Jaehoon Lee

Mitchell Wortsman, Peter J. Liu, Lechao Xiao et al. · anthropic, deepmind

Teams that have trained large Transformer-based models have reported training instabilities at large scale that did not appear when training with the same hyperparameters at smaller scales. Although the causes of such instabilities are of scientific interest, the amount of resources required to reproduce them has made investigation difficult. In this work, we seek ways to reproduce and study training stability and instability at smaller scales. First, we focus on two sources of training instability described in previous work: the growth of logits in attention layers (Dehghani et al., 2023) and divergence of the output logits from the log probabilities (Chowdhery et al., 2022). By measuring the relationship between learning rate and loss across scales, we show that these instabilities also appear in small models when training at high learning rates, and that mitigations previously employed at large scales are equally effective in this regime. This prompts us to investigate the extent to which other known optimizer and model interventions influence the sensitivity of the final loss to changes in the learning rate. To this end, we study methods such as warm-up, weight decay, and the $μ$Param (Yang et al., 2022), and combine techniques to train small models that achieve similar losses across orders of magnitude of learning rate variation. Finally, to conclude our exploration we study two cases where instabilities can be predicted before they emerge by examining the scaling behavior of model activation and gradient norms.

Katie Everett, Lechao Xiao, Mitchell Wortsman et al. · anthropic, deepmind

Robust and effective scaling of models from small to large width typically requires the precise adjustment of many algorithmic and architectural details, such as parameterization and optimizer choices. In this work, we propose a new perspective on parameterization by investigating a key assumption in prior work about the alignment between parameters and data and derive new theoretical results under weaker assumptions and a broader set of optimizers. Our extensive empirical investigation includes tens of thousands of models trained with all combinations of three optimizers, four parameterizations, several alignment assumptions, more than a dozen learning rates, and fourteen model sizes up to 26.8B parameters. We find that the best learning rate scaling prescription would often have been excluded by the assumptions in prior work. Our results show that all parameterizations, not just maximal update parameterization (muP), can achieve hyperparameter transfer; moreover, our novel per-layer learning rate prescription for standard parameterization outperforms muP. Finally, we demonstrate that an overlooked aspect of parameterization, the epsilon parameter in Adam, must be scaled correctly to avoid gradient underflow and propose Adam-atan2, a new numerically stable, scale-invariant version of Adam that eliminates the epsilon hyperparameter entirely.

Insu Han, Amir Zandieh, Jaehoon Lee et al. · deepmind

Infinite width limit has shed light on generalization and optimization aspects of deep learning by establishing connections between neural networks and kernel methods. Despite their importance, the utility of these kernel methods was limited in large-scale learning settings due to their (super-)quadratic runtime and memory complexities. Moreover, most prior works on neural kernels have focused on the ReLU activation, mainly due to its popularity but also due to the difficulty of computing such kernels for general activations. In this work, we overcome such difficulties by providing methods to work with general activations. First, we compile and expand the list of activation functions admitting exact dual activation expressions to compute neural kernels. When the exact computation is unknown, we present methods to effectively approximate them. We propose a fast sketching method that approximates any multi-layered Neural Network Gaussian Process (NNGP) kernel and Neural Tangent Kernel (NTK) matrices for a wide range of activation functions, going beyond the commonly analyzed ReLU activation. This is done by showing how to approximate the neural kernels using the truncated Hermite expansion of any desired activation functions. While most prior works require data points on the unit sphere, our methods do not suffer from such limitations and are applicable to any dataset of points in $\mathbb{R}^d$. Furthermore, we provide a subspace embedding for NNGP and NTK matrices with near input-sparsity runtime and near-optimal target dimension which applies to any \emph{homogeneous} dual activation functions with rapidly convergent Taylor expansion. Empirically, with respect to exact convolutional NTK (CNTK) computation, our method achieves $106\times$ speedup for approximate CNTK of a 5-layer Myrtle network on CIFAR-10 dataset.

