We propose a distillation scaling law that estimates distilled model performance based on a compute budget and its allocation between the student and teacher. Our findings mitigate the risks associated with large-scale distillation by enabling compute-optimal allocation for both the teacher and student to maximize student performance. We provide compute-optimal distillation recipes for two key scenarios: when a teacher already exists, and when a teacher needs training. In settings involving many students or an existing teacher, distillation outperforms supervised learning up to a compute level that scales predictably with student size. Conversely, if only one student is to be distilled and a teacher also requires training, supervised learning is generally preferable. Additionally, our large-scale study of distillation increases our understanding of the process and helps inform experimental design.

We investigate the capabilities of transformer models on relational reasoning tasks. In these tasks, models are trained on a set of strings encoding abstract relations, and are then tested out-of-distribution on data that contains symbols that did not appear in the training dataset. We prove that for any relational reasoning task in a large family of tasks, transformers learn the abstract relations and generalize to the test set when trained by gradient descent on sufficiently large quantities of training data. This is in contrast to classical fully-connected networks, which we prove fail to learn to reason. Our results inspire modifications of the transformer architecture that add only two trainable parameters per head, and that we empirically demonstrate improve data efficiency for learning to reason.

Pretrained language models are commonly aligned with human preferences and downstream tasks via reinforcement finetuning (RFT), which refers to maximizing a (possibly learned) reward function using policy gradient algorithms. This work identifies a fundamental optimization obstacle in RFT: we prove that the expected gradient for an input vanishes when its reward standard deviation under the model is small, even if the expected reward is far from optimal. Through experiments on an RFT benchmark and controlled environments, as well as a theoretical analysis, we then demonstrate that vanishing gradients due to small reward standard deviation are prevalent and detrimental, leading to extremely slow reward maximization. Lastly, we explore ways to overcome vanishing gradients in RFT. We find the common practice of an initial supervised finetuning (SFT) phase to be the most promising candidate, which sheds light on its importance in an RFT pipeline. Moreover, we show that a relatively small number of SFT optimization steps on as few as 1% of the input samples can suffice, indicating that the initial SFT phase need not be expensive in terms of compute and data labeling efforts. Overall, our results emphasize that being mindful for inputs whose expected gradient vanishes, as measured by the reward standard deviation, is crucial for successful execution of RFT.

Joint embedding (JE) architectures have emerged as a promising avenue for ac- quiring transferable data representations. A key obstacle to using JE methods, however, is the inherent challenge of evaluating learned representations without access to a downstream task, and an annotated dataset. Without efficient and re- liable evaluation, it is difficult to iterate on architectural and training choices for JE methods. In this paper, we introduce LiDAR (Linear Discriminant Analysis Rank), a metric designed to measure the quality of representations within JE archi- tectures. Our metric addresses several shortcomings of recent approaches based on feature covariance rank by discriminating between informative and uninforma- tive features. In essence, LiDAR quantifies the rank of the Linear Discriminant Analysis (LDA) matrix associated with the surrogate SSL task—a measure that intuitively captures the information content as it pertains to solving the SSL task. We empirically demonstrate that LiDAR significantly surpasses naive rank based approaches in its predictive power of optimal hyperparameters. Our proposed cri- terion presents a more robust and intuitive means of assessing the quality of rep- resentations within JE architectures, which we hope facilitates broader adoption of these powerful techniques in various domains.

Large language models exhibit surprising emergent generalization properties, yet also struggle on many simple reasoning tasks such as arithmetic and parity. In this work, we focus on length generalization, and we propose a unifying framework to understand when and how Transformers can be expected to length generalize on a given task. First, we show that there exist algorithmic tasks for which standard decoder-only Transformers trained from scratch naturally exhibit strong length generalization. For these tasks, we leverage the RASP programming language (Weiss et al., 2021) to show that the correct algorithmic solution which solves the task can be represented by a simple Transformer. We thus propose the RASP-Generalization Conjecture: Transformers tend to learn a length-generalizing solution if there exists a short RASP-L program that works for all input lengths. We present empirical evidence to support the correlation between RASP-simplicity and generalization. We leverage our insights to give new scratchpad formats which yield strong length generalization on traditionally hard tasks (such as parity and addition), and we illustrate how scratchpad can hinder generalization when it increases the complexity of the corresponding RASP-L program. Overall, our work provides a novel perspective on the mechanisms of length generalization and the algorithmic capabilities of Transformers.

