Daily Papers

Mini-batch stochastic gradient descent (SGD) is state of the art in large scale distributed training. The scheme can reach a linear speedup with respect to the number of workers, but this is rarely seen in practice as the scheme often suffers from large network delays and bandwidth limits. To overcome this communication bottleneck recent works propose to reduce the communication frequency. An algorithm of this type is local SGD that runs SGD independently in parallel on different workers and averages the sequences only once in a while. This scheme shows promising results in practice, but eluded thorough theoretical analysis. We prove concise convergence rates for local SGD on convex problems and show that it converges at the same rate as mini-batch SGD in terms of number of evaluated gradients, that is, the scheme achieves linear speedup in the number of workers and mini-batch size. The number of communication rounds can be reduced up to a factor of T^{1/2}---where T denotes the number of total steps---compared to mini-batch SGD. This also holds for asynchronous implementations. Local SGD can also be used for large scale training of deep learning models. The results shown here aim serving as a guideline to further explore the theoretical and practical aspects of local SGD in these applications.

To improve the efficiency and sustainability of learning deep models, we propose CREST, the first scalable framework with rigorous theoretical guarantees to identify the most valuable examples for training non-convex models, particularly deep networks. To guarantee convergence to a stationary point of a non-convex function, CREST models the non-convex loss as a series of quadratic functions and extracts a coreset for each quadratic sub-region. In addition, to ensure faster convergence of stochastic gradient methods such as (mini-batch) SGD, CREST iteratively extracts multiple mini-batch coresets from larger random subsets of training data, to ensure nearly-unbiased gradients with small variances. Finally, to further improve scalability and efficiency, CREST identifies and excludes the examples that are learned from the coreset selection pipeline. Our extensive experiments on several deep networks trained on vision and NLP datasets, including CIFAR-10, CIFAR-100, TinyImageNet, and SNLI, confirm that CREST speeds up training deep networks on very large datasets, by 1.7x to 2.5x with minimum loss in the performance. By analyzing the learning difficulty of the subsets selected by CREST, we show that deep models benefit the most by learning from subsets of increasing difficulty levels.

Decentralized learning provides a scalable alternative to traditional parameter-server-based training, yet its performance is often hindered by limited peer-to-peer communication. In this paper, we study how communication should be scheduled over time, including determining when and how frequently devices synchronize. Our empirical results show that concentrating communication budgets in the later stages of decentralized training markedly improves global generalization. Surprisingly, we uncover that fully connected communication at the final step, implemented by a single global merging, is sufficient to match the performance of server-based training. We further show that low communication in decentralized learning preserves the mergeability of local models throughout training. Our theoretical contributions, which explains these phenomena, are first to establish that the globally merged model of decentralized SGD can converge faster than centralized mini-batch SGD. Technically, we novelly reinterpret part of the discrepancy among local models, which were previously considered as detrimental noise, as constructive components that accelerate convergence. This work challenges the common belief that decentralized learning generalizes poorly under data heterogeneity and limited communication, while offering new insights into model merging and neural network loss landscapes.

A common way to speed up training of large convolutional networks is to add computational units. Training is then performed using data-parallel synchronous Stochastic Gradient Descent (SGD) with mini-batch divided between computational units. With an increase in the number of nodes, the batch size grows. But training with large batch size often results in the lower model accuracy. We argue that the current recipe for large batch training (linear learning rate scaling with warm-up) is not general enough and training may diverge. To overcome this optimization difficulties we propose a new training algorithm based on Layer-wise Adaptive Rate Scaling (LARS). Using LARS, we scaled Alexnet up to a batch size of 8K, and Resnet-50 to a batch size of 32K without loss in accuracy.

Language modeling on large-scale datasets leads to impressive performance gains on various downstream language tasks. The validation pre-training loss (or perplexity in autoregressive language modeling) is often used as the evaluation metric when developing language models since the pre-training loss tends to be well-correlated with downstream performance (which is itself difficult to evaluate comprehensively). Contrary to this conventional wisdom, this paper shows that 1) pre-training loss cannot fully explain downstream performance and 2) flatness of the model is well-correlated with downstream performance where pre-training loss is not. On simplified datasets, we identify three ways to produce models with the same (statistically optimal) pre-training loss but different downstream performance: continue pre-training after convergence, increasing the model size, and changing the training algorithm. These experiments demonstrate the existence of implicit bias of pre-training algorithms/optimizers -- among models with the same minimal pre-training loss, they implicitly prefer more transferable ones. Toward understanding this implicit bias, we prove that SGD with standard mini-batch noise implicitly prefers flatter minima in language models, and empirically observe a strong correlation between flatness and downstream performance among models with the same minimal pre-training loss. We also prove in a synthetic language setting that among the models with the minimal pre-training loss, the flattest model transfers to downstream tasks.

We study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian. We also derive similar results for the Generalised Gauss-Newton matrix approximation of the Hessian. As a consequence of our theorems we derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. %For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. Based on our investigations of the sub-sampled Hessian we develop a stochastic Lanczos quadrature based on the fly learning rate and momentum learner, which avoids the need for expensive multiple evaluations for these key hyper-parameters and shows good preliminary results on the Pre-Residual Architecure for CIFAR-100.

Deep learning thrives with large neural networks and large datasets. However, larger networks and larger datasets result in longer training times that impede research and development progress. Distributed synchronous SGD offers a potential solution to this problem by dividing SGD minibatches over a pool of parallel workers. Yet to make this scheme efficient, the per-worker workload must be large, which implies nontrivial growth in the SGD minibatch size. In this paper, we empirically show that on the ImageNet dataset large minibatches cause optimization difficulties, but when these are addressed the trained networks exhibit good generalization. Specifically, we show no loss of accuracy when training with large minibatch sizes up to 8192 images. To achieve this result, we adopt a hyper-parameter-free linear scaling rule for adjusting learning rates as a function of minibatch size and develop a new warmup scheme that overcomes optimization challenges early in training. With these simple techniques, our Caffe2-based system trains ResNet-50 with a minibatch size of 8192 on 256 GPUs in one hour, while matching small minibatch accuracy. Using commodity hardware, our implementation achieves ~90% scaling efficiency when moving from 8 to 256 GPUs. Our findings enable training visual recognition models on internet-scale data with high efficiency.

SGD does not produce robust results on datasets with label noise. Because the gradients calculated according to the losses of the noisy samples cause the optimization process to go in the wrong direction. In this paper, as an alternative to SGD, we recommend using samples with loss less than a threshold value determined during the optimization process, instead of using all samples in the mini-batch. Our proposed method, Adaptive-k, aims to exclude label noise samples from the optimization process and make the process robust. On noisy datasets, we found that using a threshold-based approach, such as Adaptive-k, produces better results than using all samples or a fixed number of low-loss samples in the mini-batch. Based on our theoretical analysis and experimental results, we show that the Adaptive-k method is closest to the performance of the oracle, in which noisy samples are entirely removed from the dataset. Adaptive-k is a simple but effective method. It does not require prior knowledge of the noise ratio of the dataset, does not require additional model training, and does not increase training time significantly. The code for Adaptive-k is available at https://github.com/enesdedeoglu-TR/Adaptive-k

In this work, we consider the optimization process of minibatch stochastic gradient descent (SGD) on a 2-layer neural network with data separated by a quadratic ground truth function. We prove that with data drawn from the d-dimensional Boolean hypercube labeled by the quadratic ``XOR'' function y = -x_ix_j, it is possible to train to a population error o(1) with d :polylog(d) samples. Our result considers simultaneously training both layers of the two-layer-neural network with ReLU activations via standard minibatch SGD on the logistic loss. To our knowledge, this work is the first to give a sample complexity of O(d) for efficiently learning the XOR function on isotropic data on a standard neural network with standard training. Our main technique is showing that the network evolves in two phases: a signal-finding phase where the network is small and many of the neurons evolve independently to find features, and a signal-heavy phase, where SGD maintains and balances the features. We leverage the simultaneous training of the layers to show that it is sufficient for only a small fraction of the neurons to learn features, since those neurons will be amplified by the simultaneous growth of their second layer weights.

The generalization performance of a machine learning algorithm such as a neural network depends in a non-trivial way on the structure of the data distribution. To analyze the influence of data structure on test loss dynamics, we study an exactly solveable model of stochastic gradient descent (SGD) on mean square loss which predicts test loss when training on features with arbitrary covariance structure. We solve the theory exactly for both Gaussian features and arbitrary features and we show that the simpler Gaussian model accurately predicts test loss of nonlinear random-feature models and deep neural networks trained with SGD on real datasets such as MNIST and CIFAR-10. We show that the optimal batch size at a fixed compute budget is typically small and depends on the feature correlation structure, demonstrating the computational benefits of SGD with small batch sizes. Lastly, we extend our theory to the more usual setting of stochastic gradient descent on a fixed subsampled training set, showing that both training and test error can be accurately predicted in our framework on real data.

In deep learning, mini-batch training is commonly used to optimize network parameters. However, the traditional mini-batch method may not learn the under-represented samples and complex patterns in the data, leading to a longer time for generalization. To address this problem, a variant of the traditional algorithm has been proposed, which trains the network focusing on mini-batches with high loss. The study evaluates the effectiveness of the proposed training using various deep neural networks trained on three benchmark datasets (CIFAR-10, CIFAR-100, and STL-10). The deep neural networks used in the study are ResNet-18, ResNet-50, Efficient Net B4, EfficientNetV2-S, and MobilenetV3-S. The experimental results showed that the proposed method can significantly improve the test accuracy and speed up the convergence compared to the traditional mini-batch training method. Furthermore, we introduce a hyper-parameter delta ({\delta}) that decides how many mini-batches are considered for training. Experiments on various values of {\delta} found that the performance of the proposed method for smaller {\delta} values generally results in similar test accuracy and faster generalization. We show that the proposed method generalizes in 26.47% less number of epochs than the traditional mini-batch method in EfficientNet-B4 on STL-10. The proposed method also improves the test top-1 accuracy by 7.26% in ResNet-18 on CIFAR-100.

This paper investigates scaling laws for local SGD in LLM training, a distributed optimization algorithm that facilitates training on loosely connected devices. Through extensive experiments, we show that local SGD achieves competitive results compared to conventional methods, given equivalent model parameters, datasets, and computational resources. Furthermore, we explore the application of local SGD in various practical scenarios, including multi-cluster setups and edge computing environments. Our findings elucidate the necessary conditions for effective multi-cluster LLM training and examine the potential and limitations of leveraging edge computing resources in the LLM training process. This demonstrates its viability as an alternative to single large-cluster training.

Conventional wisdom dictates that small batch sizes make language model pretraining and fine-tuning unstable, motivating gradient accumulation, which trades off the number of optimizer steps for a proportional increase in batch size. While it is common to decrease the learning rate for smaller batch sizes, other hyperparameters are often held fixed. In this work, we revisit small batch sizes all the way down to batch size one, and we propose a rule for scaling Adam hyperparameters to small batch sizes. We find that small batch sizes (1) train stably, (2) are consistently more robust to hyperparameter choices, (3) achieve equal or better per-FLOP performance than larger batch sizes, and (4) notably enable stable language model training with vanilla SGD, even without momentum, despite storing no optimizer state. Building on these results, we provide practical recommendations for selecting a batch size and setting optimizer hyperparameters. We further recommend against gradient accumulation unless training on multiple devices with multiple model replicas, bottlenecked by inter-device bandwidth.

Training large neural networks is known to be time-consuming, with the learning duration taking days or even weeks. To address this problem, large-batch optimization was introduced. This approach demonstrated that scaling mini-batch sizes with appropriate learning rate adjustments can speed up the training process by orders of magnitude. While long training time was not typically a major issue for model-free deep offline RL algorithms, recently introduced Q-ensemble methods achieving state-of-the-art performance made this issue more relevant, notably extending the training duration. In this work, we demonstrate how this class of methods can benefit from large-batch optimization, which is commonly overlooked by the deep offline RL community. We show that scaling the mini-batch size and naively adjusting the learning rate allows for (1) a reduced size of the Q-ensemble, (2) stronger penalization of out-of-distribution actions, and (3) improved convergence time, effectively shortening training duration by 3-4x times on average.

When training neural networks, it has been widely observed that a large step size is essential in stochastic gradient descent (SGD) for obtaining superior models. However, the effect of large step sizes on the success of SGD is not well understood theoretically. Several previous works have attributed this success to the stochastic noise present in SGD. However, we show through a novel set of experiments that the stochastic noise is not sufficient to explain good non-convex training, and that instead the effect of a large learning rate itself is essential for obtaining best performance.We demonstrate the same effects also in the noise-less case, i.e. for full-batch GD. We formally prove that GD with large step size -- on certain non-convex function classes -- follows a different trajectory than GD with a small step size, which can lead to convergence to a global minimum instead of a local one. Our settings provide a framework for future analysis which allows comparing algorithms based on behaviors that can not be observed in the traditional settings.

Deep neural networks are vulnerable to universal adversarial perturbation (UAP), an instance-agnostic perturbation capable of fooling the target model for most samples. Compared to instance-specific adversarial examples, UAP is more challenging as it needs to generalize across various samples and models. In this paper, we examine the serious dilemma of UAP generation methods from a generalization perspective -- the gradient vanishing problem using small-batch stochastic gradient optimization and the local optima problem using large-batch optimization. To address these problems, we propose a simple and effective method called Stochastic Gradient Aggregation (SGA), which alleviates the gradient vanishing and escapes from poor local optima at the same time. Specifically, SGA employs the small-batch training to perform multiple iterations of inner pre-search. Then, all the inner gradients are aggregated as a one-step gradient estimation to enhance the gradient stability and reduce quantization errors. Extensive experiments on the standard ImageNet dataset demonstrate that our method significantly enhances the generalization ability of UAP and outperforms other state-of-the-art methods. The code is available at https://github.com/liuxuannan/Stochastic-Gradient-Aggregation.

