Xiao-Tong Yuan

Yunlong Zhou, Chen Zhao, Danyang Peng et al.

Accurate precipitation nowcasting is vital for disaster mitigation, but deep learning methods face a key trade-off: regression models produce over-smoothed, spectrally decaying predictions that blur convective details and violate turbulence power laws; diffusion models generate realistic yet unanchored hallucinations lacking physical grounding. We propose Spectral-Decoupled Iterative Refinement (SDIR), a deterministic framework that reformulates nowcasting as progressive frequency-decoupled refinement. SDIR first extracts a stable low-frequency synoptic skeleton, then iteratively refines high-frequency textures under physical constraints, eliminating both blurring and hallucinations. It features a dual-path design: the Synoptic Frequency-Guided Former (SFG-Former) with Scale-Adaptive Transformers for global structure, and the Fourier Residual Refiner (FR-Refiner) with Scale-Conditioned Fourier Neural Operators for fine residuals. A Physically Consistent Power Spectral Density (PCPSD) loss with dynamic masking enforces a turbulence-consistent spectral distribution. Experiments on three benchmarks show SDIR significantly outperforms SOTA methods in spatial accuracy while achieving spectral fidelity competitive with diffusion-based methods, enabling reliable high-resolution operational nowcasting. Code link: https://github.com/RuntimeWarning/SDIR.

Xiao-Tong Yuan, Ping Li

The FedProx algorithm is a simple yet powerful distributed proximal point optimization method widely used for federated learning (FL) over heterogeneous data. Despite its popularity and remarkable success witnessed in practice, the theoretical understanding of FedProx is largely underinvestigated: the appealing convergence behavior of FedProx is so far characterized under certain non-standard and unrealistic dissimilarity assumptions of local functions, and the results are limited to smooth optimization problems. In order to remedy these deficiencies, we develop a novel local dissimilarity invariant convergence theory for FedProx and its minibatch stochastic extension through the lens of algorithmic stability. As a result, we contribute to derive several new and deeper insights into FedProx for non-convex federated optimization including: 1) convergence guarantees independent on local dissimilarity type conditions; 2) convergence guarantees for non-smooth FL problems; and 3) linear speedup with respect to size of minibatch and number of sampled devices. Our theory for the first time reveals that local dissimilarity and smoothness are not must-have for FedProx to get favorable complexity bounds. Preliminary experimental results on a series of benchmark FL datasets are reported to demonstrate the benefit of minibatching for improving the sample efficiency of FedProx.

Xiao-Tong Yuan, Ping Li

In this paper, we analyze the generalization performance of the Iterative Hard Thresholding (IHT) algorithm widely used for sparse recovery problems. The parameter estimation and sparsity recovery consistency of IHT has long been known in compressed sensing. From the perspective of statistical learning, another fundamental question is how well the IHT estimation would predict on unseen data. This paper makes progress towards answering this open question by introducing a novel sparse generalization theory for IHT under the notion of algorithmic stability. Our theory reveals that: 1) under natural conditions on the empirical risk function over $n$ samples of dimension $p$, IHT with sparsity level $k$ enjoys an $\mathcal{\tilde O}(n^{-1/2}\sqrt{k\log(n)\log(p)})$ rate of convergence in sparse excess risk; 2) a tighter $\mathcal{\tilde O}(n^{-1/2}\sqrt{\log(n)})$ bound can be established by imposing an additional iteration stability condition on a hypothetical IHT procedure invoked to the population risk; and 3) a fast rate of order $\mathcal{\tilde O}\left(n^{-1}k(\log^3(n)+\log(p))\right)$ can be derived for strongly convex risk function under proper strong-signal conditions. The results have been substantialized to sparse linear regression and sparse logistic regression models to demonstrate the applicability of our theory. Preliminary numerical evidence is provided to confirm our theoretical predictions.

