The ever-increasing computational complexity of deep learning models makes their training and deployment difficult on various cloud and edge platforms. Replacing floating-point arithmetic with low-bit integer arithmetic is a promising approach to save energy, memory footprint, and latency of deep learning models. As such, quantization has attracted the attention of researchers in recent years. However, using integer numbers to form a fully functional integer training pipeline including forward pass, back-propagation, and stochastic gradient descent is not studied in detail. Our empirical and mathematical results reveal that integer arithmetic seems to be enough to train deep learning models. Unlike recent proposals, instead of quantization, we directly switch the number representation of computations. Our novel training method forms a fully integer training pipeline that does not change the trajectory of the loss and accuracy compared to floating-point, nor does it need any special hyper-parameter tuning, distribution adjustment, or gradient clipping. Our experimental results show that our proposed method is effective in a wide variety of tasks such as classification (including vision transformers), object detection, and semantic segmentation.

As illustrated by the success of integer linear programming, linear integer arithmetics is a powerful tool for modelling combinatorial problems. Furthermore, the probabilistic extension of linear programming has been used to formulate problems in neurosymbolic AI. However, two key problems persist that prevent the adoption of neurosymbolic techniques beyond toy problems. First, probabilistic inference is inherently hard, #P-hard to be precise. Second, the discrete nature of integers renders the construction of meaningful gradients challenging, which is problematic for learning.

The ever-increasing computational complexity of deep learning models makes their training and deployment difficult on various cloud and edge platforms. Replacing floating-point arithmetic with low-bit integer arithmetic is a promising approach to save energy, memory footprint, and latency of deep learning models. As such, quantization has attracted the attention of researchers in recent years. However, using integer numbers to form a fully functional integer training pipeline including forward pass, back-propagation, and stochastic gradient descent is not studied in detail. Our empirical and mathematical results reveal that integer arithmetic seems to be enough to train deep learning models. Unlike recent proposals, instead of quantization, we directly switch the number representation of computations.

As illustrated by the success of integer linear programming, linear integer arithmetic is a powerful tool for modelling combinatorial problems. Furthermore, the probabilistic extension of linear programming has been used to formulate problems in neurosymbolic AI. However, two key problems persist that prevent the adoption of neurosymbolic techniques beyond toy problems. First, probabilistic inference is inherently hard, #P-hard to be precise. Second, the discrete nature of integers renders the construction of meaningful gradients challenging, which is problematic for learning. In order to mitigate these issues, we formulate linear arithmetic over integer-valued random variables as tensor manipulations that can be implemented in a straightforward fashion using modern deep learning libraries. At the core of our formulation lies the observation that the addition of two integer-valued random variables can be performed by adapting the fast Fourier transform to probabilities in the log-domain. By relying on tensor operations we obtain a differentiable data structure, which unlocks, virtually for free, gradient-based learning. In our experimental validation we show that tensorising probabilistic linear integer arithmetic and leveraging the fast Fourier transform allows us to push the state of the art by several orders of magnitude in terms of inference and learning times.

The large number of parameters of some prominent language models, such as BERT, makes their fine-tuning on downstream tasks computationally intensive and energy hungry. Previously researchers were focused on lower bit-width integer data types for the forward propagation of language models to save memory and computation. As for the backward propagation, however, only 16-bit floating-point data type has been used for the fine-tuning of BERT. In this work, we use integer arithmetic for both forward and back propagation in the fine-tuning of BERT. We study the effects of varying the integer bit-width on the model's metric performance. Our integer fine-tuning uses integer arithmetic to perform forward propagation and gradient computation of linear, layer-norm, and embedding layers of BERT. We fine-tune BERT using our integer training method on SQuAD v1.1 and SQuAD v2., and GLUE benchmark. We demonstrate that metric performance of fine-tuning 16-bit integer BERT matches both 16-bit and 32-bit floating-point baselines. Furthermore, using the faster and more memory efficient 8-bit integer data type, integer fine-tuning of BERT loses an average of 3.1 points compared to the FP32 baseline.

The ever-increasing computational complexity of deep learning models makes their training and deployment difficult on various cloud and edge platforms. Replacing floating-point arithmetic with low-bit integer arithmetic is a promising approach to save energy, memory footprint, and latency of deep learning models. As such, quantization has attracted the attention of researchers in recent years. However, using integer numbers to form a fully functional integer training pipeline including forward pass, back-propagation, and stochastic gradient descent is not studied in detail. Our empirical and mathematical results reveal that integer arithmetic seems to be enough to train deep learning models. Unlike recent proposals, instead of quantization, we directly switch the number representation of computations. Our novel training method forms a fully integer training pipeline that does not change the trajectory of the loss and accuracy compared to floating-point, nor does it need any special hyper-parameter tuning, distribution adjustment, or gradient clipping. Our experimental results show that our proposed method is effective in a wide variety of tasks such as classification (including vision transformers), object detection, and semantic segmentation.

Neural networks are very resource intensive algorithms. They not only incur significant computational costs, they also consume a lot of memory in addition. Even though the commercially available computational resources increase day by day, optimizing the training and inference of deep neural networks is extremely important. If we run our models in the cloud, we want to minimize the infrastructure costs and the carbon footprint. When we are running our models on the edge, network optimization becomes even more significant. If we have to run our models on smartphones or embedded devices, hardware limitations are immediately apparent.

Deep neural networks (DNN) are powerful models for many pattern recognition tasks, yet their high computational complexity and memory requirement limit them to applications on high-performance computing platforms. In this paper, we propose a new method to evaluate DNNs trained with 32bit floating point (float32) accuracy using only low precision integer arithmetics in combination with binary shift and clipping operations. Because hardware implementation of these operations is much simpler than high precision floating point calculation, our method can be used for an efficient DNN inference on dedicated hardware. In experiments on MNIST, we demonstrate that DNNs trained with float32 can be evaluated using a combination of 2bit integer arithmetics and a few float32 calculations in each layer or only 3bit integer arithmetics in combination with binary shift and clipping without significant performance degradation.