We propose a new variant of AMSGrad, a popular adaptive gradient based optimization algorithm widely used for training deep neural networks. Our algorithm adds prior knowledge about the sequence of consecutive mini-batch gradients and leverages its underlying structure making the gradients sequentially predictable. By exploiting the predictability and ideas from optimistic online learning, the proposed algorithm can accelerate the convergence and increase sample efficiency. After establishing a tighter upper bound under some convexity conditions on the regret, we offer a complimentary view of our algorithm which generalizes the offline and stochastic version of nonconvex optimization. In the nonconvex case, we establish a non-asymptotic convergence bound independently of the initialization. We illustrate the practical speedup on several deep learning models via numerical experiments.

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

Local stochastic gradient descent (Local-SGD), also referred to as federated averaging, is an approach to distributed optimization where each device performs more than one SGD update per communication. This work presents an empirical study of {\it asynchronous} Local-SGD for training language models; that is, each worker updates the global parameters as soon as it has finished its SGD steps. We conduct a comprehensive investigation by examining how worker hardware heterogeneity, model size, number of workers, and optimizer could impact the learning performance. We find that with naive implementations, asynchronous Local-SGD takes more iterations to converge than its synchronous counterpart despite updating the (global) model parameters more frequently. We identify momentum acceleration on the global parameters when worker gradients are stale as a key challenge. We propose a novel method that utilizes a delayed Nesterov momentum update and adjusts the workers' local training steps based on their computation speed. This approach, evaluated with models up to 150M parameters on the C4 dataset, matches the performance of synchronous Local-SGD in terms of perplexity per update step, and significantly surpasses it in terms of wall clock time.

We study the SAM (Sharpness-Aware Minimization) optimizer which has recently attracted a lot of interest due to its increased performance over more classical variants of stochastic gradient descent. Our main contribution is the derivation of continuous-time models (in the form of SDEs) for SAM and two of its variants, both for the full-batch and mini-batch settings. We demonstrate that these SDEs are rigorous approximations of the real discrete-time algorithms (in a weak sense, scaling linearly with the learning rate). Using these models, we then offer an explanation of why SAM prefers flat minima over sharp ones~--~by showing that it minimizes an implicitly regularized loss with a Hessian-dependent noise structure. Finally, we prove that SAM is attracted to saddle points under some realistic conditions. Our theoretical results are supported by detailed experiments.

Stochastic Gradient Descent (SGD) is one of the many iterative optimization methods that are widely used in solving machine learning problems. These methods display valuable properties and attract researchers and industrial machine learning engineers with their simplicity. However, one of the weaknesses of this type of methods is the necessity to tune learning rate (step-size) for every loss function and dataset combination to solve an optimization problem and get an efficient performance in a given time budget. Stochastic Gradient Descent with Polyak Step-size (SPS) is a method that offers an update rule that alleviates the need of fine-tuning the learning rate of an optimizer. In this paper, we propose an extension of SPS that employs preconditioning techniques, such as Hutchinson's method, Adam, and AdaGrad, to improve its performance on badly scaled and/or ill-conditioned datasets.

It is widely believed that the implicit regularization of SGD is fundamental to the impressive generalization behavior we observe in neural networks. In this work, we demonstrate that non-stochastic full-batch training can achieve comparably strong performance to SGD on CIFAR-10 using modern architectures. To this end, we show that the implicit regularization of SGD can be completely replaced with explicit regularization even when comparing against a strong and well-researched baseline. Our observations indicate that the perceived difficulty of full-batch training may be the result of its optimization properties and the disproportionate time and effort spent by the ML community tuning optimizers and hyperparameters for small-batch training.

Decentralized stochastic gradient descent (D-SGD) allows collaborative learning on massive devices simultaneously without the control of a central server. However, existing theories claim that decentralization invariably undermines generalization. In this paper, we challenge the conventional belief and present a completely new perspective for understanding decentralized learning. We prove that D-SGD implicitly minimizes the loss function of an average-direction Sharpness-aware minimization (SAM) algorithm under general non-convex non-beta-smooth settings. This surprising asymptotic equivalence reveals an intrinsic regularization-optimization trade-off and three advantages of decentralization: (1) there exists a free uncertainty evaluation mechanism in D-SGD to improve posterior estimation; (2) D-SGD exhibits a gradient smoothing effect; and (3) the sharpness regularization effect of D-SGD does not decrease as total batch size increases, which justifies the potential generalization benefit of D-SGD over centralized SGD (C-SGD) in large-batch scenarios.

Approximating Stochastic Gradient Descent (SGD) as a Stochastic Differential Equation (SDE) has allowed researchers to enjoy the benefits of studying a continuous optimization trajectory while carefully preserving the stochasticity of SGD. Analogous study of adaptive gradient methods, such as RMSprop and Adam, has been challenging because there were no rigorously proven SDE approximations for these methods. This paper derives the SDE approximations for RMSprop and Adam, giving theoretical guarantees of their correctness as well as experimental validation of their applicability to common large-scaling vision and language settings. A key practical result is the derivation of a square root scaling rule to adjust the optimization hyperparameters of RMSprop and Adam when changing batch size, and its empirical validation in deep learning settings.

This work studies training instabilities of behavior cloning with deep neural networks. We observe that minibatch SGD updates to the policy network during training result in sharp oscillations in long-horizon rewards, despite negligibly affecting the behavior cloning loss. We empirically disentangle the statistical and computational causes of these oscillations, and find them to stem from the chaotic propagation of minibatch SGD noise through unstable closed-loop dynamics. While SGD noise is benign in the single-step action prediction objective, it results in catastrophic error accumulation over long horizons, an effect we term gradient variance amplification (GVA). We show that many standard mitigation techniques do not alleviate GVA, but find an exponential moving average (EMA) of iterates to be surprisingly effective at doing so. We illustrate the generality of this phenomenon by showing the existence of GVA and its amelioration by EMA in both continuous control and autoregressive language generation. Finally, we provide theoretical vignettes that highlight the benefits of EMA in alleviating GVA and shed light on the extent to which classical convex models can help in understanding the benefits of iterate averaging in deep learning.

Federated learning (FL) is a distributed machine learning framework where the global model of a central server is trained via multiple collaborative steps by participating clients without sharing their data. While being a flexible framework, where the distribution of local data, participation rate, and computing power of each client can greatly vary, such flexibility gives rise to many new challenges, especially in the hyperparameter tuning on the client side. We propose Delta-SGD, a simple step size rule for SGD that enables each client to use its own step size by adapting to the local smoothness of the function each client is optimizing. We provide theoretical and empirical results where the benefit of the client adaptivity is shown in various FL scenarios.

Training with larger mini-batches improves the convergence rate and can yield superior performance. However, training with large mini-batches becomes prohibitive for Large Language Models (LLMs), due to the large GPU memory requirement. To address this problem, an effective approach is finding small mini-batch coresets that closely match the gradient of larger mini-batches. However, this approach becomes infeasible and ineffective for LLMs, due to the highly imbalanced mixture of sources in language data, use of the Adam optimizer, and the very large gradient dimensionality of LLMs. In this work, we address the above challenges by proposing Coresets for Training LLMs (CoLM). First, we show that mini-batch coresets found by gradient matching do not contain representative examples of the small sources w.h.p., and thus including all examples of the small sources in the mini-batch coresets is crucial for optimal performance. Second, we normalize the gradients by their historical exponential to find mini-batch coresets for training with Adam. Finally, we leverage zeroth-order methods to find smooth gradient of the last V-projection matrix and sparsify it to keep the dimensions with the largest normalized gradient magnitude. We apply CoLM to fine-tuning Phi-2, Phi-3, Zephyr, and Llama-3 models with LoRA on MathInstruct and SuperGLUE benchmark. Remarkably, CoLM reduces the memory requirement of fine-tuning by 2x and even outperforms training with 4x larger mini-batches. Moreover, CoLM seamlessly integrates with existing memory-efficient training methods like LoRA, further reducing the memory requirements of training LLMs. Our code is available at https://github.com/BigML-CS-UCLA/CoLM.

This is a handbook of simple proofs of the convergence of gradient and stochastic gradient descent type methods. We consider functions that are Lipschitz, smooth, convex, strongly convex, and/or Polyak-{\L}ojasiewicz functions. Our focus is on ``good proofs'' that are also simple. Each section can be consulted separately. We start with proofs of gradient descent, then on stochastic variants, including minibatching and momentum. Then move on to nonsmooth problems with the subgradient method, the proximal gradient descent and their stochastic variants. Our focus is on global convergence rates and complexity rates. Some slightly less common proofs found here include that of SGD (Stochastic gradient descent) with a proximal step, with momentum, and with mini-batching without replacement.

We provide new lower bounds on the privacy guarantee of the multi-epoch Adaptive Batch Linear Queries (ABLQ) mechanism with shuffled batch sampling, demonstrating substantial gaps when compared to Poisson subsampling; prior analysis was limited to a single epoch. Since the privacy analysis of Differentially Private Stochastic Gradient Descent (DP-SGD) is obtained by analyzing the ABLQ mechanism, this brings into serious question the common practice of implementing shuffling-based DP-SGD, but reporting privacy parameters as if Poisson subsampling was used. To understand the impact of this gap on the utility of trained machine learning models, we introduce a practical approach to implement Poisson subsampling at scale using massively parallel computation, and efficiently train models with the same. We compare the utility of models trained with Poisson-subsampling-based DP-SGD, and the optimistic estimates of utility when using shuffling, via our new lower bounds on the privacy guarantee of ABLQ with shuffling.

Mini-batch optimal transport (m-OT) has been successfully used in practical applications that involve probability measures with a very high number of supports. The m-OT solves several smaller optimal transport problems and then returns the average of their costs and transportation plans. Despite its scalability advantage, the m-OT does not consider the relationship between mini-batches which leads to undesirable estimation. Moreover, the m-OT does not approximate a proper metric between probability measures since the identity property is not satisfied. To address these problems, we propose a novel mini-batch scheme for optimal transport, named Batch of Mini-batches Optimal Transport (BoMb-OT), that finds the optimal coupling between mini-batches and it can be seen as an approximation to a well-defined distance on the space of probability measures. Furthermore, we show that the m-OT is a limit of the entropic regularized version of the BoMb-OT when the regularized parameter goes to infinity. Finally, we carry out experiments on various applications including deep generative models, deep domain adaptation, approximate Bayesian computation, color transfer, and gradient flow to show that the BoMb-OT can be widely applied and performs well in various applications.

New hardware can substantially increase the speed and efficiency of deep neural network training. To guide the development of future hardware architectures, it is pertinent to explore the hardware and machine learning properties of alternative training algorithms. In this work we evaluate the use of small batch, fine-grained Pipelined Backpropagation, an asynchronous pipeline parallel training algorithm that has significant hardware advantages. We introduce two methods, Spike Compensation and Linear Weight Prediction, that effectively mitigate the downsides caused by the asynchronicity of Pipelined Backpropagation and outperform existing techniques in our setting. We show that appropriate normalization and small batch sizes can also aid training. With our methods, fine-grained Pipelined Backpropagation using a batch size of one can match the accuracy of SGD for multiple networks trained on CIFAR-10 and ImageNet. Simple scaling rules allow the use of existing hyperparameters for traditional training without additional tuning.

In training neural networks, it is common practice to use partial gradients computed over batches, mostly very small subsets of the training set. This approach is motivated by the argument that such a partial gradient is close to the true one, with precision growing only with the square root of the batch size. A theoretical justification is with the help of stochastic approximation theory. However, the conditions for the validity of this theory are not satisfied in the usual learning rate schedules. Batch processing is also difficult to combine with efficient second-order optimization methods. This proposal is based on another hypothesis: the loss minimum of the training set can be expected to be well-approximated by the minima of its subsets. Such subset minima can be computed in a fraction of the time necessary for optimizing over the whole training set. This hypothesis has been tested with the help of the MNIST, CIFAR-10, and CIFAR-100 image classification benchmarks, optionally extended by training data augmentation. The experiments have confirmed that results equivalent to conventional training can be reached. In summary, even small subsets are representative if the overdetermination ratio for the given model parameter set sufficiently exceeds unity. The computing expense can be reduced to a tenth or less.

We showcase important features of the dynamics of the Stochastic Gradient Descent (SGD) in the training of neural networks. We present empirical observations that commonly used large step sizes (i) lead the iterates to jump from one side of a valley to the other causing loss stabilization, and (ii) this stabilization induces a hidden stochastic dynamics orthogonal to the bouncing directions that biases it implicitly toward sparse predictors. Furthermore, we show empirically that the longer large step sizes keep SGD high in the loss landscape valleys, the better the implicit regularization can operate and find sparse representations. Notably, no explicit regularization is used so that the regularization effect comes solely from the SGD training dynamics influenced by the step size schedule. Therefore, these observations unveil how, through the step size schedules, both gradient and noise drive together the SGD dynamics through the loss landscape of neural networks. We justify these findings theoretically through the study of simple neural network models as well as qualitative arguments inspired from stochastic processes. Finally, this analysis allows us to shed a new light on some common practice and observed phenomena when training neural networks. The code of our experiments is available at https://github.com/tml-epfl/sgd-sparse-features.

