Function-Space Regularization in Neural Networks: A Probabilistic Perspective

Parameter-space regularization in neural network optimization is a fundamental tool for improving generalization. However, standard parameter-space regularization methods make it challenging to encode explicit preferences about desired predictive functions into neural network training. In this work, we approach regularization in neural networks from a probabilistic perspective and show that by viewing parameter-space regularization as specifying an empirical prior distribution over the model parameters, we can derive a probabilistically well-motivated regularization technique that allows explicitly encoding information about desired predictive functions into neural network training. This method -- which we refer to as function-space empirical Bayes (FSEB) -- includes both parameter- and function-space regularization, is mathematically simple, easy to implement, and incurs only minimal computational overhead compared to standard regularization techniques. We evaluate the utility of this regularization technique empirically and demonstrate that the proposed method leads to near-perfect semantic shift detection, highly-calibrated predictive uncertainty estimates, successful task adaption from pre-trained models, and improved generalization under covariate shift.

Should We Learn Most Likely Functions or Parameters?

Standard regularized training procedures correspond to maximizing a posterior distribution over parameters, known as maximum a posteriori (MAP) estimation. However, model parameters are of interest only insomuch as they combine with the functional form of a model to provide a function that can make good predictions. Moreover, the most likely parameters under the parameter posterior do not generally correspond to the most likely function induced by the parameter posterior. In fact, we can re-parametrize a model such that any setting of parameters can maximize the parameter posterior. As an alternative, we investigate the benefits and drawbacks of directly estimating the most likely function implied by the model and the data. We show that this procedure leads to pathological solutions when using neural networks and prove conditions under which the procedure is well-behaved, as well as a scalable approximation. Under these conditions, we find that function-space MAP estimation can lead to flatter minima, better generalization, and improved robustness to overfitting.

PAC-Bayes Compression Bounds So Tight That They Can Explain Generalization

While there has been progress in developing non-vacuous generalization bounds for deep neural networks, these bounds tend to be uninformative about why deep learning works. In this paper, we develop a compression approach based on quantizing neural network parameters in a linear subspace, profoundly improving on previous results to provide state-of-the-art generalization bounds on a variety of tasks, including transfer learning. We use these tight bounds to better understand the role of model size, equivariance, and the implicit biases of optimization, for generalization in deep learning. Notably, we find large models can be compressed to a much greater extent than previously known, encapsulating Occam's razor. We also argue for data-independent bounds in explaining generalization.

Pre-Train Your Loss: Easy Bayesian Transfer Learning with Informative Priors

Deep learning is increasingly moving towards a transfer learning paradigm whereby large foundation models are fine-tuned on downstream tasks, starting from an initialization learned on the source task. But an initialization contains relatively little information about the source task. Instead, we show that we can learn highly informative posteriors from the source task, through supervised or self-supervised approaches, which then serve as the basis for priors that modify the whole loss surface on the downstream task. This simple modular approach enables significant performance gains and more data-efficient learning on a variety of downstream classification and segmentation tasks, serving as a drop-in replacement for standard pre-training strategies. These highly informative priors also can be saved for future use, similar to pre-trained weights, and stand in contrast to the zero-mean isotropic uninformative priors that are typically used in Bayesian deep learning.

On Uncertainty, Tempering, and Data Augmentation in Bayesian Classification

Aleatoric uncertainty captures the inherent randomness of the data, such as measurement noise. In Bayesian regression, we often use a Gaussian observation model, where we control the level of aleatoric uncertainty with a noise variance parameter. By contrast, for Bayesian classification we use a categorical distribution with no mechanism to represent our beliefs about aleatoric uncertainty. Our work shows that explicitly accounting for aleatoric uncertainty significantly improves the performance of Bayesian neural networks. We note that many standard benchmarks, such as CIFAR, have essentially no aleatoric uncertainty. Moreover, we show data augmentation in approximate inference has the effect of softening the likelihood, leading to underconfidence and profoundly misrepresenting our honest beliefs about aleatoric uncertainty. Accordingly, we find that a cold posterior, tempered by a power greater than one, often more honestly reflects our beliefs about aleatoric uncertainty than no tempering -- providing an explicit link between data augmentation and cold posteriors. We show that we can match or exceed the performance of posterior tempering by using a Dirichlet observation model, where we explicitly control the level of aleatoric uncertainty, without any need for tempering.

