We propose a cheaper alternative bilinear operator to matrix-multiplication in deep neural networks (DNNs). Unlike many stubborn attempts to accelerate MatMuls in DNN inference, this operator is supported by capabilities of existing GPU hardware, most notably NVIDIA TensorCores. To our knowledge, this is the first GPU-native acceleration technique which \emph{does not decrease} (in fact, increases) the number of trainable parameters of the network, mitigating the accuracy-loss of compression-based techniques. Hence, this operator is at the same time more expressive than MatMul, yet requires substantially \emph{fewer} FLOPs to evaluate. We term this new operator \emph{Strassen-Tile} (STL). The main idea behind STL$(X,W)$ is a \emph{local} change-of-basis (learnable encoder) on weights and activation \emph{tiles}, after which we perform batched \emph{elementwise} products between tiles, and a final decoding transformation (inspired by algebraic pipelines from fast matrix and polynomial multiplication). We compare STL against two benchmarks. The first one is SoTA T2T-ViT on Imagenet-1K. Here we show that replacing \emph{all} linear layers with STL and training from scratch, results in factor x2.7 reduction in FLOPs with a 0.5 \emph{accuracy improvement}. Our second speed-accuracy comparison benchmark for pretrained LLMs is the most practical GPU-acceleration technique, \twofour structured Sparsity. Finetuning TinyLlama \cite{tinyllama24} with STL layers on the Slim Pajama dataset, achieves similar accuracy to 2:4, with x2.2 FLOP speedup compared to x1.7 of the latter. Finally, we discuss a group-theoretic approach for discovering \emph{universal} encoders for STL, which could lead to fast \emph{black-box} acceleration via approximate matrix-multiplication (AMM).

Large language models (LLMs) have demonstrated remarkable capabilities, but their adoption is limited by high computational costs during inference. While increasing parameter counts enhances accuracy, it also widens the gap between state-of-the-art capabilities and practical deployability. We present Puzzle, a framework to accelerate LLM inference on specific hardware while preserving their capabilities. Through an innovative application of neural architecture search (NAS) at an unprecedented scale, Puzzle systematically optimizes models with tens of billions of parameters under hardware constraints. Our approach utilizes blockwise local knowledge distillation (BLD) for parallel architecture exploration and employs mixed-integer programming for precise constraint optimization. We demonstrate the real-world impact of our framework through Llama-3.1-Nemotron-51B-Instruct (Nemotron-51B), a publicly available model derived from Llama-3.1-70B-Instruct. Nemotron-51B achieves a 2.17x inference throughput speedup, fitting on a single NVIDIA H100 GPU while preserving 98.4% of the original model's capabilities. Nemotron-51B currently stands as the most accurate language model capable of inference on a single GPU with large batch sizes. Remarkably, this transformation required just 45B training tokens, compared to over 15T tokens used for the 70B model it was derived from. This establishes a new paradigm where powerful models can be optimized for efficient deployment with only negligible compromise of their capabilities, demonstrating that inference performance, not parameter count alone, should guide model selection. With the release of Nemotron-51B and the presentation of the Puzzle framework, we provide practitioners immediate access to state-of-the-art language modeling capabilities at significantly reduced computational costs.

Fast Fourier transform, Wavelets, and other well-known transforms in signal processing have a structured representation as a product of sparse matrices which are referred to as butterfly structures. Research in the recent past have used such structured linear networks along with randomness as pre-conditioners to improve the computational performance of large scale linear algebraic operations. With the advent of deep learning and AI and the computational efficiency of such structured matrices, it is natural to study sparse linear deep networks in which the location of the non-zero weights are predetermined by the butterfly structure. This work studies, both theoretically and empirically, the feasibility of training such networks in different scenarios. Unlike convolutional neural networks, which are structured sparse networks designed to recognize local patterns in lattices representing a spatial or a temporal structure, the butterfly architecture used in this work can replace any dense linear operator with a gadget consisting of a sequence of logarithmically (in the network width) many sparse layers, containing a total of near linear number of weights. This improves on the quadratic number of weights required in a standard dense layer, with little compromise in expressibility of the resulting operator. We show in a collection of empirical experiments that our proposed architecture not only produces results that match and often outperform existing known architectures, but it also offers faster training and prediction in deployment. This empirical phenomenon is observed in a wide variety of experiments that we report, including both supervised prediction on NLP and vision data, as well as in unsupervised representation learning using autoencoders. Preliminary theoretical results presented in the paper explain why training speed and outcome are not compromised by our proposed approach.