We consider minimizing functions for which it is expensive to compute the (possibly stochastic) gradient. Such functions are prevalent in reinforcement learning, imitation learning and adversarial training. Our target optimization framework uses the (expensive) gradient computation to construct surrogate functions in a target space (e.g. the logits output by a linear model for classification) that can be minimized efficiently. This allows for multiple parameter updates to the model, amortizing the cost of gradient computation. In the full-batch setting, we prove that our surrogate is a global upper-bound on the loss, and can be (locally) minimized using a black-box optimization algorithm. We prove that the resulting majorization-minimization algorithm ensures convergence to a stationary point of the loss. Next, we instantiate our framework in the stochastic setting and propose the SSO algorithm, which can be viewed as projected stochastic gradient descent in the target space. This connection enables us to prove theoretical guarantees for SSO when minimizing convex functions. Our framework allows the use of standard stochastic optimization algorithms to construct surrogates which can be minimized by any deterministic optimization method. To evaluate our framework, we consider a suite of supervised learning and imitation learning problems. Our experiments indicate the benefits of target optimization and the effectiveness of SSO.

The burgeoning interest in developing Large Language Models (LLMs) with up to trillion parameters has been met with concerns regarding resource efficiency and practical expense, particularly given the immense cost of experimentation. This scenario underscores the importance of exploring the potential of Small Language Models (SLMs) as a resource-efficient alternative. In this context, we introduce MiniCPM, specifically the 1.2B and 2.4B non-embedding parameter variants, not only excel in their respective categories but also demonstrate capabilities on par with 7B-13B LLMs. While focusing on SLMs, our approach exhibits scalability in both model and data dimensions for future LLM research. Regarding model scaling, we employ extensive model wind tunnel experiments for stable and optimal scaling. For data scaling, we introduce a Warmup-Stable-Decay (WSD) learning rate scheduler (LRS), conducive to continuous training and domain adaptation. We present an in-depth analysis of the intriguing training dynamics that occurred in the WSD LRS. With WSD LRS, we are now able to efficiently study data-model scaling law without extensive retraining experiments on both axes of model and data, from which we derive the much higher compute optimal data-model ratio than Chinchilla Optimal. Additionally, we introduce MiniCPM family, including MiniCPM-DPO, MiniCPM-MoE and MiniCPM-128K, whose excellent performance further cementing MiniCPM's foundation in diverse SLM applications. MiniCPM models are available publicly at https://github.com/OpenBMB/MiniCPM .

The vast majority of successful deep neural networks are trained using variants of stochastic gradient descent (SGD) algorithms. Recent attempts to improve SGD can be broadly categorized into two approaches: (1) adaptive learning rate schemes, such as AdaGrad and Adam, and (2) accelerated schemes, such as heavy-ball and Nesterov momentum. In this paper, we propose a new optimization algorithm, Lookahead, that is orthogonal to these previous approaches and iteratively updates two sets of weights. Intuitively, the algorithm chooses a search direction by looking ahead at the sequence of fast weights generated by another optimizer. We show that Lookahead improves the learning stability and lowers the variance of its inner optimizer with negligible computation and memory cost. We empirically demonstrate Lookahead can significantly improve the performance of SGD and Adam, even with their default hyperparameter settings on ImageNet, CIFAR-10/100, neural machine translation, and Penn Treebank.

Mini-batch optimal transport (m-OT) has been widely used recently to deal with the memory issue of OT in large-scale applications. Despite their practicality, m-OT suffers from misspecified mappings, namely, mappings that are optimal on the mini-batch level but are partially wrong in the comparison with the optimal transportation plan between the original measures. Motivated by the misspecified mappings issue, we propose a novel mini-batch method by using partial optimal transport (POT) between mini-batch empirical measures, which we refer to as mini-batch partial optimal transport (m-POT). Leveraging the insight from the partial transportation, we explain the source of misspecified mappings from the m-OT and motivate why limiting the amount of transported masses among mini-batches via POT can alleviate the incorrect mappings. Finally, we carry out extensive experiments on various applications such as deep domain adaptation, partial domain adaptation, deep generative model, color transfer, and gradient flow to demonstrate the favorable performance of m-POT compared to current mini-batch methods.

We revisit the theoretical properties of Hamiltonian stochastic differential equations (SDES) for Bayesian posterior sampling, and we study the two types of errors that arise from numerical SDE simulation: the discretization error and the error due to noisy gradient estimates in the context of data subsampling. Our main result is a novel analysis for the effect of mini-batches through the lens of differential operator splitting, revising previous literature results. The stochastic component of a Hamiltonian SDE is decoupled from the gradient noise, for which we make no normality assumptions. This leads to the identification of a convergence bottleneck: when considering mini-batches, the best achievable error rate is O(eta^2), with eta being the integrator step size. Our theoretical results are supported by an empirical study on a variety of regression and classification tasks for Bayesian neural networks.

In shuffle privacy, each user sends a collection of randomized messages to a trusted shuffler, the shuffler randomly permutes these messages, and the resulting shuffled collection of messages must satisfy differential privacy. Prior work in this model has largely focused on protocols that use a single round of communication to compute algorithmic primitives like means, histograms, and counts. We present interactive shuffle protocols for stochastic convex optimization. Our protocols rely on a new noninteractive protocol for summing vectors of bounded ell_2 norm. By combining this sum subroutine with mini-batch stochastic gradient descent, accelerated gradient descent, and Nesterov's smoothing method, we obtain loss guarantees for a variety of convex loss functions that significantly improve on those of the local model and sometimes match those of the central model.

Diffusion models achieve high-quality sample generation at the cost of a lengthy multistep inference procedure. To overcome this, diffusion distillation techniques produce student generators capable of matching or surpassing the teacher in a single step. However, the student model's inference speed is limited by the size of the teacher architecture, preventing real-time generation for computationally heavy applications. In this work, we introduce Multi-Student Distillation (MSD), a framework to distill a conditional teacher diffusion model into multiple single-step generators. Each student generator is responsible for a subset of the conditioning data, thereby obtaining higher generation quality for the same capacity. MSD trains multiple distilled students, allowing smaller sizes and, therefore, faster inference. Also, MSD offers a lightweight quality boost over single-student distillation with the same architecture. We demonstrate MSD is effective by training multiple same-sized or smaller students on single-step distillation using distribution matching and adversarial distillation techniques. With smaller students, MSD gets competitive results with faster inference for single-step generation. Using 4 same-sized students, MSD significantly outperforms single-student baseline counterparts and achieves remarkable FID scores for one-step image generation: 1.20 on ImageNet-64x64 and 8.20 on zero-shot COCO2014.

We introduce the Ministral 3 series, a family of parameter-efficient dense language models designed for compute and memory constrained applications, available in three model sizes: 3B, 8B, and 14B parameters. For each model size, we release three variants: a pretrained base model for general-purpose use, an instruction finetuned, and a reasoning model for complex problem-solving. In addition, we present our recipe to derive the Ministral 3 models through Cascade Distillation, an iterative pruning and continued training with distillation technique. Each model comes with image understanding capabilities, all under the Apache 2.0 license.

Large language models (LLMs) have seen significant adoption for natural language tasks, owing their success to massive numbers of model parameters (e.g., 70B+); however, LLM inference incurs significant computation and memory costs. Recent approaches propose parallel decoding strategies, such as Skeleton-of-Thought (SoT), to improve performance by breaking prompts down into sub-problems that can be decoded in parallel; however, they often suffer from reduced response quality. Our key insight is that we can request additional information, specifically dependencies and difficulty, when generating the sub-problems to improve both response quality and performance. In this paper, we propose Skeleton Graph Decoding (SGD), which uses dependencies exposed between sub-problems to support information forwarding between dependent sub-problems for improved quality while exposing parallelization opportunities for decoding independent sub-problems. Additionally, we leverage difficulty estimates for each sub-problem to select an appropriately-sized model, improving performance without significantly reducing quality. Compared to standard autoregressive generation and SoT, SGD achieves a 1.69x speedup while improving quality by up to 51%.

Test-Time Training (TTT) models context dependencies by adapting part of the model's weights (referred to as fast weights) during inference. This fast weight, akin to recurrent states in RNNs, stores temporary memories of past tokens in the current sequence. Existing TTT methods struggled to show effectiveness in handling long-context data, due to their inefficiency on modern GPUs. The TTT layers in many of these approaches operate with extremely low FLOPs utilization (often <5%) because they deliberately apply small online minibatch sizes (e.g., updating fast weights every 16 or 64 tokens). Moreover, a small minibatch implies fine-grained block-wise causal dependencies in the data, unsuitable for data beyond 1D ordered sequences, like sets or N-dimensional grids such as images or videos. In contrast, we pursue the opposite direction by using an extremely large chunk update, ranging from 2K to 1M tokens across tasks of varying modalities, which we refer to as Large Chunk Test-Time Training (LaCT). It improves hardware utilization by orders of magnitude, and more importantly, facilitates scaling of nonlinear state size (up to 40% of model parameters), hence substantially improving state capacity, all without requiring cumbersome and error-prone kernel implementations. It also allows easy integration of sophisticated optimizers, e.g. Muon for online updates. We validate our approach across diverse modalities and tasks, including novel view synthesis with image set, language models, and auto-regressive video diffusion. Our approach can scale up to 14B-parameter AR video diffusion model on sequences up to 56K tokens. In our longest sequence experiment, we perform novel view synthesis with 1 million context length. We hope this work will inspire and accelerate new research in the field of long-context modeling and test-time training. Website: https://tianyuanzhang.com/projects/ttt-done-right

We present a novel approach to accelerate stochastic gradient descent (SGD) by utilizing curvature information obtained from Hessian-vector products or finite differences of parameters and gradients, similar to the BFGS algorithm. Our approach involves two preconditioners: a matrix-free preconditioner and a low-rank approximation preconditioner. We update both preconditioners online using a criterion that is robust to stochastic gradient noise and does not require line search or damping. To preserve the corresponding symmetry or invariance, our preconditioners are constrained to certain connected Lie groups. The Lie group's equivariance property simplifies the preconditioner fitting process, while its invariance property eliminates the need for damping, which is commonly required in second-order optimizers. As a result, the learning rate for parameter updating and the step size for preconditioner fitting are naturally normalized, and their default values work well in most scenarios. Our proposed approach offers a promising direction for improving the convergence of SGD with low computational overhead. We demonstrate that Preconditioned SGD (PSGD) outperforms SoTA on Vision, NLP, and RL tasks across multiple modern deep-learning architectures. We have provided code for reproducing toy and large scale experiments in this paper.

With the increase in the number of parameters in large language models, the process of pre-training and fine-tuning increasingly demands larger volumes of GPU memory. A significant portion of this memory is typically consumed by the optimizer state. To overcome this challenge, recent approaches such as low-rank adaptation (LoRA (Hu et al., 2021)), low-rank gradient projection (GaLore (Zhao et al., 2024)), and blockwise optimization (BAdam (Luo et al., 2024)) have been proposed. However, in all these algorithms, the effective rank of the weight updates remains low-rank, which can lead to a substantial loss of information from the gradient. This loss can be critically important, especially during the pre-training stage. In this paper, we introduce FRUGAL (Full-Rank Updates with GrAdient spLitting), a new memory-efficient optimization framework. FRUGAL leverages gradient splitting to perform low-dimensional updates using advanced algorithms (such as Adam), while updates along the remaining directions are executed via state-free methods like SGD or signSGD (Bernstein et al., 2018). Our framework can be integrated with various low-rank update selection techniques, including GaLore and BAdam. We provide theoretical convergence guarantees for our framework when using SGDM for low-dimensional updates and SGD for state-free updates. Additionally, our method consistently outperforms concurrent approaches across various fixed memory budgets, achieving state-of-the-art results in pre-training and fine-tuning tasks while balancing memory efficiency and performance metrics.

We say an algorithm is batch size-invariant if changes to the batch size can largely be compensated for by changes to other hyperparameters. Stochastic gradient descent is well-known to have this property at small batch sizes, via the learning rate. However, some policy optimization algorithms (such as PPO) do not have this property, because of how they control the size of policy updates. In this work we show how to make these algorithms batch size-invariant. Our key insight is to decouple the proximal policy (used for controlling policy updates) from the behavior policy (used for off-policy corrections). Our experiments help explain why these algorithms work, and additionally show how they can make more efficient use of stale data.

We propose a tuning-free dynamic SGD step size formula, which we call Distance over Gradients (DoG). The DoG step sizes depend on simple empirical quantities (distance from the initial point and norms of gradients) and have no ``learning rate'' parameter. Theoretically, we show that a slight variation of the DoG formula enjoys strong parameter-free convergence guarantees for stochastic convex optimization assuming only locally bounded stochastic gradients. Empirically, we consider a broad range of vision and language transfer learning tasks, and show that DoG's performance is close to that of SGD with tuned learning rate. We also propose a per-layer variant of DoG that generally outperforms tuned SGD, approaching the performance of tuned Adam. A PyTorch implementation is available at https://github.com/formll/dog

Over the past few years, softmax and SGD have become a commonly used component and the default training strategy in CNN frameworks, respectively. However, when optimizing CNNs with SGD, the saturation behavior behind softmax always gives us an illusion of training well and then is omitted. In this paper, we first emphasize that the early saturation behavior of softmax will impede the exploration of SGD, which sometimes is a reason for model converging at a bad local-minima, then propose Noisy Softmax to mitigating this early saturation issue by injecting annealed noise in softmax during each iteration. This operation based on noise injection aims at postponing the early saturation and further bringing continuous gradients propagation so as to significantly encourage SGD solver to be more exploratory and help to find a better local-minima. This paper empirically verifies the superiority of the early softmax desaturation, and our method indeed improves the generalization ability of CNN model by regularization. We experimentally find that this early desaturation helps optimization in many tasks, yielding state-of-the-art or competitive results on several popular benchmark datasets.

