# Adventures with Randomized Algebra for Extreme-Scale Signal Processing

Anshumali Shrivastava (Rice University)

Sampling and Projection (or sketching) are the two fundamental hammers in Randomized Numerical Linear Algebra for reducing the size and scale of the problem at hand. At extreme scale, conventional sampling and projections themselves are prohibitively expensive.

In this talk, I will discuss some of my recent and surprising findings on the use of hashing algorithms for large-scale estimations. Locality Sensitive Hashing (LSH) is a hugely popular algorithm for sub-linear near neighbor search. However, it turns out that fundamentally LSH is a constant time (amortized) adaptive sampler from which efficient near-neighbor search is one of the many possibilities. LSH offers a unique capability to do smart sampling and statistical estimations at the cost of few hash lookups. Our observation bridges data structures (probabilistic hash tables) with efficient unbiased statistical estimations. I will demonstrate how this dynamic and efficient sampling beak the computational barriers in adaptive estimations where it is possible that we pay roughly the cost of uniform sampling but get the benefits of adaptive sampling. I will demonstrate the power of one simple idea for three favorite problems 1) Partition function estimation for large NLP models such as word2vec, 2) Adaptive Gradient Estimations for efficient SGD and 3) Sub-Linear Deep Learning with Huge Parameter Space.

In the end, if time permits, I will switch to memory cost show a simple hashing algorithm, which is a count-min sketch in disguise, can shrink memory requirements associated with k-class classification exponentially! Using our algorithms, we can train 100,000 classes with 400,000 features, on a single Titan X while only needing 5% or less memory required to store all the weights. Running a simple logistic regression on this data, the model size of 320GB is unavoidable.