Zhenyu Liao

Zenan Ling, Zhenyu Liao, Robert C. Qiu

Implicit neural networks have demonstrated remarkable success in various tasks. However, there is a lack of theoretical analysis of the connections and differences between implicit and explicit networks. In this paper, we study high-dimensional implicit neural networks and provide the high dimensional equivalents for the corresponding conjugate kernels and neural tangent kernels. Built upon this, we establish the equivalence between implicit and explicit networks in high dimensions.

Zhanbo Feng, Yuanjie Wang, Jie Li et al.

Modern machine learning (ML) models have grown to a scale where training them on a single machine becomes impractical. As a result, there is a growing trend to leverage federated learning (FL) techniques to train large ML models in a distributed and collaborative manner. These models, however, when deployed on new devices, might struggle to generalize well due to domain shifts. In this context, federated domain adaptation (FDA) emerges as a powerful approach to address this challenge. Most existing FDA approaches typically focus on aligning the distributions between source and target domains by minimizing their (e.g., MMD) distance. Such strategies, however, inevitably introduce high communication overheads and can be highly sensitive to network reliability. In this paper, we introduce RF-TCA, an enhancement to the standard Transfer Component Analysis approach that significantly accelerates computation without compromising theoretical and empirical performance. Leveraging the computational advantage of RF-TCA, we further extend it to FDA setting with FedRF-TCA. The proposed FedRF-TCA protocol boasts communication complexity that is independent of the sample size, while maintaining performance that is either comparable to or even surpasses state-of-the-art FDA methods. We present extensive experiments to showcase the superior performance and robustness (to network condition) of FedRF-TCA.

Lingyu Gu, Yongqi Du, Yuan Zhang et al.

Modern deep neural networks (DNNs) are extremely powerful; however, this comes at the price of increased depth and having more parameters per layer, making their training and inference more computationally challenging. In an attempt to address this key limitation, efforts have been devoted to the compression (e.g., sparsification and/or quantization) of these large-scale machine learning models, so that they can be deployed on low-power IoT devices. In this paper, building upon recent advances in neural tangent kernel (NTK) and random matrix theory (RMT), we provide a novel compression approach to wide and fully-connected \emph{deep} neural nets. Specifically, we demonstrate that in the high-dimensional regime where the number of data points $n$ and their dimension $p$ are both large, and under a Gaussian mixture model for the data, there exists \emph{asymptotic spectral equivalence} between the NTK matrices for a large family of DNN models. This theoretical result enables "lossless" compression of a given DNN to be performed, in the sense that the compressed network yields asymptotically the same NTK as the original (dense and unquantized) network, with its weights and activations taking values \emph{only} in $\{ 0, \pm 1 \}$ up to a scaling. Experiments on both synthetic and real-world data are conducted to support the advantages of the proposed compression scheme, with code available at \url{https://github.com/Model-Compression/Lossless_Compression}.

Yingwei Li, Song Bai, Cihang Xie et al.

This paper focuses on learning transferable adversarial examples specifically against defense models (models to defense adversarial attacks). In particular, we show that a simple universal perturbation can fool a series of state-of-the-art defenses. Adversarial examples generated by existing attacks are generally hard to transfer to defense models. We observe the property of regional homogeneity in adversarial perturbations and suggest that the defenses are less robust to regionally homogeneous perturbations. Therefore, we propose an effective transforming paradigm and a customized gradient transformer module to transform existing perturbations into regionally homogeneous ones. Without explicitly forcing the perturbations to be universal, we observe that a well-trained gradient transformer module tends to output input-independent gradients (hence universal) benefiting from the under-fitting phenomenon. Thorough experiments demonstrate that our work significantly outperforms the prior art attacking algorithms (either image-dependent or universal ones) by an average improvement of 14.0% when attacking 9 defenses in the transfer-based attack setting. In addition to the cross-model transferability, we also verify that regionally homogeneous perturbations can well transfer across different vision tasks (attacking with the semantic segmentation task and testing on the object detection task). The code is available here: https://github.com/LiYingwei/Regional-Homogeneity.

Zhenyu Liao, Yuanqian Xia, Chengmei Niu et al.

