Scaling laws are useful guides for developing language models, but there are still gaps between current scaling studies and how language models are ultimately trained and evaluated. For instance, scaling is usually studied in the compute-optimal training regime (i.e., "Chinchilla optimal" regime); however, in practice, models are often over-trained to reduce inference costs. Moreover, scaling laws mostly predict loss on next-token prediction, but ultimately models are compared based on downstream task performance. In this paper, we address both shortcomings. To do so, we create a testbed of 104 models with 0.011B to 6.9B parameters trained with various numbers of tokens on three data distributions. First, we investigate scaling in the over-trained regime. We fit scaling laws that extrapolate in both the number of model parameters and the ratio of training tokens to parameters. This enables us to predict the validation loss of a 1.4B parameter, 900B token run (i.e., 32$\times$ over-trained) and a 6.9B parameter, 138B token run$\unicode{x2014}$each from experiments that take 300$\times$ less compute. Second, we relate the perplexity of a language model to its downstream task performance via a power law. We use this law to predict top-1 error averaged over downstream tasks for the two aforementioned models using experiments that take 20$\times$ less compute. Our experiments are available at //github.com/mlfoundations/scaling.
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We show how, using linear-algebraic tools developed to prove Tverberg's theorem in combinatorial geometry, we can design new models of multi-class support vector machines (SVMs). These supervised learning protocols require fewer conditions to classify sets of points, and can be computed using existing binary SVM algorithms in higher-dimensional spaces, including soft-margin SVM algorithms. We describe how the theoretical guarantees of standard support vector machines transfer to these new classes of multi-class support vector machines. We give a new simple proof of a geometric characterization of support vectors for largest margin SVMs by Veelaert.
Many interesting physical problems described by systems of hyperbolic conservation laws are stiff, and thus impose a very small time-step because of the restrictive CFL stability condition. In this case, one can exploit the superior stability properties of implicit time integration which allows to choose the time-step only from accuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systems without tailoring it to a specific problem. The nonlinearity of high order schemes, due to space- and time-limiting procedures which control nonphysical oscillations, makes the implicit time integration difficult, e.g.~because the discrete system is nonlinear also on linear problems. This nonlinearity of the scheme is circumvented as proposed in (Puppo et al., Comm.~Appl.~Math.~\& Comput., 2023) for scalar conservation laws, where a first order implicit predictor is computed to freeze the nonlinear coefficients of the essentially non-oscillatory space reconstruction, and also to assist limiting in time. In addition, we propose a novel conservative flux-centered a-posteriori time-limiting procedure using numerical entropy indicators to detect troubled cells. The numerical tests involve classical and artificially devised stiff problems using the Euler's system of gas-dynamics.
計算學習理論
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Activation patching is a popular mechanistic interpretability technique, but has many subtleties regarding how it is applied and how one may interpret the results. We provide a summary of advice and best practices, based on our experience using this technique in practice. We include an overview of the different ways to apply activation patching and a discussion on how to interpret the results. We focus on what evidence patching experiments provide about circuits, and on the choice of metric and associated pitfalls.
Financial firms often rely on factor models to explain correlations among asset returns. These models are important for managing risk, for example by modeling the probability that many assets will simultaneously lose value. Yet after major events, e.g., COVID-19, analysts may reassess whether existing models continue to fit well: specifically, after accounting for the factor exposures, are the residuals of the asset returns independent? With this motivation, we introduce the mosaic permutation test, a nonparametric goodness-of-fit test for preexisting factor models. Our method allows analysts to use nearly any machine learning technique to detect model violations while provably controlling the false positive rate, i.e., the probability of rejecting a well-fitting model. Notably, this result does not rely on asymptotic approximations and makes no parametric assumptions. This property helps prevent analysts from unnecessarily rebuilding accurate models, which can waste resources and increase risk. We illustrate our methodology by applying it to the Blackrock Fundamental Equity Risk (BFRE) model. Using the mosaic permutation test, we find that the BFRE model generally explains the most significant correlations among assets. However, we find evidence of unexplained correlations among certain real estate stocks, and we show that adding new factors improves model fit. We implement our methods in the python package mosaicperm.
Missing data is a common problem in practical settings. Various imputation methods have been developed to deal with missing data. However, even though the label is usually available in the training data, the common practice of imputation usually only relies on the input and ignores the label. In this work, we illustrate how stacking the label into the input can significantly improve the imputation of the input. In addition, we propose a classification strategy that initializes the predicted test label with missing values and stacks the label with the input for imputation. This allows imputing the label and the input at the same time. Also, the technique is capable of handling data training with missing labels without any prior imputation and is applicable to continuous, categorical, or mixed-type data. Experiments show promising results in terms of accuracy.
