Consistency models, a new class of one-step generative models, have shown competitive performance with multi-step diffusion models. The most challenging part of consistency models is the training process, which discretizes the continuous diffusion process into $K$ steps and trains a one-step mapping function on these discretized timepoints. Despite the empirical success, only a few works focus on the discretization complexity $K$, and their setting is far from that of empirical works. More specifically, the current theoretical works analyze the variance preserving (VP) diffusion process with a uniform stepsize, while empirical works adopt a variance exploding (VE) process with a decay discretization stepsize. As a result, these works suffer from large discretization complexity and fail to explain the empirical success of consistency models. To close the gap between theory and application, we analyze consistency models with (1) VE process and (2) decay stepsize and prove the state-of-the-art discretization complexity for consistency models. This result is competitive with the results of diffusion models and shows the potential of consistency models. To balance the computation and performance, previous empirical work further proposes a $2$-step consistency algorithm. In this work, we also analyze the role of $2$-step sampling and show that it improves the discretization complexity compared with one-step generation.

Abstract Unbalanced dimensions are crucial characteristics in various minimax optimization problems, such as few-shot learning (Cortes and Mohri in Adv Neural Inf Process Syst 16, 2003; Ying et al. in Adv Neural Inf Process Syst 29, 2016) and fairness-aware machine learning (Lowd and Meek, in: Proceedings of the eleventh ACM SIGKDD international conference on knowledge discovery in data mining, 2005; Zhang et al., in: Proceedings of the 2018 AAAI/ACM conference on AI, ethics, and society, 2018). In this paper, we propose a communication-efficient second-order method named (Partially Approximate Newton methods for Distributed minimAx) to solve problems with unbalanced dimensions. requires almost the same per-iteration communication cost as the first-order methods by utilizing the special problem structure in its design for data exchange between the client and server. More importantly, it exhibits a superior linear-quadratic convergence rate and significantly reduces the total number of communication rounds through the efficient use of second-order information. We also develop based on the framework of , which further reduces the computation cost of the latter one by performing sketching operations on each client. Through comprehensive theoretical analysis and empirical evaluations, we demonstrate the superior performance of the proposed methods compared to existing state-of-the-art methods.

Deep Reinforcement Learning (DRL) has gained prominence as an effective approach for control systems. However, its practical deployment is impeded by state perturbations that can severely impact system performance. Addressing this critical challenge requires robustness verification about system performance, which involves tackling two quantitative questions: (i) how to establish guaranteed bounds for expected cumulative rewards, and (ii) how to determine tail bounds for cumulative rewards. In this work, we present the first approach for robustness verification of DRL-based control systems by introducing reward martingales, which offer a rigorous mathematical foundation to characterize the impact of state perturbations on system performance in terms of cumulative rewards. Our verified results provide provably quantitative certificates for the two questions. We then show that reward martingales can be implemented and trained via neural networks, against different types of control policies. Experimental results demonstrate that our certified bounds tightly enclose simulation outcomes on various DRL-based control systems, indicating the effectiveness and generality of the proposed approach.

This paper studies the problem of solving nonconvex nonsmooth optimization over a closed convex set. Most previous works tackle such problems by transforming the constrained problem into an unconstrained problem that can be solved by the techniques developed in the unconstrained setting. However, they only provide asymptotic convergence analysis for their methods. In this work, we provide the non-asymptotic analysis for solving constrained nonconvex nonsmooth optimization. We first generalize classical gradient mapping and the Frank–Wolfe gap in the nonsmooth setting. Then we introduce novel notions of approximate stationarity concerning such generalized quantities. We also propose several stochastic zeroth-order algorithms for the problem, along with their non-asymptotic convergence guarantees of obtaining the proposed approximate stationarity. Finally, we conduct numerical experiments that demonstrate the effectiveness of our algorithms.

