Federated learning enables isolated clients to train a shared model collaboratively by aggregating the locally-computed gradient updates. However, privacy information could be leaked from uploaded gradients and be exposed to malicious attackers or an honest-but-curious server. Although the additive homomorphic encryption technique guarantees the security of this process, it brings unacceptable computation and communication burdens to FL participants. To mitigate this cost of secure aggregation and maintain the learning performance, we propose a new framework called Encoded Gradient Aggregation (\emph{EGA}). In detail, EGA first encodes local gradient updates into an encoded domain with injected noises in each client before the aggregation in the server. Then, the encoded gradients aggregation results can be recovered for the global model update via a decoding function. This scheme could prevent the raw gradients of a single client from exposing on the internet and keep them unknown to the server. EGA could provide optimization and communication benefits under different noise levels and defend against gradient leakage. We further provide a theoretical analysis of the approximation error and its impacts on federated optimization. Moreover, EGA is compatible with the most federated optimization algorithms. We conduct intensive experiments to evaluate EGA in real-world federated settings, and the results have demonstrated its efficacy.

Let $V_* : \mathbb{R}^d \to \mathbb{R}$ be some (possibly non-convex) potential function, and consider the probability measure $\pi \propto e^{-V_*}$. When $\pi$ exhibits multiple modes, it is known that sampling techniques based on Wasserstein gradient flows of the Kullback-Leibler (KL) divergence (e.g. Langevin Monte Carlo) suffer poorly in the rate of convergence, where the dynamics are unable to easily traverse between modes. In stark contrast, the work of Lu et al. (2019; 2022) has shown that the gradient flow of the KL with respect to the Fisher-Rao (FR) geometry exhibits a convergence rate to $\pi$ is that \textit{independent} of the potential function. In this short note, we complement these existing results in the literature by providing an explicit expansion of $\text{KL}(\rho_t^{\text{FR}}\|\pi)$ in terms of $e^{-t}$, where $(\rho_t^{\text{FR}})_{t\geq 0}$ is the FR gradient flow of the KL divergence. In turn, we are able to provide a clean asymptotic convergence rate, where the burn-in time is guaranteed to be finite. Our proof is based on observing a similarity between FR gradient flows and simulated annealing with linear scaling, and facts about cumulant generating functions. We conclude with simple synthetic experiments that demonstrate our theoretical findings are indeed tight. Based on our numerics, we conjecture that the asymptotic rates of convergence for Wasserstein-Fisher-Rao gradient flows are possibly related to this expansion in some cases.

As machine learning becomes more widespread throughout society, aspects including data privacy and fairness must be carefully considered, and are crucial for deployment in highly regulated industries. Unfortunately, the application of privacy enhancing technologies can worsen unfair tendencies in models. In particular, one of the most widely used techniques for private model training, differentially private stochastic gradient descent (DPSGD), frequently intensifies disparate impact on groups within data. In this work we study the fine-grained causes of unfairness in DPSGD and identify gradient misalignment due to inequitable gradient clipping as the most significant source. This observation leads us to a new method for reducing unfairness by preventing gradient misalignment in DPSGD. The increasingly widespread use of machine learning throughout society has brought into focus social, ethical, and legal considerations surrounding its use. In highly regulated industries, such as healthcare and banking, regional laws and regulations require data collection and analysis to respect the privacy of individuals. Other regulations focus on the fairness of how models are developed and used. As machine learning is progressively adopted in highly regulated industries, the privacy and fairness aspects of models must be considered at all stages of the modelling lifecycle. There are many privacy enhancing technologies including differential privacy (Dwork et al., 2006), federated learning (McMahan et al., 2017), secure multiparty computation (Yao, 1986), and homomorphic encryption (Gentry, 2009) that are used separately or jointly to protect the privacy of individuals whose data is used for machine learning (Choquette-Choo et al., 2020; Adnan et al., 2022; Kalra et al., 2021).

