Training Large Language Models (LLMs) presents a significant communication bottleneck, predominantly due to the growing scale of the gradient to communicate across multi-device clusters. However, how to mitigate communication overhead in practice remains a formidable challenge due to the weakness of the methodology of the existing compression methods, especially the neglect of the characteristics of the gradient. In this paper, we consider and demonstrate the low-rank properties of gradient and Hessian observed in LLMs training dynamic, and take advantage of such natural properties to design SEPARATE, a simple low-rank projection for gradient compression in modern large-scale model training processes. SEPARATE realizes dimensional reduction by common random Gaussian variables and an improved moving average error-feedback technique. We theoretically demonstrate that SEPARATE-based optimizers maintain the original convergence rate for SGD and Adam-Type optimizers for general non-convex objectives. Experimental results show that SEPARATE accelerates training speed by up to 2× for GPT-2-Medium pre-training, and improves performance on various benchmarks for LLAMA2-7B fine-tuning.

This paper considers a canonical problem in kernel regression: how good are the model performances when it is trained by the popular online first-order algorithms, compared to the offline ones, such as ridge and ridgeless regression? In this paper, we analyze the foundational single-pass Stochastic Gradient Descent (SGD) in kernel regression under source condition where the optimal predictor can even not belong to the RKHS, i.e. the model is misspecified. Specifically, we focus on the inner product kernel over the sphere and characterize the exact orders of the excess risk curves under different scales of sample sizes $n$ concerning the input dimension $d$. Surprisingly, we show that SGD achieves min-max optimal rates up to constants among all the scales, $without$ suffering the saturation, a prevalent phenomenon observed in (ridge) regression, except when the model is highly misspecified and the learning is in a final stage where $n\gg d^\gamma$ with any constant $\gamma >0$. The main reason for SGD to overcome the curse of saturation is the exponentially decaying step size schedule, a common practice in deep neural network training. As a byproduct, we provide the $first$ provable advantage of the scheme over the iterative averaging method in the common setting.

Transformer models have driven breakthroughs across various language tasks by their strong capability to learn rich contextual representations. Scaling them to improve representation, however, often demands substantial memory and compute costs, such as the Key-Value (KV) cache used during auto-regressive decoding. Skip connections offer a promising way to improve representation without bloating resource usage, yet most prior works either improve expressivity while leaving KV costs unchanged, or reduce memory at the cost of weaker representation. In this work, we propose SkipV1Former, a Transformer variant that uses skip connections from the first layer's Value heads to strengthen model representation and reduce KV cache. Specifically, from the second block onward, each layer reuses half of its Value heads from the very first layer, while computing the other half as usual-cutting Value projections and V cache by nearly 50 \%. Theoretically, we show that routing uncompressed first-layer Values into deeper layers restores information lost to compression and accelerates the model’s implicit mesa-optimization-a key pattern of Transformer in auto-regressive tasks. Empirically, across different model scales, SkipV1Former delivers consistent reductions of approximately 25 \% in KV cache while improving perplexity relative to standard Multi-Head Attention (MHA) Transformers and some advanced variants. Moreover, we propose a recipe for uptraining existing MHA Transformer checkpoints to SkipV1Former with only 10-15\% additional compute. Finally, SkipV1Former can seamlessly combine advanced methods like Group-Query Attention and Multi-Latent Attention to achieve further KV cache savings and performance improvement. When combined with YOCO, it cuts KV cache size by nearly 50 \% while still improving performance. The code is available at: https://github.com/Zhoutong-Wu/SkipV1Former.

This paper introduces PaZO, a preconditioned accelerated zeroth-order optimization algorithm for fine-tuning large language models (LLMs). First, we theoretically demonstrate the necessity of preconditioning in zeroth-order optimization, proving that zeroth-order stochastic gradient descent (ZO-SGD) alone fails to achieve the ideal convergence rate. Building on this, we propose a Preconditioned Simultaneous Perturbation Stochastic Approximation (PSPSA) and theoretical version of PaZO, and demonstrate that setting the order of preconditioner as $-1/2$ in PSPSA yields the improved convergence rate for PaZO. Moreover, we design a practical version of PaZO that stabilizes training via diagonal Hessian estimate and moving average technique. Extensive experiments on diverse downstream tasks with models like RoBERTa-large and OPT show PaZO’s effectiveness. Compared to other zeroth-order baselines, PaZO achieves better performance across models and tasks.

