Jihao Andreas Lin

Jihao Andreas Lin, Javier Antorán, Shreyas Padhy et al. · cambridge

Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to conditioning. We explore stochastic gradient algorithms as a computationally efficient method of approximately solving these linear systems: we develop low-variance optimization objectives for sampling from the posterior and extend these to inducing points. Counterintuitively, stochastic gradient descent often produces accurate predictions, even in cases where it does not converge quickly to the optimum. We explain this through a spectral characterization of the implicit bias from non-convergence. We show that stochastic gradient descent produces predictive distributions close to the true posterior both in regions with sufficient data coverage, and in regions sufficiently far away from the data. Experimentally, stochastic gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks. Its uncertainty estimates match the performance of significantly more expensive baselines on a large-scale Bayesian optimization task.

Jihao Andreas Lin, Javier Antorán, José Miguel Hernández-Lobato · cambridge

The Laplace approximation provides a closed-form model selection objective for neural networks (NN). Online variants, which optimise NN parameters jointly with hyperparameters, like weight decay strength, have seen renewed interest in the Bayesian deep learning community. However, these methods violate Laplace's method's critical assumption that the approximation is performed around a mode of the loss, calling into question their soundness. This work re-derives online Laplace methods, showing them to target a variational bound on a mode-corrected variant of the Laplace evidence which does not make stationarity assumptions. Online Laplace and its mode-corrected counterpart share stationary points where 1. the NN parameters are a maximum a posteriori, satisfying the Laplace method's assumption, and 2. the hyperparameters maximise the Laplace evidence, motivating online methods. We demonstrate that these optima are roughly attained in practise by online algorithms using full-batch gradient descent on UCI regression datasets. The optimised hyperparameters prevent overfitting and outperform validation-based early stopping.

Jihao Andreas Lin, Gergely Flamich, José Miguel Hernández-Lobato · cambridge

This paper studies the qualitative behavior and robustness of two variants of Minimal Random Code Learning (MIRACLE) used to compress variational Bayesian neural networks. MIRACLE implements a powerful, conditionally Gaussian variational approximation for the weight posterior $Q_{\mathbf{w}}$ and uses relative entropy coding to compress a weight sample from the posterior using a Gaussian coding distribution $P_{\mathbf{w}}$. To achieve the desired compression rate, $D_{\mathrm{KL}}[Q_{\mathbf{w}} \Vert P_{\mathbf{w}}]$ must be constrained, which requires a computationally expensive annealing procedure under the conventional mean-variance (Mean-Var) parameterization for $Q_{\mathbf{w}}$. Instead, we parameterize $Q_{\mathbf{w}}$ by its mean and KL divergence from $P_{\mathbf{w}}$ to constrain the compression cost to the desired value by construction. We demonstrate that variational training with Mean-KL parameterization converges twice as fast and maintains predictive performance after compression. Furthermore, we show that Mean-KL leads to more meaningful variational distributions with heavier tails and compressed weight samples which are more robust to pruning.

Kenza Tazi, Jihao Andreas Lin, Ross Viljoen et al.

Gaussian Processes (GPs) offer an attractive method for regression over small, structured and correlated datasets. However, their deployment is hindered by computational costs and limited guidelines on how to apply GPs beyond simple low-dimensional datasets. We propose a framework to identify the suitability of GPs to a given problem and how to set up a robust and well-specified GP model. The guidelines formalise the decisions of experienced GP practitioners, with an emphasis on kernel design and options for computational scalability. The framework is then applied to a case study of glacier elevation change yielding more accurate results at test time.

Jihao Andreas Lin, Shreyas Padhy, Javier Antorán et al.

As is well known, both sampling from the posterior and computing the mean of the posterior in Gaussian process regression reduces to solving a large linear system of equations. We study the use of stochastic gradient descent for solving this linear system, and show that when \emph{done right} -- by which we mean using specific insights from the optimisation and kernel communities -- stochastic gradient descent is highly effective. To that end, we introduce a particularly simple \emph{stochastic dual descent} algorithm, explain its design in an intuitive manner and illustrate the design choices through a series of ablation studies. Further experiments demonstrate that our new method is highly competitive. In particular, our evaluations on the UCI regression tasks and on Bayesian optimisation set our approach apart from preconditioned conjugate gradients and variational Gaussian process approximations. Moreover, our method places Gaussian process regression on par with state-of-the-art graph neural networks for molecular binding affinity prediction.

Jihao Andreas Lin, Sebastian Ament, Louis C. Tiao et al.

Gaussian processes (GPs) are powerful and widely used probabilistic regression models, but their effectiveness in practice is often limited by the choice of kernel function. This kernel function is typically handcrafted from a small set of standard functions, a process that requires expert knowledge, results in limited adaptivity to data, and imposes strong assumptions on the hypothesis space. We study Empirical GPs, a principled framework for constructing flexible, data-driven GP priors that overcome these limitations. Rather than relying on standard parametric kernels, we estimate the mean and covariance functions empirically from a corpus of historical observations, enabling the prior to reflect rich, non-trivial covariance structures present in the data. Theoretically, we show that the resulting model converges to the GP that is closest (in KL-divergence sense) to the real data generating process. Practically, we formulate the problem of learning the GP prior from independent datasets as likelihood estimation and derive an Expectation-Maximization algorithm with closed-form updates, allowing the model handle heterogeneous observation locations across datasets. We demonstrate that Empirical GPs achieve competitive performance on learning curve extrapolation and time series forecasting benchmarks.

Jihao Andreas Lin, Joe Watson, Pascal Klink et al.