Random graph models are playing an increasingly important role in various fields ranging from social networks, telecommunication systems, to physiologic and biological networks. Within this landscape, the random Kronecker graph model, emerges as a prominent framework for scrutinizing intricate real-world networks. In this paper, we investigate large random Kronecker graphs, i.e., the number of graph vertices $N$ is large. Built upon recent advances in random matrix theory (RMT) and high-dimensional statistics, we prove that the adjacency of a large random Kronecker graph can be decomposed, in a spectral norm sense, into two parts: a small-rank (of rank $O(\log N)$) signal matrix that is linear in the graph parameters and a zero-mean random noise matrix. Based on this result, we propose a ``denoise-and-solve'' approach to infer the key graph parameters, with significantly reduced computational complexity. Experiments on both graph inference and classification are presented to evaluate the our proposed method. In both tasks, the proposed approach yields comparable or advantageous performance, than widely-used graph inference (e.g., KronFit) and graph neural net baselines, at a time cost that scales linearly as the graph size $N$.

Zhenyu Liao, Michael W. Mahoney

Modern Machine Learning (ML) and Deep Neural Networks (DNNs) often operate on high-dimensional data and rely on overparameterized models, where classical low-dimensional intuitions break down. In particular, the proportional regime where the data dimension, sample size, and number of model parameters are all large and comparable, gives rise to novel and sometimes counterintuitive behaviors. This paper extends traditional Random Matrix Theory (RMT) beyond eigenvalue-based analysis of linear models to address the challenges posed by nonlinear ML models such as DNNs in this regime. We introduce the concept of High-dimensional Equivalent, which unifies and generalizes both Deterministic Equivalent and Linear Equivalent, to systematically address three technical challenges: high dimensionality, nonlinearity, and the need to analyze generic eigenspectral functionals. Leveraging this framework, we provide precise characterizations of the training and generalization performance of linear models, nonlinear shallow networks, and deep networks. Our results capture rich phenomena, including scaling laws, double descent, and nonlinear learning dynamics, offering a unified perspective on the theoretical understanding of deep learning in high dimensions.

Xinjie Li, Zhimin Chen, Rui Zhao et al.

Recent unified models for joint understanding and generation have significantly advanced visual generation capabilities. However, their focus on conventional tasks like text-to-video generation has left the temporal reasoning potential of unified models largely underexplored. To address this gap, we introduce Next Scene Prediction (NSP), a new task that pushes unified video models toward temporal and causal reasoning. Unlike text-to-video generation, NSP requires predicting plausible futures from preceding context, demanding deeper understanding and reasoning. To tackle this task, we propose a unified framework combining Qwen-VL for comprehension and LTX for synthesis, bridged by a latent query embedding and a connector module. This model is trained in three stages on our newly curated, large-scale NSP dataset: text-to-video pre-training, supervised fine-tuning, and reinforcement learning (via GRPO) with our proposed causal consistency reward. Experiments demonstrate our model achieves state-of-the-art performance on our benchmark, advancing the capability of generalist multimodal systems to anticipate what happens next.

Jiachen Li, Shuo Cheng, Zhenyu Liao et al.

Improving the sample efficiency of reinforcement learning algorithms requires effective exploration. Following the principle of $\textit{optimism in the face of uncertainty}$ (OFU), we train a separate exploration policy to maximize the approximate upper confidence bound of the critics in an off-policy actor-critic framework. However, this introduces extra differences between the replay buffer and the target policy regarding their stationary state-action distributions. To mitigate the off-policy-ness, we adapt the recently introduced DICE framework to learn a distribution correction ratio for off-policy RL training. In particular, we correct the training distribution for both policies and critics. Empirically, we evaluate our proposed method in several challenging continuous control tasks and show superior performance compared to state-of-the-art methods. We also conduct extensive ablation studies to demonstrate the effectiveness and rationality of the proposed method.