Zeroth-order optimization algorithms recently emerge as a popular research theme in optimization and machine learning, playing important roles in many deep-learning related tasks such as black-box adversarial attack, deep reinforcement learning, as well as hyper-parameter tuning. Mainstream zeroth-order optimization algorithms, however, concentrate on exploiting zeroth-order-estimated first-order gradient information of the objective landscape. In this paper, we propose a novel meta-algorithm called Hessian-Aware Zeroth-Order (ZOHA) optimization algorithm, which utilizes several canonical variants of zeroth-order-estimated second-order Hessian information of the objective: power-method-based, and Gaussian-smoothing-based. We conclude theoretically that ZOHA enjoys an improved convergence rate compared with existing work without incorporating in zeroth-order optimization second-order Hessian information. Empirical studies on logistic regression as well as the black-box adversarial attack are provided to validate the effectiveness and improved success rates with reduced query complexity of the zeroth-order oracle.

Foundation Models (FMs) have demonstrated remarkable insights into the relational dynamics of the world, leading to the crucial question: *how do these models acquire an understanding of world hybrid relations?* Traditional statistical learning, particularly for prediction problems, may overlook the rich and inherently structured information from the data, especially regarding the relationships between objects. We introduce a mathematical model that formalizes relational learning as hypergraph recovery to study pre-training of FMs. In our framework, the world is represented as a hypergraph, with data abstracted as random samples from hyperedges. We theoretically examine the feasibility of a Pre-Trained Model (PTM) to recover this hypergraph and analyze the data efficiency in a minimax near-optimal style. By integrating rich graph theories into the realm of PTMs, our mathematical framework offers powerful tools for an in-depth understanding of pre-training from a unique perspective and can be used under various scenarios. As an example, we extend the framework to entity alignment in multimodal learning.

Neuro-symbolic reinforcement learning (NS-RL) has emerged as a promising paradigm for explainable decision-making, characterized by the interpretability of symbolic policies. NS-RL entails structured state representations for tasks with visual observations, but previous methods cannot refine the structured states with rewards due to a lack of efficiency. Accessibility also remains an issue, as extensive domain knowledge is required to interpret symbolic policies. In this paper, we present a neuro-symbolic framework for jointly learning structured states and symbolic policies, whose key idea is to distill the vision foundation model into an efficient perception module and refine it during policy learning. Moreover, we design a pipeline to prompt GPT-4 to generate textual explanations for the learned policies and decisions, significantly reducing users’ cognitive load to understand the symbolic policies. We verify the efficacy of our approach on nine Atari tasks and present GPT-generated explanations for policies and decisions.

Finite-sum optimization has wide applications in machine learning, covering important problems such as support vector machines, regression, etc. In this paper, we initiate the study of solving finite-sum optimization problems by quantum computing. Specifically, let $f_1,\ldots,f_n:\mathbb{R}^d\to\mathbb{R}$ be $\ell$-smooth convex functions and $\psi:\mathbb{R}^d\to\mathbb{R}$ be a $\mu$-strongly convex proximal function. The goal is to find an $\epsilon$-optimal point for $F(\mathbf{x})=\frac{1}{n}\sum_{i=1}^n f_i(\mathbf{x})+\psi(\mathbf{x})$. We give a quantum algorithm with complexity $\tilde{O}\big(n+\sqrt{d}+\sqrt{\ell/\mu}\big(n^{1/3}d^{1/3}+n^{-2/3}d^{5/6}\big)\big)$, improving the classical tight bound $\tilde{\Theta}\big(n+\sqrt{n\ell/\mu}\big)$. We also prove a quantum lower bound $\tilde{\Omega}(n+n^{3/4}(\ell/\mu)^{1/4})$ when $d$ is large enough. Both our quantum upper and lower bounds can extend to the cases where $\psi$ is not necessarily strongly convex, or each $f_i$ is Lipschitz but not necessarily smooth. In addition, when $F$ is nonconvex, our quantum algorithm can find an $\epsilon$-critial point using $\tilde{O}(n+\ell(d^{1/3}n^{1/3}+\sqrt{d})/\epsilon^2)$ queries.