C. Daniel Freeman, Laura Culp, Aaron Parisi et al. · anthropic, deepmind

We introduce and study the problem of adversarial arithmetic, which provides a simple yet challenging testbed for language model alignment. This problem is comprised of arithmetic questions posed in natural language, with an arbitrary adversarial string inserted before the question is complete. Even in the simple setting of 1-digit addition problems, it is easy to find adversarial prompts that make all tested models (including PaLM2, GPT4, Claude2) misbehave, and even to steer models to a particular wrong answer. We additionally provide a simple algorithm for finding successful attacks by querying those same models, which we name "prompt inversion rejection sampling" (PIRS). We finally show that models can be partially hardened against these attacks via reinforcement learning and via agentic constitutional loops. However, we were not able to make a language model fully robust against adversarial arithmetic attacks.

Haksoo Lim, Sewon Park, Minjung Kim et al.

The time-series anomaly detection is one of the most fundamental tasks for time-series. Unlike the time-series forecasting and classification, the time-series anomaly detection typically requires unsupervised (or self-supervised) training since collecting and labeling anomalous observations are difficult. In addition, most existing methods resort to limited forms of anomaly measurements and therefore, it is not clear whether they are optimal in all circumstances. To this end, we present a multivariate time-series anomaly detector based on score-based generative models, called MadSGM, which considers the broadest ever set of anomaly measurement factors: i) reconstruction-based, ii) density-based, and iii) gradient-based anomaly measurements. We also design a conditional score network and its denoising score matching loss for the time-series anomaly detection. Experiments on five real-world benchmark datasets illustrate that MadSGM achieves the most robust and accurate predictions.

Mitchell Wortsman, Jaehoon Lee, Justin Gilmer et al.

Previous research observed accuracy degradation when replacing the attention softmax with a point-wise activation such as ReLU. In the context of vision transformers, we find that this degradation is mitigated when dividing by sequence length. Our experiments training small to large vision transformers on ImageNet-21k indicate that ReLU-attention can approach or match the performance of softmax-attention in terms of scaling behavior as a function of compute.

Jaehoon Lee, Hankook Lee, Sungik Choi et al.

When solving forecasting problems including multiple time-series features, existing approaches often fall into two extreme categories, depending on whether to utilize inter-feature information: univariate and complete-multivariate models. Unlike univariate cases which ignore the information, complete-multivariate models compute relationships among a complete set of features. However, despite the potential advantage of leveraging the additional information, complete-multivariate models sometimes underperform univariate ones. Therefore, our research aims to explore a middle ground between these two by introducing what we term Partial-Multivariate models where a neural network captures only partial relationships, that is, dependencies within subsets of all features. To this end, we propose PMformer, a Transformer-based partial-multivariate model, with its training algorithm. We demonstrate that PMformer outperforms various univariate and complete-multivariate models, providing a theoretical rationale and empirical analysis for its superiority. Additionally, by proposing an inference technique for PMformer, the forecasting accuracy is further enhanced. Finally, we highlight other advantages of PMformer: efficiency and robustness under missing features.

Avi Singh, John D. Co-Reyes, Rishabh Agarwal et al. · anthropic, deepmind

Fine-tuning language models~(LMs) on human-generated data remains a prevalent practice. However, the performance of such models is often limited by the quantity and diversity of high-quality human data. In this paper, we explore whether we can go beyond human data on tasks where we have access to scalar feedback, for example, on math problems where one can verify correctness. To do so, we investigate a simple self-training method based on expectation-maximization, which we call ReST$^{EM}$, where we (1) generate samples from the model and filter them using binary feedback, (2) fine-tune the model on these samples, and (3) repeat this process a few times. Testing on advanced MATH reasoning and APPS coding benchmarks using PaLM-2 models, we find that ReST$^{EM}$ scales favorably with model size and significantly surpasses fine-tuning only on human data. Overall, our findings suggest self-training with feedback can substantially reduce dependence on human-generated data.

Jascha Sohl-Dickstein, Roman Novak, Samuel S. Schoenholz et al.