Bayesian deep learning approaches assume model parameters to be latent random variables and infer posterior distributions to quantify uncertainty, increase safety and trust, and prevent overconfident and unpredictable behavior. However, weight-space priors are model-specific, can be difficult to interpret and are hard to specify. Instead, we apply a Dirichlet prior in predictive space and perform approximate function-space variational inference. To this end, we interpret conventional categorical predictions from stochastic neural network classifiers as samples from an implicit Dirichlet distribution. By adapting the inference, the same function-space prior can be combined with different models without affecting model architecture or size. We illustrate the flexibility and efficacy of such a prior with toy experiments and demonstrate scalability, improved uncertainty quantification and adversarial robustness with large-scale image classification experiments.

Jihao Andreas Lin, Sebastian Ament, Maximilian Balandat et al.

A key task in AutoML is to model learning curves of machine learning models jointly as a function of model hyper-parameters and training progression. While Gaussian processes (GPs) are suitable for this task, naïve GPs require $\mathcal{O}(n^3m^3)$ time and $\mathcal{O}(n^2 m^2)$ space for $n$ hyper-parameter configurations and $\mathcal{O}(m)$ learning curve observations per hyper-parameter. Efficient inference via Kronecker structure is typically incompatible with early-stopping due to missing learning curve values. We impose $\textit{latent Kronecker structure}$ to leverage efficient product kernels while handling missing values. In particular, we interpret the joint covariance matrix of observed values as the projection of a latent Kronecker product. Combined with iterative linear solvers and structured matrix-vector multiplication, our method only requires $\mathcal{O}(n^3 + m^3)$ time and $\mathcal{O}(n^2 + m^2)$ space. We show that our GP model can match the performance of a Transformer on a learning curve prediction task.

Matthew Zhang, Jihao Andreas Lin, Krzysztof Choromanski et al.

We study the application of graph random features (GRFs) - a recently introduced stochastic estimator of graph node kernels - to scalable Gaussian processes on discrete input spaces. We prove that (under mild assumptions) Bayesian inference with GRFs enjoys $O(N^{3/2})$ time complexity with respect to the number of nodes $N$, compared to $O(N^3)$ for exact kernels. Substantial wall-clock speedups and memory savings unlock Bayesian optimisation on graphs with over $10^6$ nodes on a single computer chip, whilst preserving competitive performance.

Alan Yufei Dong, Jihao Andreas Lin, José Miguel Hernández-Lobato

Scalable Gaussian process (GP) inference is essential for sequential decision-making tasks, yet improving GP scalability remains a challenging problem with many open avenues of research. This paper focuses on iterative GPs, where iterative linear solvers, such as conjugate gradients, stochastic gradient descent or alternative projections, are used to approximate the GP posterior. We propose a new method which improves solver convergence of a large linear system by leveraging the known solution to a smaller system contained within. This is significant for tasks with incremental data additions, and we show that our technique achieves speed-ups when solving to tolerance, as well as improved Bayesian optimisation performance under a fixed compute budget.

Jihao Andreas Lin, Nicolas Mayoraz, Steffen Rendle et al.

Successive Halving is a popular algorithm for hyperparameter optimization which allocates exponentially more resources to promising candidates. However, the algorithm typically relies on intermediate performance values to make resource allocation decisions, which can cause it to prematurely prune slow starters that would eventually become the best candidate. We investigate whether guiding Successive Halving with learning curve predictions based on Latent Kronecker Gaussian Processes can overcome this limitation. In a large-scale empirical study involving different neural network architectures and a click prediction dataset, we compare this predictive approach to the standard approach based on current performance values. Our experiments show that, although the predictive approach achieves competitive performance, it is not Pareto optimal compared to investing more resources into the standard approach, because it requires fully observed learning curves as training data. However, this downside could be mitigated by leveraging existing learning curve data.

Jihao Andreas Lin

Gaussian processes are a powerful framework for uncertainty-aware function approximation and sequential decision-making. Unfortunately, their classical formulation does not scale gracefully to large amounts of data and modern hardware for massively-parallel computation, prompting many researchers to develop techniques which improve their scalability. This dissertation focuses on the powerful combination of iterative methods and pathwise conditioning to develop methodological contributions which facilitate the use of Gaussian processes in modern large-scale settings. By combining these two techniques synergistically, expensive computations are expressed as solutions to systems of linear equations and obtained by leveraging iterative linear system solvers. This drastically reduces memory requirements, facilitating application to significantly larger amounts of data, and introduces matrix multiplication as the main computational operation, which is ideal for modern hardware.

Jihao Andreas Lin, Jakob Brünker, Daniel Fährmann

While 2D object detection has improved significantly over the past, real world applications of computer vision often require an understanding of the 3D layout of a scene. Many recent approaches to 3D detection use LiDAR point clouds for prediction. We propose a method that only uses a single RGB image, thus enabling applications in devices or vehicles that do not have LiDAR sensors. By using an RGB image, we can leverage the maturity and success of recent 2D object detectors, by extending a 2D detector with a 3D detection head. In this paper we discuss different approaches and experiments, including both regression and classification methods, for designing this 3D detection head. Furthermore, we evaluate how subproblems and implementation details impact the overall prediction result. We use the KITTI dataset for training, which consists of street traffic scenes with class labels, 2D bounding boxes and 3D annotations with seven degrees of freedom. Our final architecture is based on Faster R-CNN. The outputs of the convolutional backbone are fixed sized feature maps for every region of interest. Fully connected layers within the network head then propose an object class and perform 2D bounding box regression. We extend the network head by a 3D detection head, which predicts every degree of freedom of a 3D bounding box via classification. We achieve a mean average precision of 47.3% for moderately difficult data, measured at a 3D intersection over union threshold of 70%, as required by the official KITTI benchmark; outperforming previous state-of-the-art single RGB only methods by a large margin.