We study a typical optimization model where the optimization variable is composed of multiple probability distributions. Though the model appears frequently in practice, such as for policy problems, it lacks specific analysis in the general setting. For this optimization problem, we propose a new structural condition/landscape description named generalized quasar-convexity (GQC) beyond the realms of convexity. In contrast to original quasar-convexity \citep{hinder2020near}, GQC allows an individual quasar-convex parameter $\gamma_i$ for each variable block $i$ and the smaller of $\gamma_i$ implies less block-convexity. To minimize the objective function, we consider a generalized oracle termed as the internal function that includes the standard gradient oracle as a special case. We provide optimistic mirror descent (OMD) for multiple distributions and prove that the algorithm can achieve an adaptive $\tilde{\mathcal{O}}((\sum_{i=1}^d1/\gamma_i)\epsilon^{-1})$ iteration complexity to find an $\varepsilon$-suboptimal global solution without pre-known the exact values of $\gamma_i$ when the objective admits ``polynomial-like'' structural. Notably, it achieves iteration complexity that does not explicitly depend on the number of distributions and strictly faster $(\sum_{i=1}^d 1/\gamma_i \text{ v.s. } d\max_{i\in[1:d]} 1/\gamma_i)$ than mirror decent methods. We also extend GQC to the minimax optimization problem proposing the generalized quasar-convexity-concavity (GQCC) condition and a decentralized variant of OMD with regularization. Finally, we show the applications of our algorithmic framework on discounted Markov Decision Processes problem and Markov games, which bring new insights on the landscape analysis of reinforcement learning.

The deep equilibrium model (DEQ) generalizes the conventional feedforward neural network by fixing the same weights for each layer block and extending the number of layers to infinity. This novel model directly finds the fixed points of such a forward process as features for prediction. Despite empirical evidence showcasing its efficacy compared to feedforward neural networks, a theoretical understanding for its separation and bias is still limited. In this paper, we take a step by proposing some separations and studying the bias of DEQ in its expressive power and learning dynamics. The results include: (1) A general separation is proposed, showing the existence of a width-$m$ DEQ that any fully connected neural networks (FNNs) with depth $O(m^{\alpha})$ for $\alpha \in (0,1)$ cannot approximate unless its width is sub-exponential in $m$; (2) DEQ with polynomially bounded size and magnitude can efficiently approximate certain steep functions (which has very large derivatives) in $L^{\infty}$ norm, whereas FNN with bounded depth and exponentially bounded width cannot unless its weights magnitudes are exponentially large; (3) The implicit regularization caused by gradient flow from a diagonal linear DEQ is characterized, with specific examples showing the benefits brought by such regularization. From the overall study, a high-level conjecture from our analysis and empirical validations is that DEQ has potential advantages in learning certain high-frequency components.

Models are expected to engage in invariance learning, which involves distinguishing the core relations that remain consistent across varying environments to ensure the predictions are safe, robust and fair. While existing works consider specific algorithms to realize invariance learning, we show that model has the potential to learn invariance through standard training procedures. In other words, this paper studies the implicit bias of Stochastic Gradient Descent (SGD) over heterogeneous data and shows that the implicit bias drives the model learning towards an invariant solution. We call the phenomenon the implicit invariance learning. Specifically, we theoretically investigate the multi-environment low-rank matrix sensing problem where in each environment, the signal comprises (i) a lower-rank invariant part shared across all environments; and (ii) a significantly varying environment-dependent spurious component. The key insight is, through simply employing the large step size large-batch SGD sequentially in each environment without any explicit regularization, the oscillation caused by heterogeneity can provably prevent model learning spurious signals. The model reaches the invariant solution after certain iterations. In contrast, model learned using pooled SGD over all data would simultaneously learn both the invariant and spurious signals. Overall, we unveil another implicit bias that is a result of the symbiosis between the heterogeneity of data and modern algorithms, which is, to the best of our knowledge, first in the literature.