Recent hardware developments have dramatically increased the scale of data parallelism available for neural network training. Among the simplest ways to harness next-generation hardware is to increase the batch size in standard mini-batch neural network training algorithms. In this work, we aim to experimentally characterize the effects of increasing the batch size on training time, as measured by the number of steps necessary to reach a goal out-of-sample error. We study how this relationship varies with the training algorithm, model, and data set, and find extremely large variation between workloads. Along the way, we show that disagreements in the literature on how batch size affects model quality can largely be explained by differences in metaparameter tuning and compute budgets at different batch sizes. We find no evidence that larger batch sizes degrade out-of-sample performance. Finally, we discuss the implications of our results on efforts to train neural networks much faster in the future. Our experimental data is publicly available as a database of 71,638,836 loss measurements taken over the course of training for 168,160 individual models across 35 workloads.

The rapid development of machine learning and deep learning has introduced increasingly complex optimization challenges that must be addressed. Indeed, training modern, advanced models has become difficult to implement without leveraging multiple computing nodes in a distributed environment. Distributed optimization is also fundamental to emerging fields such as federated learning. Specifically, there is a need to organize the training process to minimize the time lost due to communication. A widely used and extensively researched technique to mitigate the communication bottleneck involves performing local training before communication. This approach is the focus of our paper. Concurrently, adaptive methods that incorporate scaling, notably led by Adam, have gained significant popularity in recent years. Therefore, this paper aims to merge the local training technique with the adaptive approach to develop efficient distributed learning methods. We consider the classical Local SGD method and enhance it with a scaling feature. A crucial aspect is that the scaling is described generically, allowing us to analyze various approaches, including Adam, RMSProp, and OASIS, in a unified manner. In addition to theoretical analysis, we validate the performance of our methods in practice by training a neural network.

Real-world image super-resolution (Real-ISR) focuses on recovering high-quality images from low-resolution inputs that suffer from complex degradations like noise, blur, and compression. Recently, diffusion models (DMs) have shown great potential in this area by leveraging strong generative priors to restore fine details. However, their iterative denoising process incurs high computational overhead, posing challenges for real-time applications. Although one-step distillation methods, such as OSEDiff and TSD-SR, offer faster inference, they remain fundamentally constrained by their large, over-parameterized model architectures. In this work, we present TinySR, a compact yet effective diffusion model specifically designed for Real-ISR that achieves real-time performance while maintaining perceptual quality. We introduce a Dynamic Inter-block Activation and an Expansion-Corrosion Strategy to facilitate more effective decision-making in depth pruning. We achieve VAE compression through channel pruning, attention removal and lightweight SepConv. We eliminate time- and prompt-related modules and perform pre-caching techniques to further speed up the model. TinySR significantly reduces computational cost and model size, achieving up to 5.68x speedup and 83% parameter reduction compared to its teacher TSD-SR, while still providing high quality results.

Low-Rank Adaptation (LoRA), which introduces a product of two trainable low-rank matrices into frozen pre-trained weights, is widely used for efficient fine-tuning of language models in federated learning (FL). However, when combined with differentially private stochastic gradient descent (DP-SGD), LoRA faces substantial noise amplification: DP-SGD perturbs per-sample gradients, and the matrix multiplication of the LoRA update (BA) intensifies this effect. Freezing one matrix (e.g., A) reduces the noise but restricts model expressiveness, often resulting in suboptimal adaptation. To address this, we propose FedSVD, a simple yet effective method that introduces a global reparameterization based on singular value decomposition (SVD). In our approach, each client optimizes only the B matrix and transmits it to the server. The server aggregates the B matrices, computes the product BA using the previous A, and refactorizes the result via SVD. This yields a new adaptive A composed of the orthonormal right singular vectors of BA, and an updated B containing the remaining SVD components. This reparameterization avoids quadratic noise amplification, while allowing A to better capture the principal directions of the aggregate updates. Moreover, the orthonormal structure of A bounds the gradient norms of B and preserves more signal under DP-SGD, as confirmed by our theoretical analysis. As a result, FedSVD consistently improves stability and performance across a variety of privacy settings and benchmarks, outperforming relevant baselines under both private and non-private regimes.

The choice of batch sizes in stochastic gradient optimizers is critical for model training. However, the practice of varying batch sizes throughout the training process is less explored compared to other hyperparameters. We investigate adaptive batch size strategies derived from adaptive sampling methods, traditionally applied only in stochastic gradient descent. Given the significant interplay between learning rates and batch sizes, and considering the prevalence of adaptive gradient methods in deep learning, we emphasize the need for adaptive batch size strategies in these contexts. We introduce AdAdaGrad and its scalar variant AdAdaGradNorm, which incrementally increase batch sizes during training, while model updates are performed using AdaGrad and AdaGradNorm. We prove that AdaGradNorm converges with high probability at a rate of O(1/K) for finding a first-order stationary point of smooth nonconvex functions within K iterations. AdaGrad also demonstrates similar convergence properties when integrated with a novel coordinate-wise variant of our adaptive batch size strategies. Our theoretical claims are supported by numerical experiments on various image classification tasks, highlighting the enhanced adaptability of progressive batching protocols in deep learning and the potential of such adaptive batch size strategies with adaptive gradient optimizers in large-scale model training.

Collaborative learning algorithms, such as distributed SGD (or D-SGD), are prone to faulty machines that may deviate from their prescribed algorithm because of software or hardware bugs, poisoned data or malicious behaviors. While many solutions have been proposed to enhance the robustness of D-SGD to such machines, previous works either resort to strong assumptions (trusted server, homogeneous data, specific noise model) or impose a gradient computational cost that is several orders of magnitude higher than that of D-SGD. We present MoNNA, a new algorithm that (a) is provably robust under standard assumptions and (b) has a gradient computation overhead that is linear in the fraction of faulty machines, which is conjectured to be tight. Essentially, MoNNA uses Polyak's momentum of local gradients for local updates and nearest-neighbor averaging (NNA) for global mixing, respectively. While MoNNA is rather simple to implement, its analysis has been more challenging and relies on two key elements that may be of independent interest. Specifically, we introduce the mixing criterion of (alpha, lambda)-reduction to analyze the non-linear mixing of non-faulty machines, and present a way to control the tension between the momentum and the model drifts. We validate our theory by experiments on image classification and make our code available at https://github.com/LPD-EPFL/robust-collaborative-learning.

Stochastic Gradient Decent (SGD) is one of the core techniques behind the success of deep neural networks. The gradient provides information on the direction in which a function has the steepest rate of change. The main problem with basic SGD is to change by equal sized steps for all parameters, irrespective of gradient behavior. Hence, an efficient way of deep network optimization is to make adaptive step sizes for each parameter. Recently, several attempts have been made to improve gradient descent methods such as AdaGrad, AdaDelta, RMSProp and Adam. These methods rely on the square roots of exponential moving averages of squared past gradients. Thus, these methods do not take advantage of local change in gradients. In this paper, a novel optimizer is proposed based on the difference between the present and the immediate past gradient (i.e., diffGrad). In the proposed diffGrad optimization technique, the step size is adjusted for each parameter in such a way that it should have a larger step size for faster gradient changing parameters and a lower step size for lower gradient changing parameters. The convergence analysis is done using the regret bound approach of online learning framework. Rigorous analysis is made in this paper over three synthetic complex non-convex functions. The image categorization experiments are also conducted over the CIFAR10 and CIFAR100 datasets to observe the performance of diffGrad with respect to the state-of-the-art optimizers such as SGDM, AdaGrad, AdaDelta, RMSProp, AMSGrad, and Adam. The residual unit (ResNet) based Convolutional Neural Networks (CNN) architecture is used in the experiments. The experiments show that diffGrad outperforms other optimizers. Also, we show that diffGrad performs uniformly well for training CNN using different activation functions. The source code is made publicly available at https://github.com/shivram1987/diffGrad.

Training competitive deep video models is an order of magnitude slower than training their counterpart image models. Slow training causes long research cycles, which hinders progress in video understanding research. Following standard practice for training image models, video model training assumes a fixed mini-batch shape: a specific number of clips, frames, and spatial size. However, what is the optimal shape? High resolution models perform well, but train slowly. Low resolution models train faster, but they are inaccurate. Inspired by multigrid methods in numerical optimization, we propose to use variable mini-batch shapes with different spatial-temporal resolutions that are varied according to a schedule. The different shapes arise from resampling the training data on multiple sampling grids. Training is accelerated by scaling up the mini-batch size and learning rate when shrinking the other dimensions. We empirically demonstrate a general and robust grid schedule that yields a significant out-of-the-box training speedup without a loss in accuracy for different models (I3D, non-local, SlowFast), datasets (Kinetics, Something-Something, Charades), and training settings (with and without pre-training, 128 GPUs or 1 GPU). As an illustrative example, the proposed multigrid method trains a ResNet-50 SlowFast network 4.5x faster (wall-clock time, same hardware) while also improving accuracy (+0.8% absolute) on Kinetics-400 compared to the baseline training method. Code is available online.

In this work, we question the necessity of adaptive gradient methods for training deep neural networks. SGD-SaI is a simple yet effective enhancement to stochastic gradient descent with momentum (SGDM). SGD-SaI performs learning rate Scaling at Initialization (SaI) to distinct parameter groups, guided by their respective gradient signal-to-noise ratios (g-SNR). By adjusting learning rates without relying on adaptive second-order momentum, SGD-SaI helps prevent training imbalances from the very first iteration and cuts the optimizer's memory usage by half compared to AdamW. Despite its simplicity and efficiency, SGD-SaI consistently matches or outperforms AdamW in training a variety of Transformer-based tasks, effectively overcoming a long-standing challenge of using SGD for training Transformers. SGD-SaI excels in ImageNet-1K classification with Vision Transformers(ViT) and GPT-2 pretraining for large language models (LLMs, transformer decoder-only), demonstrating robustness to hyperparameter variations and practicality for diverse applications. We further tested its robustness on tasks like LoRA fine-tuning for LLMs and diffusion models, where it consistently outperforms state-of-the-art optimizers. From a memory efficiency perspective, SGD-SaI achieves substantial memory savings for optimizer states, reducing memory usage by 5.93 GB for GPT-2 (1.5B parameters) and 25.15 GB for Llama2-7B compared to AdamW in full-precision training settings.

Adaptive optimizers such as Adam (Kingma & Ba, 2015) have been central to the success of large language models. However, they often require to maintain optimizer states throughout training, which can result in memory requirements several times greater than the model footprint. This overhead imposes constraints on scalability and computational efficiency. Stochastic Gradient Descent (SGD), in contrast, is a stateless optimizer, as it does not track state variables during training. Consequently, it achieves optimal memory efficiency. However, its capability in LLM training is limited (Zhao et al., 2024b). In this work, we show that pre-processing SGD in a stateless manner can achieve the same performance as the Adam optimizer for LLM training, while drastically reducing the memory cost. Specifically, we propose to pre-process the instantaneous stochastic gradients using normalization and whitening. We show that normalization stabilizes gradient distributions, and whitening counteracts the local curvature of the loss landscape. This results in SWAN (SGD with Whitening And Normalization), a stochastic optimizer that eliminates the need to store any optimizer states. Empirically, SWAN has the same memory footprint as SGD, achieving approx 50% reduction on total end-to-end memory compared to Adam. In language modeling tasks, SWAN demonstrates comparable or even better performance than Adam: when pre-training the LLaMA model with 350M and 1.3B parameters, SWAN achieves a 2x speedup by reaching the same evaluation perplexity using half as many tokens.

First-order optimization (FOO) algorithms are pivotal in numerous computational domains such as machine learning and signal denoising. However, their application to complex tasks like neural network training often entails significant inefficiencies due to the need for many sequential iterations for convergence. In response, we introduce first-order optimization expedited with approximately parallelized iterations (OptEx), the first framework that enhances the efficiency of FOO by leveraging parallel computing to mitigate its iterative bottleneck. OptEx employs kernelized gradient estimation to make use of gradient history for future gradient prediction, enabling parallelization of iterations -- a strategy once considered impractical because of the inherent iterative dependency in FOO. We provide theoretical guarantees for the reliability of our kernelized gradient estimation and the iteration complexity of SGD-based OptEx, confirming that estimation errors diminish to zero as historical gradients accumulate and that SGD-based OptEx enjoys an effective acceleration rate of Omega(N) over standard SGD given parallelism of N. We also use extensive empirical studies, including synthetic functions, reinforcement learning tasks, and neural network training across various datasets, to underscore the substantial efficiency improvements achieved by OptEx.

Mixture of Experts (MoE), an ensemble of specialized models equipped with a router that dynamically distributes each input to appropriate experts, has achieved successful results in the field of machine learning. However, theoretical understanding of this architecture is falling behind due to its inherent complexity. In this paper, we theoretically study the sample and runtime complexity of MoE following the stochastic gradient descent (SGD) when learning a regression task with an underlying cluster structure of single index models. On the one hand, we prove that a vanilla neural network fails in detecting such a latent organization as it can only process the problem as a whole. This is intrinsically related to the concept of information exponent which is low for each cluster, but increases when we consider the entire task. On the other hand, we show that a MoE succeeds in dividing this problem into easier subproblems by leveraging the ability of each expert to weakly recover the simpler function corresponding to an individual cluster. To the best of our knowledge, this work is among the first to explore the benefits of the MoE framework by examining its SGD dynamics in the context of nonlinear regression.

