We consider a family of unadjusted HMC samplers, which includes standard position HMC samplers and discretizations of the underdamped Langevin process. A detailed analysis and optimization of the parameters is conducted in the Gaussian case. Then, a stochastic gradient version of the samplers is considered, for which dimension-free convergence rates are established for log-concave smooth targets, gathering in a unified framework previous results on both processes. Both results indicate that partial refreshments of the velocity are more efficient than standard full refreshments.
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We study a new two-time-scale stochastic gradient method for solving optimization problems, where the gradients are computed with the aid of an auxiliary variable under samples generated by time-varying Markov random processes parameterized by the underlying optimization variable. These time-varying samples make gradient directions in our update biased and dependent, which can potentially lead to the divergence of the iterates. In our two-time-scale approach, one scale is to estimate the true gradient from these samples, which is then used to update the estimate of the optimal solution. While these two iterates are implemented simultaneously, the former is updated "faster" (using bigger step sizes) than the latter (using smaller step sizes). Our first contribution is to characterize the finite-time complexity of the proposed two-time-scale stochastic gradient method. In particular, we provide explicit formulas for the convergence rates of this method under different structural assumptions, namely, strong convexity, convexity, the Polyak-Lojasiewicz condition, and general non-convexity. We apply our framework to two problems in control and reinforcement learning. First, we look at the standard online actor-critic algorithm over finite state and action spaces and derive a convergence rate of O(k^(-2/5)), which recovers the best known rate derived specifically for this problem. Second, we study an online actor-critic algorithm for the linear-quadratic regulator and show that a convergence rate of O(k^(-2/3)) is achieved. This is the first time such a result is known in the literature. Finally, we support our theoretical analysis with numerical simulations where the convergence rates are visualized.
Federated learning (FL) aims to minimize the communication complexity of training a model over heterogeneous data distributed across many clients. A common approach is local methods, where clients take multiple optimization steps over local data before communicating with the server (e.g., FedAvg). Local methods can exploit similarity between clients' data. However, in existing analyses, this comes at the cost of slow convergence in terms of the dependence on the number of communication rounds R. On the other hand, global methods, where clients simply return a gradient vector in each round (e.g., SGD), converge faster in terms of R but fail to exploit the similarity between clients even when clients are homogeneous. We propose FedChain, an algorithmic framework that combines the strengths of local methods and global methods to achieve fast convergence in terms of R while leveraging the similarity between clients. Using FedChain, we instantiate algorithms that improve upon previously known rates in the general convex and PL settings, and are near-optimal (via an algorithm-independent lower bound that we show) for problems that satisfy strong convexity. Empirical results support this theoretical gain over existing methods.
The problems of frictional contacts are the key to the investigation of mechanical performances of composite materials under varying service environments. The paper considers a linear elasticity system with strongly heterogeneous coefficients and quasistatic Tresca's friction law, and we study the homogenization theories under the frameworks of H-convergence and small $\epsilon$-periodicity. The qualitative result is based on H-convergence, which shows the original oscillating solutions will converge weakly to the homogenized solution, while our quantitative result provides an estimate of asymptotic errors in the $H^1$ norm for the periodic homogenization. We also design several numerical experiments to validate the convergence rates in the quantitative analysis.
This paper focuses on stochastic saddle point problems with decision-dependent distributions. These are problems whose objective is the expected value of a stochastic payoff function, where random variables are drawn from a distribution induced by a distributional map. For general distributional maps, the problem of finding saddle points is in general computationally burdensome, even if the distribution is known. To enable a tractable solution approach, we introduce the notion of equilibrium points -- which are saddle points for the stationary stochastic minimax problem that they induce -- and provide conditions for their existence and uniqueness. We demonstrate that the distance between the two solution types is bounded provided that the objective has a strongly-convex-strongly-concave payoff and a Lipschitz continuous distributional map. We develop deterministic and stochastic primal-dual algorithms and demonstrate their convergence to the equilibrium point. In particular, by modeling errors emerging from a stochastic gradient estimator as sub-Weibull random variables, we provide error bounds in expectation and in high probability that hold for each iteration. Moreover, we show convergence to a neighborhood almost surely. Finally, we investigate a condition on the distributional map -- which we call opposing mixture dominance -- that ensures that the objective is strongly-convex-strongly-concave. We tailor the convergence results for the primal-dual algorithms to this opposing mixture dominance setup.