We consider the non-convex non-concave objective function in two-player zero-sum continuous games. The existence of pure Nash equilibrium requires stringent conditions, posing a major challenge for this problem. To circumvent this difficulty, we examine the problem of identifying a mixed Nash equilibrium, where strategies are randomized and characterized by probability distributions over continuous domains. To this end, we propose PArticle-based Primal-dual ALgorithm (PAPAL) tailored for a weakly entropy-regularized min-max optimization over probability distributions. This algorithm employs the stochastic movements of particles to represent the updates of random strategies for the $\epsilon$-mixed Nash equilibrium. We offer a comprehensive convergence analysis of the proposed algorithm, demonstrating its effectiveness. In contrast to prior research that attempted to update particle importance without movements, PAPAL is the first implementable particle-based algorithm accompanied by non-asymptotic quantitative convergence results, running time, and sample complexity guarantees. Our framework contributes novel insights into the particle-based algorithms for continuous min-max optimization in the general non-convex non-concave setting.

Universal approximation capability, also referred to as universality, is an important property of deep neural networks, endowing them with the potency to accurately represent the underlying target function in learning tasks. In practice, the architecture of deep neural networks largely influences the performance of the models. However, most existing methodologies for designing neural architectures, such as the heuristic manual design or neural architecture search, ignore the universal approximation property, thus losing a potential safeguard about the performance. In this paper, we propose a unified framework to design the architectures of deep neural networks with a universality guarantee based on first-order optimization algorithms, where the forward pass is interpreted as the updates of an optimization algorithm. The (explicit or implicit) network is designed by replacing each gradient term in the algorithm with a learnable module similar to a two-layer network or its derivatives. Specifically, we explore the realm of width-bounded neural networks, a common practical scenario, showcasing their universality. Moreover, adding operations of normalization, downsampling, and upsampling does not hurt the universality. To the best of our knowledge, this is the first work that width-bounded networks with universal approximation guarantee can be designed in a principled way. Our framework can inspire a variety of neural architectures including some renowned structures such as ResNet and DenseNet, as well as novel innovations. The experimental results on image classification problems demonstrate that the newly inspired networks are competitive and surpass the baselines of ResNet, DenseNet, as well as the advanced ConvNeXt and ViT, testifying to the effectiveness of our framework.

Polyak-Łojasiewicz (PL) (Polyak, 1963) condition is a weaker condition than the strong convexity but suffices to ensure a global convergence for the Gradient Descent algorithm. In this paper, we study the lower bound of algorithms using first-order oracles to find an approximate optimal solution. We show that any first-order algorithm requires at least ${\Omega}\left(\frac{L}{\mu}\log\frac{1}{\epsilon}\right)$ gradient costs to με find an $\epsilon$-approximate optimal solution for a general $L$-smooth function that has an $\mu$-PL constant. This result demonstrates the optimality of the Gradient Descent algorithm to minimize smooth PL functions in the sense that there exists a “hard” PL function such that no first-order algorithm can be faster than Gradient Descent when ignoring a numerical constant. In contrast, it is well-known that the momentum technique, e.g. Nesterov (2003, chap. 2), can provably accelerate Gradient Descent to ${O}\left(\sqrt{\frac{L}{\hat{\mu}}}\log\frac{1}{\epsilon}\right)$ gradient costs for functions that are $L$-smooth and $\hat{\mu}$-strongly convex. Therefore, our result distinguishes the hardness of minimizing a smooth PL function and a smooth strongly convex function as the complexity of the former cannot be improved by any polynomial order in general.