Instance selection (IS) is important in machine learning for reducing dataset size while keeping key characteristics. Current IS methods often struggle with capturing complex relationships in high-dimensional spaces and scale with large datasets. This paper introduces a graph attention-based instance selection (GAIS) method that uses attention mechanisms to identify informative instances through their structural relationships in graph representations. We present two approaches for scalable graph construction: a distance-based mini-batch sampling technique that reduces computation through strategic batch processing, and a hierarchical hashing approach that allows for efficient similarity computation through random projections. The mini-batch approach keeps class distributions through stratified sampling, while the hierarchical hashing method captures relationships at multiple granularities through single-level, multi-level, and multi-view variants. Experiments across 39 datasets show that GAIS achieves reduction rates above 96\% while maintaining or improving model performance relative to state-of-the-art IS methods. The findings shows that the distance-based mini-batch approach offers an optimal balance of efficiency and effectiveness for large-scale datasets, while multi-view variants provide superior performance for complex, high-dimensional data, demonstrating that attention-based importance scoring can effectively identify instances crucial for maintaining decision boundaries without requiring exhaustive pairwise comparisons.

This paper proposes a novel self-supervised learning method for learning better representations with small batch sizes. Many self-supervised learning methods based on certain forms of the siamese network have emerged and received significant attention. However, these methods need to use large batch sizes to learn good representations and require heavy computational resources. We present a new triplet network combined with a triple-view loss to improve the performance of self-supervised representation learning with small batch sizes. Experimental results show that our method can drastically outperform state-of-the-art self-supervised learning methods on several datasets in small-batch cases. Our method provides a feasible solution for self-supervised learning with real-world high-resolution images that uses small batch sizes.

Recent advances with large language models (LLM) illustrate their diverse capabilities. We propose a novel algorithm, staged speculative decoding, to accelerate LLM inference in small-batch, on-device scenarios. We address the low arithmetic intensity of small-batch inference by improving upon previous work in speculative decoding. First, we restructure the speculative batch as a tree, which reduces generation costs and increases the expected tokens per batch. Second, we add a second stage of speculative decoding. Taken together, we reduce single-batch decoding latency by 3.16x with a 762M parameter GPT-2-L model while perfectly preserving output quality.

We study gradient compression methods to alleviate the communication bottleneck in data-parallel distributed optimization. Despite the significant attention received, current compression schemes either do not scale well or fail to achieve the target test accuracy. We propose a new low-rank gradient compressor based on power iteration that can i) compress gradients rapidly, ii) efficiently aggregate the compressed gradients using all-reduce, and iii) achieve test performance on par with SGD. The proposed algorithm is the only method evaluated that achieves consistent wall-clock speedups when benchmarked against regular SGD with an optimized communication backend. We demonstrate reduced training times for convolutional networks as well as LSTMs on common datasets. Our code is available at https://github.com/epfml/powersgd.

Differentially Private Stochastic Gradient Descent (DP-SGD) limits the amount of private information deep learning models can memorize during training. This is achieved by clipping and adding noise to the model's gradients, and thus networks with more parameters require proportionally stronger perturbation. As a result, large models have difficulties learning useful information, rendering training with DP-SGD exceedingly difficult on more challenging training tasks. Recent research has focused on combating this challenge through training adaptations such as heavy data augmentation and large batch sizes. However, these techniques further increase the computational overhead of DP-SGD and reduce its practical applicability. In this work, we propose using the principle of sparse model design to solve precisely such complex tasks with fewer parameters, higher accuracy, and in less time, thus serving as a promising direction for DP-SGD. We achieve such sparsity by design by introducing equivariant convolutional networks for model training with Differential Privacy. Using equivariant networks, we show that small and efficient architecture design can outperform current state-of-the-art models with substantially lower computational requirements. On CIFAR-10, we achieve an increase of up to 9% in accuracy while reducing the computation time by more than 85%. Our results are a step towards efficient model architectures that make optimal use of their parameters and bridge the privacy-utility gap between private and non-private deep learning for computer vision.

We uncover how SGD interacts with batch normalization and can exhibit undesirable training dynamics such as divergence. More precisely, we study how Single Shuffle (SS) and Random Reshuffle (RR) -- two widely used variants of SGD -- interact surprisingly differently in the presence of batch normalization: RR leads to much more stable evolution of training loss than SS. As a concrete example, for regression using a linear network with batch normalization, we prove that SS and RR converge to distinct global optima that are "distorted" away from gradient descent. Thereafter, for classification we characterize conditions under which training divergence for SS and RR can, and cannot occur. We present explicit constructions to show how SS leads to distorted optima in regression and divergence for classification, whereas RR avoids both distortion and divergence. We validate our results by confirming them empirically in realistic settings, and conclude that the separation between SS and RR used with batch normalization is relevant in practice.

Stochastic gradient descent method and its variants constitute the core optimization algorithms that achieve good convergence rates for solving machine learning problems. These rates are obtained especially when these algorithms are fine-tuned for the application at hand. Although this tuning process can require large computational costs, recent work has shown that these costs can be reduced by line search methods that iteratively adjust the stepsize. We propose an alternative approach to stochastic line search by using a new algorithm based on forward step model building. This model building step incorporates second-order information that allows adjusting not only the stepsize but also the search direction. Noting that deep learning model parameters come in groups (layers of tensors), our method builds its model and calculates a new step for each parameter group. This novel diagonalization approach makes the selected step lengths adaptive. We provide convergence rate analysis, and experimentally show that the proposed algorithm achieves faster convergence and better generalization in well-known test problems. More precisely, SMB requires less tuning, and shows comparable performance to other adaptive methods.

Accelerated stochastic gradient descent (ASGD) is a workhorse in deep learning and often achieves better generalization performance than SGD. However, existing optimization theory can only explain the faster convergence of ASGD, but cannot explain its better generalization. In this paper, we study the generalization of ASGD for overparameterized linear regression, which is possibly the simplest setting of learning with overparameterization. We establish an instance-dependent excess risk bound for ASGD within each eigen-subspace of the data covariance matrix. Our analysis shows that (i) ASGD outperforms SGD in the subspace of small eigenvalues, exhibiting a faster rate of exponential decay for bias error, while in the subspace of large eigenvalues, its bias error decays slower than SGD; and (ii) the variance error of ASGD is always larger than that of SGD. Our result suggests that ASGD can outperform SGD when the difference between the initialization and the true weight vector is mostly confined to the subspace of small eigenvalues. Additionally, when our analysis is specialized to linear regression in the strongly convex setting, it yields a tighter bound for bias error than the best-known result.

The sheer size of modern neural networks makes model serving a serious computational challenge. A popular class of compression techniques overcomes this challenge by pruning or sparsifying the weights of pretrained networks. While useful, these techniques often face serious tradeoffs between computational requirements and compression quality. In this work, we propose a novel optimization-based pruning framework that considers the combined effect of pruning (and updating) multiple weights subject to a sparsity constraint. Our approach, CHITA, extends the classical Optimal Brain Surgeon framework and results in significant improvements in speed, memory, and performance over existing optimization-based approaches for network pruning. CHITA's main workhorse performs combinatorial optimization updates on a memory-friendly representation of local quadratic approximation(s) of the loss function. On a standard benchmark of pretrained models and datasets, CHITA leads to significantly better sparsity-accuracy tradeoffs than competing methods. For example, for MLPNet with only 2% of the weights retained, our approach improves the accuracy by 63% relative to the state of the art. Furthermore, when used in conjunction with fine-tuning SGD steps, our method achieves significant accuracy gains over the state-of-the-art approaches.

Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this approach. In particular, we show that the orthogonalized gradient method can be seen as a first-order trust-region optimization method, where the trust-region is defined in terms of the matrix spectral norm. Motivated by this observation, we develop the stochastic non-Euclidean trust-region gradient method with momentum, which recovers the Muon optimizer (Jordan et al., 2024) as a special case, along with normalized SGD and signSGD with momentum (Cutkosky and Mehta, 2020; Sun et al., 2023). In addition, we prove state-of-the-art convergence results for the proposed algorithm in a range of scenarios, which involve arbitrary non-Euclidean norms, constrained and composite problems, and non-convex, star-convex, first- and second-order smooth functions. Finally, our theoretical findings provide an explanation for several practical observations, including the practical superiority of Muon compared to the Orthogonal-SGDM algorithm of Tuddenham et al. (2022) and the importance of weight decay in the training of large-scale language models.

Stochastic Gradient Descent (SGD) stands as a cornerstone optimization algorithm with proven real-world empirical successes but relatively limited theoretical understanding. Recent research has illuminated a key factor contributing to its practical efficacy: the implicit regularization it instigates. Several studies have investigated the linear stability property of SGD in the vicinity of a stationary point as a predictive proxy for sharpness and generalization error in overparameterized neural networks (Wu et al., 2022; Jastrzebski et al., 2019; Cohen et al., 2021). In this paper, we delve deeper into the relationship between linear stability and sharpness. More specifically, we meticulously delineate the necessary and sufficient conditions for linear stability, contingent on hyperparameters of SGD and the sharpness at the optimum. Towards this end, we introduce a novel coherence measure of the loss Hessian that encapsulates pertinent geometric properties of the loss function that are relevant to the linear stability of SGD. It enables us to provide a simplified sufficient condition for identifying linear instability at an optimum. Notably, compared to previous works, our analysis relies on significantly milder assumptions and is applicable for a broader class of loss functions than known before, encompassing not only mean-squared error but also cross-entropy loss.

Communication-reduction techniques are a popular way to improve scalability in data-parallel training of deep neural networks (DNNs). The recent emergence of large language models such as GPT has created the need for new approaches to exploit data-parallelism. Among these, fully-sharded data parallel (FSDP) training is highly popular, yet it still encounters scalability bottlenecks. One reason is that applying compression techniques to FSDP is challenging: as the vast majority of the communication involves the model's weights, direct compression alters convergence and leads to accuracy loss. We present QSDP, a variant of FSDP which supports both gradient and weight quantization with theoretical guarantees, is simple to implement and has essentially no overheads. To derive QSDP we prove that a natural modification of SGD achieves convergence even when we only maintain quantized weights, and thus the domain over which we train consists of quantized points and is, therefore, highly non-convex. We validate this approach by training GPT-family models with up to 1.3 billion parameters on a multi-node cluster. Experiments show that QSDP preserves model accuracy, while completely removing the communication bottlenecks of FSDP, providing end-to-end speedups of up to 2.2x.

We study convergence lower bounds of without-replacement stochastic gradient descent (SGD) for solving smooth (strongly-)convex finite-sum minimization problems. Unlike most existing results focusing on final iterate lower bounds in terms of the number of components n and the number of epochs K, we seek bounds for arbitrary weighted average iterates that are tight in all factors including the condition number kappa. For SGD with Random Reshuffling, we present lower bounds that have tighter kappa dependencies than existing bounds. Our results are the first to perfectly close the gap between lower and upper bounds for weighted average iterates in both strongly-convex and convex cases. We also prove weighted average iterate lower bounds for arbitrary permutation-based SGD, which apply to all variants that carefully choose the best permutation. Our bounds improve the existing bounds in factors of n and kappa and thereby match the upper bounds shown for a recently proposed algorithm called GraB.

We examine a simple stochastic strategy for adapting well-known single-point acquisition functions to allow batch active learning. Unlike acquiring the top-K points from the pool set, score- or rank-based sampling takes into account that acquisition scores change as new data are acquired. This simple strategy for adapting standard single-sample acquisition strategies can even perform just as well as compute-intensive state-of-the-art batch acquisition functions, like BatchBALD or BADGE, while using orders of magnitude less compute. In addition to providing a practical option for machine learning practitioners, the surprising success of the proposed method in a wide range of experimental settings raises a difficult question for the field: when are these expensive batch acquisition methods pulling their weight?

Classical machine learning models such as deep neural networks are usually trained by using Stochastic Gradient Descent-based (SGD) algorithms. The classical SGD can be interpreted as a discretization of the stochastic gradient flow. In this paper we propose a novel, robust and accelerated stochastic optimizer that relies on two key elements: (1) an accelerated Nesterov-like Stochastic Differential Equation (SDE) and (2) its semi-implicit Gauss-Seidel type discretization. The convergence and stability of the obtained method, referred to as NAG-GS, are first studied extensively in the case of the minimization of a quadratic function. This analysis allows us to come up with an optimal learning rate in terms of the convergence rate while ensuring the stability of NAG-GS. This is achieved by the careful analysis of the spectral radius of the iteration matrix and the covariance matrix at stationarity with respect to all hyperparameters of our method. Further, we show that NAG- GS is competitive with state-of-the-art methods such as momentum SGD with weight decay and AdamW for the training of machine learning models such as the logistic regression model, the residual networks models on standard computer vision datasets, Transformers in the frame of the GLUE benchmark and the recent Vision Transformers.

This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a local-entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the local entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.

In centralized settings, it is well known that stochastic gradient descent (SGD) avoids saddle points and converges to local minima in nonconvex problems. However, similar guarantees are lacking for distributed first-order algorithms. The paper studies distributed stochastic gradient descent (D-SGD)--a simple network-based implementation of SGD. Conditions under which D-SGD avoids saddle points and converges to local minima are studied. First, we consider the problem of computing critical points. Assuming loss functions are nonconvex and possibly nonsmooth, it is shown that, for each fixed initialization, D-SGD converges to critical points of the loss with probability one. Next, we consider the problem of avoiding saddle points. In this case, we again assume that loss functions may be nonconvex and nonsmooth, but are smooth in a neighborhood of a saddle point. It is shown that, for any fixed initialization, D-SGD avoids such saddle points with probability one. Results are proved by studying the underlying (distributed) gradient flow, using the ordinary differential equation (ODE) method of stochastic approximation, and extending classical techniques from dynamical systems theory such as stable manifolds. Results are proved in the general context of subspace-constrained optimization, of which D-SGD is a special case.