Although Federated Learning (FL) enables global model training across clients without compromising their raw data, due to the un- evenly distributed data among clients, existing Federated Averaging (FedAvg)-based methods suffer from the problem of low inference performance. Specifically, different data distributions among clients lead to various optimization directions of local models. Aggregat- ing local models usually results in a low-generalized global model, which performs worse on most of the clients. To address the above issue, inspired by the observation from a geometric perspective that a well-generalized solution is located in a flat area rather than a sharp area, we propose a novel and heuristic FL paradigm named FedMR (Federated Model Recombination). The goal of FedMR is to guide the recombined models to be trained towards a flat area. Unlike conventional FedAvg-based methods, in FedMR, the cloud server recombines collected local models by shuffling each layer of them to generate multiple recombined models for local training on clients rather than an aggregated global model. Since the area of the flat area is larger than the sharp area, when local models are located in different areas, recombined models have a higher probability of locating in a flat area. When all recombined models are located in the same flat area, they are optimized towards the same direction. We theoretically analyze the convergence of model recombination. Experimental results show that, compared with state-of-the-art FL methods, FedMR can significantly improve the inference accuracy without exposing the privacy of each client.

Large-scale streaming matrix multiplication is very common in various applications, sparking significant interest in develop efficient algorithms for approximate matrix multiplication (AMM) over streams. In addition, many practical scenarios require to process time-sensitive data and aim to compute matrix multiplication for most recent columns of the data matrices rather than the entire matrices, which motivated us to study efficient AMM algorithms over sliding windows. In this paper, we present two novel deterministic algorithms for this problem and provide corresponding error guarantees. We further reduce the space and time costs of our methods for sparse matrices by performing an approximate singular value decomposition which can utilize the sparsity of matrices. Extensive experimental results on both synthetic and real-world datasets validate our theoretical analysis and highlight the efficiency of our methods.

While diffusion models have demonstrated impressive performance, there is a growing need for generating samples tailored to specific user-defined concepts. The customized requirements promote the development of few-shot diffusion models, which use limited $n_{ta}$ target samples to fine-tune a pre-trained diffusion model trained on $n_s$ source samples. Despite the empirical success, no theoretical work specifically analyzes few-shot diffusion models. Moreover, the existing results for diffusion models without a fine-tuning phase can not explain why few-shot models generate great samples due to the curse of dimensionality. In this work, we analyze few-shot diffusion models under a linear structure distribution with a latent dimension $d$. From the approximation perspective, we prove that few-shot models have a $\widetilde{O}(n_s^{-2/d}+n_{ta}^{-1/2})$ bound to approximate the target score function, which is better than $n_{ta}^{-2/d}$ results. From the optimization perspective, we consider a latent Gaussian special case and prove that the optimization problem has a closed-form minimizer. This means few-shot models can directly obtain an approximated minimizer without a complex optimization process. Furthermore, we also provide the accuracy bound $\widetilde{O}(1/n_{ta}+1/\sqrt{n_s})$ for the empirical solution, which still has better dependence on $n_{ta}$ compared to $n_s$. The results of the real-world experiments also show that the models obtained by only fine-tuning the encoder and decoder specific to the target distribution can produce novel images with the target feature, which supports our theoretical results.

This paper studies quasi-Newton methods for solving nonlinear equations. We propose block variants of both good and bad Broyden's methods, which enjoy explicit local superlinear convergence rates. Our block good Broyden's method has faster condition-number-free convergence rate than existing Broyden's methods because it takes the advantage of multiple rank modification on the Jacobian estimator. On the other hand, our block bad Broyden's method directly estimates the inverse of the Jacobian provably, which reduces the computational cost of the iteration. Our theoretical results provide some new insights on why good Broyden's method outperforms bad Broyden's method in most of the cases. The empirical results also demonstrate the superiority of our methods and validate our theoretical analysis.

Formally verifying deep reinforcement learning (DRL) systems suffers from both inaccurate verification results and limited scalability. The major obstacle lies in the large overestimation introduced inherently during training and then transforming the inexplicable decision-making models, i.e., deep neural networks (DNNs), into easy-to-verify models. In this paper, we propose an inverse transform-then-train approach, which first encodes a DNN into an equivalent set of efficiently and tightly verifiable linear control policies and then optimizes them via reinforcement learning. We accompany our inverse approach with a novel neural network model called piece-wise linear decision neural networks (PLDNNs), which are compatible with most existing DRL training algorithms with comparable performance against conventional DNNs. Our extensive experiments show that, compared to DNN-based DRL systems, PLDNN-based systems can be more efficiently and tightly verified with up to $438$ times speedup and a significant reduction in overestimation. In particular, even a complex $12$-dimensional DRL system is efficiently verified with up to 7 times deeper computation steps.