Zeroth-order optimization is a fundamental research topic that has been a focus of various learning tasks, such as black-box adversarial attacks, bandits, and reinforcement learning. However, in theory, most complexity results assert a linear dependency on the dimension of optimization variable, which implies paralyzations of zeroth-order algorithms for high-dimensional problems and cannot explain their effectiveness in practice. In this paper, we present a novel zeroth-order optimization theory characterized by complexities that exhibit weak dependencies on dimensionality. The key contribution lies in the introduction of a new factor, denoted as $\effdim_{\alpha} = \sup_{\x\in \mathbb{R}^d} \sum_{i=1}^d \sigma_i^\alpha(\nabla^2 f(\x))$ ($\alpha>0$, $\sigma_i(\cdot)$ is the $i$-th singular value in non-increasing order), which effectively functions as a measure of dimensionality. The algorithms we propose demonstrate significantly reduced complexities when measured in terms of the factor $\effdim_{\alpha}$. Specifically, we first study a well-known zeroth-order algorithm from Nesterov and Spokoiny (2017) on quadratic objectives and show a complexity of $ \mathcal{O}\left(\frac{\effdim_1 }{\sigma_d}\log(1/\epsilon)\right) $ for the strongly convex setting. For linear regression, such a complexity is dimension-free and outperforms the traditional result by a factor of $d$ under common conditions. Furthermore, we introduce novel algorithms that leverages the Heavy-ball mechanism to enhance the optimization process. By incorporating this acceleration scheme, our proposed algorithm exhibits a complexity of $ \mathcal{O}\left(\frac{\effdim_{1/2} }{\sqrt{\sigma_d}}\cdot\log{\frac{L}{\mu}}\cdot\log(1/\epsilon)\right) $. For linear regression, under some mild conditions, it is faster than state-of-the-art algorithms by $\sqrt{d}$. We further expand the scope of the method to encompass generic smooth optimization problems, while incorporating an additional Hessian-smooth condition. By considering this extended framework, our approach becomes applicable to a broader range of optimization scenarios. The resultant algorithms demonstrate remarkable complexities, with dimension-independent dominant terms that surpass existing algorithms by an order in $d$ under appropriate conditions. Our analysis lays the foundation for investigating zeroth-order optimization methods for smooth functions within high-dimensional settings.

This paper focuses on the high-dimensional sampling of log-concave distributions with composite structures: $p^*(\mathrm{d}x)\propto \exp(-g(x)-f(x))\mathrm{d}x$. We develop a double randomization technique, which leads to a fast underdamped Langevin algorithm with a dimension-independent convergence guarantee. We prove that the algorithm enjoys an overall $\tilde{\mathcal{O}}\left(\frac{\left(\mathrm{tr}(H)\right)^{1/3}}{\epsilon^{2/3}}\right)$ iteration complexity to reach an $\epsilon$-tolerated sample whose distribution $p$ admits $W_2(p,p^*)\leq \epsilon$. Here, $H$ is an upper bound of the Hessian matrices for $f$ and does not explicitly depend on dimension $d$. For the posterior sampling over linear models with normalized data, we show a clear superiority of convergence rate which is dimension-free and outperforms the previous best-known results by a $d^{1/3}$ factor. The analysis to achieve a faster convergence rate brings new insights into high-dimensional sampling.

Pre-training has achieved remarkable success when transferred to downstream tasks. In machine learning, we care about not only the good performance of a model but also its behavior under reasonable shifts of condition. The same philosophy holds when pre-training a foundation model. However, the foundation model may not uniformly behave well for a series of related downstream tasks. This happens, for example, when conducting mask recovery regression where the recovery ability or the training instances diverge like pattern features are extracted dominantly on pre-training, but semantic features are also required on a downstream task. This paper considers pre-training a model that guarantees a uniformly good performance over the downstream tasks. We call this goal as *downstream-task robustness*. Our method first separates the upstream task into several representative ones and applies a simple minimax loss for pre-training. We then design an efficient algorithm to solve the minimax loss and prove its convergence in the convex setting. In the experiments, we show both on large-scale natural language processing and computer vision datasets our method increases the metrics on worse-case downstream tasks. Additionally, some theoretical explanations for why our loss is beneficial are provided. Specifically, we show fewer samples are inherently required for the most challenging downstream task in some cases.