Hafiz Tiomoko Ali, Zhenyu Liao, Romain Couillet

In this article, we investigate the spectral behavior of random features kernel matrices of the type ${\bf K} = \mathbb{E}_{\bf w} \left[σ\left({\bf w}^{\sf T}{\bf x}_i\right)σ\left({\bf w}^{\sf T}{\bf x}_j\right)\right]_{i,j=1}^n$, with nonlinear function $σ(\cdot)$, data ${\bf x}_1, \ldots, {\bf x}_n \in \mathbb{R}^p$, and random projection vector ${\bf w} \in \mathbb{R}^p$ having i.i.d. entries. In a high-dimensional setting where the number of data $n$ and their dimension $p$ are both large and comparable, we show, under a Gaussian mixture model for the data, that the eigenspectrum of ${\bf K}$ is independent of the distribution of the i.i.d.(zero-mean and unit-variance) entries of ${\bf w}$, and only depends on $σ(\cdot)$ via its (generalized) Gaussian moments $\mathbb{E}_{z\sim \mathcal N(0,1)}[σ'(z)]$ and $\mathbb{E}_{z\sim \mathcal N(0,1)}[σ''(z)]$. As a result, for any kernel matrix ${\bf K}$ of the form above, we propose a novel random features technique, called Ternary Random Feature (TRF), that (i) asymptotically yields the same limiting kernel as the original ${\bf K}$ in a spectral sense and (ii) can be computed and stored much more efficiently, by wisely tuning (in a data-dependent manner) the function $σ$ and the random vector ${\bf w}$, both taking values in $\{-1,0,1\}$. The computation of the proposed random features requires no multiplication, and a factor of $b$ times less bits for storage compared to classical random features such as random Fourier features, with $b$ the number of bits to store full precision values. Besides, it appears in our experiments on real data that the substantial gains in computation and storage are accompanied with somewhat improved performances compared to state-of-the-art random features compression/quantization methods.

Zhenyu Liao, Michael W. Mahoney

Given an optimization problem, the Hessian matrix and its eigenspectrum can be used in many ways, ranging from designing more efficient second-order algorithms to performing model analysis and regression diagnostics. When nonlinear models and non-convex problems are considered, strong simplifying assumptions are often made to make Hessian spectral analysis more tractable. This leads to the question of how relevant the conclusions of such analyses are for more realistic nonlinear models. In this paper, we exploit deterministic equivalent techniques from random matrix theory to make a \emph{precise} characterization of the Hessian eigenspectra for a broad family of nonlinear models, including models that generalize the classical generalized linear models, without relying on strong simplifying assumptions used previously. We show that, depending on the data properties, the nonlinear response model, and the loss function, the Hessian can have \emph{qualitatively} different spectral behaviors: of bounded or unbounded support, with single- or multi-bulk, and with isolated eigenvalues on the left- or right-hand side of the bulk. By focusing on such a simple but nontrivial nonlinear model, our analysis takes a step forward to unveil the theoretical origin of many visually striking features observed in more complex machine learning models.

Michał Dereziński, Zhenyu Liao, Edgar Dobriban et al.

For a tall $n\times d$ matrix $A$ and a random $m\times n$ sketching matrix $S$, the sketched estimate of the inverse covariance matrix $(A^\top A)^{-1}$ is typically biased: $E[(\tilde A^\top\tilde A)^{-1}]\ne(A^\top A)^{-1}$, where $\tilde A=SA$. This phenomenon, which we call inversion bias, arises, e.g., in statistics and distributed optimization, when averaging multiple independently constructed estimates of quantities that depend on the inverse covariance. We develop a framework for analyzing inversion bias, based on our proposed concept of an $(ε,δ)$-unbiased estimator for random matrices. We show that when the sketching matrix $S$ is dense and has i.i.d. sub-gaussian entries, then after simple rescaling, the estimator $(\frac m{m-d}\tilde A^\top\tilde A)^{-1}$ is $(ε,δ)$-unbiased for $(A^\top A)^{-1}$ with a sketch of size $m=O(d+\sqrt d/ε)$. This implies that for $m=O(d)$, the inversion bias of this estimator is $O(1/\sqrt d)$, which is much smaller than the $Θ(1)$ approximation error obtained as a consequence of the subspace embedding guarantee for sub-gaussian sketches. We then propose a new sketching technique, called LEverage Score Sparsified (LESS) embeddings, which uses ideas from both data-oblivious sparse embeddings as well as data-aware leverage-based row sampling methods, to get $ε$ inversion bias for sketch size $m=O(d\log d+\sqrt d/ε)$ in time $O(\text{nnz}(A)\log n+md^2)$, where nnz is the number of non-zeros. The key techniques enabling our analysis include an extension of a classical inequality of Bai and Silverstein for random quadratic forms, which we call the Restricted Bai-Silverstein inequality; and anti-concentration of the Binomial distribution via the Paley-Zygmund inequality, which we use to prove a lower bound showing that leverage score sampling sketches generally do not achieve small inversion bias.