Online learning to rank (OLTR) interactively learns to choose lists of items from a large collection based on certain click models that describe users' click behaviors. Most recent works for this problem focus on the stochastic environment where the item attractiveness is assumed to be invariant during the learning process. In many real-world scenarios, however, the environment could be dynamic or even arbitrarily changing. This work studies the OLTR problem in both stochastic and adversarial environments under the position-based model (PBM). We propose a method based on the follow-the-regularized-leader (FTRL) framework with Tsallis entropy and develop a new self-bounding constraint especially designed for PBM. We prove the proposed algorithm simultaneously achieves O(log T) regret in the stochastic environment and O(m√nT) regret in the adversarial environment, where T is the number of rounds, n is the number of items and m is the number of positions. We also provide a lower bound of order Ω(m√nT) for adversarial PBM, which matches our upper bound and improves over the state-of-the-art lower bound. The experiments show that our algorithm could simultaneously learn in both stochastic and adversarial environments and is competitive compared to existing methods that are designed for a single environment.

Active regression considers a linear regression problem where the learner receives a large number of data points but can only observe a small number of labels. Since online algorithms can deal with incremental training data and take advantage of low computational cost, we consider an online extension of the active regression problem: the learner receives data points one by one and immediately decides whether it should collect the corresponding labels. The goal is to efficiently maintain the regression of received data points with a small budget of label queries. We propose novel algorithms for this problem under $\ell_p$ loss where $p\in[1,2]$. To achieve a $(1+\epsilon)$-approximate solution, our proposed algorithms only requires $\tilde{\mathcal{O}}(d/poly(\epsilon))$ queries of labels. The numerical results verify our theoretical results and show that our methods have comparable performance with offline active regression algorithms.

We study the smooth minimax optimization problem $\min_{\bf x}\max_{\bf y} f({\bf x},{\bf y})$, where $f$ is $\ell$-smooth, strongly-concave in ${\bf y}$ but possibly nonconvex in ${\bf x}$. Most of existing works focus on finding the first-order stationary point of the function $f({\bf x},{\bf y})$ or its primal function $P({\bf x})\triangleq \max_{\bf y} f({\bf x},{\bf y})$, but few of them focus on achieving the second-order stationary point, which is essential to nonconvex problems. In this paper, we propose a novel approach for minimax optimization, called Minimax Cubic Newton (MCN), which could find an ${\mathcal O}\left(\varepsilon,\kappa^{1.5}\sqrt{\rho\varepsilon}\right)$-second-order stationary point of $P({\bf x})$ with calling ${\mathcal O}\left(\kappa^{1.5}\sqrt{\rho}\varepsilon^{-1.5}\right)$ times of second-order oracles and $\tilde{\mathcal O}\left(\kappa^{2}\sqrt{\rho}\varepsilon^{-1.5}\right)$ times of first-order oracles, where $\kappa$ is the condition number and $\rho$ is the Lipschitz continuous constant for the Hessian of $f({\bf x},{\bf y})$. In addition, we propose an inexact variant of MCN for high-dimensional problems to avoid calling the expensive second-order oracles. Instead, our method solves the cubic sub-problem inexactly via gradient descent and matrix Chebyshev expansion. This strategy still obtains the desired approximate second-order stationary point with high probability but only requires $\tilde{\mathcal O}\left(\kappa^{1.5}\ell\varepsilon^{-2}\right)$ Hessian-vector oracle calls and $\tilde{\mathcal O}\left(\kappa^{2}\sqrt{\rho}\varepsilon^{-1.5}\right)$ first-order oracle calls. To the best of our knowledge, this is the first work that considers the non-asymptotic convergence behavior of finding second-order stationary points for minimax problems without the convex-concave assumptions.

Formula retrieval is an important research topic in Mathematical Information Retrieval (MIR).Most studies have focused on comparing formulae to determine the similarity between mathematical documents.However, two similar formulae may appear in completely different knowledge domains and have different meanings.Based on N-ary Tree-based Formula Embedding Model (NTFEM), we introduce a new hybrid retrieval model combining formula with its surrounding text for more accurate retrieval.Using keywords extraction technology, we extract keywords from text around the formula which can supplement the semantic information of formula.Then we get the representation vectors of keywords by FastText N-gram embedding model, and the representation vectors of formulae by NTFEM.Finally, documents are first sorted according to the similarity of keywords, and then the ranking results are optimized by formula similarity.Experimental results show that the accuracy of top-10 results is at least 20% higher than that of NTFEM and can be 50% in some specific topics.

