This paper introduces a new stochastic optimization method based on the regularized Fisher information matrix (FIM), named SOFIM, which can efficiently utilize the FIM to approximate the Hessian matrix for finding Newton's gradient update in large-scale stochastic optimization of machine learning models. It can be viewed as a variant of natural gradient descent, where the challenge of storing and calculating the full FIM is addressed through making use of the regularized FIM and directly finding the gradient update direction via Sherman-Morrison matrix inversion. Additionally, like the popular Adam method, SOFIM uses the first moment of the gradient to address the issue of non-stationary objectives across mini-batches due to heterogeneous data. The utilization of the regularized FIM and Sherman-Morrison matrix inversion leads to the improved convergence rate with the same space and time complexities as stochastic gradient descent (SGD) with momentum. The extensive experiments on training deep learning models using several benchmark image classification datasets demonstrate that the proposed SOFIM outperforms SGD with momentum and several state-of-the-art Newton optimization methods in term of the convergence speed for achieving the pre-specified objectives of training and test losses as well as test accuracy.

Learning expressive stochastic policies instead of deterministic ones has been proposed to achieve better stability, sample complexity, and robustness. Notably, in Maximum Entropy Reinforcement Learning (MaxEnt RL), the policy is modeled as an expressive Energy-Based Model (EBM) over the Q-values. However, this formulation requires the estimation of the entropy of such EBMs, which is an open problem. To address this, previous MaxEnt RL methods either implicitly estimate the entropy, resulting in high computational complexity and variance (SQL), or follow a variational inference procedure that fits simplified actor distributions (e.g., Gaussian) for tractability (SAC). We propose Stein Soft Actor-Critic (S$^2$AC), a MaxEnt RL algorithm that learns expressive policies without compromising efficiency. Specifically, S$^2$AC uses parameterized Stein Variational Gradient Descent (SVGD) as the underlying policy. We derive a closed-form expression of the entropy of such policies. Our formula is computationally efficient and only depends on first-order derivatives and vector products. Empirical results show that S$^2$AC yields more optimal solutions to the MaxEnt objective than SQL and SAC in the multi-goal environment, and outperforms SAC and SQL on the MuJoCo benchmark. Our code is available at: https://github.com/SafaMessaoud/S2AC-Energy-Based-RL-with-Stein-Soft-Actor-Critic

In the era of noisy intermediate-scale quantum (NISQ), variational quantum circuits (VQCs) have been widely applied in various domains, advancing the superiority of quantum circuits against classic models. Similar to classic models, regular VQCs can be optimized by various gradient-based methods. However, the optimization may be initially trapped in barren plateaus or eventually entangled in saddle points during training. These gradient issues can significantly undermine the trainability of VQC. In this work, we propose a strategy that regularizes model parameters with prior knowledge of the train data and Gaussian noise diffusion. We conduct ablation studies to verify the effectiveness of our strategy across four public datasets and demonstrate that our method can improve the trainability of VQCs against the above-mentioned gradient issues.

While many methods address these two issues separately, only a few currently deal with both simultaneously. In this paper, we introduce Utility-based Perturbed Gradient Descent (UPGD) as a novel approach for the continual learning of representations. UPGD combines gradient updates with perturbations, where it applies smaller modifications to more useful units, protecting them from forgetting, and larger modifications to less useful units, rejuvenating their plasticity. We use a challenging streaming learning setup where continual learning problems have hundreds of non-stationarities and unknown task boundaries. We show that many existing methods suffer from at least one of the issues, predominantly manifested by their decreasing accuracy over tasks. On the other hand, UPGD continues to improve performance and surpasses or is competitive with all methods in all problems. Finally, in extended reinforcement learning experiments with PPO, we show that while Adam exhibits a performance drop after initial learning, UPGD avoids it by addressing both continual learning issues. Continual learning remains a significant hurdle for artificial intelligence, despite advancements in natural language processing, games, and computer vision. Catastrophic forgetting (McCloskey & Cohen 1989, Hetherington & Seidenberg 1989) in neural networks is widely recognized as a major challenge of continual learning (De Lange et al. 2021). The phenomenon manifests as the failure of gradient-based methods like SGD or Adam to retain or leverage past knowledge due to forgetting or overwriting previously learned units (Kirkpatrick et al. 2017). This issue also raises a concern for reusing large practical models, where finetuning them for new tasks causes significant forgetting of pretrained models (Chen et al. 2020, He et al. 2021). Methods for mitigating catastrophic forgetting are primarily designed for specific settings. These include settings with independently and identically distributed (i.i.d.) samples, tasks fully contained within a batch or dataset, growing memory requirements, known task boundaries, storing past samples, and offline evaluation. Such setups are often impractical in situations where continual learning is paramount, such as on-device learning. For example, retaining samples may not be possible due to the limitation of computational resources (Hayes et al. 2019, Hayes et al. 2020, Hayes & Kannan 2022, Wang et al. 2023) or concerns over data privacy (Van de Ven et al. 2020). In the challenging and practical setting of streaming learning, catastrophic forgetting is more severe and remains largely unaddressed (Hayes et al. 2019). In streaming learning, samples are presented to the learner as they arise, which is non-i.i.d. in most practical problems.

We consider an asynchronous decentralized learning system, which consists of a network of connected devices trying to learn a machine learning model without any centralized parameter server. The users in the network have their own local training data, which is used for learning across all the nodes in the network. The learning method consists of two processes, evolving simultaneously without any necessary synchronization. The first process is the model update, where the users update their local model via a fixed number of stochastic gradient descent steps. The second process is model mixing, where the users communicate with each other via randomized gossiping to exchange their models and average them to reach consensus. In this work, we investigate the staleness criteria for such a system, which is a sufficient condition for convergence of individual user models. We show that for network scaling, i.e., when the number of user devices $n$ is very large, if the gossip capacity of individual users scales as $\Omega(\log n)$, we can guarantee the convergence of user models in finite time. Furthermore, we show that the bounded staleness can only be guaranteed by any distributed opportunistic scheme by $\Omega(n)$ scaling.

