Model compression as constrained optimization, with application to neural nets. Part I: general framework

Compressing neural nets is an active research problem, given the large size of state-of-the-art nets for tasks such as object recognition, and the computational limits imposed by mobile devices. We give a general formulation of model compression as constrained optimization. This includes many types of compression: quantization, low-rank decomposition, pruning, lossless compression and others. Then, we give a general algorithm to optimize this nonconvex problem based on the augmented Lagrangian and alternating optimization. This results in a "learning-compression" algorithm, which alternates a learning step of the uncompressed model, independent of the compression type, with a compression step of the model parameters, independent of the learning task. This simple, efficient algorithm is guaranteed to find the best compressed model for the task in a local sense under standard assumptions. We present separately in several companion papers the development of this general framework into specific algorithms for model compression based on quantization, pruning and other variations, including experimental results on compressing neural nets and other models.Comment: 23 pages, 2 figure

Model compression as constrained optimization, with application to neural nets. Part II: quantization

We consider the problem of deep neural net compression by quantization: given a large, reference net, we want to quantize its real-valued weights using a codebook with $K$ entries so that the training loss of the quantized net is minimal. The codebook can be optimally learned jointly with the net, or fixed, as for binarization or ternarization approaches. Previous work has quantized the weights of the reference net, or incorporated rounding operations in the backpropagation algorithm, but this has no guarantee of converging to a loss-optimal, quantized net. We describe a new approach based on the recently proposed framework of model compression as constrained optimization \citep{Carreir17a}. This results in a simple iterative "learning-compression" algorithm, which alternates a step that learns a net of continuous weights with a step that quantizes (or binarizes/ternarizes) the weights, and is guaranteed to converge to local optimum of the loss for quantized nets. We develop algorithms for an adaptive codebook or a (partially) fixed codebook. The latter includes binarization, ternarization, powers-of-two and other important particular cases. We show experimentally that we can achieve much higher compression rates than previous quantization work (even using just 1 bit per weight) with negligible loss degradation.Comment: 33 pages, 15 figures, 3 table

Multi-task Neural Networks for QSAR Predictions

Although artificial neural networks have occasionally been used for Quantitative Structure-Activity/Property Relationship (QSAR/QSPR) studies in the past, the literature has of late been dominated by other machine learning techniques such as random forests. However, a variety of new neural net techniques along with successful applications in other domains have renewed interest in network approaches. In this work, inspired by the winning team's use of neural networks in a recent QSAR competition, we used an artificial neural network to learn a function that predicts activities of compounds for multiple assays at the same time. We conducted experiments leveraging recent methods for dealing with overfitting in neural networks as well as other tricks from the neural networks literature. We compared our methods to alternative methods reported to perform well on these tasks and found that our neural net methods provided superior performance

A Convex Duality Framework for GANs

Generative adversarial network (GAN) is a minimax game between a generator mimicking the true model and a discriminator distinguishing the samples produced by the generator from the real training samples. Given an unconstrained discriminator able to approximate any function, this game reduces to finding the generative model minimizing a divergence measure, e.g. the Jensen-Shannon (JS) divergence, to the data distribution. However, in practice the discriminator is constrained to be in a smaller class $\mathcal{F}$ such as neural nets. Then, a natural question is how the divergence minimization interpretation changes as we constrain $\mathcal{F}$. In this work, we address this question by developing a convex duality framework for analyzing GANs. For a convex set $\mathcal{F}$, this duality framework interprets the original GAN formulation as finding the generative model with minimum JS-divergence to the distributions penalized to match the moments of the data distribution, with the moments specified by the discriminators in $\mathcal{F}$. We show that this interpretation more generally holds for f-GAN and Wasserstein GAN. As a byproduct, we apply the duality framework to a hybrid of f-divergence and Wasserstein distance. Unlike the f-divergence, we prove that the proposed hybrid divergence changes continuously with the generative model, which suggests regularizing the discriminator's Lipschitz constant in f-GAN and vanilla GAN. We numerically evaluate the power of the suggested regularization schemes for improving GAN's training performance

Deep Structured Prediction with Nonlinear Output Transformations

Deep structured models are widely used for tasks like semantic segmentation, where explicit correlations between variables provide important prior information which generally helps to reduce the data needs of deep nets. However, current deep structured models are restricted by oftentimes very local neighborhood structure, which cannot be increased for computational complexity reasons, and by the fact that the output configuration, or a representation thereof, cannot be transformed further. Very recent approaches which address those issues include graphical model inference inside deep nets so as to permit subsequent non-linear output space transformations. However, optimization of those formulations is challenging and not well understood. Here, we develop a novel model which generalizes existing approaches, such as structured prediction energy networks, and discuss a formulation which maintains applicability of existing inference techniques.Comment: Appearing in NIPS 201