Formula retrieval is an important research topic in Mathematical Information Retrieval (MIR). Most studies have focused on formula comparison to determine the similarity between mathematical documents. However, two similar formulae may appear in entirely different knowledge domains and have different meanings. Based on N-ary Tree-based Formula Embedding Model (NTFEM, our previous work in [Y. Dai, L. Chen, and Z. Zhang, An N-ary tree-based model for similarity evaluation on mathematical formulae, in Proc. 2020 IEEE Int. Conf. Systems, Man, and Cybernetics, 2020, pp. 2578–2584.], we introduce a new hybrid retrieval model, NTFEM-K, which combines formulae with their surrounding keywords for more accurate retrieval. By using keywords extraction technology, we extract keywords from context, which can supplement the semantic information of the formula. Then, we get the vector representations of keywords by FastText N-gram embedding model and the vector representations of formulae by NTFEM. Finally, documents are sorted according to the similarity between keywords, and then the ranking results are optimized by formula similarity. For performance evaluation, NTFEM-K is not only compared with NTFEM but also hybrid retrieval models combining formulae with long text and hybrid retrieval models combining formulae with their keywords using other keyword extraction algorithms. Experimental results show that the accuracy of top-10 results of NTFEM-K is at least 20% higher than that of NTFEM and can be 50% in some specific topics.

The CUR matrix decomposition is an important tool for low-rank matrix approximation. It approximates a data matrix though selecting a small number of columns and rows of the matrix. Those CUR algorithms with gap-dependent approximation bounds can obtain high approximation quality for matrices with good singular value spectrum decay, but they have impractically high time complexities. In this paper, we propose a novel CUR algorithm based on truncated LU factorization with an efficient variant of complete pivoting. Our algorithm has gap-dependent approximation bounds on both spectral and Frobenius norms while maintaining high efficiency. Numerical experiments demonstrate the effectiveness of our algorithm and verify our theoretical guarantees.

The linear contextual bandits is a sequential decision-making problem where an agent decides among sequential actions given their corresponding contexts. Since large-scale data sets become more and more common, we study the linear contextual bandits in high-dimensional situations. Recent works focus on employing matrix sketching methods to accelerating contextual bandits. However, the matrix approximation error will bring additional terms to the regret bound. In this paper we first propose a novel matrix sketching method which is called Spectral Compensation Frequent Directions (SCFD). Then we propose an efficient approach for contextual bandits by adopting SCFD to approximate the covariance matrices. By maintaining and manipulating sketched matrices, our method only needs O(md) space and O(md) updating time in each round, where d is the dimensionality of the data and m is the sketching size. Theoretical analysis reveals that our method has better regret bounds than previous methods in high-dimensional cases. Experimental results demonstrate the effectiveness of our algorithm and verify our theoretical guarantees.

The Frank-Wolfe algorithm is a classic method for constrained optimization problems. It has recently been popular in many machine learning applications because its projection-free property leads to more efficient iterations. In this paper, we study projection-free algorithms for convex-strongly-concave saddle point problems with complicated constraints. Our method combines Conditional Gradient Sliding with Mirror-Prox and show that it only requires $\tilde{\cO}(1/\sqrt{\epsilon})$ gradient evaluations and $\tilde{\cO}(1/\epsilon^2)$ linear optimizations in the batch setting. We also extend our method to the stochastic setting and propose first stochastic projection-free algorithms for saddle point problems. Experimental results demonstrate the effectiveness of our algorithms and verify our theoretical guarantees.

The frequent directions (FD) technique is a deterministic approach for online sketching that has many applications in machine learning. The conventional FD is a heuristic procedure that often outputs rank deficient matrices. To overcome the rank deficiency problem, we propose a new sketching strategy called robust frequent directions (RFD) by introducing a regularization term. RFD can be derived from an optimization problem. It updates the sketch matrix and the regularization term adaptively and jointly. RFD reduces the approximation error of FD without increasing the computational cost. We also apply RFD to online learning and propose an effective hyperparameter-free online Newton algorithm. We derive a regret bound for our online Newton algorithm based on RFD, which guarantees the robustness of the algorithm. The experimental studies demonstrate that the proposed method outperforms state-of-the-art second order online learning algorithms.