iterative scheme, and (c) has the requisite symmetry properties to be used in this context. In addition, our design is efficient in terms of computational cost and facilitates scaling to large domains.
Despite the decomposition of convolutional kernels for lightweight CNNs being well studied, existing works that rely on tensor network diagrams or hyperdimensional abstraction lack geometry intuition. This work devises a new perspective by linking a 3D-reshaped kernel tensor to its various slice-wise and rank-1 decompositions, permitting a straightforward connection between various tensor approximations and efficient CNN modules. Specifically, it is discovered that a pointwise-depthwise-pointwise (PDP) configuration constitutes a viable construct for lightweight CNNs. Moreover, a novel link to the latest ShiftNet is established, inspiring a first-ever shift layer pruning that achieves nearly 50% compression with < 1% drop in accuracy for ShiftResNet.
Diffusion Probabilistic Models stand as a critical tool in generative modelling, enabling the generation of complex data distributions. This family of generative models yields record-breaking performance in tasks such as image synthesis, video generation, and molecule design. Despite their capabilities, their efficiency, especially in the reverse process, remains a challenge due to slow convergence rates and high computational costs. In this paper, we introduce an approach that leverages continuous dynamical systems to design a novel denoising network for diffusion models that is more parameter-efficient, exhibits faster convergence, and demonstrates increased noise robustness. Experimenting with Denoising Diffusion Probabilistic Models (DDPMs), our framework operates with approximately a quarter of the parameters, and $\sim$ 30\% of the Floating Point Operations (FLOPs) compared to standard U-Nets in DDPMs. Furthermore, our model is notably faster in inference than the baseline when measured in fair and equal conditions. We also provide a mathematical intuition as to why our proposed reverse process is faster as well as a mathematical discussion of the empirical tradeoffs in the denoising downstream task. Finally, we argue that our method is compatible with existing performance enhancement techniques, enabling further improvements in efficiency, quality, and speed.
This paper presents a motion planning algorithm for quadruped locomotion based on density functions. We decompose the locomotion problem into a high-level density planner and a model predictive controller (MPC). Due to density functions having a physical interpretation through the notion of occupancy, it is intuitive to represent the environment with safety constraints. Hence, there is an ease of use to constructing the planning problem with density. The proposed method uses a simplified model of the robot into an integrator system, where the high-level plan is in a feedback form formulated through an analytically constructed density function. We then use the MPC to optimize the reference trajectory, in which a low-level PID controller is used to obtain the torque level control. The overall framework is implemented in simulation, demonstrating our feedback density planner for legged locomotion. The implementation of work is available at \url{//github.com/AndrewZheng-1011/legged_planner}
Neural networks with self-attention (a.k.a. Transformers) like ViT and Swin have emerged as a better alternative to traditional convolutional neural networks (CNNs). However, our understanding of how the new architecture works is still limited. In this paper, we focus on the phenomenon that Transformers show higher robustness against corruptions than CNNs, while not being overconfident. This is contrary to the intuition that robustness increases with confidence. We resolve this contradiction by empirically investigating how the output of the penultimate layer moves in the representation space as the input data moves linearly within a small area. In particular, we show the following. (1) While CNNs exhibit fairly linear relationship between the input and output movements, Transformers show nonlinear relationship for some data. For those data, the output of Transformers moves in a curved trajectory as the input moves linearly. (2) When a data is located in a curved region, it is hard to move it out of the decision region since the output moves along a curved trajectory instead of a straight line to the decision boundary, resulting in high robustness of Transformers. (3) If a data is slightly modified to jump out of the curved region, the movements afterwards become linear and the output goes to the decision boundary directly. In other words, there does exist a decision boundary near the data, which is hard to find only because of the curved representation space. This explains the underconfident prediction of Transformers. Also, we examine mathematical properties of the attention operation that induce nonlinear response to linear perturbation. Finally, we share our additional findings, regarding what contributes to the curved representation space of Transformers, and how the curvedness evolves during training.
We propose a novel framework DropTop that suppresses the shortcut bias in online continual learning (OCL) while being adaptive to the varying degree of the shortcut bias incurred by continuously changing environment. By the observed high-attention property of the shortcut bias, highly-activated features are considered candidates for debiasing. More importantly, resolving the limitation of the online environment where prior knowledge and auxiliary data are not ready, two novel techniques -- feature map fusion and adaptive intensity shifting -- enable us to automatically determine the appropriate level and proportion of the candidate shortcut features to be dropped. Extensive experiments on five benchmark datasets demonstrate that, when combined with various OCL algorithms, DropTop increases the average accuracy by up to 10.4% and decreases the forgetting by up to 63.2%.
Emotion recognition in conversation (ERC) aims to detect the emotion label for each utterance. Motivated by recent studies which have proven that feeding training examples in a meaningful order rather than considering them randomly can boost the performance of models, we propose an ERC-oriented hybrid curriculum learning framework. Our framework consists of two curricula: (1) conversation-level curriculum (CC); and (2) utterance-level curriculum (UC). In CC, we construct a difficulty measurer based on "emotion shift" frequency within a conversation, then the conversations are scheduled in an "easy to hard" schema according to the difficulty score returned by the difficulty measurer. For UC, it is implemented from an emotion-similarity perspective, which progressively strengthens the model's ability in identifying the confusing emotions. With the proposed model-agnostic hybrid curriculum learning strategy, we observe significant performance boosts over a wide range of existing ERC models and we are able to achieve new state-of-the-art results on four public ERC datasets.
Recently, a considerable literature has grown up around the theme of Graph Convolutional Network (GCN). How to effectively leverage the rich structural information in complex graphs, such as knowledge graphs with heterogeneous types of entities and relations, is a primary open challenge in the field. Most GCN methods are either restricted to graphs with a homogeneous type of edges (e.g., citation links only), or focusing on representation learning for nodes only instead of jointly propagating and updating the embeddings of both nodes and edges for target-driven objectives. This paper addresses these limitations by proposing a novel framework, namely the Knowledge Embedding based Graph Convolutional Network (KE-GCN), which combines the power of GCNs in graph-based belief propagation and the strengths of advanced knowledge embedding (a.k.a. knowledge graph embedding) methods, and goes beyond. Our theoretical analysis shows that KE-GCN offers an elegant unification of several well-known GCN methods as specific cases, with a new perspective of graph convolution. Experimental results on benchmark datasets show the advantageous performance of KE-GCN over strong baseline methods in the tasks of knowledge graph alignment and entity classification.
We study the problem of textual relation embedding with distant supervision. To combat the wrong labeling problem of distant supervision, we propose to embed textual relations with global statistics of relations, i.e., the co-occurrence statistics of textual and knowledge base relations collected from the entire corpus. This approach turns out to be more robust to the training noise introduced by distant supervision. On a popular relation extraction dataset, we show that the learned textual relation embedding can be used to augment existing relation extraction models and significantly improve their performance. Most remarkably, for the top 1,000 relational facts discovered by the best existing model, the precision can be improved from 83.9% to 89.3%.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis.
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