6 research outputs found
Communication-Efficient Gradient Descent-Accent Methods for Distributed Variational Inequalities: Unified Analysis and Local Updates
Distributed and federated learning algorithms and techniques associated
primarily with minimization problems. However, with the increase of minimax
optimization and variational inequality problems in machine learning, the
necessity of designing efficient distributed/federated learning approaches for
these problems is becoming more apparent. In this paper, we provide a unified
convergence analysis of communication-efficient local training methods for
distributed variational inequality problems (VIPs). Our approach is based on a
general key assumption on the stochastic estimates that allows us to propose
and analyze several novel local training algorithms under a single framework
for solving a class of structured non-monotone VIPs. We present the first local
gradient descent-accent algorithms with provable improved communication
complexity for solving distributed variational inequalities on heterogeneous
data. The general algorithmic framework recovers state-of-the-art algorithms
and their sharp convergence guarantees when the setting is specialized to
minimization or minimax optimization problems. Finally, we demonstrate the
strong performance of the proposed algorithms compared to state-of-the-art
methods when solving federated minimax optimization problems
Solving Structured Hierarchical Games Using Differential Backward Induction
Many real-world systems possess a hierarchical structure where a strategic
plan is forwarded and implemented in a top-down manner. Examples include
business activities in large companies or policy making for reducing the spread
during pandemics. We introduce a novel class of games that we call structured
hierarchical games (SHGs) to capture these strategic interactions. In an SHG,
each player is represented as a vertex in a multi-layer decision tree and
controls a real-valued action vector reacting to orders from its predecessors
and influencing its descendants' behaviors strategically based on its own
subjective utility. SHGs generalize extensive form games as well as Stackelberg
games. For general SHGs with (possibly) nonconvex payoffs and high-dimensional
action spaces, we propose a new solution concept which we call local subgame
perfect equilibrium. By exploiting the hierarchical structure and strategic
dependencies in payoffs, we derive a back propagation-style gradient-based
algorithm which we call Differential Backward Induction to compute an
equilibrium. We theoretically characterize the convergence properties of DBI
and empirically demonstrate a large overlap between the stable points reached
by DBI and equilibrium solutions. Finally, we demonstrate the effectiveness of
our algorithm in finding \emph{globally} stable solutions and its scalability
for a recently introduced class of SHGs for pandemic policy making
LEAD: Least-Action Dynamics for Min-Max Optimization
Adversarial formulations such as generative adversarial networks (GANs) have
rekindled interest in two-player min-max games. A central obstacle in the
optimization of such games is the rotational dynamics that hinder their
convergence. Existing methods typically employ intuitive, carefully
hand-designed mechanisms for controlling such rotations. In this paper, we take
a novel approach to address this issue by casting min-max optimization as a
physical system. We leverage tools from physics to introduce LEAD (Least-Action
Dynamics), a second-order optimizer for min-max games. Next, using Lyapunov
stability theory and spectral analysis, we study LEAD's convergence properties
in continuous and discrete-time settings for bilinear games to demonstrate
linear convergence to the Nash equilibrium. Finally, we empirically evaluate
our method on synthetic setups and CIFAR-10 image generation to demonstrate
improvements over baseline methods