Time-varying stochastic optimization problems frequently arise in machine learning practice (e.g. gradual domain shift, object tracking, strategic classification). Although most problems are solved in discrete time, the underlying process is often continuous in nature. We exploit this underlying continuity by developing predictor-corrector algorithms for time-varying stochastic optimizations. We provide error bounds for the iterates, both in presence of pure and noisy access to the queries from the relevant derivatives of the loss function. Furthermore, we show (theoretically and empirically in several examples) that our method outperforms non-predictor corrector methods that do not exploit the underlying continuous process.

In this paper, we propose energy-based sample adaptation at test time for domain generalization. Where previous works adapt their models to target domains, we adapt the unseen target samples to source-trained models. To this end, we design a discriminative energy-based model, which is trained on source domains to jointly model the conditional distribution for classification and data distribution for sample adaptation. The model is optimized to simultaneously learn a classifier and an energy function. To adapt target samples to source distributions, we iteratively update the samples by energy minimization with stochastic gradient Langevin dynamics. Moreover, to preserve the categorical information in the sample during adaptation, we introduce a categorical latent variable into the energy-based model. The latent variable is learned from the original sample before adaptation by variational inference and fixed as a condition to guide the sample update. Experiments on six benchmarks for classification of images and microblog threads demonstrate the effectiveness of our proposal. Deep neural networks are vulnerable to domain shifts and suffer from lack of generalization on test samples that do not resemble the ones in the training distribution (Recht et al., 2019; Zhou et al., 2021; Krueger et al., 2021; Shen et al., 2022). To deal with the domain shifts, domain generalization has been proposed (Muandet et al., 2013; Gulrajani & Lopez-Paz, 2020; Cha et al., 2021). Domain generalization strives to learn a model exclusively on source domains in order to generalize well on unseen target domains. The major challenge stems from the large domain shifts and the unavailability of any target domain data during training. To address the problem, domain invariant learning has been widely studied, e.g., (Motiian et al., 2017; Zhao et al., 2020; Nguyen et al., 2021), based on the assumption that invariant representations obtained on source domains are also valid for unseen target domains. However, since the target data is inaccessible during training, it is likely an "adaptivity gap" (Dubey et al., 2021) exists between representations from the source and target domains. Therefore, recent works try to adapt the classification model with target samples at test time by further fine-tuning model parameters (Sun et al., 2020; Wang et al., 2021) or by introducing an extra network module for adaptation (Dubey et al., 2021). Rather than adapting the model to target domains, Xiao et al. (2022) adapt the classifier for each sample at test time. Nevertheless, a single sample would not be able to adjust the whole model due to the large number of model parameters and the limited information contained in the sample.

An increasing number of data science and machine learning problems rely on computation with tensors, which better capture the multi-way relationships and interactions of data than matrices. When tapping into this critical advantage, a key challenge is to develop computationally efficient and provably correct algorithms for extracting useful information from tensor data that are simultaneously robust to corruptions and ill-conditioning. This paper tackles tensor robust principal component analysis (RPCA), which aims to recover a low-rank tensor from its observations contaminated by sparse corruptions, under the Tucker decomposition. To minimize the computation and memory footprints, we propose to directly recover the low-dimensional tensor factors -- starting from a tailored spectral initialization -- via scaled gradient descent (ScaledGD), coupled with an iteration-varying thresholding operation to adaptively remove the impact of corruptions. Theoretically, we establish that the proposed algorithm converges linearly to the true low-rank tensor at a constant rate that is independent with its condition number, as long as the level of corruptions is not too large. Empirically, we demonstrate that the proposed algorithm achieves better and more scalable performance than state-of-the-art matrix and tensor RPCA algorithms through synthetic experiments and real-world applications.