We present the lifted proximal operator machine (LPOM) to train fully-connected feed-forward neural networks. LPOM represents the activation function as an equivalent proximal operator and adds the proximal operators to the objective function of a network as penalties. LPOM is block multi-convex in all layer-wise weights and activations. This allows us to develop a new block coordinate descent (BCD) method with convergence guarantee to solve it. Due to the novel formulation and solving method, LPOM only uses the activation function itself and does not require any gradient steps. Thus it avoids the gradient vanishing or exploding issues, which are often blamed in gradient-based methods. Also, it can handle various non-decreasing Lipschitz continuous activation functions. Additionally, LPOM is almost as memory-efficient as stochastic gradient descent and its parameter tuning is relatively easy. We further implement and analyze the parallel solution of LPOM. We first propose a general asynchronous-parallel BCD method with convergence guarantee. Then we use it to solve LPOM, resulting in asynchronous-parallel LPOM. For faster speed, we develop the synchronous-parallel LPOM. We validate the advantages of LPOM on various network architectures and datasets. We also apply synchronous-parallel LPOM to autoencoder training and demonstrate its fast convergence and superior performance.

The analysis of over-parameterized neural networks has drawn significant attention in recent years. It was shown that such systems behave like convex systems under various restricted settings, such as for two-layer neural networks, and when learning is only restricted locally in the so-called neural tangent kernel space around specialized initializations. However, there is a lack of powerful theoretical techniques that can analyze fully trained deep neural networks under general conditions. This paper considers this fundamental problem by investigating such overparameterized deep neural networks when fully trained. Specifically, we characterize a deep neural network by its features’ distributions and propose a metric to intuitively measure the usefulness of feature representations. Under certain regularizers that bounds the metric, we show deep neural networks can be reformulated as aconvexoptimization and the system can guarantee effective feature representations in terms of the metric. Our new analysis is more consistent with empirical observations that deep neural networks are capable of learning efficient feature representations. Empirical studies confirm that predictions of our theory are consistent with results observed in practice.

This paper proposes a new mean-field framework for over-parameterized deep neural networks (DNNs), which can be used to analyze neural network training. In this framework, a DNN is represented by probability measures and functions over its features (that is, the function values of the hidden units over the training data) in the continuous limit, instead of the neural network parameters as most existing studies have done. This new representation overcomes the degenerate situation where all the hidden units essentially have only one meaningful hidden unit in each middle layer, leading to a simpler representation of DNNs. Moreover, we construct a non-linear dynamics called neural feature flow, which captures the evolution of an over-parameterized DNN trained by Gradient Descent. We illustrate the framework via the Residual Network (Res-Net) architecture. It is shown that when the neural feature flow process converges, it reaches a global minimal solution under suitable conditions.

Deep learning has received considerable empirical success in recent years. However, while many ad hoc tricks have been discovered by practitioners, until recently, there has been a lack of theoretical understanding for tricks invented in the deep learning literature. Known by practitioners that overparameterized neural networks (NNs) are easy to learn, in the past few years, there have been important theoretical developments in the analysis of overparameterized NNs. In particular, it was shown that such systems behave like convex systems under various restricted settings, such as for two-layer NNs, and when learning is restricted locally in the so-called neural tangent kernel space around specialized initializations. This article discusses some of these recent signs of progress leading to a significantly better understanding of NNs. We will focus on the analysis of two-layer NNs and explain the key mathematical models, with their algorithmic implications. We will then discuss challenges in understanding deep NNs and some current research directions.

For many initialization schemes, parameters of two randomly initialized deep neural networks (DNNs) can be quite different, but feature distributions of the hidden nodes are similar at each layer. With the help of a new technique called {\it neural network grafting}, we demonstrate that even during the entire training process, feature distributions of differently initialized networks remain similar at each layer. In this paper, we present an explanation of this phenomenon. Specifically, we consider the loss landscape of an overparameterized convolutional neural network (CNN) in the continuous limit, where the numbers of channels/hidden nodes in the hidden layers go to infinity. Although the landscape of the overparameterized CNN is still non-convex with respect to the trainable parameters, we show that very surprisingly, it can be reformulated as a convex function with respect to the feature distributions in the hidden layers. Therefore by reparameterizing neural networks in terms of feature distributions, we obtain a much simpler characterization of the landscape of overparameterized CNNs. We further argue that training with respect to network parameters leads to a fixed trajectory in the feature distributions.

