On the Power of Decision Trees in Auto-Regressive Language Modeling

Originally proposed for handling time series data, Auto-regressive Decision Trees (ARDTs) have not yet been explored for language modeling. This paper delves into both the theoretical and practical applications of ARDTs in this new context. We theoretically demonstrate that ARDTs can compute complex functions, such as simulating automata, Turing machines, and sparse circuits, by leveraging "chain-of-thought" computations. Our analysis provides bounds on the size, depth, and computational efficiency of ARDTs, highlighting their surprising computational power. Empirically, we train ARDTs on simple language generation tasks, showing that they can learn to generate coherent and grammatically correct text on par with a smaller Transformer model. Additionally, we show that ARDTs can be used on top of transformer representations to solve complex reasoning tasks. This research reveals the unique computational abilities of ARDTs, aiming to broaden the architectural diversity in language model development.

Distributed Speculative Inference of Large Language Models

Accelerating the inference of large language models (LLMs) is an important challenge in artificial intelligence. This paper introduces distributed speculative inference (DSI), a novel distributed inference algorithm that is provably faster than speculative inference (SI) [leviathan2023fast, chen2023accelerating, miao2023specinfer] and traditional autoregressive inference (non-SI). Like other SI algorithms, DSI works on frozen LLMs, requiring no training or architectural modifications, and it preserves the target distribution. Prior studies on SI have demonstrated empirical speedups (compared to non-SI) but require a fast and accurate drafter LLM. In practice, off-the-shelf LLMs often do not have matching drafters that are sufficiently fast and accurate. We show a gap: SI gets slower than non-SI when using slower or less accurate drafters. We close this gap by proving that DSI is faster than both SI and non-SI given any drafters. By orchestrating multiple instances of the target and drafters, DSI is not only faster than SI but also supports LLMs that cannot be accelerated with SI. Our simulations show speedups of off-the-shelf LLMs in realistic settings: DSI is 1.29-1.92x faster than SI.

Centered Self-Attention Layers

The self-attention mechanism in transformers and the message-passing mechanism in graph neural networks are repeatedly applied within deep learning architectures. We show that this application inevitably leads to oversmoothing, i.e., to similar representations at the deeper layers for different tokens in transformers and different nodes in graph neural networks. Based on our analysis, we present a correction term to the aggregating operator of these mechanisms. Empirically, this simple term eliminates much of the oversmoothing problem in visual transformers, obtaining performance in weakly supervised segmentation that surpasses elaborate baseline methods that introduce multiple auxiliary networks and training phrases. In graph neural networks, the correction term enables the training of very deep architectures more effectively than many recent solutions to the same problem.

Reverse Engineering Self-Supervised Learning

Self-supervised learning (SSL) is a powerful tool in machine learning, but understanding the learned representations and their underlying mechanisms remains a challenge. This paper presents an in-depth empirical analysis of SSL-trained representations, encompassing diverse models, architectures, and hyperparameters. Our study reveals an intriguing aspect of the SSL training process: it inherently facilitates the clustering of samples with respect to semantic labels, which is surprisingly driven by the SSL objective's regularization term. This clustering process not only enhances downstream classification but also compresses the data information. Furthermore, we establish that SSL-trained representations align more closely with semantic classes rather than random classes. Remarkably, we show that learned representations align with semantic classes across various hierarchical levels, and this alignment increases during training and when moving deeper into the network. Our findings provide valuable insights into SSL's representation learning mechanisms and their impact on performance across different sets of classes.

The Probabilistic Stability of Stochastic Gradient Descent

A fundamental open problem in deep learning theory is how to define and understand the stability of stochastic gradient descent (SGD) close to a fixed point. Conventional literature relies on the convergence of statistical moments, esp., the variance, of the parameters to quantify the stability. We revisit the definition of stability for SGD and use the \textit{convergence in probability} condition to define the \textit{probabilistic stability} of SGD. The proposed stability directly answers a fundamental question in deep learning theory: how SGD selects a meaningful solution for a neural network from an enormous number of solutions that may overfit badly. To achieve this, we show that only under the lens of probabilistic stability does SGD exhibit rich and practically relevant phases of learning, such as the phases of the complete loss of stability, incorrect learning, convergence to low-rank saddles, and correct learning. When applied to a neural network, these phase diagrams imply that SGD prefers low-rank saddles when the underlying gradient is noisy, thereby improving the learning performance. This result is in sharp contrast to the conventional wisdom that SGD prefers flatter minima to sharp ones, which we find insufficient to explain the experimental data. We also prove that the probabilistic stability of SGD can be quantified by the Lyapunov exponents of the SGD dynamics, which can easily be measured in practice. Our work potentially opens a new venue for addressing the fundamental question of how the learning algorithm affects the learning outcome in deep learning.