There are currently two parameterizations used to derive fixed kernels corresponding to infinite width neural networks, the NTK (Neural Tangent Kernel) parameterization and the naive standard parameterization. However, the extrapolation of both of these parameterizations to infinite width is problematic. The standard parameterization leads to a divergent neural tangent kernel while the NTK parameterization fails to capture crucial aspects of finite width networks such as: the dependence of training dynamics on relative layer widths, the relative training dynamics of weights and biases, and overall learning rate scale. Here we propose an improved extrapolation of the standard parameterization that preserves all of these properties as width is taken to infinity and yields a well-defined neural tangent kernel. We show experimentally that the resulting kernels typically achieve similar accuracy to those resulting from an NTK parameterization, but with better correspondence to the parameterization of typical finite width networks. Additionally, with careful tuning of width parameters, the improved standard parameterization kernels can outperform those stemming from an NTK parameterization. We release code implementing this improved standard parameterization as part of the Neural Tangents library at https://github.com/google/neural-tangents.

Roman Novak, Lechao Xiao, Jiri Hron et al.

Neural Tangents is a library designed to enable research into infinite-width neural networks. It provides a high-level API for specifying complex and hierarchical neural network architectures. These networks can then be trained and evaluated either at finite-width as usual or in their infinite-width limit. Infinite-width networks can be trained analytically using exact Bayesian inference or using gradient descent via the Neural Tangent Kernel. Additionally, Neural Tangents provides tools to study gradient descent training dynamics of wide but finite networks in either function space or weight space. The entire library runs out-of-the-box on CPU, GPU, or TPU. All computations can be automatically distributed over multiple accelerators with near-linear scaling in the number of devices. Neural Tangents is available at www.github.com/google/neural-tangents. We also provide an accompanying interactive Colab notebook.

Brian Lester, Jaehoon Lee, Alex Alemi et al. · anthropic

In this paper, we explore the idea of training large language models (LLMs) over highly compressed text. While standard subword tokenizers compress text by a small factor, neural text compressors can achieve much higher rates of compression. If it were possible to train LLMs directly over neurally compressed text, this would confer advantages in training and serving efficiency, as well as easier handling of long text spans. The main obstacle to this goal is that strong compression tends to produce opaque outputs that are not well-suited for learning. In particular, we find that text naïvely compressed via Arithmetic Coding is not readily learnable by LLMs. To overcome this, we propose Equal-Info Windows, a novel compression technique whereby text is segmented into blocks that each compress to the same bit length. Using this method, we demonstrate effective learning over neurally compressed text that improves with scale, and outperforms byte-level baselines by a wide margin on perplexity and inference speed benchmarks. While our method delivers worse perplexity than subword tokenizers for models trained with the same parameter count, it has the benefit of shorter sequence lengths. Shorter sequence lengths require fewer autoregressive generation steps, and reduce latency. Finally, we provide extensive analysis of the properties that contribute to learnability, and offer concrete suggestions for how to further improve the performance of high-compression tokenizers.

Sungik Choi, Sungwoo Park, Jaehoon Lee et al.

Dramatic advances in the quality of the latent diffusion models (LDMs) also led to the malicious use of AI-generated images. While current AI-generated image detection methods assume the availability of real/AI-generated images for training, this is practically limited given the vast expressibility of LDMs. This motivates the training-free detection setup where no related data are available in advance. The existing LDM-generated image detection method assumes that images generated by LDM are easier to reconstruct using an autoencoder than real images. However, we observe that this reconstruction distance is overfitted to background information, leading the current method to underperform in detecting images with simple backgrounds. To address this, we propose a novel method called HFI. Specifically, by viewing the autoencoder of LDM as a downsampling-upsampling kernel, HFI measures the extent of aliasing, a distortion of high-frequency information that appears in the reconstructed image. HFI is training-free, efficient, and consistently outperforms other training-free methods in detecting challenging images generated by various generative models. We also show that HFI can successfully detect the images generated from the specified LDM as a means of implicit watermarking. HFI outperforms the best baseline method while achieving magnitudes of

Jaehoon Lee, Jihyeon Hyeong, Jinsung Jeon et al.