Fanghui Liu, Zhenyu Liao, Johan A. K. Suykens

In this paper, we provide a precise characterization of generalization properties of high dimensional kernel ridge regression across the under- and over-parameterized regimes, depending on whether the number of training data n exceeds the feature dimension d. By establishing a bias-variance decomposition of the expected excess risk, we show that, while the bias is (almost) independent of d and monotonically decreases with n, the variance depends on n, d and can be unimodal or monotonically decreasing under different regularization schemes. Our refined analysis goes beyond the double descent theory by showing that, depending on the data eigen-profile and the level of regularization, the kernel regression risk curve can be a double-descent-like, bell-shaped, or monotonic function of n. Experiments on synthetic and real data are conducted to support our theoretical findings.

Zhenyu Liao, Romain Couillet, Michael W. Mahoney

Given a large data matrix, sparsifying, quantizing, and/or performing other entry-wise nonlinear operations can have numerous benefits, ranging from speeding up iterative algorithms for core numerical linear algebra problems to providing nonlinear filters to design state-of-the-art neural network models. Here, we exploit tools from random matrix theory to make precise statements about how the eigenspectrum of a matrix changes under such nonlinear transformations. In particular, we show that very little change occurs in the informative eigenstructure even under drastic sparsification/quantization, and consequently that very little downstream performance loss occurs with very aggressively sparsified or quantized spectral clustering. We illustrate how these results depend on the nonlinearity, we characterize a phase transition beyond which spectral clustering becomes possible, and we show when such nonlinear transformations can introduce spurious non-informative eigenvectors.

Michał Dereziński, Feynman Liang, Zhenyu Liao et al.

It is often desirable to reduce the dimensionality of a large dataset by projecting it onto a low-dimensional subspace. Matrix sketching has emerged as a powerful technique for performing such dimensionality reduction very efficiently. Even though there is an extensive literature on the worst-case performance of sketching, existing guarantees are typically very different from what is observed in practice. We exploit recent developments in the spectral analysis of random matrices to develop novel techniques that provide provably accurate expressions for the expected value of random projection matrices obtained via sketching. These expressions can be used to characterize the performance of dimensionality reduction in a variety of common machine learning tasks, ranging from low-rank approximation to iterative stochastic optimization. Our results apply to several popular sketching methods, including Gaussian and Rademacher sketches, and they enable precise analysis of these methods in terms of spectral properties of the data. Empirical results show that the expressions we derive reflect the practical performance of these sketching methods, down to lower-order effects and even constant factors.

Zhenyu Liao, Romain Couillet, Michael W. Mahoney

This article characterizes the exact asymptotics of random Fourier feature (RFF) regression, in the realistic setting where the number of data samples $n$, their dimension $p$, and the dimension of feature space $N$ are all large and comparable. In this regime, the random RFF Gram matrix no longer converges to the well-known limiting Gaussian kernel matrix (as it does when $N \to \infty$ alone), but it still has a tractable behavior that is captured by our analysis. This analysis also provides accurate estimates of training and test regression errors for large $n,p,N$. Based on these estimates, a precise characterization of two qualitatively different phases of learning, including the phase transition between them, is provided; and the corresponding double descent test error curve is derived from this phase transition behavior. These results do not depend on strong assumptions on the data distribution, and they perfectly match empirical results on real-world data sets.

Qing Jin, Linjie Yang, Zhenyu Liao

Quantization reduces computation costs of neural networks but suffers from performance degeneration. Is this accuracy drop due to the reduced capacity, or inefficient training during the quantization procedure? After looking into the gradient propagation process of neural networks by viewing the weights and intermediate activations as random variables, we discover two critical rules for efficient training. Recent quantization approaches violates the two rules and results in degenerated convergence. To deal with this problem, we propose a simple yet effective technique, named scale-adjusted training (SAT), to comply with the discovered rules and facilitates efficient training. We also analyze the quantization error introduced in calculating the gradient in the popular parameterized clipping activation (PACT) technique. Through SAT together with gradient-calibrated PACT, quantized models obtain comparable or even better performance than their full-precision counterparts, achieving state-of-the-art accuracy with consistent improvement over previous quantization methods on a wide spectrum of models including MobileNet-V1/V2 and PreResNet-50.

