This study extends the problem settings of Invariant Risk Minimization(IRM) for Out-of-Distribution generalization problems to unknown clustered environments settings. In this scenario, where a given set of environments exhibits an unknown clustered structure, our objective is to identify a single invariant feature extractor and per-cluster regressors (or classifiers) built on top of the feature extractor. To achieve this, we propose a new framework called Clustered IRM for simultaneously identifying the cluster structure and the invariant features. Our theoretical analysis demonstrates that the required number of training environments for such identification is only $O(d_\mathrm{sp} + K^2)$, where $d_\mathrm{sp}$ represents the dimensionality of the spurious features, and $K$ is the number of clusters. Numerical experiments validate the effectiveness of our proposed framework.

In this study, we consider nonconvex federated learning problems with the existence of Byzantine workers. We propose a new simple Byzantine robust algorithm called Momentum Screening. The algorithm is adaptive to the Byzantine fraction, i.e., all its hyperparameters do not depend on the number of Byzantine workers. We show that our method achieves the best optimization error of $O(\delta^2\zeta_\mathrm{max}^2)$ for nonconvex smooth local objectives satisfying $\zeta_\mathrm{max}$-uniform gradient heterogeneity condition under $\delta$-Byzantine fraction, which can be better than the best known error rate of $O(\delta\zeta_\mathrm{mean}^2)$ for local objectives satisfying $\zeta_\mathrm{mean}$-mean heterogeneity condition when $\delta \leq (\zeta_\mathrm{max}/\zeta_\mathrm{mean})^2$. Furthermore, we derive an algorithm independent lower bound for local objectives satisfying $\zeta_\mathrm{max}$-uniform gradient heterogeneity condition and show the minimax optimality of our proposed method on this class. In numerical experiments, we validate the superiority of our method over the existing robust aggregation algorithms and verify our theoretical results.

Most variance reduction methods require multiple times of full gradient computation, which is time-consuming and hence a bottleneck in application to distributed optimization. We present a single-loop variance-reduced gradient estimator named SILVER (SIngle-Loop VariancE-Reduction) for the finite-sum non-convex optimization, which does not require multiple full gradients but nevertheless achieves the optimal gradient complexity. Notably, unlike existing methods, SILVER provably reaches second-order optimality, with exponential convergence in the Polyak-Łojasiewicz (PL) region, and achieves further speedup depending on the data heterogeneity. Owing to these advantages, SILVER serves as a new base method to design communication-efficient federated learning algorithms: we combine SILVER with local updates which gives the best communication rounds and number of communicated gradients across all range of Hessian heterogeneity, and, at the same time, guarantees second-order optimality and exponential convergence in the PL region.

A novel algorithm is proposed for highly privacy-preserving decentralized federated learning (FL). Several studies have reported security risks in decentralized FL by reconstructing data even from model update differences. A common approach to overcome this issue is to use the diffusion process following differential privacy (DP), i.e., message passing between nodes is hidden by noise. However, this often makes the learning process unstable, leading to degraded results compared to without using DP diffusion process. In this paper, we propose a primal-dual DP algorithm with denoising normalization (DP-Norm) for less sensitivity to noise/interference, such as DP diffusion and heterogeneous data allocation. For DP-Norm, privacy analysis to determine minimal noise level and convergence analysis are conducted. Through image classification benchmark tests, we confirmed that DP-Norm performed close to the single-node reference score, even when statistically heterogeneous data was allocated on six nodes.

Differential private optimization for nonconvex smooth objective is considered. In the previous work, the best known utility bound is $\widetilde O(\sqrt{d}/(n\varepsilon_\mathrm{DP}))$ in terms of the squared full gradient norm, which is achieved by Differential Private Gradient Descent (DP-GD) as an instance, where $n$ is the sample size, $d$ is the problem dimensionality and $\varepsilon_\mathrm{DP}$ is the differential privacy parameter. To improve the best known utility bound, we propose a new differential private optimization framework called DIFF2 (DIFFerential private optimization via gradient DIFFerences) that constructs a differential private global gradient estimator with possibly quite small variance based on communicated gradient differences rather than gradients themselves. It is shown that DIFF2 with a gradient descent subroutine achieves the utility of $\widetilde O(d^{2/3}/(n\varepsilon_\mathrm{DP})^{4/3})$, which can be significantly better than the previous one in terms of the dependence on the sample size $n$. To the best of our knowledge, this is the first fundamental result to improve the standard utility $\widetilde O(\sqrt{d}/(n\varepsilon_\mathrm{DP}))$ for nonconvex objectives. Additionally, a more computational and communication efficient subroutine is combined with DIFF2 and its theoretical analysis is also given. Numerical experiments are conducted to validate the superiority of DIFF2 framework.

In recent centralized nonconvex distributed learning and federated learning, local methods are one of the promising approaches to reduce communication time. However, existing work has mainly focused on studying first-order optimality guarantees. On the other side, second-order optimality guaranteed algorithms, i.e., algorithms escaping saddle points, have been extensively studied in the non-distributed optimization literature. In this paper, we study a new local algorithm called Bias-Variance Reduced Local Perturbed SGD (BVR-L-PSGD), that combines the existing bias-variance reduced gradient estimator with parameter perturbation to find second-order optimal points in centralized nonconvex distributed optimization. BVR-L-PSGD enjoys second-order optimality with nearly the same communication complexity as the best known one of BVR-L-SGD to find first-order optimality. Particularly, the communication complexity is better than non-local methods when the local datasets heterogeneity is smaller than the smoothness of the local loss. In an extreme case, the communication complexity approaches to $\widetilde \Theta(1)$ when the local datasets heterogeneity goes to zero. Numerical results validate our theoretical findings.