Optimization techniques in deep learning are predominantly led by first-order gradient methodologies, such as SGD. However, neural network training can greatly benefit from the rapid convergence characteristics of second-order optimization. Newton's GD stands out in this category, by rescaling the gradient using the inverse Hessian. Nevertheless, one of its major bottlenecks is matrix inversion, which is notably time-consuming in $O(N^3)$ time with weak scalability. Matrix inversion can be translated into solving a series of linear equations. Given that quantum linear solver algorithms (QLSAs), leveraging the principles of quantum superposition and entanglement, can operate within a $\text{polylog}(N)$ time frame, they present a promising approach with exponential acceleration. Specifically, one of the most recent QLSAs demonstrates a complexity scaling of $O(d\cdot\kappa \log(N\cdot\kappa/\epsilon))$, depending on: {size~$N$, condition number~$\kappa$, error tolerance~$\epsilon$, quantum oracle sparsity~$d$} of the matrix. However, this also implies that their potential exponential advantage may be hindered by certain properties (i.e. $\kappa$ and $d$). We propose Q-Newton, a hybrid quantum-classical scheduler for accelerating neural network training with Newton's GD. Q-Newton utilizes a streamlined scheduling module that coordinates between quantum and classical linear solvers, by estimating & reducing $\kappa$ and constructing $d$ for the quantum solver. Our evaluation showcases the potential for Q-Newton to significantly reduce the total training time compared to commonly used optimizers like SGD. We hypothesize a future scenario where the gate time of quantum machines is reduced, possibly realized by attoseconds physics. Our evaluation establishes an ambitious and promising target for the evolution of quantum computing.

We consider the robust estimation of the parameters of multivariate Gaussian linear regression models. To this aim we consider robust version of the usual (Mahalanobis) least-square criterion, with or without Ridge regularization. We introduce two methods each considered contrast: (i) online stochastic gradient descent algorithms and their averaged versions and (ii) offline fix-point algorithms. Under weak assumptions, we prove the asymptotic normality of the resulting estimates. Because the variance matrix of the noise is usually unknown, we propose to plug a robust estimate of it in the Mahalanobis-based stochastic gradient descent algorithms. We show, on synthetic data, the dramatic gain in terms of robustness of the proposed estimates as compared to the classical least-square ones. Well also show the computational efficiency of the online versions of the proposed algorithms. All the proposed algorithms are implemented in the R package RobRegression available on CRAN.

Constrained optimization plays a crucial role in the fields of quantum physics and quantum information science and becomes especially challenging for high-dimensional complex structure problems. One specific issue is that of quantum process tomography, in which the goal is to retrieve the underlying quantum process based on a given set of measurement data. In this paper, we introduce a modified version of stochastic gradient descent on a Riemannian manifold that integrates recent advancements in numerical methods for Riemannian optimization. This approach inherently supports the physically driven constraints of a quantum process, takes advantage of state-of-the-art large-scale stochastic objective optimization, and has superior performance to traditional approaches such as maximum likelihood estimation and projected least squares. The data-driven approach enables accurate, order-of-magnitude faster results, and works with incomplete data. We demonstrate our approach on simulations of quantum processes and in hardware by characterizing an engineered process on quantum computers.

Second-order optimization techniques have the potential to achieve faster convergence rates compared to first-order methods through the incorporation of second-order derivatives or statistics. However, their utilization in deep learning is limited due to their computational inefficiency. Various approaches have been proposed to address this issue, primarily centered on minimizing the size of the matrix to be inverted. Nevertheless, the necessity of performing the inverse operation iteratively persists. In this work, we present a fast natural gradient descent (FNGD) method that only requires inversion during the first epoch. Specifically, it is revealed that natural gradient descent (NGD) is essentially a weighted sum of per-sample gradients. Our novel approach further proposes to share these weighted coefficients across epochs without affecting empirical performance. Consequently, FNGD exhibits similarities to the average sum in first-order methods, leading to the computational complexity of FNGD being comparable to that of first-order methods. Extensive experiments on image classification and machine translation tasks demonstrate the efficiency of the proposed FNGD. For training ResNet-18 on CIFAR-100, FNGD can achieve a speedup of 2.07$\times$ compared with KFAC. For training Transformer on Multi30K, FNGD outperforms AdamW by 24 BLEU score while requiring almost the same training time.

Federated learning (FL) is a distributed learning paradigm wherein users exchange FL models with a server instead of raw datasets, thereby preserving data privacy and reducing communication overhead. However, the increased number of FL users may hinder completing large-scale FL over wireless networks due to high imposed latency. Cell-free massive multiple-input multiple-output~(CFmMIMO) is a promising architecture for implementing FL because it serves many users on the same time/frequency resources. While CFmMIMO enhances energy efficiency through spatial multiplexing and collaborative beamforming, it remains crucial to meticulously allocate uplink transmission powers to the FL users. In this paper, we propose an uplink power allocation scheme in FL over CFmMIMO by considering the effect of each user's power on the energy and latency of other users to jointly minimize the users' uplink energy and the latency of FL training. The proposed solution algorithm is based on the coordinate gradient descent method. Numerical results show that our proposed method outperforms the well-known max-sum rate by increasing up to~$27$\% and max-min energy efficiency of the Dinkelbach method by increasing up to~$21$\% in terms of test accuracy while having limited uplink energy and latency budget for FL over CFmMIMO.