Tabular data synthesis has received wide attention in the literature. This is because available data is often limited, incomplete, or cannot be obtained easily, and data privacy is becoming increasingly important. In this work, we present a generalized GAN framework for tabular synthesis, which combines the adversarial training of GANs and the negative log-density regularization of invertible neural networks. The proposed framework can be used for two distinctive objectives. First, we can further improve the synthesis quality, by decreasing the negative log-density of real records in the process of adversarial training. On the other hand, by increasing the negative log-density of real records, realistic fake records can be synthesized in a way that they are not too much close to real records and reduce the chance of potential information leakage. We conduct experiments with real-world datasets for classification, regression, and privacy attacks. In general, the proposed method demonstrates the best synthesis quality (in terms of task-oriented evaluation metrics, e.g., F1) when decreasing the negative log-density during the adversarial training. If increasing the negative log-density, our experimental results show that the distance between real and fake records increases, enhancing robustness against privacy attacks.

Timothy Nguyen, Roman Novak, Lechao Xiao et al.

The effectiveness of machine learning algorithms arises from being able to extract useful features from large amounts of data. As model and dataset sizes increase, dataset distillation methods that compress large datasets into significantly smaller yet highly performant ones will become valuable in terms of training efficiency and useful feature extraction. To that end, we apply a novel distributed kernel based meta-learning framework to achieve state-of-the-art results for dataset distillation using infinitely wide convolutional neural networks. For instance, using only 10 datapoints (0.02% of original dataset), we obtain over 65% test accuracy on CIFAR-10 image classification task, a dramatic improvement over the previous best test accuracy of 40%. Our state-of-the-art results extend across many other settings for MNIST, Fashion-MNIST, CIFAR-10, CIFAR-100, and SVHN. Furthermore, we perform some preliminary analyses of our distilled datasets to shed light on how they differ from naturally occurring data.

Yasaman Bahri, Ethan Dyer, Jared Kaplan et al.

The population loss of trained deep neural networks often follows precise power-law scaling relations with either the size of the training dataset or the number of parameters in the network. We propose a theory that explains the origins of and connects these scaling laws. We identify variance-limited and resolution-limited scaling behavior for both dataset and model size, for a total of four scaling regimes. The variance-limited scaling follows simply from the existence of a well-behaved infinite data or infinite width limit, while the resolution-limited regime can be explained by positing that models are effectively resolving a smooth data manifold. In the large width limit, this can be equivalently obtained from the spectrum of certain kernels, and we present evidence that large width and large dataset resolution-limited scaling exponents are related by a duality. We exhibit all four scaling regimes in the controlled setting of large random feature and pretrained models and test the predictions empirically on a range of standard architectures and datasets. We also observe several empirical relationships between datasets and scaling exponents under modifications of task and architecture aspect ratio. Our work provides a taxonomy for classifying different scaling regimes, underscores that there can be different mechanisms driving improvements in loss, and lends insight into the microscopic origins of and relationships between scaling exponents.

Daniel S. Park, Jaehoon Lee, Daiyi Peng et al.

The predictions of wide Bayesian neural networks are described by a Gaussian process, known as the Neural Network Gaussian Process (NNGP). Analytic forms for NNGP kernels are known for many models, but computing the exact kernel for convolutional architectures is prohibitively expensive. One can obtain effective approximations of these kernels through Monte-Carlo estimation using finite networks at initialization. Monte-Carlo NNGP inference is orders-of-magnitude cheaper in FLOPs compared to gradient descent training when the dataset size is small. Since NNGP inference provides a cheap measure of performance of a network architecture, we investigate its potential as a signal for neural architecture search (NAS). We compute the NNGP performance of approximately 423k networks in the NAS-bench 101 dataset on CIFAR-10 and compare its utility against conventional performance measures obtained by shortened gradient-based training. We carry out a similar analysis on 10k randomly sampled networks in the mobile neural architecture search (MNAS) space for ImageNet. We discover comparative advantages of NNGP-based metrics, and discuss potential applications. In particular, we propose that NNGP performance is an inexpensive signal independent of metrics obtained from training that can either be used for reducing big search spaces, or improving training-based performance measures.