Gradient clipping is commonly used in training deep neural networks partly due to its practicability in relieving the exploding gradient problem. Recently, \citet{zhang2019gradient} show that clipped (stochastic) Gradient Descent (GD) converges faster than vanilla GD via introducing a new assumption called $(L_0, L_1)$-smoothness, which characterizes the violent fluctuation of gradients typically encountered in deep neural networks. However, their iteration complexities on the problem-dependent parameters are rather pessimistic, and theoretical justification of clipping combined with other crucial techniques, e.g. momentum acceleration, are still lacking. In this paper, we bridge the gap by presenting a general framework to study the clipping algorithms, which also takes momentum methods into consideration.We provide convergence analysis of the framework in both deterministic and stochastic setting, and demonstrate the tightness of our results by comparing them with existing lower bounds. Our results imply that the efficiency of clipping methods will not degenerate even in highly non-smooth regions of the landscape. Experiments confirm the superiority of clipping-based methods in deep learning tasks.

Numerical optimization serves as one of the pillars of machine learning. To meet the demands of big data applications, lots of efforts have been put on designing theoretically and practically fast algorithms. This article provides a comprehensive survey on accelerated first-order algorithms with a focus on stochastic algorithms. Specifically, this article starts with reviewing the basic accelerated algorithms on deterministic convex optimization, then concentrates on their extensions to stochastic convex optimization, and at last introduces some recent developments on acceleration for nonconvex optimization.

We propose a new optimization method for training feedforward neural networks. By rewriting the activation function as an equivalent proximal operator, we approximate a feedforward neural network by adding the proximal operators to the objective function as penalties, hence we call the lifted proximal operator machine (LPOM). LPOM is block multiconvex in all layer-wise weights and activations. This allows us to use block coordinate descent to update the layer-wise weights and activations. Most notably, we only use the mapping of the activation function itself, rather than its derivative, thus avoiding the gradient vanishing or blow-up issues in gradient based training methods. So our method is applicable to various non-decreasing Lipschitz continuous activation functions, which can be saturating and non-differentiable. LPOM does not require more auxiliary variables than the layer-wise activations, thus using roughly the same amount of memory as stochastic gradient descent (SGD) does. Its parameter tuning is also much simpler. We further prove the convergence of updating the layer-wise weights and activations and point out that the optimization could be made parallel by asynchronous update. Experiments on MNIST and CIFAR-10 datasets testify to the advantages of LPOM.

For constrained nonconvex minimization problems, we propose a meta stochastic projection-free optimization algorithm, named Normalized Frank Wolfe Updating, that can take any Gradient Estimator (GE) as input. For this algorithm, we prove its convergence rate, regardless of the choice of GE. Using a sophisticated GE, this algorithm can significantly improve the Stochastic First order Oracle (SFO) complexity. Further, a new second order GE strategy is proposed to incorporate curvature information, which enjoys theoretical advantage over the first order ones. Besides, this paper also provides a lower bound of Linear-optimization Oracle (LO) queried to achieve an approximate stationary point. Simulation studies validate our analysis under various parameter settings.

In this paper, we give a sharp analysis for Stochastic Gradient Descent (SGD) and prove that SGD is able to efficiently escape from saddle points and find an $(\epsilon, O(\epsilon^{0.5}))$-approximate second-order stationary point in $\tilde{O}(\epsilon^{-3.5})$ stochastic gradient computations for generic nonconvex optimization problems, when the objective function satisfies gradient-Lipschitz, Hessian-Lipschitz, and dispersive noise assumptions. This result subverts the classical belief that SGD requires at least $O(\epsilon^{-4})$ stochastic gradient computations for obtaining an $(\epsilon,O(\epsilon^{0.5}))$-approximate second-order stationary point. Such SGD rate matches, up to a polylogarithmic factor of problem-dependent parameters, the rate of most accelerated nonconvex stochastic optimization algorithms that adopt additional techniques, such as Nesterov’s momentum acceleration, negative curvature search, as well as quadratic and cubic regularization tricks. Our novel analysis gives new insights into nonconvex SGD and can be potentially generalized to a broad class of stochastic optimization algorithms.

