On the Benefits of Rank in Attention Layers

Attention-based mechanisms are widely used in machine learning, most prominently in transformers. However, hyperparameters such as the rank of the attention matrices and the number of heads are scaled nearly the same way in all realizations of this architecture, without theoretical justification. In this work we show that there are dramatic trade-offs between the rank and number of heads of the attention mechanism. Specifically, we present a simple and natural target function that can be represented using a single full-rank attention head for any context length, but that cannot be approximated by low-rank attention unless the number of heads is exponential in the embedding dimension, even for short context lengths. Moreover, we prove that, for short context lengths, adding depth allows the target to be approximated by low-rank attention. For long contexts, we conjecture that full-rank attention is necessary. Finally, we present experiments with off-the-shelf transformers that validate our theoretical findings.

Reconstructing Training Data From Real World Models Trained with Transfer Learning

Current methods for reconstructing training data from trained classifiers are restricted to very small models, limited training set sizes, and low-resolution images. Such restrictions hinder their applicability to real-world scenarios. In this paper, we present a novel approach enabling data reconstruction in realistic settings for models trained on high-resolution images. Our method adapts the reconstruction scheme of arXiv:2206.07758 to real-world scenarios -- specifically, targeting models trained via transfer learning over image embeddings of large pre-trained models like DINO-ViT and CLIP. Our work employs data reconstruction in the embedding space rather than in the image space, showcasing its applicability beyond visual data. Moreover, we introduce a novel clustering-based method to identify good reconstructions from thousands of candidates. This significantly improves on previous works that relied on knowledge of the training set to identify good reconstructed images. Our findings shed light on a potential privacy risk for data leakage from models trained using transfer learning.

When Can Transformers Count to n?

Large language models based on the transformer architectures can solve highly complex tasks. But are there simple tasks that such models cannot solve? Here we focus on very simple counting tasks, that involve counting how many times a token in the vocabulary have appeared in a string. We show that if the dimension of the transformer state is linear in the context length, this task can be solved. However, the solution we propose does not scale beyond this limit, and we provide theoretical arguments for why it is likely impossible for a size limited transformer to implement this task. Our empirical results demonstrate the same phase-transition in performance, as anticipated by the theoretical argument. Our results demonstrate the importance of understanding how transformers can solve simple tasks.

MALT Powers Up Adversarial Attacks

Current adversarial attacks for multi-class classifiers choose the target class for a given input naively, based on the classifier's confidence levels for various target classes. We present a novel adversarial targeting method, \textit{MALT - Mesoscopic Almost Linearity Targeting}, based on medium-scale almost linearity assumptions. Our attack wins over the current state of the art AutoAttack on the standard benchmark datasets CIFAR-100 and ImageNet and for a variety of robust models. In particular, our attack is \emph{five times faster} than AutoAttack, while successfully matching all of AutoAttack's successes and attacking additional samples that were previously out of reach. We then prove formally and demonstrate empirically that our targeting method, although inspired by linear predictors, also applies to standard non-linear models.

RedEx: Beyond Fixed Representation Methods via Convex Optimization

Optimizing Neural networks is a difficult task which is still not well understood. On the other hand, fixed representation methods such as kernels and random features have provable optimization guarantees but inferior performance due to their inherent inability to learn the representations. In this paper, we aim at bridging this gap by presenting a novel architecture called RedEx (Reduced Expander Extractor) that is as expressive as neural networks and can also be trained in a layer-wise fashion via a convex program with semi-definite constraints and optimization guarantees. We also show that RedEx provably surpasses fixed representation methods, in the sense that it can efficiently learn a family of target functions which fixed representation methods cannot.

Locally Optimal Descent for Dynamic Stepsize Scheduling

We introduce a novel dynamic learning-rate scheduling scheme grounded in theory with the goal of simplifying the manual and time-consuming tuning of schedules in practice. Our approach is based on estimating the locally-optimal stepsize, guaranteeing maximal descent in the direction of the stochastic gradient of the current step. We first establish theoretical convergence bounds for our method within the context of smooth non-convex stochastic optimization, matching state-of-the-art bounds while only assuming knowledge of the smoothness parameter. We then present a practical implementation of our algorithm and conduct systematic experiments across diverse datasets and optimization algorithms, comparing our scheme with existing state-of-the-art learning-rate schedulers. Our findings indicate that our method needs minimal tuning when compared to existing approaches, removing the need for auxiliary manual schedules and warm-up phases and achieving comparable performance with drastically reduced parameter tuning.

Deconstructing Data Reconstruction: Multiclass, Weight Decay and General Losses

Memorization of training data is an active research area, yet our understanding of the inner workings of neural networks is still in its infancy. Recently, Haim et al. (2022) proposed a scheme to reconstruct training samples from multilayer perceptron binary classifiers, effectively demonstrating that a large portion of training samples are encoded in the parameters of such networks. In this work, we extend their findings in several directions, including reconstruction from multiclass and convolutional neural networks. We derive a more general reconstruction scheme which is applicable to a wider range of loss functions such as regression losses. Moreover, we study the various factors that contribute to networks' susceptibility to such reconstruction schemes. Intriguingly, we observe that using weight decay during training increases reconstructability both in terms of quantity and quality. Additionally, we examine the influence of the number of neurons relative to the number of training samples on the reconstructability.

From Tempered to Benign Overfitting in ReLU Neural Networks

Overparameterized neural networks (NNs) are observed to generalize well even when trained to perfectly fit noisy data. This phenomenon motivated a large body of work on "benign overfitting", where interpolating predictors achieve near-optimal performance. Recently, it was conjectured and empirically observed that the behavior of NNs is often better described as "tempered overfitting", where the performance is non-optimal yet also non-trivial, and degrades as a function of the noise level. However, a theoretical justification of this claim for non-linear NNs has been lacking so far. In this work, we provide several results that aim at bridging these complementing views. We study a simple classification setting with 2-layer ReLU NNs, and prove that under various assumptions, the type of overfitting transitions from tempered in the extreme case of one-dimensional data, to benign in high dimensions. Thus, we show that the input dimension has a crucial role on the type of overfitting in this setting, which we also validate empirically for intermediate dimensions. Overall, our results shed light on the intricate connections between the dimension, sample size, architecture and training algorithm on the one hand, and the type of resulting overfitting on the other hand.

Reconstructing Training Data from Multiclass Neural Networks

Reconstructing samples from the training set of trained neural networks is a major privacy concern. Haim et al. (2022) recently showed that it is possible to reconstruct training samples from neural network binary classifiers, based on theoretical results about the implicit bias of gradient methods. In this work, we present several improvements and new insights over this previous work. As our main improvement, we show that training-data reconstruction is possible in the multi-class setting and that the reconstruction quality is even higher than in the case of binary classification. Moreover, we show that using weight-decay during training increases the vulnerability to sample reconstruction. Finally, while in the previous work the training set was of size at most $1000$ from $10$ classes, we show preliminary evidence of the ability to reconstruct from a model trained on $5000$ samples from $100$ classes.

Adversarial Examples Exist in Two-Layer ReLU Networks for Low Dimensional Data Manifolds

Despite a great deal of research, it is still not well-understood why trained neural networks are highly vulnerable to adversarial examples. In this work we focus on two-layer neural networks trained using data which lie on a low dimensional linear subspace. We show that standard gradient methods lead to non-robust neural networks, namely, networks which have large gradients in directions orthogonal to the data subspace, and are susceptible to small adversarial $L_2$-perturbations in these directions. Moreover, we show that decreasing the initialization scale of the training algorithm, or adding $L_2$ regularization, can make the trained network more robust to adversarial perturbations orthogonal to the data.