LiDAR: Sensing Linear Probing Performance in Joint Embedding SSL Architectures

Joint embedding (JE) architectures have emerged as a promising avenue for acquiring 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 reliable 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 architectures. Our metric addresses several shortcomings of recent approaches based on feature covariance rank by discriminating between informative and uninformative 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 criterion presents a more robust and intuitive means of assessing the quality of representations within JE architectures, which we hope facilitates broader adoption of these powerful techniques in various domains.

Vanishing Gradients in Reinforcement Finetuning of Language Models

Pretrained language models are commonly aligned with human preferences and downstream tasks via reinforcement finetuning (RFT), which entails maximizing a (possibly learned) reward function using policy gradient algorithms. This work highlights 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.

What Algorithms can Transformers Learn? A Study in Length Generalization

Large language models exhibit surprising emergent generalization properties, yet also struggle on many simple reasoning tasks such as arithmetic and parity. This raises the question of if and when Transformer models can learn the true algorithm for solving a task. We study the scope of Transformers' abilities in the specific setting of length generalization on algorithmic tasks. Here, we propose a unifying framework to understand when and how Transformers can exhibit strong length generalization on a given task. Specifically, we leverage RASP (Weiss et al., 2021) -- a programming language designed for the computational model of a Transformer -- and introduce the RASP-Generalization Conjecture: Transformers tend to length generalize on a task if the task can be solved by a short RASP program which works for all input lengths. This simple conjecture remarkably captures most known instances of length generalization on algorithmic tasks. Moreover, we leverage our insights to drastically improve generalization performance on traditionally hard tasks (such as parity and addition). On the theoretical side, we give a simple example where the "min-degree-interpolator" model of learning from Abbe et al. (2023) does not correctly predict Transformers' out-of-distribution behavior, but our conjecture does. Overall, our work provides a novel perspective on the mechanisms of compositional generalization and the algorithmic capabilities of Transformers.

When can transformers reason with abstract symbols?

We investigate the capabilities of transformer large language models (LLMs) on relational reasoning tasks involving abstract symbols. Such tasks have long been studied in the neuroscience literature as fundamental building blocks for more complex abilities in programming, mathematics, and verbal reasoning. For (i) regression tasks, we prove that transformers generalize when trained, but require astonishingly large quantities of training data. For (ii) next-token-prediction tasks with symbolic labels, we show an "inverse scaling law": transformers fail to generalize as their embedding dimension increases. For both settings (i) and (ii), we propose subtle transformer modifications which can reduce the amount of data needed by adding two trainable parameters per head.

Adaptivity and Modularity for Efficient Generalization Over Task Complexity

Can transformers generalize efficiently on problems that require dealing with examples with different levels of difficulty? We introduce a new task tailored to assess generalization over different complexities and present results that indicate that standard transformers face challenges in solving these tasks. These tasks are variations of pointer value retrieval previously introduced by Zhang et al. (2021). We investigate how the use of a mechanism for adaptive and modular computation in transformers facilitates the learning of tasks that demand generalization over the number of sequential computation steps (i.e., the depth of the computation graph). Based on our observations, we propose a transformer-based architecture called Hyper-UT, which combines dynamic function generation from hyper networks with adaptive depth from Universal Transformers. This model demonstrates higher accuracy and a fairer allocation of computational resources when generalizing to higher numbers of computation steps. We conclude that mechanisms for adaptive depth and modularity complement each other in improving efficient generalization concerning example complexity. Additionally, to emphasize the broad applicability of our findings, we illustrate that in a standard image recognition task, Hyper- UT's performance matches that of a ViT model but with considerably reduced computational demands (achieving over 70\% average savings by effectively using fewer layers).

Tensor Programs IVb: Adaptive Optimization in the Infinite-Width Limit

Going beyond stochastic gradient descent (SGD), what new phenomena emerge in wide neural networks trained by adaptive optimizers like Adam? Here we show: The same dichotomy between feature learning and kernel behaviors (as in SGD) holds for general optimizers as well, including Adam -- albeit with a nonlinear notion of "kernel." We derive the corresponding "neural tangent" and "maximal update" limits for any architecture. Two foundational advances underlie the above results: 1) A new Tensor Program language, NEXORT, that can express how adaptive optimizers process gradients into updates. 2) The introduction of bra-ket notation to drastically simplify expressions and calculations in Tensor Programs. This work summarizes and generalizes all previous results in the Tensor Programs series of papers.

Transformers learn through gradual rank increase

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.

The NTK approximation is valid for longer than you think

We study when the neural tangent kernel (NTK) approximation is valid for training a model with the square loss. In the lazy training setting of Chizat et al. 2019, we show that rescaling the model by a factor of $\alpha = O(T)$ suffices for the NTK approximation to be valid until training time $T$. Our bound is tight and improves on the previous bound of Chizat et al. 2019, which required a larger rescaling factor of $\alpha = O(T^2)$.

Stabilizing Transformer Training by Preventing Attention Entropy Collapse

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 the proposed reparameterization 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, automatic speech recognition, and language modeling tasks, across Transformer architectures. 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.