We provide a clustering algorithm that approximately optimizes the k-means objective, in the one-pass streaming setting. We make no assumptions about the data, and our algorithm is very light-weight in terms of memory, and computation. This setting is applicable to unsupervised learning on massive data sets, or resource-constrained devices. The two main ingredients of our theoretical work are: a derivation of an extremely simple pseudo-approximation batch algorithm for k-means, in which the algorithm is allowed to output more than k centers (based on the recent k-means "), and a streaming clustering algorithm in which batch clustering algorithms are performed on small inputs (fitting in memory) and combined in a hierarchical manner. Empirical evaluations on real and simulated data reveal the practical utility of our method."

Convolutional networks have marked their place over the last few years as the best performing model for various visual tasks. They are, however, most suited for supervised learning from large amounts of labeled data. Previous attempts have been made to use unlabeled data to improve model performance by applying unsupervised techniques. These attempts require different architectures and training methods. In this work we present a novel approach for unsupervised training of Convolutional networks that is based on contrasting between spatial regions within images. This criterion can be employed within conventional neural networks and trained using standard techniques such as SGD and back-propagation, thus complementing supervised methods.

Convolutional networks have marked their place over the last few years as the best performing model for various visual tasks. They are, however, most suited for supervised learning from large amounts of labeled data. Previous attempts have been made to use unlabeled data to improve model performance by applying unsupervised techniques. These attempts require different architectures and training methods. In this work we present a novel approach for unsupervised training of Convolutional networks that is based on contrasting between spatial regions within images. This criterion can be employed within conventional neural networks and trained using standard techniques such as SGD and back-propagation, thus complementing supervised methods.

Deep learning has proven itself as a successful set of models for learning useful semantic representations of data. These, however, are mostly implicitly learned as part of a classification task. In this paper we propose the triplet network model, which aims to learn useful representations by distance comparisons. A similar model was defined by Wang et al. (2014), tailor made for learning a ranking for image information retrieval. Here we demonstrate using various datasets that our model learns a better representation than that of its immediate competitor, the Siamese network. We also discuss future possible usage as a framework for unsupervised learning.

Given a set $V$ of $n$ objects, an online ranking system outputs at each time step a full ranking of the set, observes a feedback of some form and suffers a loss. We study the setting in which the (adversarial) feedback is an element in $V$, and the loss is the position (0th, 1st, 2nd...) of the item in the outputted ranking. More generally, we study a setting in which the feedback is a subset $U$ of at most $k$ elements in $V$, and the loss is the sum of the positions of those elements. We present an algorithm of expected regret $O(n^{3/2}\sqrt{Tk})$ over a time horizon of $T$ steps with respect to the best single ranking in hindsight. This improves previous algorithms and analyses either by a factor of either $\Omega(\sqrt{k})$, a factor of $\Omega(\sqrt{\log n})$ or by improving running time from quadratic to $O(n\log n)$ per round. We also prove a matching lower bound. Our techniques also imply an improved regret bound for online rank aggregation over the Spearman correlation measure, and to other more complex ranking loss functions.

This paper investigates graph clustering in the planted cluster model in the presence of {\em small clusters}. Traditional results dictate that for an algorithm to provably correctly recover the clusters, {\em all} clusters must be sufficiently large (in particular, $\tilde{\Omega}(\sqrt{n})$ where $n$ is the number of nodes of the graph). We show that this is not really a restriction: by a more refined analysis of the trace-norm based recovery approach proposed in Jalali et al. (2011) and Chen et al. (2012), we prove that small clusters, under certain mild assumptions, do not hinder recovery of large ones. Based on this result, we further devise an iterative algorithm to recover {\em almost all clusters} via a "peeling strategy", i.e., recover large clusters first, leading to a reduced problem, and repeat this procedure. These results are extended to the {\em partial observation} setting, in which only a (chosen) part of the graph is observed.The peeling strategy gives rise to an active learning algorithm, in which edges adjacent to smaller clusters are queried more often as large clusters are learned (and removed). From a high level, this paper sheds novel insights on high-dimensional statistics and learning structured data, by presenting a structured matrix learning problem for which a one shot convex relaxation approach necessarily fails, but a carefully constructed sequence of convex relaxationsdoes the job.

Given a set $V$ of $n$ elements we wish to linearly order them using pairwise preference labels which may be non-transitive (due to irrationality or arbitrary noise). The goal is to linearly order the elements while disagreeing with as few pairwise preference labels as possible. Our performance is measured by two parameters: The number of disagreements (loss) and the query complexity (number of pairwise preference labels). Our algorithm adaptively queries at most $O(n\poly(\log n,\eps^{-1}))$ preference labels for a regret of $\eps$ times the optimal loss. This is strictly better, and often significantly better than what non-adaptive sampling could achieve. Our main result helps settle an open problem posed by learning-to-rank (from pairwise information) theoreticians and practitioners: What is a provably correct way to sample preference labels?