Xiao-Tong Yuan, Ping Li

The stochastic proximal point (SPP) methods have gained recent attention for stochastic optimization, with strong convergence guarantees and superior robustness to the classic stochastic gradient descent (SGD) methods showcased at little to no cost of computational overhead added. In this article, we study a minibatch variant of SPP, namely M-SPP, for solving convex composite risk minimization problems. The core contribution is a set of novel excess risk bounds of M-SPP derived through the lens of algorithmic stability theory. Particularly under smoothness and quadratic growth conditions, we show that M-SPP with minibatch-size $n$ and iteration count $T$ enjoys an in-expectation fast rate of convergence consisting of an $\mathcal{O}\left(\frac{1}{T^2}\right)$ bias decaying term and an $\mathcal{O}\left(\frac{1}{nT}\right)$ variance decaying term. In the small-$n$-large-$T$ setting, this result substantially improves the best known results of SPP-type approaches by revealing the impact of noise level of model on convergence rate. In the complementary small-$T$-large-$n$ regime, we provide a two-phase extension of M-SPP to achieve comparable convergence rates. Moreover, we derive a near-tight high probability (over the randomness of data) bound on the parameter estimation error of a sampling-without-replacement variant of M-SPP. Numerical evidences are provided to support our theoretical predictions when substantialized to Lasso and logistic regression models.

Xiao-Tong Yuan, Ping Li

Exponential generalization bounds with near-tight rates have recently been established for uniformly stable learning algorithms. The notion of uniform stability, however, is stringent in the sense that it is invariant to the data-generating distribution. Under the weaker and distribution dependent notions of stability such as hypothesis stability and $L_2$-stability, the literature suggests that only polynomial generalization bounds are possible in general cases. The present paper addresses this long standing tension between these two regimes of results and makes progress towards relaxing it inside a classic framework of confidence-boosting. To this end, we first establish an in-expectation first moment generalization error bound for potentially randomized learning algorithms with $L_2$-stability, based on which we then show that a properly designed subbagging process leads to near-tight exponential generalization bounds over the randomness of both data and algorithm. We further substantialize these generic results to stochastic gradient descent (SGD) to derive improved high-probability generalization bounds for convex or non-convex optimization problems with natural time decaying learning rates, which have not been possible to prove with the existing hypothesis stability or uniform stability based results.

William de Vazelhes, Hualin Zhang, Huimin Wu et al.

$\ell_0$ constrained optimization is prevalent in machine learning, particularly for high-dimensional problems, because it is a fundamental approach to achieve sparse learning. Hard-thresholding gradient descent is a dominant technique to solve this problem. However, first-order gradients of the objective function may be either unavailable or expensive to calculate in a lot of real-world problems, where zeroth-order (ZO) gradients could be a good surrogate. Unfortunately, whether ZO gradients can work with the hard-thresholding operator is still an unsolved problem. To solve this puzzle, in this paper, we focus on the $\ell_0$ constrained black-box stochastic optimization problems, and propose a new stochastic zeroth-order gradient hard-thresholding (SZOHT) algorithm with a general ZO gradient estimator powered by a novel random support sampling. We provide the convergence analysis of SZOHT under standard assumptions. Importantly, we reveal a conflict between the deviation of ZO estimators and the expansivity of the hard-thresholding operator, and provide a theoretical minimal value of the number of random directions in ZO gradients. In addition, we find that the query complexity of SZOHT is independent or weakly dependent on the dimensionality under different settings. Finally, we illustrate the utility of our method on a portfolio optimization problem as well as black-box adversarial attacks.

William de Vazelhes, Xiao-Tong Yuan, Bin Gu

In sparse optimization, enforcing hard constraints using the $\ell_0$ pseudo-norm offers advantages like controlled sparsity compared to convex relaxations. However, many real-world applications demand not only sparsity constraints but also some extra constraints. While prior algorithms have been developed to address this complex scenario with mixed combinatorial and convex constraints, they typically require the closed form projection onto the mixed constraints which might not exist, and/or only provide local guarantees of convergence which is different from the global guarantees commonly sought in sparse optimization. To fill this gap, in this paper, we study the problem of sparse optimization with extra support-preserving constraints commonly encountered in the literature. We present a new variant of iterative hard-thresholding algorithm equipped with a two-step consecutive projection operator customized for these mixed constraints, serving as a simple alternative to the Euclidean projection onto the mixed constraint. By introducing a novel trade-off between sparsity relaxation and sub-optimality, we provide global guarantees in objective value for the output of our algorithm, in the deterministic, stochastic, and zeroth-order settings, under the conventional restricted strong-convexity/smoothness assumptions. As a fundamental contribution in proof techniques, we develop a novel extension of the classic three-point lemma to the considered two-step non-convex projection operator, which allows us to analyze the convergence in objective value in an elegant way that has not been possible with existing techniques. In the zeroth-order case, such technique also improves upon the state-of-the-art result from de Vazelhes et. al. (2022), even in the case without additional constraints, by allowing us to remove a non-vanishing system error present in their work.