Learning Partial Differential Equations (LPDEs) from training data for particular tasks has been successfully applied to many image processing problems. In this paper, we aim to learn compact Partial Differential Equations (LCPDEs) for color image tasks by proposing a more effective algorithm. The LCPDEs system is formulated as a linear combination of fundamental differential invariants and simplified by omitting the PDE which works as an indicate function. We replace L2-norm with Ll-norm to regularize the coefficients with respect to the invariants. As the objective function is non-smooth, we resort to proximal algorithm to optimize it, which ensures convergence in an at least sub-linear rate. Experiments demonstrate the advantages of the proposed method over other PDE-based methods in terms of both quality and efficiency.

In this paper, we propose a new technique named \textit{Stochastic Path-Integrated Differential EstimatoR} (SPIDER), which can be used to track many deterministic quantities of interests with significantly reduced computational cost. Combining SPIDER with the method of normalized gradient descent, we propose SPIDER-SFO that solve non-convex stochastic optimization problems using stochastic gradients only. We provide a few error-bound results on its convergence rates. Specially, we prove that the SPIDER-SFO algorithm achieves a gradient computation cost of $\mathcal{O}\left( \min( n^{1/2} \epsilon^{-2}, \epsilon^{-3} ) \right)$ to find an $\epsilon$-approximate first-order stationary point. In addition, we prove that SPIDER-SFO nearly matches the algorithmic lower bound for finding stationary point under the gradient Lipschitz assumption in the finite-sum setting. Our SPIDER technique can be further applied to find an $(\epsilon, \mathcal{O}(\ep^{0.5}))$-approximate second-order stationary point at a gradient computation cost of $\tilde{\mathcal{O}}\left( \min( n^{1/2} \epsilon^{-2}+\epsilon^{-2.5}, \epsilon^{-3} ) \right)$.

Nowadays, asynchronous parallel algorithms have received much attention in the optimization field due to the crucial demands for modern large-scale optimization problems. However, most asynchronous algorithms focus on convex problems. Analysis on nonconvex problems is lacking. For the Asynchronous Stochastic Descent (ASGD) algorithm, the best result from (Lian et al., 2015) can only achieve an asymptotic O(\frac{1}{\epsilon^2}) rate (convergence to the stationary points) on nonconvex problems. In this paper, we study Stochastic Variance Reduced Gradient (SVRG) in the asynchronous setting. We propose the Asynchronous Stochastic Variance Reduced Gradient (ASVRG) algorithm for nonconvex finite-sum problems. We develop two schemes for ASVRG, depending on whether the parameters are updated as an atom or not. We prove that both of the two schemes can achieve linear speed up (a non-asymptotic O(\frac{n^\frac{2}{3}}{\epsilon}) rate to the stationary points) for nonconvex problems when the delay parameter \tau\leq n^{\frac{1}{3}}, where n is the number of training samples. We also establish a non-asymptotic O(\frac{n^\frac{2}{3}\tau^\frac{1}{3}}{\epsilon}) rate (convergence to the stationary points) for our algorithm without assumptions on \tau. This further demonstrates that even with asynchronous updating, SVRG has less number of Incremental First-order Oracles (IFOs) compared with Stochastic Gradient Descent and Gradient Descent. We also experiment on a shared memory multi-core system to demonstrate the efficiency of our algorithm.

We study stochastic convex optimization subjected to linear equality constraints. Traditional Stochastic Alternating Direction Method of Multipliers and its Nesterov's acceleration scheme can only achieve ergodic O(1/\sqrt{K}) convergence rates, where K is the number of iteration. By introducing Variance Reduction (VR) techniques, the convergence rates improve to ergodic O(1/K). In this paper, we propose a new stochastic ADMM which elaborately integrates Nesterov's extrapolation and VR techniques. With Nesterov’s extrapolation, our algorithm can achieve a non-ergodic O(1/K) convergence rate which is optimal for separable linearly constrained non-smooth convex problems, while the convergence rates of VR based ADMM methods are actually tight O(1/\sqrt{K}) in non-ergodic sense. To the best of our knowledge, this is the first work that achieves a truly accelerated, stochastic convergence rate for constrained convex problems. The experimental results demonstrate that our algorithm is significantly faster than the existing state-of-the-art stochastic ADMM methods.