Diffusion models excel in high-quality generation but suffer from slow inference due to iterative sampling. While recent methods have successfully transformed diffusion models into one-step generators, they neglect model size reduction, limiting their applicability in compute-constrained scenarios. This paper aims to develop small, efficient one-step diffusion models based on the powerful rectified flow framework, by exploring joint compression of inference steps and model size. The rectified flow framework trains one-step generative models using two operations, reflow and distillation. Compared with the original framework, squeezing the model size brings two new challenges: (1) the initialization mismatch between large teachers and small students during reflow; (2) the underperformance of naive distillation on small student models. To overcome these issues, we propose Annealing Reflow and Flow-Guided Distillation, which together comprise our SlimFlow framework. With our novel framework, we train a one-step diffusion model with an FID of 5.02 and 15.7M parameters, outperforming the previous state-of-the-art one-step diffusion model (FID=6.47, 19.4M parameters) on CIFAR10. On ImageNet 64times64 and FFHQ 64times64, our method yields small one-step diffusion models that are comparable to larger models, showcasing the effectiveness of our method in creating compact, efficient one-step diffusion models.

Mixture-of-Experts (MoE) has emerged as a promising paradigm for foundation models due to its efficient and powerful scalability. In this work, we present Sigma-MoE-Tiny, an MoE language model that achieves the highest sparsity compared to existing open-source models. Sigma-MoE-Tiny employs fine-grained expert segmentation with up to 96 experts per layer, while activating only one expert for each token, resulting in 20B total parameters with just 0.5B activated. The major challenge introduced by such extreme sparsity lies in expert load balancing. We find that the widely-used load balancing loss tends to become ineffective in the lower layers under this setting. To address this issue, we propose a progressive sparsification schedule aiming to balance expert utilization and training stability. Sigma-MoE-Tiny is pre-trained on a diverse and high-quality corpus, followed by post-training to further unlock its capabilities. The entire training process remains remarkably stable, with no occurrence of irrecoverable loss spikes. Comprehensive evaluations reveal that, despite activating only 0.5B parameters, Sigma-MoE-Tiny achieves top-tier performance among counterparts of comparable or significantly larger scale. In addition, we provide an in-depth discussion of load balancing in highly sparse MoE models, offering insights for advancing sparsity in future MoE architectures. Project page: https://qghuxmu.github.io/Sigma-MoE-Tiny Code: https://github.com/microsoft/ltp-megatron-lm

Large-batch training has become a commonly used technique when training neural networks with a large number of GPU/TPU processors. As batch size increases, stochastic optimizers tend to converge to sharp local minima, leading to degraded test performance. Current methods usually use extensive data augmentation to increase the batch size, but we found the performance gain with data augmentation decreases as batch size increases, and data augmentation will become insufficient after certain point. In this paper, we propose to use adversarial learning to increase the batch size in large-batch training. Despite being a natural choice for smoothing the decision surface and biasing towards a flat region, adversarial learning has not been successfully applied in large-batch training since it requires at least two sequential gradient computations at each step, which will at least double the running time compared with vanilla training even with a large number of processors. To overcome this issue, we propose a novel Concurrent Adversarial Learning (ConAdv) method that decouple the sequential gradient computations in adversarial learning by utilizing staled parameters. Experimental results demonstrate that ConAdv can successfully increase the batch size on ResNet-50 training on ImageNet while maintaining high accuracy. In particular, we show ConAdv along can achieve 75.3\% top-1 accuracy on ImageNet ResNet-50 training with 96K batch size, and the accuracy can be further improved to 76.2\% when combining ConAdv with data augmentation. This is the first work successfully scales ResNet-50 training batch size to 96K.

We prove new convergence rates for a generalized version of stochastic Nesterov acceleration under interpolation conditions. Unlike previous analyses, our approach accelerates any stochastic gradient method which makes sufficient progress in expectation. The proof, which proceeds using the estimating sequences framework, applies to both convex and strongly convex functions and is easily specialized to accelerated SGD under the strong growth condition. In this special case, our analysis reduces the dependence on the strong growth constant from rho to rho as compared to prior work. This improvement is comparable to a square-root of the condition number in the worst case and address criticism that guarantees for stochastic acceleration could be worse than those for SGD.

In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve mgg 1 lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling I blocks and sampling B samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of O(mepsilon^{-3I(I<m)}{II} + mepsilon^{-3}{IB}) for finding an epsilon-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.

A critical approach for efficiently deploying computationally demanding large language models (LLMs) is Key-Value (KV) caching. The KV cache stores key-value states of previously generated tokens, significantly reducing the need for repetitive computations and thereby lowering latency in autoregressive generation. However, the size of the KV cache grows linearly with sequence length, posing challenges for applications requiring long context input and extensive sequence generation. In this paper, we present a simple yet effective approach, called MiniCache, to compress the KV cache across layers from a novel depth perspective, significantly reducing the memory footprint for LLM inference. Our approach is based on the observation that KV cache states exhibit high similarity between the adjacent layers in the middle-to-deep portion of LLMs. To facilitate merging, we propose disentangling the states into the magnitude and direction components, interpolating the directions of the state vectors while preserving their lengths unchanged. Furthermore, we introduce a token retention strategy to keep highly distinct state pairs unmerged, thus preserving the information with minimal additional storage overhead. Our MiniCache is training-free and general, complementing existing KV cache compression strategies, such as quantization and sparsity. We conduct a comprehensive evaluation of MiniCache utilizing various models including LLaMA-2, LLaMA-3, Phi-3, Mistral, and Mixtral across multiple benchmarks, demonstrating its exceptional performance in achieving superior compression ratios and high throughput. On the ShareGPT dataset, LLaMA-2-7B with 4-bit MiniCache achieves a remarkable compression ratio of up to 5.02x, enhances inference throughput by approximately 5x, and reduces the memory footprint by 41% compared to the FP16 full cache baseline, all while maintaining near-lossless performance.

The non-convexity of the artificial neural network (ANN) training landscape brings inherent optimization difficulties. While the traditional back-propagation stochastic gradient descent (SGD) algorithm and its variants are effective in certain cases, they can become stuck at spurious local minima and are sensitive to initializations and hyperparameters. Recent work has shown that the training of an ANN with ReLU activations can be reformulated as a convex program, bringing hope to globally optimizing interpretable ANNs. However, naively solving the convex training formulation has an exponential complexity, and even an approximation heuristic requires cubic time. In this work, we characterize the quality of this approximation and develop two efficient algorithms that train ANNs with global convergence guarantees. The first algorithm is based on the alternating direction method of multiplier (ADMM). It solves both the exact convex formulation and the approximate counterpart. Linear global convergence is achieved, and the initial several iterations often yield a solution with high prediction accuracy. When solving the approximate formulation, the per-iteration time complexity is quadratic. The second algorithm, based on the "sampled convex programs" theory, is simpler to implement. It solves unconstrained convex formulations and converges to an approximately globally optimal classifier. The non-convexity of the ANN training landscape exacerbates when adversarial training is considered. We apply the robust convex optimization theory to convex training and develop convex formulations that train ANNs robust to adversarial inputs. Our analysis explicitly focuses on one-hidden-layer fully connected ANNs, but can extend to more sophisticated architectures.

In an increasing number of domains it has been demonstrated that deep learning models can be trained using relatively large batch sizes without sacrificing data efficiency. However the limits of this massive data parallelism seem to differ from domain to domain, ranging from batches of tens of thousands in ImageNet to batches of millions in RL agents that play the game Dota 2. To our knowledge there is limited conceptual understanding of why these limits to batch size differ or how we might choose the correct batch size in a new domain. In this paper, we demonstrate that a simple and easy-to-measure statistic called the gradient noise scale predicts the largest useful batch size across many domains and applications, including a number of supervised learning datasets (MNIST, SVHN, CIFAR-10, ImageNet, Billion Word), reinforcement learning domains (Atari and Dota), and even generative model training (autoencoders on SVHN). We find that the noise scale increases as the loss decreases over a training run and depends on the model size primarily through improved model performance. Our empirically-motivated theory also describes the tradeoff between compute-efficiency and time-efficiency, and provides a rough model of the benefits of adaptive batch-size training.

Production deployments in complex systems require ML architectures to be highly efficient and usable against multiple tasks. Particularly demanding are classification problems in which data arrives in a streaming fashion and each class is presented separately. Recent methods with stochastic gradient learning have been shown to struggle in such setups or have limitations like memory buffers, and being restricted to specific domains that disable its usage in real-world scenarios. For this reason, we present a fully differentiable architecture based on the Mixture of Experts model, that enables the training of high-performance classifiers when examples from each class are presented separately. We conducted exhaustive experiments that proved its applicability in various domains and ability to learn online in production environments. The proposed technique achieves SOTA results without a memory buffer and clearly outperforms the reference methods.

As models for nature language processing (NLP), computer vision (CV) and recommendation systems (RS) require surging computation, a large number of GPUs/TPUs are paralleled as a large batch (LB) to improve training throughput. However, training such LB tasks often meets large generalization gap and downgrades final precision, which limits enlarging the batch size. In this work, we develop the variance reduced gradient descent technique (VRGD) based on the gradient signal to noise ratio (GSNR) and apply it onto popular optimizers such as SGD/Adam/LARS/LAMB. We carry out a theoretical analysis of convergence rate to explain its fast training dynamics, and a generalization analysis to demonstrate its smaller generalization gap on LB training. Comprehensive experiments demonstrate that VRGD can accelerate training (1sim 2 times), narrow generalization gap and improve final accuracy. We push the batch size limit of BERT pretraining up to 128k/64k and DLRM to 512k without noticeable accuracy loss. We improve ImageNet Top-1 accuracy at 96k by 0.52pp than LARS. The generalization gap of BERT and ImageNet training is significantly reduce by over 65%.

This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of our study is that large-scale machine learning represents a distinctive setting in which the stochastic gradient (SG) method has traditionally played a central role while conventional gradient-based nonlinear optimization techniques typically falter. Based on this viewpoint, we present a comprehensive theory of a straightforward, yet versatile SG algorithm, discuss its practical behavior, and highlight opportunities for designing algorithms with improved performance. This leads to a discussion about the next generation of optimization methods for large-scale machine learning, including an investigation of two main streams of research on techniques that diminish noise in the stochastic directions and methods that make use of second-order derivative approximations.

Differential privacy (DP) is widely being employed in the industry as a practical standard for privacy protection. While private training of computer vision or natural language processing applications has been studied extensively, the computational challenges of training of recommender systems (RecSys) with DP have not been explored. In this work, we first present our detailed characterization of private RecSys training using DP-SGD, root-causing its several performance bottlenecks. Specifically, we identify DP-SGD's noise sampling and noisy gradient update stage to suffer from a severe compute and memory bandwidth limitation, respectively, causing significant performance overhead in training private RecSys. Based on these findings, we propose LazyDP, an algorithm-software co-design that addresses the compute and memory challenges of training RecSys with DP-SGD. Compared to a state-of-the-art DP-SGD training system, we demonstrate that LazyDP provides an average 119x training throughput improvement while also ensuring mathematically equivalent, differentially private RecSys models to be trained.

Diffusion models have been proven highly effective at generating high-quality images. However, as these models grow larger, they require significantly more memory and suffer from higher latency, posing substantial challenges for deployment. In this work, we aim to accelerate diffusion models by quantizing their weights and activations to 4 bits. At such an aggressive level, both weights and activations are highly sensitive, where conventional post-training quantization methods for large language models like smoothing become insufficient. To overcome this limitation, we propose SVDQuant, a new 4-bit quantization paradigm. Different from smoothing which redistributes outliers between weights and activations, our approach absorbs these outliers using a low-rank branch. We first consolidate the outliers by shifting them from activations to weights, then employ a high-precision low-rank branch to take in the weight outliers with Singular Value Decomposition (SVD). This process eases the quantization on both sides. However, na\"{\i}vely running the low-rank branch independently incurs significant overhead due to extra data movement of activations, negating the quantization speedup. To address this, we co-design an inference engine Nunchaku that fuses the kernels of the low-rank branch into those of the low-bit branch to cut off redundant memory access. It can also seamlessly support off-the-shelf low-rank adapters (LoRAs) without the need for re-quantization. Extensive experiments on SDXL, PixArt-Sigma, and FLUX.1 validate the effectiveness of SVDQuant in preserving image quality. We reduce the memory usage for the 12B FLUX.1 models by 3.5times, achieving 3.0times speedup over the 4-bit weight-only quantized baseline on the 16GB laptop 4090 GPU, paving the way for more interactive applications on PCs. Our quantization library and inference engine are open-sourced.

Attention encoder-decoder model architecture is the backbone of several recent top performing foundation speech models: Whisper, Seamless, OWSM, and Canary-1B. However, the reported data and compute requirements for their training are prohibitive for many in the research community. In this work, we focus on the efficiency angle and ask the questions of whether we are training these speech models efficiently, and what can we do to improve? We argue that a major, if not the most severe, detrimental factor for training efficiency is related to the sampling strategy of sequential data. We show that negligence in mini-batch sampling leads to more than 50% computation being spent on padding. To that end, we study, profile, and optimize Canary-1B training to show gradual improvement in GPU utilization leading up to 5x increase in average batch sizes versus its original training settings. This in turn allows us to train an equivalent model using 4x less GPUs in the same wall time, or leverage the original resources and train it in 2x shorter wall time. Finally, we observe that the major inference bottleneck lies in the autoregressive decoder steps. We find that adjusting the model architecture to transfer model parameters from the decoder to the encoder results in a 3x inference speedup as measured by inverse real-time factor (RTFx) while preserving the accuracy and compute requirements for convergence. The training code and models will be available as open-source.