We study the distributed minimum spanning tree (MST) problem, a fundamental problem in distributed computing. It is well-known that distributed MST can be solved in $\tilde{O}(D+\sqrt{n})$ rounds in the standard CONGEST model (where $n$ is the network size and $D$ is the network diameter) and this is essentially the best possible round complexity (up to logarithmic factors). However, in resource-constrained networks such as ad hoc wireless and sensor networks, nodes spending so much time can lead to significant spending of resources such as energy. Motivated by the above consideration, we study distributed algorithms for MST under the \emph{sleeping model} [Chatterjee et al., PODC 2020], a model for design and analysis of resource-efficient distributed algorithms. In the sleeping model, a node can be in one of two modes in any round -- \emph{sleeping} or \emph{awake} (unlike the traditional model where nodes are always awake). Only the rounds in which a node is \emph{awake} are counted, while \emph{sleeping} rounds are ignored. A node spends resources only in the awake rounds and hence the main goal is to minimize the \emph{awake complexity} of a distributed algorithm, the worst-case number of rounds any node is awake. We present deterministic and randomized distributed MST algorithms that have an \emph{optimal} awake complexity of $O(\log n)$ time with a matching lower bound. We also show that our randomized awake-optimal algorithm has essentially the best possible round complexity by presenting a lower bound of $\tilde{\Omega}(n)$ on the product of the awake and round complexity of any distributed algorithm (including randomized) that outputs an MST, where $\tilde{\Omega}$ hides a $1/(\text{polylog } n)$ factor.
Momentum methods, including heavy-ball~(HB) and Nesterov's accelerated gradient~(NAG), are widely used in training neural networks for their fast convergence. However, there is a lack of theoretical guarantees for their convergence and acceleration since the optimization landscape of the neural network is non-convex. Nowadays, some works make progress towards understanding the convergence of momentum methods in an over-parameterized regime, where the number of the parameters exceeds that of the training instances. Nonetheless, current results mainly focus on the two-layer neural network, which are far from explaining the remarkable success of the momentum methods in training deep neural networks. Motivated by this, we investigate the convergence of NAG with constant learning rate and momentum parameter in training two architectures of deep linear networks: deep fully-connected linear neural networks and deep linear ResNets. Based on the over-parameterization regime, we first analyze the residual dynamics induced by the training trajectory of NAG for a deep fully-connected linear neural network under the random Gaussian initialization. Our results show that NAG can converge to the global minimum at a $(1 - \mathcal{O}(1/\sqrt{\kappa}))^t$ rate, where $t$ is the iteration number and $\kappa > 1$ is a constant depending on the condition number of the feature matrix. Compared to the $(1 - \mathcal{O}(1/{\kappa}))^t$ rate of GD, NAG achieves an acceleration over GD. To the best of our knowledge, this is the first theoretical guarantee for the convergence of NAG to the global minimum in training deep neural networks. Furthermore, we extend our analysis to deep linear ResNets and derive a similar convergence result.
Stochastic optimization algorithms implemented on distributed computing architectures are increasingly used to tackle large-scale machine learning applications. A key bottleneck in such distributed systems is the communication overhead for exchanging information such as stochastic gradients between different workers. Sparse communication with memory and the adaptive aggregation methodology are two successful frameworks among the various techniques proposed to address this issue. In this paper, we exploit the advantages of Sparse communication and Adaptive aggregated Stochastic Gradients to design a communication-efficient distributed algorithm named SASG. Specifically, we determine the workers who need to communicate with the parameter server based on the adaptive aggregation rule and then sparsify the transmitted information. Therefore, our algorithm reduces both the overhead of communication rounds and the number of communication bits in the distributed system. We define an auxiliary sequence and provide convergence results of the algorithm with the help of Lyapunov function analysis. Experiments on training deep neural networks show that our algorithm can significantly reduce the communication overhead compared to the previous methods, with little impact on training and testing accuracy.
The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.
Statistical-Computational Trade-offs in Tensor PCA and Related Problems via Communication Complexity
Tensor PCA is a stylized statistical inference problem introduced by Montanari and Richard to study the computational difficulty of estimating an unknown parameter from higher-order moment tensors. Unlike its matrix counterpart, Tensor PCA exhibits a statistical-computational gap, i.e., a sample size regime where the problem is information-theoretically solvable but conjectured to be computationally hard. This paper derives computational lower bounds on the run-time of memory bounded algorithms for Tensor PCA using communication complexity. These lower bounds specify a trade-off among the number of passes through the data sample, the sample size, and the memory required by any algorithm that successfully solves Tensor PCA. While the lower bounds do not rule out polynomial-time algorithms, they do imply that many commonly-used algorithms, such as gradient descent and power method, must have a higher iteration count when the sample size is not large enough. Similar lower bounds are obtained for Non-Gaussian Component Analysis, a family of statistical estimation problems in which low-order moment tensors carry no information about the unknown parameter. Finally, stronger lower bounds are obtained for an asymmetric variant of Tensor PCA and related statistical estimation problems. These results explain why many estimators for these problems use a memory state that is significantly larger than the effective dimensionality of the parameter of interest.
We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov accelerated gradient descent (PAGD) method. We discretize these problems with a Fourier collocation method in space, and employ various second-order schemes in time. We observe a significant speedup with this solver when compared to the preconditioned gradient descent (PGD) method. With the PAGD solver, fully implicit, second-order-in-time schemes are not only feasible to solve the PFC and FCH equations, but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account. Benchmark computations of five different schemes for the PFC and FCH equations are conducted and the results indicate that, for the FCH experiments, the fully implicit schemes (midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver) perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision. For the PFC, the results are not as conclusive as in the FCH experiments, which, we believe, is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation. We also discuss some practical matters in applying the PAGD. We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters. The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.
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