Timothy Nguyen, Zhourong Chen, Jaehoon Lee

One of the most fundamental aspects of any machine learning algorithm is the training data used by the algorithm. We introduce the novel concept of $ε$-approximation of datasets, obtaining datasets which are much smaller than or are significant corruptions of the original training data while maintaining similar model performance. We introduce a meta-learning algorithm called Kernel Inducing Points (KIP) for obtaining such remarkable datasets, inspired by the recent developments in the correspondence between infinitely-wide neural networks and kernel ridge-regression (KRR). For KRR tasks, we demonstrate that KIP can compress datasets by one or two orders of magnitude, significantly improving previous dataset distillation and subset selection methods while obtaining state of the art results for MNIST and CIFAR-10 classification. Furthermore, our KIP-learned datasets are transferable to the training of finite-width neural networks even beyond the lazy-training regime, which leads to state of the art results for neural network dataset distillation with potential applications to privacy-preservation.

Ben Adlam, Jaehoon Lee, Lechao Xiao et al.

Modern deep learning models have achieved great success in predictive accuracy for many data modalities. However, their application to many real-world tasks is restricted by poor uncertainty estimates, such as overconfidence on out-of-distribution (OOD) data and ungraceful failing under distributional shift. Previous benchmarks have found that ensembles of neural networks (NNs) are typically the best calibrated models on OOD data. Inspired by this, we leverage recent theoretical advances that characterize the function-space prior of an ensemble of infinitely-wide NNs as a Gaussian process, termed the neural network Gaussian process (NNGP). We use the NNGP with a softmax link function to build a probabilistic model for multi-class classification and marginalize over the latent Gaussian outputs to sample from the posterior. This gives us a better understanding of the implicit prior NNs place on function space and allows a direct comparison of the calibration of the NNGP and its finite-width analogue. We also examine the calibration of previous approaches to classification with the NNGP, which treat classification problems as regression to the one-hot labels. In this case the Bayesian posterior is exact, and we compare several heuristics to generate a categorical distribution over classes. We find these methods are well calibrated under distributional shift. Finally, we consider an infinite-width final layer in conjunction with a pre-trained embedding. This replicates the important practical use case of transfer learning and allows scaling to significantly larger datasets. As well as achieving competitive predictive accuracy, this approach is better calibrated than its finite width analogue.

Jaehoon Lee, Samuel S. Schoenholz, Jeffrey Pennington et al.

We perform a careful, thorough, and large scale empirical study of the correspondence between wide neural networks and kernel methods. By doing so, we resolve a variety of open questions related to the study of infinitely wide neural networks. Our experimental results include: kernel methods outperform fully-connected finite-width networks, but underperform convolutional finite width networks; neural network Gaussian process (NNGP) kernels frequently outperform neural tangent (NT) kernels; centered and ensembled finite networks have reduced posterior variance and behave more similarly to infinite networks; weight decay and the use of a large learning rate break the correspondence between finite and infinite networks; the NTK parameterization outperforms the standard parameterization for finite width networks; diagonal regularization of kernels acts similarly to early stopping; floating point precision limits kernel performance beyond a critical dataset size; regularized ZCA whitening improves accuracy; finite network performance depends non-monotonically on width in ways not captured by double descent phenomena; equivariance of CNNs is only beneficial for narrow networks far from the kernel regime. Our experiments additionally motivate an improved layer-wise scaling for weight decay which improves generalization in finite-width networks. Finally, we develop improved best practices for using NNGP and NT kernels for prediction, including a novel ensembling technique. Using these best practices we achieve state-of-the-art results on CIFAR-10 classification for kernels corresponding to each architecture class we consider.