Two competing paradigms exist for self-supervised learning of data representations. Joint Embedding Predictive Architectures (JEPAs) is a class of architectures in which semantically similar inputs are encoded into representations that are predictive of each other. A recent successful approach that falls under the JEPA framework is self-distillation, where an online encoder is trained to predict the output of the target encoder, sometimes with a lightweight predictor network. This is contrasted with the Masked Auto Encoder (MAE) paradigm, where an encoder and decoder are trained to reconstruct missing parts of the input in ambient space rather than its latent representation. A common motivation for using the JEPA approach over MAE is that the JEPA objective prioritizes abstract features over fine-grained pixel information (which can be unpredictable and uninformative). In this work, we seek to understand the mechanism behind this empirical observation by analyzing deep linear models. We uncover a surprising mechanism: in a simplified linear setting where both approaches learn similar representations, JEPAs are biased to learn high influence features, or features characterized by having high regression coefficients. Our results point to a distinct implicit bias of predicting in latent space that may shed light on its success in practice.

Training stability is of great importance to Transformers. In this work, we investigate the training dynamics of Transformers by examining the evolution of the attention layers. In particular, we track the attention entropy for each attention head during the course of training, which is a proxy for model sharpness. We identify a common pattern across different architectures and tasks, where low attention entropy is accompanied by high training instability, which can take the form of oscillating loss or divergence. We denote the pathologically low attention entropy, corresponding to highly concentrated attention scores, as $\textit{entropy collapse}$. As a remedy, we propose $\sigma$Reparam, a simple and efficient solution where we reparametrize all linear layers with spectral normalization and an additional learned scalar. We demonstrate that $\sigma$Reparam successfully prevents entropy collapse in the attention layers, promoting more stable training. Additionally, we prove a tight lower bound of the attention entropy, which decreases exponentially fast with the spectral norm of the attention logits, providing additional motivation for our approach. We conduct experiments with $\sigma$Reparam on image classification, image self-supervised learning, machine translation, speech recognition, and language modeling tasks. We show that $\sigma$Reparam provides stability and robustness with respect to the choice of hyperparameters, going so far as enabling training (a) a Vision Transformer to competitive performance without warmup, weight decay, layer normalization or adaptive optimizers; (b) deep architectures in machine translation and (c) speech recognition to competitive performance without warmup and adaptive optimizers. Code is available at https://github.com/apple/ml-sigma-reparam.

We identify incremental learning dynamics in transformers, where the difference between trained and initial weights progressively increases in rank. We rigorously prove this occurs under the simplifying assumptions of diagonal weight matrices and small initialization. Our experiments support the theory and also show that phenomenon can occur in practice without the simplifying assumptions.

Yang (2020) recently showed that the Neural Tangent Kernel (NTK) at initialization has an infinite-width limit for a large class of architectures including modern staples such as ResNet and Transformers. However, their analysis does not apply to training. Here, we show the same neural networks (in the so-called NTK parametrization) during training follow a kernel gradient descent dynamics in function space, where the kernel is the infinite-width NTK. This completes the proof of the architectural universality of NTK behavior. To achieve this result, we apply the Tensor Programs technique: Write the entire SGD dynamics inside a Tensor Program and analyze it via the Master Theorem. To facilitate this proof, we develop a graphical notation for Tensor Programs, which we believe is also an important contribution toward the pedagogy and exposition of the Tensor Programs technique.

We analyze the finite corrections to the neural tangent kernel (NTK) of residual and densely connected networks, as a function of both depth and width. Surprisingly, our analysis reveals that given a fixed depth, residual networks provide the best tradeoff between the parameter complexity and the coefficient of variation (normalized variance), followed by densely connected networks and vanilla MLPs. While in networks that do not use skip connections, convergence to the NTK requires one to fix the depth, while increasing the layers’ width. Our findings show that in ResNets, convergence to the NTK may occur when depth and width simultaneously tend to infinity, provided with a proper initialization. In DenseNets, however, the convergence of the NTK to its limit as the width tends to infinity is guaranteed, at a rate that is independent of both the depth and scale of the weights. Our experiments validate the theoretical results and demonstrate the advantage of deep ResNets and DenseNets for kernel regression with random gradient features.

