Federated learning (FL) is a subfield of machine learning where multiple clients try to collaboratively learn a model over a network under communication constraints. We consider finite-sum federated optimization under a second-order function similarity condition and strong convexity, and propose two new algorithms: SVRP and Catalyzed SVRP. This second-order similarity condition has grown popular recently, and is satisfied in many applications including distributed statistical learning and differentially private empirical risk minimization. The first algorithm, SVRP, combines approximate stochastic proximal point evaluations, client sampling, and variance reduction. We show that SVRP is communication efficient and achieves superior performance to many existing algorithms when function similarity is high enough. Our second algorithm, Catalyzed SVRP, is a Catalyst-accelerated variant of SVRP that achieves even better performance and uniformly improves upon existing algorithms for federated optimization under second-order similarity and strong convexity. In the course of analyzing these algorithms, we provide a new analysis of the Stochastic Proximal Point Method (SPPM) that might be of independent interest. Our analysis of SPPM is simple, allows for approximate proximal point evaluations, does not require any smoothness assumptions, and shows a clear benefit in communication complexity over ordinary distributed stochastic gradient descent.

We consider the problem of unconstrained minimization of finite sums of functions. We propose a simple, yet, practical way to incorporate variance reduction techniques into SignSGD, guaranteeing convergence that is similar to the full sign gradient descent. The core idea is first instantiated on the problem of minimizing sums of convex and Lipschitz functions and is then extended to the smooth case via variance reduction. Our analysis is elementary and much simpler than the typical proof for variance reduction methods. We show that for smooth functions our method gives $\mathcal{O}(1 / \sqrt{T})$ rate for expected norm of the gradient and $\mathcal{O}(1/T)$ rate in the case of smooth convex functions, recovering convergence results of deterministic methods, while preserving computational advantages of SignSGD.

Differentially private (stochastic) gradient descent is the workhorse of DP private machine learning in both the convex and non-convex settings. Without privacy constraints, second-order methods, like Newton's method, converge faster than first-order methods like gradient descent. In this work, we investigate the prospect of using the second-order information from the loss function to accelerate DP convex optimization. We first develop a private variant of the regularized cubic Newton method of Nesterov and Polyak, and show that for the class of strongly convex loss functions, our algorithm has quadratic convergence and achieves the optimal excess loss. We then design a practical second-order DP algorithm for the unconstrained logistic regression problem. We theoretically and empirically study the performance of our algorithm. Empirical results show our algorithm consistently achieves the best excess loss compared to other baselines and is 10-40x faster than DP-GD/DP-SGD.

The classical analysis of Stochastic Gradient Descent (SGD) with polynomially decaying stepsize $\eta_t = \eta/\sqrt{t}$ relies on well-tuned $\eta$ depending on problem parameters such as Lipschitz smoothness constant, which is often unknown in practice. In this work, we prove that SGD with arbitrary $\eta > 0$, referred to as untuned SGD, still attains an order-optimal convergence rate $\widetilde{O}(T^{-1/4})$ in terms of gradient norm for minimizing smooth objectives. Unfortunately, it comes at the expense of a catastrophic exponential dependence on the smoothness constant, which we show is unavoidable for this scheme even in the noiseless setting. We then examine three families of adaptive methods $\unicode{x2013}$ Normalized SGD (NSGD), AMSGrad, and AdaGrad $\unicode{x2013}$ unveiling their power in preventing such exponential dependency in the absence of information about the smoothness parameter and boundedness of stochastic gradients. Our results provide theoretical justification for the advantage of adaptive methods over untuned SGD in alleviating the issue with large gradients.

Importance sampling (IS) is a popular technique in off-policy evaluation, which re-weights the return of trajectories in the replay buffer to boost sample efficiency. However, training with IS can be unstable and previous attempts to address this issue mainly focus on analyzing the variance of IS. In this paper, we reveal that the instability is also related to a new notion of Reuse Bias of IS -- the bias in off-policy evaluation caused by the reuse of the replay buffer for evaluation and optimization. We theoretically show that the off-policy evaluation and optimization of the current policy with the data from the replay buffer result in an overestimation of the objective, which may cause an erroneous gradient update and degenerate the performance. We further provide a high-probability upper bound of the Reuse Bias, and show that controlling one term of the upper bound can control the Reuse Bias by introducing the concept of stability for off-policy algorithms. Based on these analyses, we finally present a novel Bias-Regularized Importance Sampling (BIRIS) framework along with practical algorithms, which can alleviate the negative impact of the Reuse Bias. Experimental results show that our BIRIS-based methods can significantly improve the sample efficiency on a series of continuous control tasks in MuJoCo.