William de Vazelhes, Bhaskar Mukhoty, Xiao-Tong Yuan et al.

Sparse recovery is ubiquitous in machine learning and signal processing. Due to the NP-hard nature of sparse recovery, existing methods are known to suffer either from restrictive (or even unknown) applicability conditions, or high computational cost. Recently, iterative regularization methods have emerged as a promising fast approach because they can achieve sparse recovery in one pass through early stopping, rather than the tedious grid-search used in the traditional methods. However, most of those iterative methods are based on the $\ell_1$ norm which requires restrictive applicability conditions and could fail in many cases. Therefore, achieving sparse recovery with iterative regularization methods under a wider range of conditions has yet to be further explored. To address this issue, we propose a novel iterative regularization algorithm, IRKSN, based on the $k$-support norm regularizer rather than the $\ell_1$ norm. We provide conditions for sparse recovery with IRKSN, and compare them with traditional conditions for recovery with $\ell_1$ norm regularizers. Additionally, we give an early stopping bound on the model error of IRKSN with explicit constants, achieving the standard linear rate for sparse recovery. Finally, we illustrate the applicability of our algorithm on several experiments, including a support recovery experiment with a correlated design matrix.

Pan Zhou, Caiming Xiong, Xiao-Tong Yuan et al.

For an image query, unsupervised contrastive learning labels crops of the same image as positives, and other image crops as negatives. Although intuitive, such a native label assignment strategy cannot reveal the underlying semantic similarity between a query and its positives and negatives, and impairs performance, since some negatives are semantically similar to the query or even share the same semantic class as the query. In this work, we first prove that for contrastive learning, inaccurate label assignment heavily impairs its generalization for semantic instance discrimination, while accurate labels benefit its generalization. Inspired by this theory, we propose a novel self-labeling refinement approach for contrastive learning. It improves the label quality via two complementary modules: (i) self-labeling refinery (SLR) to generate accurate labels and (ii) momentum mixup (MM) to enhance similarity between query and its positive. SLR uses a positive of a query to estimate semantic similarity between a query and its positive and negatives, and combines estimated similarity with vanilla label assignment in contrastive learning to iteratively generate more accurate and informative soft labels. We theoretically show that our SLR can exactly recover the true semantic labels of label-corrupted data, and supervises networks to achieve zero prediction error on classification tasks. MM randomly combines queries and positives to increase semantic similarity between the generated virtual queries and their positives so as to improves label accuracy. Experimental results on CIFAR10, ImageNet, VOC and COCO show the effectiveness of our method. PyTorch code and model will be released online.

Hongduan Tian, Bo Liu, Xiao-Tong Yuan et al.

Meta-learning is a powerful paradigm for few-shot learning. Although with remarkable success witnessed in many applications, the existing optimization based meta-learning models with over-parameterized neural networks have been evidenced to ovetfit on training tasks. To remedy this deficiency, we propose a network pruning based meta-learning approach for overfitting reduction via explicitly controlling the capacity of network. A uniform concentration analysis reveals the benefit of network capacity constraint for reducing generalization gap of the proposed meta-learner. We have implemented our approach on top of Reptile assembled with two network pruning routines: Dense-Sparse-Dense (DSD) and Iterative Hard Thresholding (IHT). Extensive experimental results on benchmark datasets with different over-parameterized deep networks demonstrate that our method not only effectively alleviates meta-overfitting but also in many cases improves the overall generalization performance when applied to few-shot classification tasks.