Qing Jin, Linjie Yang, Zhenyu Liao

Deep neural networks with adaptive configurations have gained increasing attention due to the instant and flexible deployment of these models on platforms with different resource budgets. In this paper, we investigate a novel option to achieve this goal by enabling adaptive bit-widths of weights and activations in the model. We first examine the benefits and challenges of training quantized model with adaptive bit-widths, and then experiment with several approaches including direct adaptation, progressive training and joint training. We discover that joint training is able to produce comparable performance on the adaptive model as individual models. We further propose a new technique named Switchable Clipping Level (S-CL) to further improve quantized models at the lowest bit-width. With our proposed techniques applied on a bunch of models including MobileNet-V1/V2 and ResNet-50, we demonstrate that bit-width of weights and activations is a new option for adaptively executable deep neural networks, offering a distinct opportunity for improved accuracy-efficiency trade-off as well as instant adaptation according to the platform constraints in real-world applications.

Zhenyu Liao, Romain Couillet

This article investigates the eigenspectrum of the inner product-type kernel matrix $\sqrt{p} \mathbf{K}=\{f( \mathbf{x}_i^{\sf T} \mathbf{x}_j/\sqrt{p})\}_{i,j=1}^n $ under a binary mixture model in the high dimensional regime where the number of data $n$ and their dimension $p$ are both large and comparable. Based on recent advances in random matrix theory, we show that, for a wide range of nonlinear functions $f$, the eigenspectrum behavior is asymptotically equivalent to that of an (at most) cubic function. This sheds new light on the understanding of nonlinearity in large dimensional problems. As a byproduct, we propose a simple function prototype valued in $ (-1,0,1) $ that, while reducing both storage memory and running time, achieves the same (asymptotic) classification performance as any arbitrary function $f$.

Yuexiang Zhai, Zitong Yang, Zhenyu Liao et al.

This paper considers the fundamental problem of learning a complete (orthogonal) dictionary from samples of sparsely generated signals. Most existing methods solve the dictionary (and sparse representations) based on heuristic algorithms, usually without theoretical guarantees for either optimality or complexity. The recent $\ell^1$-minimization based methods do provide such guarantees but the associated algorithms recover the dictionary one column at a time. In this work, we propose a new formulation that maximizes the $\ell^4$-norm over the orthogonal group, to learn the entire dictionary. We prove that under a random data model, with nearly minimum sample complexity, the global optima of the $\ell^4$ norm are very close to signed permutations of the ground truth. Inspired by this observation, we give a conceptually simple and yet effective algorithm based on "matching, stretching, and projection" (MSP). The algorithm provably converges locally at a superlinear (cubic) rate and cost per iteration is merely an SVD. In addition to strong theoretical guarantees, experiments show that the new algorithm is significantly more efficient and effective than existing methods, including KSVD and $\ell^1$-based methods. Preliminary experimental results on mixed real imagery data clearly demonstrate advantages of so learned dictionary over classic PCA bases.

Xiaoyi Mai, Zhenyu Liao

This article provides, through theoretical analysis, an in-depth understanding of the classification performance of the empirical risk minimization framework, in both ridge-regularized and unregularized cases, when high dimensional data are considered. Focusing on the fundamental problem of separating a two-class Gaussian mixture, the proposed analysis allows for a precise prediction of the classification error for a set of numerous data vectors $\mathbf{x} \in \mathbb R^p$ of sufficiently large dimension $p$. This precise error depends on the loss function, the number of training samples, and the statistics of the mixture data model. It is shown to hold beyond Gaussian distribution under some additional non-sparsity condition of the data statistics. Building upon this quantitative error analysis, we identify the simple square loss as the optimal choice for high dimensional classification in both ridge-regularized and unregularized cases, regardless of the number of training samples.

Yacine Chitour, Zhenyu Liao, Romain Couillet

In this paper, we propose a geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. We translate a well-known empirical observation of linear neural nets into a conjecture that we call the \emph{overfitting conjecture} which states that, for almost all training data and initial conditions, the trajectory of the corresponding gradient descent system converges to a global minimum. This would imply that the solution achieved by vanilla gradient descent algorithms is equivalent to that of the least-squares estimation, for linear neural networks of an arbitrary number of hidden layers. Built upon a key invariance property induced by the network structure, we first establish convergence of gradient descent trajectories to critical points of the square loss function in the case of linear networks of arbitrary depth. Our second result is the proof of the \emph{overfitting conjecture} in the case of single-hidden-layer linear networks with an argument based on the notion of normal hyperbolicity and under a generic property on the training data (i.e., holding for almost all training data).