This note continues and extends the study from Spokoiny (2023a) about estimation for parametric models with possibly large or even infinite parameter dimension. We consider a special class of stochastically linear smooth (SLS) models satisfying three major conditions: the stochastic component of the log-likelihood is linear in the model parameter, while the expected log-likelihood is a smooth and concave function. For the penalized maximum likelihood estimators (pMLE), we establish several finite sample bounds about its concentration and large deviations as well as the Fisher and Wilks expansions and risk bounds. In all results, the remainder is given explicitly and can be evaluated in terms of the effective sample size $ n $ and effective parameter dimension $ \mathbb{p} $ which allows us to identify the so-called \emph{critical parameter dimension}. The results are also dimension and coordinate-free. Despite generality, all the presented bounds are nearly sharp and the classical asymptotic results can be obtained as simple corollaries. Our results indicate that the use of advanced fourth-order expansions allows to relax the critical dimension condition $ \mathbb{p}^{3} \ll n $ from Spokoiny (2023a) to $ \mathbb{p}^{3/2} \ll n $. Examples for classical models like logistic regression, log-density and precision matrix estimation illustrate the applicability of general results.
Soft targets combined with the cross-entropy loss have shown to improve generalization performance of deep neural networks on supervised classification tasks. The standard cross-entropy loss however assumes data to be categorically distributed, which may often not be the case in practice. In contrast, InfoNCE does not rely on such an explicit assumption but instead implicitly estimates the true conditional through negative sampling. Unfortunately, it cannot be combined with soft targets in its standard formulation, hindering its use in combination with sophisticated training strategies. In this paper, we address this limitation by proposing a principled loss function that is compatible with probabilistic targets. Our new soft target InfoNCE loss is conceptually simple, efficient to compute, and can be derived within the framework of noise contrastive estimation. Using a toy example, we demonstrate shortcomings of the categorical distribution assumption of cross-entropy, and discuss implications of sampling from soft distributions. We observe that soft target InfoNCE performs on par with strong soft target cross-entropy baselines and outperforms hard target NLL and InfoNCE losses on popular benchmarks, including ImageNet. Finally, we provide a simple implementation of our loss, geared towards supervised classification and fully compatible with deep classification model trained with cross-entropy.
In-context learning (ICL) ability has emerged with the increasing scale of large language models (LLMs), enabling them to learn input-label mappings from demonstrations and perform well on downstream tasks. However, under the standard ICL setting, LLMs may sometimes neglect query-related information in demonstrations, leading to incorrect predictions. To address this limitation, we propose a new paradigm called Hint-enhanced In-Context Learning (HICL) to explore the power of ICL in open-domain question answering, an important form in knowledge-intensive tasks. HICL leverages LLMs' reasoning ability to extract query-related knowledge from demonstrations, then concatenates the knowledge to prompt LLMs in a more explicit way. Furthermore, we track the source of this knowledge to identify specific examples, and introduce a Hint-related Example Retriever (HER) to select informative examples for enhanced demonstrations. We evaluate HICL with HER on 3 open-domain QA benchmarks, and observe average performance gains of 2.89 EM score and 2.52 F1 score on gpt-3.5-turbo, 7.62 EM score and 7.27 F1 score on LLaMA-2-Chat-7B compared with standard setting.
Learning interpretable representations of data generative latent factors is an important topic for the development of artificial intelligence. With the rise of the large multimodal model, it can align images with text to generate answers. In this work, we propose a framework to comprehensively explain each latent variable in the generative models using a large multimodal model. We further measure the uncertainty of our generated explanations, quantitatively evaluate the performance of explanation generation among multiple large multimodal models, and qualitatively visualize the variations of each latent variable to learn the disentanglement effects of different generative models on explanations. Finally, we discuss the explanatory capabilities and limitations of state-of-the-art large multimodal models.
The inference stage of diffusion models can be seen as running a reverse-time diffusion stochastic differential equation, where samples from a Gaussian latent distribution are transformed into samples from a target distribution that usually reside on a low-dimensional manifold, e.g., an image manifold. The intermediate values between the initial latent space and the image manifold can be interpreted as noisy images, with the amount of noise determined by the forward diffusion process noise schedule. We utilize this interpretation to present Boomerang, an approach for local sampling of image manifolds. As implied by its name, Boomerang local sampling involves adding noise to an input image, moving it closer to the latent space, and then mapping it back to the image manifold through a partial reverse diffusion process. Thus, Boomerang generates images on the manifold that are ``similar,'' but nonidentical, to the original input image. We can control the proximity of the generated images to the original by adjusting the amount of noise added. Furthermore, due to the stochastic nature of the reverse diffusion process in Boomerang, the generated images display a certain degree of stochasticity, allowing us to obtain local samples from the manifold without encountering any duplicates. Boomerang offers the flexibility to work seamlessly with any pretrained diffusion model, such as Stable Diffusion, without necessitating any adjustments to the reverse diffusion process. We present three applications for Boomerang. First, we provide a framework for constructing privacy-preserving datasets having controllable degrees of anonymity. Second, we show that using Boomerang for data augmentation increases generalization performance and outperforms state-of-the-art synthetic data augmentation. Lastly, we introduce a perceptual image enhancement framework, which enables resolution enhancement.
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