Xiao-Tong Yuan, Ping Li

The DANE algorithm is an approximate Newton method popularly used for communication-efficient distributed machine learning. Reasons for the interest in DANE include scalability and versatility. Convergence of DANE, however, can be tricky; its appealing convergence rate is only rigorous for quadratic objective, and for more general convex functions the known results are no stronger than those of the classic first-order methods. To remedy these drawbacks, we propose in this paper some new alternatives of DANE which are more suitable for analysis. We first introduce a simple variant of DANE equipped with backtracking line search, for which global asymptotic convergence and sharper local non-asymptotic convergence rate guarantees can be proved for both quadratic and non-quadratic strongly convex functions. Then we propose a heavy-ball method to accelerate the convergence of DANE, showing that nearly tight local rate of convergence can be established for strongly convex functions, and with proper modification of algorithm the same result applies globally to linear prediction models. Numerical evidence is provided to confirm the theoretical and practical advantages of our methods.

Bo Liu, Xiao-Tong Yuan, Lezi Wang et al.

Iterative Hard Thresholding (IHT) is a class of projected gradient descent methods for optimizing sparsity-constrained minimization models, with the best known efficiency and scalability in practice. As far as we know, the existing IHT-style methods are designed for sparse minimization in primal form. It remains open to explore duality theory and algorithms in such a non-convex and NP-hard problem setting. In this paper, we bridge this gap by establishing a duality theory for sparsity-constrained minimization with $\ell_2$-regularized loss function and proposing an IHT-style algorithm for dual maximization. Our sparse duality theory provides a set of sufficient and necessary conditions under which the original NP-hard/non-convex problem can be equivalently solved in a dual formulation. The proposed dual IHT algorithm is a super-gradient method for maximizing the non-smooth dual objective. An interesting finding is that the sparse recovery performance of dual IHT is invariant to the Restricted Isometry Property (RIP), which is required by virtually all the existing primal IHT algorithms without sparsity relaxation. Moreover, a stochastic variant of dual IHT is proposed for large-scale stochastic optimization. Numerical results demonstrate the superiority of dual IHT algorithms to the state-of-the-art primal IHT-style algorithms in model estimation accuracy and computational efficiency.

Xiao-Tong Yuan, Ping Li, Tong Zhang

Hard Thresholding Pursuit (HTP) is an iterative greedy selection procedure for finding sparse solutions of underdetermined linear systems. This method has been shown to have strong theoretical guarantee and impressive numerical performance. In this paper, we generalize HTP from compressive sensing to a generic problem setup of sparsity-constrained convex optimization. The proposed algorithm iterates between a standard gradient descent step and a hard thresholding step with or without debiasing. We prove that our method enjoys the strong guarantees analogous to HTP in terms of rate of convergence and parameter estimation accuracy. Numerical evidences show that our method is superior to the state-of-the-art greedy selection methods in sparse logistic regression and sparse precision matrix estimation tasks.

Xiao-Tong Yuan, Ping Li, Tong Zhang

We investigate a generic problem of learning pairwise exponential family graphical models with pairwise sufficient statistics defined by a global mapping function, e.g., Mercer kernels. This subclass of pairwise graphical models allow us to flexibly capture complex interactions among variables beyond pairwise product. We propose two $\ell_1$-norm penalized maximum likelihood estimators to learn the model parameters from i.i.d. samples. The first one is a joint estimator which estimates all the parameters simultaneously. The second one is a node-wise conditional estimator which estimates the parameters individually for each node. For both estimators, we show that under proper conditions the extra flexibility gained in our model comes at almost no cost of statistical and computational efficiency. We demonstrate the advantages of our model over state-of-the-art methods on synthetic and real datasets.

Xiao-Tong Yuan, Tong Zhang

This paper studies the partial estimation of Gaussian graphical models from high-dimensional empirical observations. We derive a convex formulation for this problem using $\ell_1$-regularized maximum-likelihood estimation, which can be solved via a block coordinate descent algorithm. Statistical estimation performance can be established for our method. The proposed approach has competitive empirical performance compared to existing methods, as demonstrated by various experiments on synthetic and real datasets.