In the paper, we propose a class of faster adaptive Gradient Descent Ascent (GDA) methods for solving the nonconvex-strongly-concave minimax problems by using the unified adaptive matrices, which include almost all existing coordinate-wise and global adaptive learning rates. In particular, we provide an effective convergence analysis framework for our adaptive GDA methods. Specifically, we propose a fast Adaptive Gradient Descent Ascent (AdaGDA) method based on the basic momentum technique, which reaches a lower gradient complexity of $\tilde{O}(\kappa^4\epsilon^{-4})$ for finding an $\epsilon$-stationary point without large batches, which improves the existing results of the adaptive GDA methods by a factor of $O(\sqrt{\kappa})$. Moreover, we propose an accelerated version of AdaGDA (VR-AdaGDA) method based on the momentum-based variance reduced technique, which achieves a lower gradient complexity of $\tilde{O}(\kappa^{4.5}\epsilon^{-3})$ for finding an $\epsilon$-stationary point without large batches, which improves the existing results of the adaptive GDA methods by a factor of $O(\epsilon^{-1})$. Moreover, we prove that our VR-AdaGDA method can reach the best known gradient complexity of $\tilde{O}(\kappa^{3}\epsilon^{-3})$ with the mini-batch size $O(\kappa^3)$. The experiments on policy evaluation and fair classifier learning tasks are conducted to verify the efficiency of our new algorithms.

Recently, researchers observed that gradient descent for deep neural networks operates in an ``edge-of-stability'' (EoS) regime: the sharpness (maximum eigenvalue of the Hessian) is often larger than stability threshold $2/\eta$ (where $\eta$ is the step size). Despite this, the loss oscillates and converges in the long run, and the sharpness at the end is just slightly below $2/\eta$. While many other well-understood nonconvex objectives such as matrix factorization or two-layer networks can also converge despite large sharpness, there is often a larger gap between sharpness of the endpoint and $2/\eta$. In this paper, we study EoS phenomenon by constructing a simple function that has the same behavior. We give rigorous analysis for its training dynamics in a large local region and explain why the final converging point has sharpness close to $2/\eta$. Globally we observe that the training dynamics for our example has an interesting bifurcating behavior, which was also observed in the training of neural nets.

This paper considers a convex composite optimization problem with affine constraints, which includes problems that take the form of minimizing a smooth convex objective function over the intersection of (simple) convex sets, or regularized with multiple (simple) functions. Motivated by high-dimensional applications in which exact projection/proximal computations are not tractable, we propose a \textit{projection-free} augmented Lagrangian-based method, in which primal updates are carried out using a \textit{weak proximal oracle} (WPO). In an earlier work, WPO was shown to be more powerful than the standard \textit{linear minimization oracle} (LMO) that underlies conditional gradient-based methods (aka Frank-Wolfe methods). Moreover, WPO is computationally tractable for many high-dimensional problems of interest, including those motivated by recovery of low-rank matrices and tensors, and optimization over polytopes which admit efficient LMOs. The main result of this paper shows that under a certain curvature assumption (which is weaker than strong convexity), our WPO-based algorithm achieves an ergodic rate of convergence of $O(1/T)$ for both the objective residual and feasibility gap. This result, to the best of our knowledge, improves upon the $O(1/\sqrt{T})$ rate for existing LMO-based projection-free methods for this class of problems. Empirical experiments on a low-rank and sparse covariance matrix estimation task and the Max Cut semidefinite relaxation demonstrate that of our method can outperform state-of-the-art LMO-based Lagrangian-based methods.

Stochastic gradient descent-ascent (SGDA) is one of the main workhorses for solving finite-sum minimax optimization problems. Most practical implementations of SGDA randomly reshuffle components and sequentially use them (i.e., without-replacement sampling); however, there are few theoretical results on this approach for minimax algorithms, especially outside the easier-to-analyze (strongly-)monotone setups. To narrow this gap, we study the convergence bounds of SGDA with random reshuffling (SGDA-RR) for smooth nonconvex-nonconcave objectives with Polyak-{\L}ojasiewicz (P{\L}) geometry. We analyze both simultaneous and alternating SGDA-RR for nonconvex-P{\L} and primal-P{\L}-P{\L} objectives, and obtain convergence rates faster than with-replacement SGDA. Our rates extend to mini-batch SGDA-RR, recovering known rates for full-batch gradient descent-ascent (GDA). Lastly, we present a comprehensive lower bound for GDA with an arbitrary step-size ratio, which matches the full-batch upper bound for the primal-P{\L}-P{\L} case.