When are Iterative Gaussian Processes Reliably Accurate?

While recent work on conjugate gradient methods and Lanczos decompositions have achieved scalable Gaussian process inference with highly accurate point predictions, in several implementations these iterative methods appear to struggle with numerical instabilities in learning kernel hyperparameters, and poor test likelihoods. By investigating CG tolerance, preconditioner rank, and Lanczos decomposition rank, we provide a particularly simple prescription to correct these issues: we recommend that one should use a small CG tolerance ($\epsilon \leq 0.01$) and a large root decomposition size ($r \geq 5000$). Moreover, we show that L-BFGS-B is a compelling optimizer for Iterative GPs, achieving convergence with fewer gradient updates.

A Simple and Fast Baseline for Tuning Large XGBoost Models

XGBoost, a scalable tree boosting algorithm, has proven effective for many prediction tasks of practical interest, especially using tabular datasets. Hyperparameter tuning can further improve the predictive performance, but unlike neural networks, full-batch training of many models on large datasets can be time consuming. Owing to the discovery that (i) there is a strong linear relation between dataset size & training time, (ii) XGBoost models satisfy the ranking hypothesis, and (iii) lower-fidelity models can discover promising hyperparameter configurations, we show that uniform subsampling makes for a simple yet fast baseline to speed up the tuning of large XGBoost models using multi-fidelity hyperparameter optimization with data subsets as the fidelity dimension. We demonstrate the effectiveness of this baseline on large-scale tabular datasets ranging from $15-70\mathrm{GB}$ in size.

SKIing on Simplices: Kernel Interpolation on the Permutohedral Lattice for Scalable Gaussian Processes

State-of-the-art methods for scalable Gaussian processes use iterative algorithms, requiring fast matrix vector multiplies (MVMs) with the covariance kernel. The Structured Kernel Interpolation (SKI) framework accelerates these MVMs by performing efficient MVMs on a grid and interpolating back to the original space. In this work, we develop a connection between SKI and the permutohedral lattice used for high-dimensional fast bilateral filtering. Using a sparse simplicial grid instead of a dense rectangular one, we can perform GP inference exponentially faster in the dimension than SKI. Our approach, Simplex-GP, enables scaling SKI to high dimensions, while maintaining strong predictive performance. We additionally provide a CUDA implementation of Simplex-GP, which enables significant GPU acceleration of MVM based inference.

Variational Auto-Regressive Gaussian Processes for Continual Learning

This paper proposes Variational Auto-Regressive Gaussian Process (VAR-GP), a principled Bayesian updating mechanism to incorporate new data for sequential tasks in the context of continual learning. It relies on a novel auto-regressive characterization of the variational distribution and inference is made scalable using sparse inducing point approximations. Experiments on standard continual learning benchmarks demonstrate the ability of VAR-GPs to perform well at new tasks without compromising performance on old ones, yielding competitive results to state-of-the-art methods. In addition, we qualitatively show how VAR-GP improves the predictive entropy estimates as we train on new tasks. Further, we conduct a thorough ablation study to verify the effectiveness of inferential choices.

First-Order Preconditioning via Hypergradient Descent

Standard gradient descent methods are susceptible to a range of issues that can impede training, such as high correlations and different scaling in parameter space. These difficulties can be addressed by second-order approaches that apply a preconditioning matrix to the gradient to improve convergence. Unfortunately, such algorithms typically struggle to scale to high-dimensional problems, in part because the calculation of specific preconditioners such as the inverse Hessian or Fisher information matrix is highly expensive. We introduce first-order preconditioning (FOP), a fast, scalable approach that generalizes previous work on hypergradient descent (Almeida et al., 1998; Maclaurin et al., 2015; Baydin et al., 2017) to learn a preconditioning matrix that only makes use of first-order information. Experiments show that FOP is able to improve the performance of standard deep learning optimizers on several visual classification tasks with minimal computational overhead. We also investigate the properties of the learned preconditioning matrices and perform a preliminary theoretical analysis of the algorithm.