A flexible, extensible software framework for model compression based on the LC algorithm

We propose a software framework based on the ideas of the Learning-Compression (LC) algorithm, that allows a user to compress a neural network or other machine learning model using different compression schemes with minimal effort. Currently, the supported compressions include pruning, quantization, low-rank methods (including automatically learning the layer ranks), and combinations of those, and the user can choose different compression types for different parts of a neural network. The LC algorithm alternates two types of steps until convergence: a learning (L) step, which trains a model on a dataset (using an algorithm such as SGD); and a compression (C) step, which compresses the model parameters (using a compression scheme such as low-rank or quantization). This decoupling of the "machine learning" aspect from the "signal compression" aspect means that changing the model or the compression type amounts to calling the corresponding subroutine in the L or C step, respectively. The library fully supports this by design, which makes it flexible and extensible. This does not come at the expense of performance: the runtime needed to compress a model is comparable to that of training the model in the first place; and the compressed model is competitive in terms of prediction accuracy and compression ratio with other algorithms (which are often specialized for specific models or compression schemes). The library is written in Python and PyTorch and available in Github.Comment: 15 pages, 4 figures, 2 table

Trading-off Accuracy and Energy of Deep Inference on Embedded Systems: A Co-Design Approach

Deep neural networks have seen tremendous success for different modalities of data including images, videos, and speech. This success has led to their deployment in mobile and embedded systems for real-time applications. However, making repeated inferences using deep networks on embedded systems poses significant challenges due to constrained resources (e.g., energy and computing power). To address these challenges, we develop a principled co-design approach. Building on prior work, we develop a formalism referred to as Coarse-to-Fine Networks (C2F Nets) that allow us to employ classifiers of varying complexity to make predictions. We propose a principled optimization algorithm to automatically configure C2F Nets for a specified trade-off between accuracy and energy consumption for inference. The key idea is to select a classifier on-the-fly whose complexity is proportional to the hardness of the input example: simple classifiers for easy inputs and complex classifiers for hard inputs. We perform comprehensive experimental evaluation using four different C2F Net architectures on multiple real-world image classification tasks. Our results show that optimized C2F Net can reduce the Energy Delay Product (EDP) by 27 to 60 percent with no loss in accuracy when compared to the baseline solution, where all predictions are made using the most complex classifier in C2F Net.Comment: Published in IEEE Trans. on CAD of Integrated Circuits and System

An Essay on Optimization Mystery of Deep Learning

Despite the huge empirical success of deep learning, theoretical understanding of neural networks learning process is still lacking. This is the reason, why some of its features seem "mysterious". We emphasize two mysteries of deep learning: generalization mystery, and optimization mystery. In this essay we review and draw connections between several selected works concerning the latter

Imposing Hard Constraints on Deep Networks: Promises and Limitations

Imposing constraints on the output of a Deep Neural Net is one way to improve the quality of its predictions while loosening the requirements for labeled training data. Such constraints are usually imposed as soft constraints by adding new terms to the loss function that is minimized during training. An alternative is to impose them as hard constraints, which has a number of theoretical benefits but has not been explored so far due to the perceived intractability of the problem. In this paper, we show that imposing hard constraints can in fact be done in a computationally feasible way and delivers reasonable results. However, the theoretical benefits do not materialize and the resulting technique is no better than existing ones relying on soft constraints. We analyze the reasons for this and hope to spur other researchers into proposing better solutions

Applying Spiking Neural Nets to Noise Shaping

-In recent years, there has been an increased focus on the mechanics of information transmission in spiking neural networks. Especially the Noise Shaping properties of these networks and their similarity to Delta-Sigma Modulators has received a lot of attention. However, very little of the research done in this area has focused on the effect the weights in these networks have on the Noise Shaping properties and on post- processing of the network output signal. This paper concerns itself with the various modes of network operation and beneficial as well as detrimental effects which the systematic generation of network weights can effect. Also, a method for post-processing of the spiking output signal is introduced, bringing the output signal more in line with conventional Delta-Sigma Modulators. Relevancy of this research to industrial application of neural nets as building blocks of oversampled A/D converters is shown. Also, further points of contention are listed, which must be thoroughly researched to add to the above mentioned applicability of spiking neural nets