{\em Hypernetworks} are architectures that produce the weights of a task-specific {\em primary network}. A notable application of hypernetworks in the recent literature involves learning to output functional representations. In these scenarios, the hypernetwork learns a representation corresponding to the weights of a shallow MLP, which typically encodes shape or image information. While such representations have seen considerable success in practice, they remain lacking in the theoretical guarantees in the wide regime of the standard architectures. In this work, we study wide over-parameterized hypernetworks. We show that unlike typical architectures, infinitely wide hypernetworks do not guarantee convergence to a global minima under gradient descent. We further show that convexity can be achieved by increasing the dimensionality of the hypernetwork's output, to represent wide MLPs. In the dually infinite-width regime, we identify the functional priors of these architectures by deriving their corresponding GP and NTK kernels, the latter of which we refer to as the {\em hyperkernel}. As part of this study, we make a mathematical contribution by deriving tight bounds on high order Taylor expansion terms of standard fully connected ReLU networks.

Modern neural network performance typically improves as model size increases. A recent line of research on the Neural Tangent Kernel (NTK) of over-parameterized networks indicates that the improvement with size increase is a product of a better conditioned loss landscape. In this work, we investigate a form of over-parameterization achieved through ensembling, where we define collegial ensembles (CE) as the aggregation of multiple independent models with identical architectures, trained as a single model. We show that the optimization dynamics of CE simplify dramatically when the number of models in the ensemble is large, resembling the dynamics of wide models, yet scale much more favorably. We use recent theoretical results on the finite width corrections of the NTK to perform efficient architecture search in a space of finite width CE that aims to either minimize capacity, or maximize trainability under a set of constraints. The resulting ensembles can be efficiently implemented in practical architectures using group convolutions and block diagonal layers. Finally, we show how our framework can be used to analytically derive optimal group convolution modules originally found using expensive grid searches, without having to train a single model.

Normalization techniques play an important role in supporting efficient and often more effective training of deep neural networks. While conventional methods explicitly normalize the activations, we suggest to add a loss term instead. This new loss term encourages the variance of the activations to be stable and not vary from one random mini-batch to the next. As we prove, this encourages the activations to be distributed around a few distinct modes. We also show that if the inputs are from a mixture of two Gaussians, the new loss would either join the two together, or separate between them optimally in the LDA sense, depending on the prior probabilities. Finally, we are able to link the new regularization term to the batchnorm method, which provides it with a regularization perspective. Our experiments demonstrate an improvement in accuracy over the batchnorm technique for both CNNs and fully connected networks.

Deep learning techniques are renowned for supporting effective transfer learning. However, as we demonstrate, the transferred representations support only a few modes of separation and much of its dimensionality is unutilized. In this work, we suggest to learn, in the source domain, multiple orthogonal classifiers. We prove that this leads to a reduced rank representation, which, however, supports more discriminative directions. Interestingly, the softmax probabilities produced by the multiple classifiers are likely to be identical. Experimental results, on CIFAR-100 and LFW, further demonstrate the effectiveness of our method.

Multiverse networks were recently proposed as a method for promoting more effective transfer learning. While an extensive analysis was proposed, this analysis failed to capture two main aspects of these networks: (i) the rank of the representation is much lower than the rank predicted by the analysis; and (ii) the contribution of increased multiplicity in such networks diminishes quickly. In this work, we propose additional analysis of multiverse networks which addresses both deficits. A major contribution of our work is quantifying the Rademacher complexity of the multiverse network. It is shown that the complexity upper bound of multiverse networks is significantly lower than that of conventional networks, and diminishes by a factor of √k, k being the multiplicity. In addition, we generalize the notion of multiverse networks to multilayer multiverse networks. We derive the Rademacher complexity formula to such networks and present experimental results.

In this paper, we introduce a spherical embedding technique to position a given set of silhouettes of an object as observed from a set of cameras arbitrarily positioned around the object. Our technique estimates dissimilarities among the silhouettes and embeds them directly in the rotation space SO(3). The embedding is obtained by an optimization scheme applied over the rotations represented with exponential maps. Since the measure for inter-silhouette dissimilarities contains many outliers, our key idea is to perform the embedding by only using a subset of the estimated dissimilarities. We present a technique that carefully screens for inlier-distances, and the pairwise scaled dissimilarities are embedded in a spherical space, diffeomorphic to SO(3). We show that our method outperforms spherical MDS embedding, demonstrate its performance on various multi-view sets, and highlight its robustness to outliers.