Deep neural networks are typically trained by optimizing a loss function with an SGD variant, in conjunction with a decaying learning rate, until convergence. We show that simple averaging of multiple points along the trajectory of SGD, with a cyclical or constant learning rate, leads to better generalization than conventional training. We also show that this Stochastic Weight Averaging (SWA) procedure finds much flatter solutions than SGD, and approximates the recent Fast Geometric Ensembling (FGE) approach with a single model. Using SWA we achieve notable improvement in test accuracy over conventional SGD training on a range of state-of-the-art residual networks, PyramidNets, DenseNets, and Shake-Shake networks on CIFAR-10, CIFAR-100, and ImageNet. In short, SWA is extremely easy to implement, improves generalization, and has almost no computational overhead.

Training neural networks requires optimizing a loss function that may be highly irregular, and in particular neither convex nor smooth. Popular training algorithms are based on stochastic gradient descent with momentum (SGDM), for which classical analysis applies only if the loss is either convex or smooth. We show that a very small modification to SGDM closes this gap: simply scale the update at each time point by an exponentially distributed random scalar. The resulting algorithm achieves optimal convergence guarantees. Intriguingly, this result is not derived by a specific analysis of SGDM: instead, it falls naturally out of a more general framework for converting online convex optimization algorithms to non-convex optimization algorithms.

We propose a method that achieves near-optimal rates for smooth stochastic convex optimization and requires essentially no prior knowledge of problem parameters. This improves on prior work which requires knowing at least the initial distance to optimality d0. Our method, U-DoG, combines UniXGrad (Kavis et al., 2019) and DoG (Ivgi et al., 2023) with novel iterate stabilization techniques. It requires only loose bounds on d0 and the noise magnitude, provides high probability guarantees under sub-Gaussian noise, and is also near-optimal in the non-smooth case. Our experiments show consistent, strong performance on convex problems and mixed results on neural network training.

Modern deep neural networks (DNNs) have achieved state-of-the-art performances but are typically over-parameterized. The over-parameterization may result in undesirably large generalization error in the absence of other customized training strategies. Recently, a line of research under the name of Sharpness-Aware Minimization (SAM) has shown that minimizing a sharpness measure, which reflects the geometry of the loss landscape, can significantly reduce the generalization error. However, SAM-like methods incur a two-fold computational overhead of the given base optimizer (e.g. SGD) for approximating the sharpness measure. In this paper, we propose Sharpness-Aware Training for Free, or SAF, which mitigates the sharp landscape at almost zero additional computational cost over the base optimizer. Intuitively, SAF achieves this by avoiding sudden drops in the loss in the sharp local minima throughout the trajectory of the updates of the weights. Specifically, we suggest a novel trajectory loss, based on the KL-divergence between the outputs of DNNs with the current weights and past weights, as a replacement of the SAM's sharpness measure. This loss captures the rate of change of the training loss along the model's update trajectory. By minimizing it, SAF ensures the convergence to a flat minimum with improved generalization capabilities. Extensive empirical results show that SAF minimizes the sharpness in the same way that SAM does, yielding better results on the ImageNet dataset with essentially the same computational cost as the base optimizer.

This paper introduces MiniCPM4, a highly efficient large language model (LLM) designed explicitly for end-side devices. We achieve this efficiency through systematic innovation in four key dimensions: model architecture, training data, training algorithms, and inference systems. Specifically, in terms of model architecture, we propose InfLLM v2, a trainable sparse attention mechanism that accelerates both prefilling and decoding phases for long-context processing. Regarding training data, we propose UltraClean, an efficient and accurate pre-training data filtering and generation strategy, and UltraChat v2, a comprehensive supervised fine-tuning dataset. These datasets enable satisfactory model performance to be achieved using just 8 trillion training tokens. Regarding training algorithms, we propose ModelTunnel v2 for efficient pre-training strategy search, and improve existing post-training methods by introducing chunk-wise rollout for load-balanced reinforcement learning and data-efficient tenary LLM, BitCPM. Regarding inference systems, we propose CPM.cu that integrates sparse attention, model quantization, and speculative sampling to achieve efficient prefilling and decoding. To meet diverse on-device requirements, MiniCPM4 is available in two versions, with 0.5B and 8B parameters, respectively. Sufficient evaluation results show that MiniCPM4 outperforms open-source models of similar size across multiple benchmarks, highlighting both its efficiency and effectiveness. Notably, MiniCPM4-8B demonstrates significant speed improvements over Qwen3-8B when processing long sequences. Through further adaptation, MiniCPM4 successfully powers diverse applications, including trustworthy survey generation and tool use with model context protocol, clearly showcasing its broad usability.

The concept of Critical Batch Size, as pioneered by OpenAI, has long served as a foundational principle for large-scale pre-training. However, with the paradigm shift towards the Warmup-Stable-Decay (WSD) learning rate scheduler, we observe that the original theoretical framework and its underlying mechanisms fail to align with new pre-training dynamics. To bridge this gap between theory and practice, this paper derives a revised E(S) relationship tailored for WSD scheduler, characterizing the trade-off between training data consumption E and steps S during pre-training. Our theoretical analysis reveals two fundamental properties of WSD-based pre-training: 1) B_min, the minimum batch size threshold required to achieve a target loss, and 2) B_opt, the optimal batch size that maximizes data efficiency by minimizing total tokens. Building upon these properties, we propose a dynamic Batch Size Scheduler. Extensive experiments demonstrate that our revised formula precisely captures the dynamics of large-scale pre-training, and the resulting scheduling strategy significantly enhances both training efficiency and final model quality.

Restart techniques are common in gradient-free optimization to deal with multimodal functions. Partial warm restarts are also gaining popularity in gradient-based optimization to improve the rate of convergence in accelerated gradient schemes to deal with ill-conditioned functions. In this paper, we propose a simple warm restart technique for stochastic gradient descent to improve its anytime performance when training deep neural networks. We empirically study its performance on the CIFAR-10 and CIFAR-100 datasets, where we demonstrate new state-of-the-art results at 3.14% and 16.21%, respectively. We also demonstrate its advantages on a dataset of EEG recordings and on a downsampled version of the ImageNet dataset. Our source code is available at https://github.com/loshchil/SGDR

We propose a novel framework to study asynchronous federated learning optimization with delays in gradient updates. Our theoretical framework extends the standard FedAvg aggregation scheme by introducing stochastic aggregation weights to represent the variability of the clients update time, due for example to heterogeneous hardware capabilities. Our formalism applies to the general federated setting where clients have heterogeneous datasets and perform at least one step of stochastic gradient descent (SGD). We demonstrate convergence for such a scheme and provide sufficient conditions for the related minimum to be the optimum of the federated problem. We show that our general framework applies to existing optimization schemes including centralized learning, FedAvg, asynchronous FedAvg, and FedBuff. The theory here provided allows drawing meaningful guidelines for designing a federated learning experiment in heterogeneous conditions. In particular, we develop in this work FedFix, a novel extension of FedAvg enabling efficient asynchronous federated training while preserving the convergence stability of synchronous aggregation. We empirically demonstrate our theory on a series of experiments showing that asynchronous FedAvg leads to fast convergence at the expense of stability, and we finally demonstrate the improvements of FedFix over synchronous and asynchronous FedAvg.

Compact neural networks are specially designed for applications on edge devices with faster inference speed yet modest performance. However, training strategies of compact models are borrowed from that of conventional models at present, which ignores their difference in model capacity and thus may impede the performance of compact models. In this paper, by systematically investigating the impact of different training ingredients, we introduce a strong training strategy for compact models. We find that the appropriate designs of re-parameterization and knowledge distillation are crucial for training high-performance compact models, while some commonly used data augmentations for training conventional models, such as Mixup and CutMix, lead to worse performance. Our experiments on ImageNet-1K dataset demonstrate that our specialized training strategy for compact models is applicable to various architectures, including GhostNetV2, MobileNetV2 and ShuffleNetV2. Specifically, equipped with our strategy, GhostNetV3 1.3times achieves a top-1 accuracy of 79.1% with only 269M FLOPs and a latency of 14.46ms on mobile devices, surpassing its ordinarily trained counterpart by a large margin. Moreover, our observation can also be extended to object detection scenarios. PyTorch code and checkpoints can be found at https://github.com/huawei-noah/Efficient-AI-Backbones/tree/master/ghostnetv3_pytorch.

We present EGN, a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges to an epsilon-stationary point at a linear rate. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, and SGN optimizers across various supervised and reinforcement learning tasks.

The Mixture of Experts (MoE) architecture has emerged as a powerful paradigm for scaling large language models (LLMs) while maintaining inference efficiency. However, their enormous memory requirements make them prohibitively expensive to fine-tune or deploy in resource-constrained environments. To address this challenge, we introduce SlimMoE, a multi-stage compression framework for transforming large MoE models into much smaller, efficient variants without incurring the prohibitive costs of training from scratch. Our method systematically reduces parameter counts by slimming experts and transferring knowledge through intermediate stages, effectively mitigating the performance degradation common in one-shot pruning approaches. Using this framework, we compress Phi 3.5-MoE (41.9B total/6.6B activated parameters) to create Phi-mini-MoE (7.6B total/2.4B activated parameters) and Phi-tiny-MoE (3.8B total/1.1B activated parameters) using only 400B tokens--less than 10% of the original model's training data. These compressed models can be fine-tuned on a single GPU (A100 for Phi-mini-MoE, A6000 for Phi-tiny-MoE), making them highly suitable for academic and resource-limited settings. Our experiments demonstrate that these compressed models outperform others of similar size and remain competitive with larger models. For instance, Phi-mini-MoE achieves similar or better performance to Phi-3-mini using only 2/3 of the activated parameters and yields comparable MMLU scores to Llama 3.1 8B despite having significantly lower latency. Our findings demonstrate that structured pruning combined with staged distillation offers an effective path to creating high-quality, compact MoE models, paving the way for broader adoption of MoE architectures. We make our models publicly available at https://huggingface.co/microsoft/Phi-mini-MoE-instruct and https://huggingface.co/microsoft/Phi-tiny-MoE-instruct .

Score-based generative models (SGMs) sample from a target distribution by iteratively transforming noise using the score function of the perturbed target. For any finite training set, this score function can be evaluated in closed form, but the resulting SGM memorizes its training data and does not generate novel samples. In practice, one approximates the score by training a neural network via score-matching. The error in this approximation promotes generalization, but neural SGMs are costly to train and sample, and the effective regularization this error provides is not well-understood theoretically. In this work, we instead explicitly smooth the closed-form score to obtain an SGM that generates novel samples without training. We analyze our model and propose an efficient nearest-neighbor-based estimator of its score function. Using this estimator, our method achieves competitive sampling times while running on consumer-grade CPUs.

Previous results have shown that a two time-scale update rule (TTUR) using different learning rates, such as different constant rates or different decaying rates, is useful for training generative adversarial networks (GANs) in theory and in practice. Moreover, not only the learning rate but also the batch size is important for training GANs with TTURs and they both affect the number of steps needed for training. This paper studies the relationship between batch size and the number of steps needed for training GANs with TTURs based on constant learning rates. We theoretically show that, for a TTUR with constant learning rates, the number of steps needed to find stationary points of the loss functions of both the discriminator and generator decreases as the batch size increases and that there exists a critical batch size minimizing the stochastic first-order oracle (SFO) complexity. Then, we use the Fr'echet inception distance (FID) as the performance measure for training and provide numerical results indicating that the number of steps needed to achieve a low FID score decreases as the batch size increases and that the SFO complexity increases once the batch size exceeds the measured critical batch size. Moreover, we show that measured critical batch sizes are close to the sizes estimated from our theoretical results.

Diffusion (score-based) generative models have been widely used for modeling various types of complex data, including images, audios, and point clouds. Recently, the deep connection between forward-backward stochastic differential equations (SDEs) and diffusion-based models has been revealed, and several new variants of SDEs are proposed (e.g., sub-VP, critically-damped Langevin) along this line. Despite the empirical success of the hand-crafted fixed forward SDEs, a great quantity of proper forward SDEs remain unexplored. In this work, we propose a general framework for parameterizing the diffusion model, especially the spatial part of the forward SDE. An abstract formalism is introduced with theoretical guarantees, and its connection with previous diffusion models is leveraged. We demonstrate the theoretical advantage of our method from an optimization perspective. Numerical experiments on synthetic datasets, MINIST and CIFAR10 are also presented to validate the effectiveness of our framework.

Recent studies suggest that with sufficiently wide models, most SGD solutions can, up to permutation, converge into the same basin. This phenomenon, known as the model re-basin regime, has significant implications for model averaging by ensuring the linear mode connectivity. However, current re-basin strategies are ineffective in many scenarios due to a lack of comprehensive understanding of underlying mechanisms. Addressing this gap, this paper provides novel insights into understanding and improving the standard practice. Firstly, we decompose re-normalization into rescaling and reshift, uncovering that rescaling plays a crucial role in re-normalization while re-basin performance is sensitive to shifts in model activation. The finding calls for a more nuanced handling of the activation shift. Secondly, we identify that the merged model suffers from the issue of activation collapse and magnitude collapse. Varying the learning rate, weight decay, and initialization method can mitigate the issues and improve model performance. Lastly, we propose a new perspective to unify the re-basin and pruning, under which a lightweight yet effective post-pruning technique is derived, which can significantly improve the model performance after pruning. Our implementation is available at https://github.com/XingyuQu/rethink-re-basin.