We propose an online learning algorithm for a class of machine learning models under a separable stochastic approximation framework. The essence of our idea lies in the observation that certain parameters in the models are easier to optimize than others. In this paper, we focus on models where some parameters have a linear nature, which is common in machine learning. In one routine of the proposed algorithm, the linear parameters are updated by the recursive least squares (RLS) algorithm, which is equivalent to a stochastic Newton method; then, based on the updated linear parameters, the nonlinear parameters are updated by the stochastic gradient method (SGD). The proposed algorithm can be understood as a stochastic approximation version of block coordinate gradient descent approach in which one part of the parameters is updated by a second-order SGD method while the other part is updated by a first-order SGD. Global convergence of the proposed online algorithm for non-convex cases is established in terms of the expected violation of a first-order optimality condition. Numerical experiments show that the proposed method accelerates convergence significantly and produces more robust training and test performance when compared to other popular learning algorithms. Moreover, our algorithm is less sensitive to the learning rate and outperforms the recently proposed slimTrain algorithm (Newman et al., 2022). The code has been uploaded to GitHub for validation.

Recently, several studies consider the stochastic optimization problem but in a heavy-tailed noise regime, i.e., the difference between the stochastic gradient and the true gradient is assumed to have a finite $p$-th moment (say being upper bounded by $\sigma^{p}$ for some $\sigma\geq0$) where $p\in(1,2]$, which not only generalizes the traditional finite variance assumption ($p=2$) but also has been observed in practice for several different tasks. Under this challenging assumption, lots of new progress has been made for either convex or nonconvex problems, however, most of which only consider smooth objectives. In contrast, people have not fully explored and well understood this problem when functions are nonsmooth. This paper aims to fill this crucial gap by providing a comprehensive analysis of stochastic nonsmooth convex optimization with heavy-tailed noises. We revisit a simple clipping-based algorithm, whereas, which is only proved to converge in expectation but under the additional strong convexity assumption. Under appropriate choices of parameters, for both convex and strongly convex functions, we not only establish the first high-probability rates but also give refined in-expectation bounds compared with existing works. Remarkably, all of our results are optimal (or nearly optimal up to logarithmic factors) with respect to the time horizon $T$ even when $T$ is unknown in advance. Additionally, we show how to make the algorithm parameter-free with respect to $\sigma$, in other words, the algorithm can still guarantee convergence without any prior knowledge of $\sigma$. Furthermore, an initial distance adaptive convergence rate is provided if $\sigma$ is assumed to be known.

Whenever applicable, the Stochastic Gradient Descent (SGD) has shown itself to be unreasonably effective. Instead of underperforming and getting trapped in local minima due to the batch noise, SGD leverages it to learn to generalize better and find minima that are good enough for the entire dataset. This led to numerous theoretical and experimental investigations, especially in the context of Artificial Neural Networks (ANNs), leading to better machine learning algorithms. However, SGD is not applicable in a non-differentiable setting, leaving all that prior research off the table. In this paper, we show that a class of evolutionary algorithms (EAs) inspired by the Gillespie-Orr Mutational Landscapes model for natural evolution is formally equivalent to SGD in certain settings and, in practice, is well adapted to large ANNs. We refer to such EAs as Gillespie-Orr EA class (GO-EAs) and empirically show how an insight transfer from SGD can work for them. We then show that for ANNs trained to near-optimality or in the transfer learning setting, the equivalence also allows transferring the insights from the Mutational Landscapes model to SGD. We then leverage this equivalence to experimentally show how SGD and GO-EAs can provide mutual insight through examples of minima flatness, transfer learning, and mixing of individuals in EAs applied to large models.

Sharpness-Aware Minimization (SAM) is a recent optimization framework aiming to improve the deep neural network generalization, through obtaining flatter (i.e. less sharp) solutions. As SAM has been numerically successful, recent papers have studied the theoretical aspects of the framework and have shown SAM solutions are indeed flat. However, there has been limited theoretical exploration regarding statistical properties of SAM. In this work, we directly study the statistical performance of SAM, and present a new theoretical explanation of why SAM generalizes well. To this end, we study two statistical problems, neural networks with a hidden layer and kernel regression, and prove under certain conditions, SAM has smaller prediction error over Gradient Descent (GD). Our results concern both convex and non-convex settings, and show that SAM is particularly well-suited for non-convex problems. Additionally, we prove that in our setup, SAM solutions are less sharp as well, showing our results are in agreement with the previous work. Our theoretical findings are validated using numerical experiments on numerous scenarios, including deep neural networks.

Machine learning techniques have a long history of success in denoising tasks. The recent breakthrough of diffusionbased generation [1, 2] has further revived the interest in denoising networks, demonstrating how they can also be leveraged, beyond denoising, for generative tasks. However, this rapidly expanding range of applications stands in sharp contrast to the relatively scarce theoretical understanding of denoising neural networks, even for the simplest instance thereof - namely Denoising Auto Encoders (DAEs) [3]. Theoretical studies of autoencoders have hitherto almost exclusively focused on data compression tasks using Reconstruction Auto Encoders (RAEs), where the goal is to learn a concise latent representation of the data. A majority of this body of work addresses linear autoencoders [4-7]. The authors of [8, 9] analyze the gradient-based training of non-linear autoencoders with online stochastic gradient descent or in population, thus implicitly assuming the availability of an infinite number of training samples. Furthermore, two-layer RAEs were shown to learn to essentially perform Principal Component Analysis (PCA) [10-12], i.e. to learn a linear model. Ref. [13] shows that this is also true for infinite-width architectures. Learning in DAEs has been the object of theoretical investigations only in the linear case [14], while the case of non-linear DAEs remains theoretically largely unexplored.