Modified K-means Algorithm with Local Optimality Guarantees

The K-means algorithm is one of the most widely studied clustering algorithms in machine learning. While extensive research has focused on its ability to achieve a globally optimal solution, there still lacks a rigorous analysis of its local optimality guarantees. In this paper, we first present conditions under which the K-means algorithm converges to a locally optimal solution. Based on this, we propose simple modifications to the K-means algorithm which ensure local optimality in both the continuous and discrete sense, with the same computational complexity as the original K-means algorithm. As the dissimilarity measure, we consider a general Bregman divergence, which is an extension of the squared Euclidean distance often used in the K-means algorithm. Numerical experiments confirm that the K-means algorithm does not always find a locally optimal solution in practice, while our proposed methods provide improved locally optimal solutions with reduced clustering loss. Our code is available at https://github.com/lmingyi/LO-K-means.

Batch-Max: Higher LLM Throughput using Larger Batch Sizes and KV Cache Compression

Several works have developed eviction policies to remove key-value (KV) pairs from the KV cache for more efficient inference. The focus has been on compressing the KV cache after the input prompt has been processed for faster token generation. In settings with limited GPU memory, and when the input context is longer than the generation length, we show that by also compressing the KV cache during the input processing phase, larger batch sizes can be used resulting in significantly higher throughput while still maintaining the original model's accuracy.

Draft on the Fly: Adaptive Self-Speculative Decoding using Cosine Similarity

We present a simple on the fly method for faster inference of large language models. Unlike other (self-)speculative decoding techniques, our method does not require fine-tuning or black-box optimization to generate a fixed draft model, relying instead on simple rules to generate varying draft models adapted to the input context. We show empirically that our light-weight algorithm is competitive with the current SOTA for self-speculative decoding, while being a truly plug-and-play method.

Mathematical Challenges in Deep Learning

Deep models are dominating the artificial intelligence (AI) industry since the ImageNet challenge in 2012. The size of deep models is increasing ever since, which brings new challenges to this field with applications in cell phones, personal computers, autonomous cars, and wireless base stations. Here we list a set of problems, ranging from training, inference, generalization bound, and optimization with some formalism to communicate these challenges with mathematicians, statisticians, and theoretical computer scientists. This is a subjective view of the research questions in deep learning that benefits the tech industry in long run.

Variants of SGD for Lipschitz Continuous Loss Functions in Low-Precision Environments

Motivated by neural network training in low-bit floating and fixed-point environments, this work studies the convergence of variants of SGD with computational error. Considering a general stochastic Lipschitz continuous loss function, a novel convergence result to a Clarke stationary point is presented assuming that only an approximation of its stochastic gradient can be computed as well as error in computing the SGD step itself. Different variants of SGD are then tested empirically in a variety of low-precision arithmetic environments, with improved test set accuracy achieved compared to SGD for two image recognition tasks.