Labeling cost is often expensive and is a fundamental limitation of supervised learning. In this paper, we study importance labeling problem, in which we are given many unlabeled data and select a limited number of data to be labeled from the unlabeled data, and then a learning algorithm is executed on the selected one. We propose a new importance labeling scheme that can effectively select an informative subset of unlabeled data in least squares regression in Reproducing Kernel Hilbert Spaces (RKHS). We analyze the generalization error of gradient descent combined with our labeling scheme and show that the proposed algorithm achieves the optimal rate of convergence in much wider settings and especially gives much better generalization ability in a small noise setting than the usual uniform sampling scheme. Numerical experiments verify our theoretical findings.

Recently, local SGD has got much attention and been extensively studied in the distributed learning community to overcome the communication bottleneck problem. However, the superiority of local SGD to minibatch SGD only holds in quite limited situations. In this paper, we study a new local algorithm called Bias-Variance Reduced Local SGD (BVR-L-SGD) for nonconvex distributed optimization. Algorithmically, our proposed bias and variance reduced local gradient estimator fully utilizes small second-order heterogeneity of local objectives and suggests randomly picking up one of the local models instead of taking the average of them when workers are synchronized. Theoretically, under small heterogeneity of local objectives, we show that BVR-L-SGD achieves better communication complexity than both the previous non-local and local methods under mild conditions, and particularly BVR-L-SGD is the first method that breaks the barrier of communication complexity $\Theta(1/\varepsilon)$ for general nonconvex smooth objectives when the heterogeneity is small and the local computation budget is large. Numerical results are given to verify the theoretical findings and give empirical evidence of the superiority of our method.

Compression techniques for deep neural network models are becoming very important for the efficient execution of high-performance deep learning systems on edge-computing devices. The concept of model compression is also important for analyzing the generalization error of deep learning, known as the compression-based error bound. However, there is still huge gap between a practically effective compression method and its rigorous background of statistical learning theory. To resolve this issue, we develop a new theoretical framework for model compression and propose a new pruning method called {\it spectral pruning} based on this framework. We define the ``degrees of freedom'' to quantify the intrinsic dimensionality of a model by using the eigenvalue distribution of the covariance matrix across the internal nodes and show that the compression ability is essentially controlled by this quantity. Moreover, we present a sharp generalization error bound of the compressed model and characterize the bias--variance tradeoff induced by the compression procedure. We apply our method to several datasets to justify our theoretical analyses and show the superiority of the the proposed method.

The minibatching technique has been extensively adopted to facilitate stochastic first-order methods because of their computational efficiency in parallel computing for large-scale machine learning and data mining. However, the optimal minibatch size determination for accelerated stochastic gradient methods is not completely understood. Actually, there appears trade-off between the iteration complexity and the total computational complexity; that is, the number of iterations (minibatch queries) can be decreased by increasing the minibatch size, but too large minibatch size would result in an unnecessarily large total computational cost. In this study, we give a sharp characterization of the minimax optimal minibatch size to achieve the optimal iteration complexity by providing a reachable lower bound for minimizing finite sum of convex functions and, surprisingly, show that the optimal method with the minimax optimal minibatch size can achieve both of the optimal iteration complexity and the optimal total computational complexity simultaneously. Finally, this feature is verified experimentally.

We develop new stochastic gradient methods for efficiently solving sparse linear regression in a partial attribute observation setting, where learners are only allowed to observe a fixed number of actively chosen attributes per example at training and prediction times. It is shown that the methods achieve essentially a sample complexity of $O(1/\varepsilon)$ to attain an error of $\varepsilon$ under a variant of restricted eigenvalue condition, and the rate has better dependency on the problem dimension than existing methods. Particularly, if the smallest magnitude of the non-zero components of the optimal solution is not too small, the rate of our proposed {\it Hybrid} algorithm can be boosted to near the minimax optimal sample complexity of {\it full information} algorithms. The core ideas are (i) efficient construction of an unbiased gradient estimator by the iterative usage of the hard thresholding operator for configuring an exploration algorithm; and (ii) an adaptive combination of the exploration and an exploitation algorithms for quickly identifying the support of the optimum and efficiently searching the optimal parameter in its support. Experimental results are presented to validate our theoretical findings and the superiority of our proposed methods.

We develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings. The use of mini-batches has become a golden standard in the machine learning community, because the mini-batch techniques stabilize the gradient estimate and can easily make good use of parallel computing. The core of our proposed method is the incorporation of our new ``double acceleration'' technique and variance reduction technique. We theoretically analyze our proposed method and show that our method much improves the mini-batch efficiencies of previous accelerated stochastic methods, and essentially only needs size $\sqrt{n}$ mini-batches for achieving the optimal iteration complexities for both non-strongly and strongly convex objectives, where $n$ is the training set size. Further, we show that even in non-mini-batch settings, our method achieves the best known convergence rate for non-strongly convex and strongly convex objectives.