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Title: LSH Softmax: Sub-Linear Learning and Inference of the Softmax Layer in Deep Architectures

Paper content:
Log-linear models models are widely used in machine learning, and in particular are ubiquitous in deep learning architectures in the form of the softmax.
While exact inference and learning of these requires linear time, it can be done approximately in sub-linear time with strong concentrations guarantees.
In this work, we present LSH Softmax, a method to perform sub-linear learning and inference of the softmax layer in the deep learning setting.
Our method relies on the popular Locality-Sensitive Hashing to build a well-concentrated gradient estimator, using nearest neighbors and uniform samples.
We also present an inference scheme in sub-linear time for LSH Softmax using the Gumbel distribution.
On language modeling, we show that Recurrent Neural Networks trained with LSH Softmax perform on-par with computing the exact softmax while requiring sub-linear computations.
Deep neural networks have achieved impressive successes in tasks spanning vision BID9 BID16 , language BID3 , speech BID6 BID27 and videos BID1 .
While these models can vastly differ in architecture, activation functions, and presence of recurrence, they (almost) all share a common trait: the softmax layer.
The softmax layer, or log-linear model, is a widely used model in machine learning and statistics that transforms a feature vector into a distribution over the output space, modeling log-probabilities as a linear function of the feature vector.
For example, in object classification, the softmax layer at the end of a deep convolutional network transforms a feature vector into a probability distribution over classes for the image; in language modeling using recurrent neural networks, it maps the hidden state to a distribution over next words.While parameterizing for logits offers modeling flexibility, inference and learning have linear runtime in the number of classes.
Indeed, both of these require computing the un-normalized probability for every class to compute the partition function and retrieve an actual probability distribution.
Problems with large output spaces arise naturally in many areas like natural language processing (NLP), where the output space is a language's vocabulary and can be on the order of hundreds of thousands of elements BID15 ; BID12 .
This can also occur in computer vision BID14 when attempting tag prediction on massive, weakly-labeled datasets such as Flickr100M BID31 .Many
solutions have been proposed to address this bottleneck, all revolving around two themes: approximation of the softmax probabilities or computation of exact probabilities for an approximate model. Canonical
examples of the former are importance sampling (IS) or noise contrastive estimation (NCE; BID8 ). Instead of
computing probabilities over the whole output space, these methods compute the softmax over a smaller, sampled vocabulary and re-weight the probabilities, providing an unbiased estimator. An illustration
of the latter is Hierarchical Softmax BID24 , where the output classes are first clustered such that you only need to compute the softmax over a smaller output space. While the former
is an unbiased estimate, it comes with no concentration guarantees, and it is often more art than science to craft proposal distributions which will provide low-variance estimators. The latter, while
efficient, requires carefully hand-crafted clustering of the output space, at the risk of making mistakes from which there is no recovery.More recently, estimators based on nearest neighbor search have been proposed for inference and learning in log-linear models BID25 BID26 . These estimators
hinge on Maximum Inner Product Search using Locality-Sensitive to retrieve the largest logits of the distribution and account for the tail with uniformly sampled classes. They boast strong
theoretical guarantees and well-established concentration bounds. However, they were
constrained to toy settings and not directly applicable to real-world, large-scale, machine learning. In this work, we build
upon these estimators to make them amenable to deep learning practitioners, without losing any theoretical guarantees. We first show how they
can be extended to be usable within training of deep learning models, then present our efficient implementation, adapted to deep learning hardware and frameworks. Finally, we show the applicability
and efficiency of our method by evaluating on a real-world task: language modeling. We show significant perplexity gains
against competing methods with significant speed-ups.Our contributions are as follows:• We present a new deep learning layer, LSH Softmax, an efficient replacement for the softmax layer based on Locality-Sensitive Hashing and the Gumbel distribution, for any deep learning architecture, with strong theoretical guarantees for sub-linear learning and inference.• We provide details for efficient implementation
on deep learning hardware (GPUs) and modern deep learning frameworks BID0 BID19 ).• Empirically, we show, on several datasets, that
training and sampling from LSH Softmax performs similarly to an exact softmax while requiring significantly less FLOPS.
In this work, we presented LSH Softmax, a softmax approximation layer for large output spaces with sub-linear learning and inference cost (in the number of states) and strong theoretical guarantees.
We showcased both its applicability and efficiency by evaluating LSH on a common NLP task, language modeling.
On several datasets for this task, we report perplexity closest to exact training among all baselines, as well as significant speed-ups.
Our hope is that, for any architecture, this layer could be chosen in lieu of softmax, when the output space is sufficiently large to warrant the approximation.To that end, we plan to release source-code with the camera-ready version.

assistant: text: we present LSH Softmax, a softmax approximation layer for sub-linear learning and inference with strong theoretical guarantees; we showcase both its applicability and efficiency by evaluating on a real-world task: language modeling.