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Minimax problems have gained tremendous attentions across the optimization and machine learning community recently. In this paper, we introduce a new quasi-Newton method for minimax problems, which we call $J$-symmetric quasi-Newton method. The method is obtained by exploiting the $J$-symmetric structure of the second-order derivative of the objective function in minimax problem. We show that the Hessian estimation (as well as its inverse) can be updated by a rank-2 operation, and it turns out that the update rule is a natural generalization of the classic Powell symmetric Broyden (PSB) method from minimization problems to minimax problems. In theory, we show that our proposed quasi-Newton algorithm enjoys local Q-superlinear convergence to a desirable solution under standard regularity conditions. Furthermore, we introduce a trust-region variant of the algorithm that enjoys global R-superlinear convergence. Finally, we present numerical experiments that verify our theory and show the effectiveness of our proposed algorithms compared to Broyden's method and the extragradient method on three classes of minimax problems.

In this paper, we study the challenging task of Byzantine-robust decentralized training on arbitrary communication graphs. Unlike federated learning where workers communicate through a server, workers in the decentralized environment can only talk to their neighbors, making it harder to reach consensus. We identify a novel dissensus attack in which few malicious nodes can take advantage of information bottlenecks in the topology to poison the collaboration. To address these issues, we propose a Self-Centered Clipping (SCClip) algorithm for Byzantine-robust consensus and optimization, which is the first to provably converge to a $O(\delta_{\max}\zeta^2/\gamma^2)$ neighborhood of the stationary point for non-convex objectives under standard assumptions. Finally, we demonstrate the encouraging empirical performance of SCClip under a large number of attacks.

Recently a machine learning approach to Monte-Carlo simulations called Neural Markov Chain Monte-Carlo (NMCMC) is gaining traction. In its most popular form it uses the neural networks to construct normalizing flows which are then trained to approximate the desired target distribution. As this distribution is usually defined via a Hamiltonian or action, the standard learning algorithm requires estimation of the action gradient with respect to the fields. In this contribution we present another gradient estimator (and the corresponding [PyTorch implementation) that avoids this calculation, thus potentially speeding up training for models with more complicated actions. We also study the statistical properties of several gradient estimators and show that our formulation leads to better training results.

We show how probabilistic numerics can be used to convert an initial value problem into a Gauss--Markov process parametrised by the dynamics of the initial value problem. Consequently, the often difficult problem of parameter estimation in ordinary differential equations is reduced to hyperparameter estimation in Gauss--Markov regression, which tends to be considerably easier. The method's relation and benefits in comparison to classical numerical integration and gradient matching approaches is elucidated. In particular, the method can, in contrast to gradient matching, handle partial observations, and has certain routes for escaping local optima not available to classical numerical integration. Experimental results demonstrate that the method is on par or moderately better than competing approaches.

High computational time is one of the most important operational issues in centralized on-demand shared mobility systems. To resolve this issue, we propose a decentralized ride-matching system that is based on vehicle to infrastructure (V2I) and infrastructure to infrastructure (I2I) communication. Application on the downtown Toronto road network demonstrated that the decentralized system resulted in a speed-up of 125 times in terms of computational time and showed high scalability. Moreover, the service rate in the proposed system improved by 7% compared to the centralized. However, the centralized system showed 29% and 17% improvement in wait time and detour time.

Contextual bandit algorithms have been recently studied under the federated learning setting to satisfy the demand of keeping data decentralized and pushing the learning of bandit models to the client side. But limited by the required communication efficiency, existing solutions are restricted to linear models to exploit their closed-form solutions for parameter estimation. Such a restricted model choice greatly hampers these algorithms' practical utility. In this paper, we take the first step to addressing this challenge by studying generalized linear bandit models under a federated learning setting. We propose a communication-efficient solution framework that employs online regression for local update and offline regression for global update. We rigorously proved that, though the setting is more general and challenging, our algorithm can attain sub-linear rate in both regret and communication cost, which is also validated by our extensive empirical evaluations.

Softmax policy gradient is a popular algorithm for policy optimization in single-agent reinforcement learning, particularly since projection is not needed for each gradient update. However, in multi-agent systems, the lack of central coordination introduces significant additional difficulties in the convergence analysis. Even for a stochastic game with identical interest, there can be multiple Nash Equilibria (NEs), which disables proof techniques that rely on the existence of a unique global optimum. Moreover, the softmax parameterization introduces non-NE policies with zero gradient, making NE-seeking difficult for gradient-based algorithms. In this paper, we study the finite time convergence of decentralized softmax gradient play in a special form of game, Markov Potential Games (MPGs), which includes the identical interest game as a special case. We investigate both gradient play and natural gradient play, with and without $\log$-barrier regularization. Establishing convergence for the unregularized cases relies on an assumption that the stationary policies are isolated, and yields convergence bounds that contain a trajectory dependent constant that can be arbitrarily large. We introduce the $\log$-barrier regularization to overcome these drawbacks, with the cost of slightly worse dependence on other factors such as the action set size. An empirical study on an identical interest matrix game confirms the theoretical findings.

The distributed convex optimization problem over the multi-agent system is considered in this paper, and it is assumed that each agent possesses its own cost function and communicates with its neighbours over a sequence of time-varying directed graphs. However, due to some reasons there exist communication delays while agents receive information from other agents, and we are going to seek the optimal value of the sum of agents' loss functions in this case. We desire to handle this problem with the push-sum distributed dual averaging (PS-DDA) algorithm. It is proved that this algorithm converges and the error decays at a rate $\mathcal{O}\left(T^{-0.5}\right)$ with proper step size, where $T$ is iteration span. The main result presented in this paper also illustrates the convergence of the proposed algorithm is related to the maximum value of the communication delay on one edge. We finally apply the theoretical results to numerical simulations to show the PS-DDA algorithm's performance.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.