Among the several paradigms of artificial intelligence (AI) or machine learning (ML), a remarkably successful paradigm is deep learning. Deep learning's phenomenal success has been hoped to be interpreted via fundamental research on the theory of deep learning. Accordingly, applied research on deep learning has spurred the theory of deep learning-oriented depth and breadth of developments. Inspired by such developments, we pose these fundamental questions: can we accurately approximate an arbitrary matrix-vector product using deep rectified linear unit (ReLU) feedforward neural networks (FNNs)? If so, can we bound the resulting approximation error? In light of these questions, we derive error bounds in Lebesgue and Sobolev norms that comprise our developed deep approximation theory. Guided by this theory, we have successfully trained deep ReLU FNNs whose test results justify our developed theory. The developed theory is also applicable for guiding and easing the training of teacher deep ReLU FNNs in view of the emerging teacher-student AI or ML paradigms that are essential for solving several AI or ML problems in wireless communications and signal processing; network science and graph signal processing; and network neuroscience and brain physics.
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Processing-in-memory (PIM) architectures have demonstrated great potential in accelerating numerous deep learning tasks. Particularly, resistive random-access memory (RRAM) devices provide a promising hardware substrate to build PIM accelerators due to their abilities to realize efficient in-situ vector-matrix multiplications (VMMs). However, existing PIM accelerators suffer from frequent and energy-intensive analog-to-digital (A/D) conversions, severely limiting their performance. This paper presents a new PIM architecture to efficiently accelerate deep learning tasks by minimizing the required A/D conversions with analog accumulation and neural approximated peripheral circuits. We first characterize the different dataflows employed by existing PIM accelerators, based on which a new dataflow is proposed to remarkably reduce the required A/D conversions for VMMs by extending shift and add (S+A) operations into the analog domain before the final quantizations. We then leverage a neural approximation method to design both analog accumulation circuits (S+A) and quantization circuits (ADCs) with RRAM crossbar arrays in a highly-efficient manner. Finally, we apply them to build an RRAM-based PIM accelerator (i.e., \textbf{Neural-PIM}) upon the proposed analog dataflow and evaluate its system-level performance. Evaluations on different benchmarks demonstrate that Neural-PIM can improve energy efficiency by 5.36x (1.73x) and speed up throughput by 3.43x (1.59x) without losing accuracy, compared to the state-of-the-art RRAM-based PIM accelerators, i.e., ISAAC (CASCADE).
We study the conjectured relationship between the implicit regularization in neural networks, trained with gradient-based methods, and rank minimization of their weight matrices. Previously, it was proved that for linear networks (of depth 2 and vector-valued outputs), gradient flow (GF) w.r.t. the square loss acts as a rank minimization heuristic. However, understanding to what extent this generalizes to nonlinear networks is an open problem. In this paper, we focus on nonlinear ReLU networks, providing several new positive and negative results. On the negative side, we prove (and demonstrate empirically) that, unlike the linear case, GF on ReLU networks may no longer tend to minimize ranks, in a rather strong sense (even approximately, for "most" datasets of size 2). On the positive side, we reveal that ReLU networks of sufficient depth are provably biased towards low-rank solutions in several reasonable settings.
We present a novel neural-networks-based algorithm to compute optimal transport maps and plans for strong and weak transport costs. To justify the usage of neural networks, we prove that they are universal approximators of transport plans between probability distributions. We evaluate the performance of our optimal transport algorithm on toy examples and on the unpaired image-to-image style translation task.
We study the overparametrization bounds required for the global convergence of stochastic gradient descent algorithm for a class of one hidden layer feed-forward neural networks, considering most of the activation functions used in practice, including ReLU. We improve the existing state-of-the-art results in terms of the required hidden layer width. We introduce a new proof technique combining nonlinear analysis with properties of random initializations of the network. First, we establish the global convergence of continuous solutions of the differential inclusion being a nonsmooth analogue of the gradient flow for the MSE loss. Second, we provide a technical result (working also for general approximators) relating solutions of the aforementioned differential inclusion to the (discrete) stochastic gradient descent sequences, hence establishing linear convergence towards zero loss for the stochastic gradient descent iterations.
Multi-access edge computing (MEC) is a key enabler to reduce the latency of vehicular network. Due to the vehicles mobility, their requested services (e.g., infotainment services) should frequently be migrated across different MEC servers to guarantee their stringent quality of service requirements. In this paper, we study the problem of service migration in a MEC-enabled vehicular network in order to minimize the total service latency and migration cost. This problem is formulated as a nonlinear integer program and is linearized to help obtaining the optimal solution using off-the-shelf solvers. Then, to obtain an efficient solution, it is modeled as a multi-agent Markov decision process and solved by leveraging deep Q learning (DQL) algorithm. The proposed DQL scheme performs a proactive services migration while ensuring their continuity under high mobility constraints. Finally, simulations results show that the proposed DQL scheme achieves close-to-optimal performance.
In 1954, Alston S. Householder published Principles of Numerical Analysis, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields.
Binary neural networks (BNNs) represent original full-precision weights and activations into 1-bit with sign function. Since the gradient of the conventional sign function is almost zero everywhere which cannot be used for back-propagation, several attempts have been proposed to alleviate the optimization difficulty by using approximate gradient. However, those approximations corrupt the main direction of factual gradient. To this end, we propose to estimate the gradient of sign function in the Fourier frequency domain using the combination of sine functions for training BNNs, namely frequency domain approximation (FDA). The proposed approach does not affect the low-frequency information of the original sign function which occupies most of the overall energy, and high-frequency coefficients will be ignored to avoid the huge computational overhead. In addition, we embed a noise adaptation module into the training phase to compensate the approximation error. The experiments on several benchmark datasets and neural architectures illustrate that the binary network learned using our method achieves the state-of-the-art accuracy. Code will be available at \textit{//gitee.com/mindspore/models/tree/master/research/cv/FDA-BNN}.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
Proximal Policy Optimization (PPO) is a highly popular model-free reinforcement learning (RL) approach. However, in continuous state and actions spaces and a Gaussian policy -- common in computer animation and robotics -- PPO is prone to getting stuck in local optima. In this paper, we observe a tendency of PPO to prematurely shrink the exploration variance, which naturally leads to slow progress. Motivated by this, we borrow ideas from CMA-ES, a black-box optimization method designed for intelligent adaptive Gaussian exploration, to derive PPO-CMA, a novel proximal policy optimization approach that can expand the exploration variance on objective function slopes and shrink the variance when close to the optimum. This is implemented by using separate neural networks for policy mean and variance and training the mean and variance in separate passes. Our experiments demonstrate a clear improvement over vanilla PPO in many difficult OpenAI Gym MuJoCo tasks.
This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks.
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