Despite this, several operations are still performed in a sequential manner.

Neural Information Processing Systems http://nips.cc/

Parallelization techniques have become ubiquitous for accelerating inference and training of deep neural networks. Despite this, several operations are still performed in a sequential manner. For instance, the forward and backward passes are executed layer-by-layer, and the output of diffusion models is produced by applying a sequence of denoising steps. This sequential approach results in a computational cost proportional to the number of steps involved, presenting a potential bottleneck as the number of steps increases. In this work, we introduce DeepPCR, a novel algorithm which parallelizes typically sequential operations in order to speed up inference and training of neural networks. DeepPCR is based on interpreting a sequence of $L$ steps as the solution of a specific system of equations, which we recover using the Parallel Cyclic Reduction algorithm. This reduces the complexity of computing the sequential operations from $\mathcal{O}(L)$ to $\mathcal{O}(\log_2L)$, thus yielding a speedup for large $L$. To verify the theoretical lower complexity of the algorithm, and to identify regimes for speedup, we test the effectiveness of DeepPCR in parallelizing the forward and backward pass in multi-layer perceptrons, and reach speedups of up to $30\times$ for the forward and $200\times$ for the backward pass. We additionally showcase the flexibility of DeepPCR by parallelizing training of ResNets with as many as 1024 layers, and generation in diffusion models, enabling up to $7\times$ faster training and $11\times$ faster generation, respectively, when compared to the sequential approach.

Deep learning succeeds by doing hierarchical feature learning, yet tuning hyper-parameters (HP) such as initialization scales, learning rates etc., only give indirect control over this behavior. In this paper, we introduce a key notion to predict and control feature learning: the angle $\theta_\ell$ between the feature updates and the backward pass (at layer index $\ell$). We show that the magnitude of feature updates after one GD step, at any training time, can be expressed via a simple and general *feature speed formula* in terms of this angle $\theta_\ell$, the loss decay, and the magnitude of the backward pass. This angle $\theta_\ell$ is controlled by the conditioning of the layer-to-layer Jacobians and at random initialization, it is determined by the spectrum of a certain kernel, which coincides with the Neural Tangent Kernel when $\ell=\text{depth}$. Given $\theta_\ell$, the feature speed formula provides us with rules to adjust HPs (scales and learning rates) so as to satisfy certain dynamical properties, such as feature learning and loss decay. We investigate the implications of our approach for ReLU MLPs and ResNets in the large width-then-depth limit. Relying on prior work, we show that in ReLU MLPs with iid initialization, the angle degenerates with depth as $\cos(\theta_\ell)=\Theta(1/\sqrt{\ell})$. In contrast, ResNets with branch scale $O(1/\sqrt{\text{depth}})$ maintain a non-degenerate angle $\cos(\theta_\ell)=\Theta(1)$. We use these insights to recover key properties of known HP scalings (such as $\mu$P), and also introduce a new HP scaling for large depth ReLU MLPs with favorable theoretical properties.

Recent techniques that integrate solver layers into Deep Neural Networks (DNNs) have shown promise in bridging a long-standing gap between inductive learning and symbolic reasoning techniques. In this paper we present a set of techniques for integrating Satisfiability Modulo Theories (SMT) solvers into the forward and backward passes of a deep network layer, called SMTLayer.Using this approach, one can encode rich domain knowledge into the network in the form of mathematical formulas.In the forward pass, the solver uses symbols produced by prior layers, along with these formulas, to construct inferences; in the backward pass, the solver informs updates to the network, driving it towards representations that are compatible with the solver's theory.Notably, the solver need not be differentiable. We implement SMTLayer as a Pytorch module, and our empirical results show that it leads to models that 1) require fewer training samples than conventional models, 2) that are robust to certain types of covariate shift, and 3) that ultimately learn representations that are consistent with symbolic knowledge, and thus naturally interpretable.

