We consider stochastic optimization problems where data is drawn from a Markov chain. Existing methods for this setting crucially rely on knowing the mixing time of the chain, which in real-world applications is usually unknown. We propose the first optimization method that does not require the knowledge of the mixing time, yet obtains the optimal asymptotic convergence rate when applied to convex problems. We further show that our approach can be extended to: (i) finding stationary points in non-convex optimization with Markovian data, and (ii) obtaining better dependence on the mixing time in temporal difference (TD) learning; in both cases, our method is completely oblivious to the mixing time. Our method relies on a novel combination of multi-level Monte Carlo (MLMC) gradient estimation together with an adaptive learning method.

As a prevalent distributed learning paradigm, Federated Learning (FL) trains a global model on a massive amount of devices with infrequent communication. This paper investigates a class of composite optimization and statistical recovery problems in the FL setting, whose loss function consists of a data-dependent smooth loss and a non-smooth regularizer. Examples include sparse linear regression using Lasso, low-rank matrix recovery using nuclear norm regularization, etc. In the existing literature, federated composite optimization algorithms are designed only from an optimization perspective without any statistical guarantees. In addition, they do not consider commonly used (restricted) strong convexity in statistical recovery problems. We advance the frontiers of this problem from both optimization and statistical perspectives. From optimization upfront, we propose a new algorithm named \textit{Fast Federated Dual Averaging} for strongly convex and smooth loss and establish state-of-the-art iteration and communication complexity in the composite setting. In particular, we prove that it enjoys a fast rate, linear speedup, and reduced communication rounds. From statistical upfront, for restricted strongly convex and smooth loss, we design another algorithm, namely \textit{Multi-stage Federated Dual Averaging}, and prove a high probability complexity bound with linear speedup up to optimal statistical precision. Numerical experiments in both synthetic and real data demonstrate that our methods perform better than other baselines. To the best of our knowledge, this is the first work providing fast optimization algorithms and statistical recovery guarantees for composite problems in FL.

We study oracle complexity of gradient based methods for stochastic approximation problems. Though in many settings optimal algorithms and tight lower bounds are known for such problems, these optimal algorithms do not achieve the best performance when used in practice. We address this theory-practice gap by focusing on \emph{instance-dependent complexity} instead of worst case complexity. In particular, we first summarize known instance-dependent complexity results and categorize them into three levels. We identify the domination relation between different levels and propose a fourth instance-dependent bound that dominates existing ones. We then provide a sufficient condition according to which an adaptive algorithm with moment estimation can achieve the proposed bound without knowledge of noise levels. Our proposed algorithm and its analysis provide a theoretical justification for the success of moment estimation as it achieves improved instance complexity.

Large-scale machine learning systems often involve data distributed across a collection of users. Federated learning algorithms leverage this structure by communicating model updates to a central server, rather than entire datasets. In this paper, we study stochastic optimization algorithms for a personalized federated learning setting involving local and global models subject to user-level (joint) differential privacy. While learning a private global model induces a cost of privacy, local learning is perfectly private. We provide generalization guarantees showing that coordinating local learning with private centralized learning yields a generically useful and improved tradeoff between accuracy and privacy. We illustrate our theoretical results with experiments on synthetic and real-world datasets.

In this paper, we investigate the problem of stochastic multi-level compositional optimization, where the objective function is a composition of multiple smooth but possibly non-convex functions. Existing methods for solving this problem either suffer from sub-optimal sample complexities or need a huge batch size. To address this limitation, we propose a Stochastic Multi-level Variance Reduction method (SMVR), which achieves the optimal sample complexity of $\mathcal{O}\left(1 / \epsilon^{3}\right)$ to find an $\epsilon$-stationary point for non-convex objectives. Furthermore, when the objective function satisfies the convexity or Polyak-Łojasiewicz (PL) condition, we propose a stage-wise variant of SMVR and improve the sample complexity to $\mathcal{O}\left(1 / \epsilon^{2}\right)$ for convex functions or $\mathcal{O}\left(1 /(\mu\epsilon)\right)$ for non-convex functions satisfying the $\mu$-PL condition. The latter result implies the same complexity for $\mu$-strongly convex functions. To make use of adaptive learning rates, we also develop Adaptive SMVR, which achieves the same optimal complexities but converges faster in practice. All our complexities match the lower bounds not only in terms of $\epsilon$ but also in terms of $\mu$ (for PL or strongly convex functions), without using a large batch size in each iteration.

This paper studies stochastic optimization for a sum of compositional functions, where the inner-level function of each summand is coupled with the corresponding summation index. We refer to this family of problems as finite-sum coupled compositional optimization (FCCO). It has broad applications in machine learning for optimizing non-convex or convex compositional measures/objectives such as average precision (AP), p-norm push, listwise ranking losses, neighborhood component analysis (NCA), deep survival analysis, deep latent variable models, etc., which deserves finer analysis. Yet, existing algorithms and analyses are restricted in one or other aspects. The contribution of this paper is to provide a comprehensive convergence analysis of a simple stochastic algorithm for both non-convex and convex objectives. Our key result is the improved oracle complexity with the parallel speed-up by using the moving-average based estimator with mini-batching. Our theoretical analysis also exhibits new insights for improving the practical implementation by sampling the batches of equal size for the outer and inner levels. Numerical experiments on AP maximization, NCA, and p-norm push corroborate some aspects of the theory.

