We present a novel universal gradient method for solving convex optimization problems. Our algorithm -- Dual Averaging with Distance Adaptation (DADA) -- is based on the classical scheme of dual averaging and dynamically adjusts its coefficients based on observed gradients and the distance between iterates and the starting point, eliminating the need for problem-specific parameters. DADA is a universal algorithm that simultaneously works for a broad spectrum of problem classes, provided the local growth of the objective function around its minimizer can be bounded. Particular examples of such problem classes are nonsmooth Lipschitz functions, Lipschitz-smooth functions, H\"older-smooth functions, functions with high-order Lipschitz derivative, quasi-self-concordant functions, and $(L_0,L_1)$-smooth functions. Crucially, DADA is applicable to both unconstrained and constrained problems, even when the domain is unbounded, without requiring prior knowledge of the number of iterations or desired accuracy.

Modern distributed training relies heavily on communication compression to reduce the communication overhead. In this work, we study algorithms employing a popular class of contractive compressors in order to reduce communication overhead. However, the naive implementation often leads to unstable convergence or even exponential divergence due to the compression bias. Error Compensation (EC) is an extremely popular mechanism to mitigate the aforementioned issues during the training of models enhanced by contractive compression operators. Compared to the effectiveness of EC in the data homogeneous regime, the understanding of the practicality and theoretical foundations of EC in the data heterogeneous regime is limited. Existing convergence analyses typically rely on strong assumptions such as bounded gradients, bounded data heterogeneity, or large batch accesses, which are often infeasible in modern machine learning applications. We resolve the majority of current issues by proposing EControl, a novel mechanism that can regulate error compensation by controlling the strength of the feedback signal. We prove fast convergence for EControl in standard strongly convex, general convex, and nonconvex settings without any additional assumptions on the problem or data heterogeneity. We conduct extensive numerical evaluations to illustrate the efficacy of our method and support our theoretical findings.

We introduce ProxSkip -- a surprisingly simple and provably efficient method for minimizing the sum of a smooth ($f$) and an expensive nonsmooth proximable ($\psi$) function. The canonical approach to solving such problems is via the proximal gradient descent (ProxGD) algorithm, which is based on the evaluation of the gradient of $f$ and the prox operator of $\psi$ in each iteration. In this work we are specifically interested in the regime in which the evaluation of prox is costly relative to the evaluation of the gradient, which is the case in many applications. ProxSkip allows for the expensive prox operator to be skipped in most iterations: while its iteration complexity is $\mathcal{O}\left(\kappa \log \frac{1}{\varepsilon}\right)$, where $\kappa$ is the condition number of $f$, the number of prox evaluations is $\mathcal{O}\left(\sqrt{\kappa} \log \frac{1}{\varepsilon}\right)$ only. Our main motivation comes from federated learning, where evaluation of the gradient operator corresponds to taking a local GD step independently on all devices, and evaluation of prox corresponds to (expensive) communication in the form of gradient averaging. In this context, ProxSkip offers an effective acceleration of communication complexity. Unlike other local gradient-type methods, such as FedAvg, SCAFFOLD, S-Local-GD and FedLin, whose theoretical communication complexity is worse than, or at best matching, that of vanilla GD in the heterogeneous data regime, we obtain a provable and large improvement without any heterogeneity-bounding assumptions.

When training neural networks, it has been widely observed that a large step size is essential in stochastic gradient descent (SGD) for obtaining superior models. However, the effect of large step sizes on the success of SGD is not well understood theoretically. Several previous works have attributed this success to the stochastic noise present in SGD. However, we show through a novel set of experiments that the stochastic noise is not sufficient to explain good non-convex training, and that instead the effect of a large learning rate itself is essential for obtaining best performance.We demonstrate the same effects also in the noise-less case, i.e. for full-batch GD. We formally prove that GD with large step size -- on certain non-convex function classes -- follows a different trajectory than GD with a small step size, which can lead to convergence to a global minimum instead of a local one. Our settings provide a framework for future analysis which allows comparing algorithms based on behaviors that can not be observed in the traditional settings.

We consider a stochastic continuum armed bandit problem where the arms are indexed by the $\ell_2$ ball $B_{d}(1+\nu)$ of radius $1+\nu$ in $\mathbb{R}^d$. The reward functions $r :B_{d}(1+\nu) \rightarrow \mathbb{R}$ are considered to intrinsically depend on $k \ll d$ unknown linear parameters so that $r(\mathbf{x}) = g(\mathbf{A} \mathbf{x})$ where $\mathbf{A}$ is a full rank $k \times d$ matrix. Assuming the mean reward function to be smooth we make use of results from low-rank matrix recovery literature and derive an efficient randomized algorithm which achieves a regret bound of $O(C(k,d) n^{\frac{1+k}{2+k}} (\log n)^{\frac{1}{2+k}})$ with high probability. Here $C(k,d)$ is at most polynomial in $d$ and $k$ and $n$ is the number of rounds or the sampling budget which is assumed to be known beforehand.