We rigorously study the joint evolution of training dynamics via stochastic gradient descent (SGD) and the spectra of empirical Hessian and gradient matrices. We prove that in two canonical classification tasks for multi-class high-dimensional mixtures and either 1 or 2-layer neural networks, the SGD trajectory rapidly aligns with emerging low-rank outlier eigenspaces of the Hessian and gradient matrices. Moreover, in multi-layer settings this alignment occurs per layer, with the final layer's outlier eigenspace evolving over the course of training, and exhibiting rank deficiency when the SGD converges to sub-optimal classifiers. This establishes some of the rich predictions that have arisen from extensive numerical studies in the last decade about the spectra of Hessian and information matrices over the course of training in overparametrized networks.

Minibatch optimal transport coupling straightens paths in unconditional flow matching. This leads to computationally less demanding inference as fewer integration steps and less complex numerical solvers can be employed when numerically solving an ordinary differential equation at test time. However, in the conditional setting, minibatch optimal transport falls short. This is because the default optimal transport mapping disregards conditions, resulting in a conditionally skewed prior distribution during training. In contrast, at test time, we have no access to the skewed prior, and instead sample from the full, unbiased prior distribution. This gap between training and testing leads to a subpar performance. To bridge this gap, we propose conditional optimal transport C^2OT that adds a conditional weighting term in the cost matrix when computing the optimal transport assignment. Experiments demonstrate that this simple fix works with both discrete and continuous conditions in 8gaussians-to-moons, CIFAR-10, ImageNet-32x32, and ImageNet-256x256. Our method performs better overall compared to the existing baselines across different function evaluation budgets. Code is available at https://hkchengrex.github.io/C2OT

We propose a new score-based model with one-step sampling. Previously, score-based models were burdened with heavy computations due to iterative sampling. For substituting the iterative process, we train a standalone generator to compress all the time steps with the gradient backpropagated from the score network. In order to produce meaningful gradients for the generator, the score network is trained to simultaneously match the real data distribution and mismatch the fake data distribution. This model has the following advantages: 1) For sampling, it generates a fake image with only one step forward. 2) For training, it only needs 10 diffusion steps.3) Compared with consistency model, it is free of the ill-posed problem caused by consistency loss. On the popular CIFAR-10 dataset, our model outperforms Consistency Model and Denoising Score Matching, which demonstrates the potential of the framework. We further provide more examples on the MINIST and LSUN datasets. The code is available on GitHub.

Very deep convolutional networks with hundreds of layers have led to significant reductions in error on competitive benchmarks. Although the unmatched expressiveness of the many layers can be highly desirable at test time, training very deep networks comes with its own set of challenges. The gradients can vanish, the forward flow often diminishes, and the training time can be painfully slow. To address these problems, we propose stochastic depth, a training procedure that enables the seemingly contradictory setup to train short networks and use deep networks at test time. We start with very deep networks but during training, for each mini-batch, randomly drop a subset of layers and bypass them with the identity function. This simple approach complements the recent success of residual networks. It reduces training time substantially and improves the test error significantly on almost all data sets that we used for evaluation. With stochastic depth we can increase the depth of residual networks even beyond 1200 layers and still yield meaningful improvements in test error (4.91% on CIFAR-10).

As inference on Large Language Models (LLMs) emerges as an important workload in machine learning applications, weight quantization has become a standard technique for efficient GPU deployment. Quantization not only reduces model size, but has also been shown to yield substantial speedups for single-user inference, due to reduced memory movement, with low accuracy impact. Yet, it remains open whether speedups are achievable also in batched settings with multiple parallel clients, which are highly relevant for practical serving. It is unclear whether GPU kernels can be designed to remain practically memory-bound, while supporting the substantially increased compute requirements of batched workloads. This paper resolves this question positively by describing the design of Mixed-precision Auto-Regressive LINear kernels, called MARLIN. Concretely, given a model whose weights are compressed via quantization to, e.g., 4 bits per element, MARLIN shows that batchsizes up to 16-32 can be supported with close to maximum (4times) quantization speedup, and larger batchsizes up to 64-128 with gradually decreasing, but still significant, acceleration. MARLIN accomplishes this via a combination of techniques, such as asynchronous memory access, complex task scheduling and pipelining, and bespoke quantization support. Our experiments show that MARLIN's near-optimal performance on individual LLM layers across different scenarios can also lead to end-to-end LLM inference speedups (of up to 2.8times) when integrated with the popular vLLM serving engine. Finally, MARLIN is extensible to further compression techniques, like NVIDIA 2:4 sparsity, leading to additional speedups.

Diffusion models (DMs) have been adopted across diverse fields with its remarkable abilities in capturing intricate data distributions. In this paper, we propose a Fast Diffusion Model (FDM) to significantly speed up DMs from a stochastic optimization perspective for both faster training and sampling. We first find that the diffusion process of DMs accords with the stochastic optimization process of stochastic gradient descent (SGD) on a stochastic time-variant problem. Then, inspired by momentum SGD that uses both gradient and an extra momentum to achieve faster and more stable convergence than SGD, we integrate momentum into the diffusion process of DMs. This comes with a unique challenge of deriving the noise perturbation kernel from the momentum-based diffusion process. To this end, we frame the process as a Damped Oscillation system whose critically damped state -- the kernel solution -- avoids oscillation and yields a faster convergence speed of the diffusion process. Empirical results show that our FDM can be applied to several popular DM frameworks, e.g., VP, VE, and EDM, and reduces their training cost by about 50% with comparable image synthesis performance on CIFAR-10, FFHQ, and AFHQv2 datasets. Moreover, FDM decreases their sampling steps by about 3x to achieve similar performance under the same samplers. The code is available at https://github.com/sail-sg/FDM.

While frontier large language models (LLMs) continue to push capability boundaries, their deployment remains confined to GPU-powered cloud infrastructure. We challenge this paradigm with SmallThinker, a family of LLMs natively designed - not adapted - for the unique constraints of local devices: weak computational power, limited memory, and slow storage. Unlike traditional approaches that mainly compress existing models built for clouds, we architect SmallThinker from the ground up to thrive within these limitations. Our innovation lies in a deployment-aware architecture that transforms constraints into design principles. First, We introduce a two-level sparse structure combining fine-grained Mixture-of-Experts (MoE) with sparse feed-forward networks, drastically reducing computational demands without sacrificing model capacity. Second, to conquer the I/O bottleneck of slow storage, we design a pre-attention router that enables our co-designed inference engine to prefetch expert parameters from storage while computing attention, effectively hiding storage latency that would otherwise cripple on-device inference. Third, for memory efficiency, we utilize NoPE-RoPE hybrid sparse attention mechanism to slash KV cache requirements. We release SmallThinker-4B-A0.6B and SmallThinker-21B-A3B, which achieve state-of-the-art performance scores and even outperform larger LLMs. Remarkably, our co-designed system mostly eliminates the need for expensive GPU hardware: with Q4_0 quantization, both models exceed 20 tokens/s on ordinary consumer CPUs, while consuming only 1GB and 8GB of memory respectively. SmallThinker is publicly available at hf.co/PowerInfer/SmallThinker-4BA0.6B-Instruct and hf.co/PowerInfer/SmallThinker-21BA3B-Instruct.

We first propose a decentralized proximal stochastic gradient tracking method (DProxSGT) for nonconvex stochastic composite problems, with data heterogeneously distributed on multiple workers in a decentralized connected network. To save communication cost, we then extend DProxSGT to a compressed method by compressing the communicated information. Both methods need only O(1) samples per worker for each proximal update, which is important to achieve good generalization performance on training deep neural networks. With a smoothness condition on the expected loss function (but not on each sample function), the proposed methods can achieve an optimal sample complexity result to produce a near-stationary point. Numerical experiments on training neural networks demonstrate the significantly better generalization performance of our methods over large-batch training methods and momentum variance-reduction methods and also, the ability of handling heterogeneous data by the gradient tracking scheme.

We investigate a general formulation for clustering and transductive few-shot learning, which integrates prototype-based objectives, Laplacian regularization and supervision constraints from a few labeled data points. We propose a concave-convex relaxation of the problem, and derive a computationally efficient block-coordinate bound optimizer, with convergence guarantee. At each iteration,our optimizer computes independent (parallel) updates for each point-to-cluster assignment. Therefore, it could be trivially distributed for large-scale clustering and few-shot tasks. Furthermore, we provides a thorough convergence analysis based on point-to-set maps. Were port comprehensive clustering and few-shot learning experiments over various data sets, showing that our method yields competitive performances, in term of accuracy and optimization quality, while scaling up to large problems. Using standard training on the base classes, without resorting to complex meta-learning and episodic-training strategies, our approach outperforms state-of-the-art few-shot methods by significant margins, across various models, settings and data sets. Surprisingly, we found that even standard clustering procedures (e.g., K-means), which correspond to particular, non-regularized cases of our general model, already achieve competitive performances in comparison to the state-of-the-art in few-shot learning. These surprising results point to the limitations of the current few-shot benchmarks, and question the viability of a large body of convoluted few-shot learning techniques in the recent literature.

Scaling foundation model training with Distributed Data Parallel (DDP) methods is bandwidth-limited. Existing infrequent communication methods like Local SGD were designed to synchronize only model parameters and cannot be trivially applied to adaptive optimizers due to additional optimizer states. Current approaches extending Local SGD either lack convergence guarantees or require synchronizing all optimizer states, tripling communication costs. We propose Desynced Low Communication Adaptive Optimizers (DES-LOC), a family of optimizers assigning independent synchronization periods to parameters and momenta, enabling lower communication costs while preserving convergence. Through extensive experiments on language models of up to 1.7B, we show that DES-LOC can communicate 170x less than DDP and 2x less than the previous state-of-the-art Local ADAM. Furthermore, unlike previous heuristic approaches, DES-LOC is suited for practical training scenarios prone to system failures. DES-LOC offers a scalable, bandwidth-efficient, and fault-tolerant solution for foundation model training.

Learning rate schedules used in practice bear little resemblance to those recommended by theory. We close much of this theory/practice gap, and as a consequence are able to derive new problem-adaptive learning rate schedules. Our key technical contribution is a refined analysis of learning rate schedules for a wide class of optimization algorithms (including SGD). In contrast to most prior works that study the convergence of the average iterate, we study the last iterate, which is what most people use in practice. When considering only worst-case analysis, our theory predicts that the best choice is the linear decay schedule: a popular choice in practice that sets the stepsize proportionally to 1 - t/T, where t is the current iteration and T is the total number of steps. To go beyond this worst-case analysis, we use the observed gradient norms to derive schedules refined for any particular task. These refined schedules exhibit learning rate warm-up and rapid learning rate annealing near the end of training. Ours is the first systematic approach to automatically yield both of these properties. We perform the most comprehensive evaluation of learning rate schedules to date, evaluating across 10 diverse deep learning problems, a series of LLMs, and a suite of logistic regression problems. We validate that overall, the linear-decay schedule matches or outperforms all commonly used default schedules including cosine annealing, and that our schedule refinement method gives further improvements.

As the size of large language models grows exponentially, GPU memory has become a bottleneck for adapting these models to downstream tasks. In this paper, we aim to push the limits of memory-efficient training by minimizing memory usage on model weights, gradients, and optimizer states, within a unified framework. Our idea is to eliminate both gradients and optimizer states using zeroth-order optimization, which approximates gradients by perturbing weights during forward passes to identify gradient directions. To minimize memory usage on weights, we employ model quantization, e.g., converting from bfloat16 to int4. However, directly applying zeroth-order optimization to quantized weights is infeasible due to the precision gap between discrete weights and continuous gradients, which would otherwise require de-quantization and re-quantization. To overcome this challenge, we propose Quantized Zeroth-order Optimization (QZO), a novel approach that perturbs the continuous quantization scale for gradient estimation and uses a directional derivative clipping method to stabilize training. QZO is orthogonal to both scalar-based and codebook-based post-training quantization methods. Compared to full-parameter fine-tuning in bfloat16, QZO can reduce the total memory cost by more than 18times for 4-bit LLMs, and enables fine-tuning Llama-2-13B and Stable Diffusion 3.5 Large within a single 24GB GPU.

We introduce new differentially private (DP) mechanisms for gradient-based machine learning (ML) with multiple passes (epochs) over a dataset, substantially improving the achievable privacy-utility-computation tradeoffs. We formalize the problem of DP mechanisms for adaptive streams with multiple participations and introduce a non-trivial extension of online matrix factorization DP mechanisms to our setting. This includes establishing the necessary theory for sensitivity calculations and efficient computation of optimal matrices. For some applications like >!! 10,000 SGD steps, applying these optimal techniques becomes computationally expensive. We thus design an efficient Fourier-transform-based mechanism with only a minor utility loss. Extensive empirical evaluation on both example-level DP for image classification and user-level DP for language modeling demonstrate substantial improvements over all previous methods, including the widely-used DP-SGD . Though our primary application is to ML, our main DP results are applicable to arbitrary linear queries and hence may have much broader applicability.

It is widely believed that a neural network can fit a training set containing at least as many samples as it has parameters, underpinning notions of overparameterized and underparameterized models. In practice, however, we only find solutions accessible via our training procedure, including the optimizer and regularizers, limiting flexibility. Moreover, the exact parameterization of the function class, built into an architecture, shapes its loss surface and impacts the minima we find. In this work, we examine the ability of neural networks to fit data in practice. Our findings indicate that: (1) standard optimizers find minima where the model can only fit training sets with significantly fewer samples than it has parameters; (2) convolutional networks are more parameter-efficient than MLPs and ViTs, even on randomly labeled data; (3) while stochastic training is thought to have a regularizing effect, SGD actually finds minima that fit more training data than full-batch gradient descent; (4) the difference in capacity to fit correctly labeled and incorrectly labeled samples can be predictive of generalization; (5) ReLU activation functions result in finding minima that fit more data despite being designed to avoid vanishing and exploding gradients in deep architectures.