We aim to make stochastic gradient descent (SGD) adaptive to (i) the noise $\sigma^2$ in the stochastic gradients and (ii) problem-dependent constants. When minimizing smooth, strongly-convex functions with condition number $\kappa$, we prove that $T$ iterations of SGD with exponentially decreasing step-sizes and knowledge of the smoothness can achieve an $\tilde{O} \left(\exp \left( \nicefrac{-T}{\kappa} \right) + \nicefrac{\sigma^2}{T} \right)$ rate, without knowing $\sigma^2$. In order to be adaptive to the smoothness, we use a stochastic line-search (SLS) and show (via upper and lower-bounds) that SGD with SLS converges at the desired rate, but only to a neighbourhood of the solution. On the other hand, we prove that SGD with an offline estimate of the smoothness converges to the minimizer. However, its rate is slowed down proportional to the estimation error. Next, we prove that SGD with Nesterov acceleration and exponential step-sizes (referred to as ASGD) can achieve the near-optimal $\tilde{O} \left(\exp \left( \nicefrac{-T}{\sqrt{\kappa}} \right) + \nicefrac{\sigma^2}{T} \right)$ rate, without knowledge of $\sigma^2$. When used with offline estimates of the smoothness and strong-convexity, ASGD still converges to the solution, albeit at a slower rate. Finally, we empirically demonstrate the effectiveness of exponential step-sizes coupled with a novel variant of SLS.

The implicit stochastic gradient descent (ISGD), a proximal version of SGD, is gaining interest in the literature due to its stability over (explicit) SGD. In this paper, we conduct an in-depth analysis of the two modes of ISGD for smooth convex functions, namely proximal Robbins-Monro (proxRM) and proximal Poylak-Ruppert (proxPR) procedures, for their use in statistical inference on model parameters. Specifically, we derive non-asymptotic point estimation error bounds of both proxRM and proxPR iterates and their limiting distributions, and propose on-line estimators of their asymptotic covariance matrices that require only a single run of ISGD. The latter estimators are used to construct valid confidence intervals for the model parameters. Our analysis is free of the generalized linear model assumption that has limited the preceding analyses, and employs feasible procedures. Our on-line covariance matrix estimators appear to be the first of this kind in the ISGD literature.

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 $\mathcal{O}(\kappa \log \nicefrac{1}{\varepsilon})$, where $\kappa$ is the condition number of $f$, the number of prox evaluations is $\mathcal{O}(\sqrt{\kappa} \log \nicefrac{1}{\varepsilon})$ 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 {\em 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.

Federated learning is a machine learning training paradigm that enables clients to jointly train models without sharing their own localized data. However, the implementation of federated learning in practice still faces numerous challenges, such as the large communication overhead due to the repetitive server-client synchronization and the lack of adaptivity by SGD-based model updates. Despite that various methods have been proposed for reducing the communication cost by gradient compression or quantization, and the federated versions of adaptive optimizers such as FedAdam are proposed to add more adaptivity, the current federated learning framework still cannot solve the aforementioned challenges all at once. In this paper, we propose a novel communication-efficient adaptive federated learning method (FedCAMS) with theoretical convergence guarantees. We show that in the nonconvex stochastic optimization setting, our proposed FedCAMS achieves the same convergence rate of $O(\frac{1}{\sqrt{TKm}})$ as its non-compressed counterparts. Extensive experiments on various benchmarks verify our theoretical analysis.

The accelerated proximal point method (APPA), also known as "Catalyst", is a well-established reduction from convex optimization to approximate proximal point computation (i.e., regularized minimization). This reduction is conceptually elegant and yields strong convergence rate guarantees. However, these rates feature an extraneous logarithmic term arising from the need to compute each proximal point to high accuracy. In this work, we propose a novel Relaxed Error Criterion for Accelerated Proximal Point (RECAPP) that eliminates the need for high accuracy subproblem solutions. We apply RECAPP to two canonical problems: finite-sum and max-structured minimization. For finite-sum problems, we match the best known complexity, previously obtained by carefully-designed problem-specific algorithms. For minimizing max_y f(x,y) where f is convex in x and strongly-concave in y, we improve on the best known (Catalyst-based) bound by a logarithmic factor.

Recently, some accelerated stochastic variance reduction algorithms such as Katyusha and ASVRG-ADMM achieve faster convergence than non-accelerated methods such as SVRG and SVRG-ADMM. However, there are still some gaps between the oracle complexities and their lower bounds. To fill in these gaps, this paper proposes a novel Directly Accelerated stochastic Variance reductIon (DAVIS) algorithm with two Snapshots for non-strongly convex (non-SC) unconstrained problems. Our theoretical results show that DAVIS achieves the optimal convergence rate O(1/(nS^2)) and optimal gradient complexity O(n+\sqrt{nL/\epsilon}), which is identical to its lower bound. To the best of our knowledge, this is the first directly accelerated algorithm that attains the optimal lower bound and improves the convergence rate from O(1/S^2) to O(1/(nS^2)). Moreover, we extend DAVIS and theoretical results to non-SC problems with a structured regularizer, and prove that the proposed algorithm with double-snapshots also attains the optimal convergence rate O(1/(nS)) and optimal oracle complexity O(n+L/\epsilon) for such problems, and it is at least a factor n/S faster than existing accelerated stochastic algorithms, where n\gg S in general.