Linjian Ma

Linjian Ma, Edgar Solomonik

This work discusses tensor network embeddings, which are random matrices ($S$) with tensor network structure. These embeddings have been used to perform dimensionality reduction of tensor network structured inputs $x$ and accelerate applications such as tensor decomposition and kernel regression. Existing works have designed embeddings for inputs $x$ with specific structures, such that the computational cost for calculating $Sx$ is efficient. We provide a systematic way to design tensor network embeddings consisting of Gaussian random tensors, such that for inputs with more general tensor network structures, both the sketch size (row size of $S$) and the sketching computational cost are low. We analyze general tensor network embeddings that can be reduced to a sequence of sketching matrices. We provide a sufficient condition to quantify the accuracy of such embeddings and derive sketching asymptotic cost lower bounds using embeddings that satisfy this condition and have a sketch size lower than any input dimension. We then provide an algorithm to efficiently sketch input data using such embeddings. The sketch size of the embedding used in the algorithm has a linear dependence on the number of sketching dimensions of the input. Assuming tensor contractions are performed with classical dense matrix multiplication algorithms, this algorithm achieves asymptotic cost within a factor of $O(\sqrt{m})$ of our cost lower bound, where $m$ is the sketch size. Further, when each tensor in the input has a dimension that needs to be sketched, this algorithm yields the optimal sketching asymptotic cost. We apply our sketching analysis to inexact tensor decomposition optimization algorithms. We provide a sketching algorithm for CP decomposition that is asymptotically faster than existing work in multiple regimes, and show optimality of an existing algorithm for tensor train rounding.

Zhewei Yao, Xiaoxia Wu, Linjian Ma et al.

Pruning is an effective method to reduce the memory footprint and computational cost associated with large natural language processing models. However, current pruning algorithms either only focus on one pruning category, e.g., structured pruning and unstructured, or need extensive hyperparameter tuning in order to get reasonable accuracy performance. To address these challenges, we propose LEArnable Pruning (LEAP), an effective method to gradually prune the model based on thresholds learned by gradient descent. Different than previous learnable pruning methods, which utilize $L_0$ or $L_1$ penalty to indirectly affect the final pruning ratio, LEAP introduces a novel regularization function, that directly interacts with the preset target pruning ratio. Moreover, in order to reduce hyperparameter tuning, a novel adaptive regularization coefficient is deployed to control the regularization penalty adaptively. With the new regularization term and its associated adaptive regularization coefficient, LEAP is able to be applied for different pruning granularity, including unstructured pruning, structured pruning, and hybrid pruning, with minimal hyperparameter tuning. We apply LEAP for BERT models on QQP/MNLI/SQuAD for different pruning settings. Our result shows that for all datasets, pruning granularity, and pruning ratios, LEAP achieves on-par or better results as compared to previous heavily hand-tuned methods.