Zhenyu Liao, Romain Couillet

Understanding the learning dynamics of neural networks is one of the key issues for the improvement of optimization algorithms as well as for the theoretical comprehension of why deep neural nets work so well today. In this paper, we introduce a random matrix-based framework to analyze the learning dynamics of a single-layer linear network on a binary classification problem, for data of simultaneously large dimension and size, trained by gradient descent. Our results provide rich insights into common questions in neural nets, such as overfitting, early stopping and the initialization of training, thereby opening the door for future studies of more elaborate structures and models appearing in today's neural networks.

Zhenyu Liao, Romain Couillet

Random feature maps are ubiquitous in modern statistical machine learning, where they generalize random projections by means of powerful, yet often difficult to analyze nonlinear operators. In this paper, we leverage the "concentration" phenomenon induced by random matrix theory to perform a spectral analysis on the Gram matrix of these random feature maps, here for Gaussian mixture models of simultaneously large dimension and size. Our results are instrumental to a deeper understanding on the interplay of the nonlinearity and the statistics of the data, thereby allowing for a better tuning of random feature-based techniques.

Cosme Louart, Zhenyu Liao, Romain Couillet

This article studies the Gram random matrix model $G=\frac1TΣ^{\rm T}Σ$, $Σ=σ(WX)$, classically found in the analysis of random feature maps and random neural networks, where $X=[x_1,\ldots,x_T]\in{\mathbb R}^{p\times T}$ is a (data) matrix of bounded norm, $W\in{\mathbb R}^{n\times p}$ is a matrix of independent zero-mean unit variance entries, and $σ:{\mathbb R}\to{\mathbb R}$ is a Lipschitz continuous (activation) function --- $σ(WX)$ being understood entry-wise. By means of a key concentration of measure lemma arising from non-asymptotic random matrix arguments, we prove that, as $n,p,T$ grow large at the same rate, the resolvent $Q=(G+γI_T)^{-1}$, for $γ>0$, has a similar behavior as that met in sample covariance matrix models, involving notably the moment $Φ=\frac{T}n{\mathbb E}[G]$, which provides in passing a deterministic equivalent for the empirical spectral measure of $G$. Application-wise, this result enables the estimation of the asymptotic performance of single-layer random neural networks. This in turn provides practical insights into the underlying mechanisms into play in random neural networks, entailing several unexpected consequences, as well as a fast practical means to tune the network hyperparameters.

Zhenyu Liao, Romain Couillet

In this article, a large dimensional performance analysis of kernel least squares support vector machines (LS-SVMs) is provided under the assumption of a two-class Gaussian mixture model for the input data. Building upon recent advances in random matrix theory, we show, when the dimension of data $p$ and their number $n$ are both large, that the LS-SVM decision function can be well approximated by a normally distributed random variable, the mean and variance of which depend explicitly on a local behavior of the kernel function. This theoretical result is then applied to the MNIST and Fashion-MNIST datasets which, despite their non-Gaussianity, exhibit a convincingly close behavior. Most importantly, our analysis provides a deeper understanding of the mechanism into play in SVM-type methods and in particular of the impact on the choice of the kernel function as well as some of their theoretical limits in separating high dimensional Gaussian vectors.

Zhenyu Liao, Romain Couillet

This article proposes a performance analysis of kernel least squares support vector machines (LS-SVMs) based on a random matrix approach, in the regime where both the dimension of data $p$ and their number $n$ grow large at the same rate. Under a two-class Gaussian mixture model for the input data, we prove that the LS-SVM decision function is asymptotically normal with means and covariances shown to depend explicitly on the derivatives of the kernel function. This provides improved understanding along with new insights into the internal workings of SVM-type methods for large datasets.

Zeyuan Allen-Zhu, Zhenyu Liao, Lorenzo Orecchia

In this paper, we provide a novel construction of the linear-sized spectral sparsifiers of Batson, Spielman and Srivastava [BSS14]. While previous constructions required $Ω(n^4)$ running time [BSS14, Zou12], our sparsification routine can be implemented in almost-quadratic running time $O(n^{2+\varepsilon})$. The fundamental conceptual novelty of our work is the leveraging of a strong connection between sparsification and a regret minimization problem over density matrices. This connection was known to provide an interpretation of the randomized sparsifiers of Spielman and Srivastava [SS11] via the application of matrix multiplicative weight updates (MWU) [CHS11, Vis14]. In this paper, we explain how matrix MWU naturally arises as an instance of the Follow-the-Regularized-Leader framework and generalize this approach to yield a larger class of updates. This new class allows us to accelerate the construction of linear-sized spectral sparsifiers, and give novel insights on the motivation behind Batson, Spielman and Srivastava [BSS14].