Norm-based Generalization Bounds for Compositionally Sparse Neural Networks

In this paper, we investigate the Rademacher complexity of deep sparse neural networks, where each neuron receives a small number of inputs. We prove generalization bounds for multilayered sparse ReLU neural networks, including convolutional neural networks. These bounds differ from previous ones, as they consider the norms of the convolutional filters instead of the norms of the associated Toeplitz matrices, independently of weight sharing between neurons. As we show theoretically, these bounds may be orders of magnitude better than standard norm-based generalization bounds and empirically, they are almost non-vacuous in estimating generalization in various simple classification problems. Taken together, these results suggest that compositional sparsity of the underlying target function is critical to the success of deep neural networks.

Exploring the Approximation Capabilities of Multiplicative Neural Networks for Smooth Functions

Multiplication layers are a key component in various influential neural network modules, including self-attention and hypernetwork layers. In this paper, we investigate the approximation capabilities of deep neural networks with intermediate neurons connected by simple multiplication operations. We consider two classes of target functions: generalized bandlimited functions, which are frequently used to model real-world signals with finite bandwidth, and Sobolev-Type balls, which are embedded in the Sobolev Space $\mathcal{W}^{r,2}$. Our results demonstrate that multiplicative neural networks can approximate these functions with significantly fewer layers and neurons compared to standard ReLU neural networks, with respect to both input dimension and approximation error. These findings suggest that multiplicative gates can outperform standard feed-forward layers and have potential for improving neural network design.

Generalization Bounds for Transfer Learning with Pretrained Classifiers

We study the ability of foundation models to learn representations for classification that are transferable to new, unseen classes. Recent results in the literature show that representations learned by a single classifier over many classes are competitive on few-shot learning problems with representations learned by special-purpose algorithms designed for such problems. We offer an explanation for this phenomenon based on the concept of class-features variability collapse, which refers to the training dynamics of deep classification networks where the feature embeddings of samples belonging to the same class tend to concentrate around their class means. More specifically, we examine the few-shot error of the learned feature map, which is the classification error of the nearest class-center classifier using centers learned from a small number of random samples from each class. Assuming that the classes appearing in the data are selected independently from a distribution, we show that the few-shot error generalizes from the training data to unseen test data, and we provide an upper bound on the expected few-shot error for new classes (selected from the same distribution) using the average few-shot error for the source classes. Additionally, we show that the few-shot error on the training data can be upper bounded using the degree of class-features variability collapse. This suggests that foundation models can provide feature maps that are transferable to new downstream tasks even with limited data available.

SGD Noise and Implicit Low-Rank Bias in Deep Neural Networks

We analyze deep ReLU neural networks trained with mini-batch Stochastic Gradient Descent (SGD) and weight decay. We study the source of SGD noise and prove that when training with weight decay, the only solutions of SGD at convergence are zero functions. Furthermore, we show, both theoretically and empirically, that when training a neural network using SGD with weight decay and small batch size, the resulting weight matrices are expected to be of small rank. Our analysis relies on a minimal set of assumptions and the neural networks may be arbitrarily wide or deep, and may include residual connections, as well as batch normalization layers.

On the Implicit Bias Towards Minimal Depth of Deep Neural Networks

We study the implicit bias of gradient based training methods to favor low-depth solutions when training deep neural networks. Recent results in the literature suggest that penultimate layer representations learned by a classifier over multiple classes exhibit a clustering property, called neural collapse. We demonstrate empirically that the neural collapse property extends beyond the penultimate layer and tends to emerge in intermediate layers as well. In this regards, we hypothesize that gradient based methods are implicitly biased towards selecting neural networks of minimal depth for achieving this clustering property.