Neural network training is computationally and memory intensive. Sparse training can reduce the burden on emerging hardware platforms designed to accelerate sparse computations, but it can also affect network convergence. In this work, we propose a novel CNN training algorithm called Sparse Weight Activation Training (SWAT). SWAT is more computation and memory-efficient than conventional training. SWAT modifies back-propagation based on the empirical insight that convergence during training tends to be robust to the elimination of (i) small magnitude weights during the forward pass and (ii) both small magnitude weights and activations during the backward pass. We evaluate SWAT on recent CNN architectures such as ResNet, VGG, DenseNet and WideResNet using CIFAR-10, CIFAR-100 and ImageNet datasets. For ResNet-50 on ImageNet SWAT reduces total floating-point operations (FLOPs) during training by 80% resulting in a 3.3x training speedup when run on a simulated sparse learning accelerator representative of emerging platforms while incurring only 1.63% reduction in validation accuracy. Moreover, SWAT reduces memory footprint during the backward pass by 23% to 50% for activations and 50% to 90% for weights.

Recent research in deep learning has investigated two very different forms of ''implicitness'': implicit representations model high-frequency data such as images or 3D shapes directly via a low-dimensional neural network (often using e.g., sinusoidal bases or nonlinearities); implicit layers, in contrast, refer to techniques where the forward pass of a network is computed via non-linear dynamical systems, such as fixed-point or differential equation solutions, with the backward pass computed via the implicit function theorem. In this work, we demonstrate that these two seemingly orthogonal concepts are remarkably well-suited for each other. In particular, we show that by exploiting fixed-point implicit layer to model implicit representations, we can substantially improve upon the performance of the conventional explicit-layer-based approach. Additionally, as implicit representation networks are typically trained in large-batch settings, we propose to leverage the property of implicit layers to amortize the cost of fixed-point forward/backward passes over training steps -- thereby addressing one of the primary challenges with implicit layers (that many iterations are required for the black-box fixed-point solvers). We empirically evaluated our method on learning multiple implicit representations for images, videos and audios, showing that our $(\textrm{Implicit})^2$ approach substantially improve upon existing models while being both faster to train and much more memory efficient.

Convex relaxations have emerged as a promising approach for verifying properties of neural networks, but widely used using Linear Programming (LP) relaxations only provide meaningful certificates when networks are specifically trained to facilitate verification. This precludes many important applications which involve \emph{verification-agnostic} networks that are not trained specifically to promote verifiability. On the other hand, semidefinite programming (SDP) relaxations have shown success on verification-agnostic networks, such as adversarially trained image classifiers without additional regularization, but do not currently scale beyond small networks due to poor time and space asymptotics. In this work, we propose a first-order dual SDP algorithm that provides (1) any-time bounds (2) requires memory only linear in the total number of network activations and (3) has per-iteration complexity that scales linearly with the complexity of a forward/backward pass through the network. By exploiting iterative eigenvector methods, we express all solver operations in terms of forward and backward passes through the network, enabling efficient use of hardware optimized for deep learning. This allows us to dramatically improve the magnitude of $\ell_\infty$ perturbations for which we can verify robustness verification-agnostic networks ($1\% \to 88\%$ on MNIST, $6\%\to 40\%$ on CIFAR-10). We also demonstrate tight verification for a quadratic stability specification for the decoder of a variational autoencoder.

Training and inference of Large Language Models (LLMs) with tensor-parallelism requires substantial communication to synchronize activations. Our findings suggest that with a few minor adjustments to current practices, LLMs can be trained without fully synchronizing activations, reducing bandwidth demands. We name this "Communication-Aware Architecture for Tensor-parallelism" (CAAT-Net). We train a 7B parameter CAAT-Net model and show that tensor-parallel communication can be reduced by up to 50% with no significant drop in pretraining accuracy across nearly all evaluated benchmarks. We also experiment with smaller 130M and 1.1B models to show the robustness and scalability of our method. We find that, in some scenarios, validation loss can even improve when reducing communication. Finally, we demonstrate how CAAT-Net accelerates both training and inference workloads across various settings and model sizes.