Post-training quantization (PTQ) is a promising approach to reducing the storage and computational requirements of large language models (LLMs) without additional training cost. Recent PTQ studies have primarily focused on quantizing only weights to sub-8-bits while maintaining activations at 8-bits or higher. Accurate sub-8-bit quantization for both weights and activations without relying on quantization-aware training remains a significant challenge. We propose a novel quantization method called block clustered quantization (BCQ) wherein each operand tensor is decomposed into blocks (a block is a group of contiguous scalars), blocks are clustered based on their statistics, and a dedicated optimal quantization codebook is designed for each cluster. As a specific embodiment of this approach, we propose a PTQ algorithm called Locally-Optimal BCQ (LO-BCQ) that iterates between the steps of block clustering and codebook design to greedily minimize the quantization mean squared error. When weight and activation scalars are encoded to W4A4 format (with 0.5-bits of overhead for storing scaling factors and codebook selectors), we advance the current state-of-the-art by demonstrating <1% loss in inference accuracy across several LLMs and downstream tasks.

Although Large Language Models (LLMs) exhibit superior performance across diverse applications, their empirical deployment remains challenging due to their associated considerable model size and high inference costs. To mitigate these emerging challenges, model compression research such as post-training compression (Ashkboos et al., 2024; Ma et al., 2023) and compression-aware training (Alvarez & Salzmann, 2017; Lym et al., 2019; Liu et al., 2024, 2023c) has been extensively explored to reduce the computational resource demands of serving LLMs (Zhu et al., 2023). However, most existing methods either incur significant accuracy degradation compared to uncompressed models or have high training time. Additionally, their flexibility is often limited by a discrete set of compression formats (e.g., 2:4 sparsity, 3/4-bit quantization), making it challenging to meet the diverse capacity and efficiency requirements of different users. To overcome the above flexibility limitation, we re-formulate the model compression problem into the customized compensation problem: Given a compressed model, we aim to introduce residual low-rank paths to compensate for compression errors under customized requirements from users, such as tasks, compression ratios, etc. Rather than focusing solely on producing compressed models with minimal performance degradation, by incorporating these residual paths, the compensated model gains greater flexibility in adjusting overall capacity, without being constrained by specific compression formats.

We propose ESPACE, an LLM compression technique based on dimensionality reduction of activations. Unlike prior works on weight-centric tensor decomposition, ESPACE projects activations onto a pre-calibrated set of principal components. The activation-centrality of the approach enables retraining LLMs with no loss of expressivity; while at inference, weight decomposition is obtained as a byproduct of matrix multiplication associativity. Theoretical results on the construction of projection matrices with optimal computational accuracy are provided. Experimentally, we find ESPACE enables 50% compression of GPT3, Llama2, and Nemotron4 models with small accuracy degradation, as low as a 0.18 perplexity increase on GPT3-22B. At lower compression rates of 20% to 40%, ESPACE drives GPT3 models to outperforming their baseline, by up to a 0.38 decrease in perplexity for GPT3-8B. ESPACE also reduces GEMM execution time and prefill inference latency on existing hardware. Comparison with related works on compressing Llama2-7B via matrix factorization shows that ESPACE is a first step in advancing the state-of-the-art in tensor decomposition compression of LLMs.

High-dimensional motion generation requires numerical precision for smooth, collision-free solutions. Typically, double-precision or single-precision floating-point (FP) formats are utilized. Using these for big tensors imposes a strain on the memory bandwidth provided by the devices and alters the memory footprint, hence limiting their applicability to low-power edge devices needed for mobile robots. The uniform application of reduced precision can be advantageous but severely degrades solutions. Using decreased precision data types for important tensors, we propose to accelerate motion generation by removing memory bottlenecks. We propose variable-precision (VaPr) search optimization to determine the appropriate precision for large tensors from a vast search space of approximately 4 million unique combinations for FP data types across the tensors. To obtain the efficiency gains, we exploit existing platform support for an out-of-the-box GPU speedup and evaluate prospective precision converter units for GPU types that are not currently supported. Our experimental results on 800 planning problems for the Franka Panda robot on the MotionBenchmaker dataset across 8 environments show that a 4-bit FP format is sufficient for the largest set of tensors in the motion generation stack. With the software-only solution, VaPr achieves 6.3% and 6.3% speedups on average for a significant portion of motion generation over the SOTA solution (CuRobo) on Jetson Orin and RTX2080 Ti GPU, respectively, and 9.9%, 17.7% speedups with the FP converter.

Efforts to reduce the numerical precision of computations in deep learning training have yielded systems that aggressively quantize weights and activations, yet employ wide high-precision accumulators for partial sums in inner-product operations to preserve the quality of convergence. The absence of any framework to analyze the precision requirements of partial sum accumulations results in conservative design choices. This imposes an upper-bound on the reduction of complexity of multiply-accumulate units. We present a statistical approach to analyze the impact of reduced accumulation precision on deep learning training. Observing that a bad choice for accumulation precision results in loss of information that manifests itself as a reduction in variance in an ensemble of partial sums, we derive a set of equations that relate this variance to the length of accumulation and the minimum number of bits needed for accumulation. We apply our analysis to three benchmark networks: CIFAR-10 ResNet 32, ImageNet ResNet 18 and ImageNet AlexNet. In each case, with accumulation precision set in accordance with our proposed equations, the networks successfully converge to the single precision floating-point baseline. We also show that reducing accumulation precision further degrades the quality of the trained network, proving that our equations produce tight bounds. Overall this analysis enables precise tailoring of computation hardware to the application, yielding area- and power-optimal systems.

The high computational and parameter complexity of neural networks makes their training very slow and difficult to deploy on energy and storage-constrained computing systems. Many network complexity reduction techniques have been proposed including fixed-point implementation. However, a systematic approach for designing full fixed-point training and inference of deep neural networks remains elusive. We describe a precision assignment methodology for neural network training in which all network parameters, i.e., activations and weights in the feedforward path, gradients and weight accumulators in the feedback path, are assigned close to minimal precision. The precision assignment is derived analytically and enables tracking the convergence behavior of the full precision training, known to converge a priori. Thus, our work leads to a systematic methodology of determining suitable precision for fixed-point training. The near optimality (minimality) of the resulting precision assignment is validated empirically for four networks on the CIFAR-10, CIFAR-100, and SVHN datasets. The complexity reduction arising from our approach is compared with other fixed-point neural network designs.

It is well-known that the precision of data, hyperparameters, and internal representations employed in learning systems directly impacts its energy, throughput, and latency. The precision requirements for the training algorithm are also important for systems that learn on-the-fly. Prior work has shown that the data and hyperparameters can be quantized heavily without incurring much penalty in classification accuracy when compared to floating point implementations. These works suffer from two key limitations. First, they assume uniform precision for the classifier and for the training algorithm and thus miss out on the opportunity to further reduce precision. Second, prior works are empirical studies. In this article, we overcome both these limitations by deriving analytical lower bounds on the precision requirements of the commonly employed stochastic gradient descent (SGD) on-line learning algorithm in the specific context of a support vector machine (SVM). Lower bounds on the data precision are derived in terms of the the desired classification accuracy and precision of the hyperparameters used in the classifier. Additionally, lower bounds on the hyperparameter precision in the SGD training algorithm are obtained. These bounds are validated using both synthetic and the UCI breast cancer dataset. Additionally, the impact of these precisions on the energy consumption of a fixed-point SVM with on-line training is studied.