Patrick O'Driscoll, Jaehoon Lee, Bo Fu

We propose that intelligently combining models from the domains of Artificial Intelligence or Machine Learning with Physical and Expert models will yield a more "trustworthy" model than any one model from a single domain, given a complex and narrow enough problem. Based on mean-variance portfolio theory and bias-variance trade-off analysis, we prove combining models from various domains produces a model that has lower risk, increasing user trust. We call such combined models - physics enhanced artificial intelligence (PEAI), and suggest use cases for PEAI.

Christopher J. Shallue, Jaehoon Lee, Joseph Antognini et al.

Recent hardware developments have dramatically increased the scale of data parallelism available for neural network training. Among the simplest ways to harness next-generation hardware is to increase the batch size in standard mini-batch neural network training algorithms. In this work, we aim to experimentally characterize the effects of increasing the batch size on training time, as measured by the number of steps necessary to reach a goal out-of-sample error. We study how this relationship varies with the training algorithm, model, and data set, and find extremely large variation between workloads. Along the way, we show that disagreements in the literature on how batch size affects model quality can largely be explained by differences in metaparameter tuning and compute budgets at different batch sizes. We find no evidence that larger batch sizes degrade out-of-sample performance. Finally, we discuss the implications of our results on efforts to train neural networks much faster in the future. Our experimental data is publicly available as a database of 71,638,836 loss measurements taken over the course of training for 168,160 individual models across 35 workloads.

Roman Novak, Lechao Xiao, Jaehoon Lee et al.

There is a previously identified equivalence between wide fully connected neural networks (FCNs) and Gaussian processes (GPs). This equivalence enables, for instance, test set predictions that would have resulted from a fully Bayesian, infinitely wide trained FCN to be computed without ever instantiating the FCN, but by instead evaluating the corresponding GP. In this work, we derive an analogous equivalence for multi-layer convolutional neural networks (CNNs) both with and without pooling layers, and achieve state of the art results on CIFAR10 for GPs without trainable kernels. We also introduce a Monte Carlo method to estimate the GP corresponding to a given neural network architecture, even in cases where the analytic form has too many terms to be computationally feasible. Surprisingly, in the absence of pooling layers, the GPs corresponding to CNNs with and without weight sharing are identical. As a consequence, translation equivariance, beneficial in finite channel CNNs trained with stochastic gradient descent (SGD), is guaranteed to play no role in the Bayesian treatment of the infinite channel limit - a qualitative difference between the two regimes that is not present in the FCN case. We confirm experimentally, that while in some scenarios the performance of SGD-trained finite CNNs approaches that of the corresponding GPs as the channel count increases, with careful tuning SGD-trained CNNs can significantly outperform their corresponding GPs, suggesting advantages from SGD training compared to fully Bayesian parameter estimation.

Jaehoon Lee, Yasaman Bahri, Roman Novak et al.

It has long been known that a single-layer fully-connected neural network with an i.i.d. prior over its parameters is equivalent to a Gaussian process (GP), in the limit of infinite network width. This correspondence enables exact Bayesian inference for infinite width neural networks on regression tasks by means of evaluating the corresponding GP. Recently, kernel functions which mimic multi-layer random neural networks have been developed, but only outside of a Bayesian framework. As such, previous work has not identified that these kernels can be used as covariance functions for GPs and allow fully Bayesian prediction with a deep neural network. In this work, we derive the exact equivalence between infinitely wide deep networks and GPs. We further develop a computationally efficient pipeline to compute the covariance function for these GPs. We then use the resulting GPs to perform Bayesian inference for wide deep neural networks on MNIST and CIFAR-10. We observe that trained neural network accuracy approaches that of the corresponding GP with increasing layer width, and that the GP uncertainty is strongly correlated with trained network prediction error. We further find that test performance increases as finite-width trained networks are made wider and more similar to a GP, and thus that GP predictions typically outperform those of finite-width networks. Finally we connect the performance of these GPs to the recent theory of signal propagation in random neural networks.