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This paper studies policy optimization algorithms for multi-agent reinforcement learning. We begin by proposing an algorithm framework for two-player zero-sum Markov Games in the full-information setting, where each iteration consists of a policy update step at each state using a certain matrix game algorithm, and a value update step with a certain learning rate. This framework unifies many existing and new policy optimization algorithms. We show that the state-wise average policy of this algorithm converges to an approximate Nash equilibrium (NE) of the game, as long as the matrix game algorithms achieve low weighted regret at each state, with respect to weights determined by the speed of the value updates. Next, we show that this framework instantiated with the Optimistic Follow-The-Regularized-Leader (OFTRL) algorithm at each state (and smooth value updates) can find an $\mathcal{\widetilde{O}}(T^{-5/6})$ approximate NE in $T$ iterations, and a similar algorithm with slightly modified value update rule achieves a faster $\mathcal{\widetilde{O}}(T^{-1})$ convergence rate. These improve over the current best $\mathcal{\widetilde{O}}(T^{-1/2})$ rate of symmetric policy optimization type algorithms. We also extend this algorithm to multi-player general-sum Markov Games and show an $\mathcal{\widetilde{O}}(T^{-3/4})$ convergence rate to Coarse Correlated Equilibria (CCE). Finally, we provide a numerical example to verify our theory and investigate the importance of smooth value updates, and find that using "eager" value updates instead (equivalent to the independent natural policy gradient algorithm) may significantly slow down the convergence, even on a simple game with $H=2$ layers.

In our time cybersecurity has grown to be a topic of massive proportion at the national and enterprise levels. Our thesis is that the economic perspective and investment decision-making are vital factors in determining the outcome of the struggle. To build our economic framework, we borrow from the pioneering work of Gordon and Loeb in which the Defender optimally trades-off investments for lower likelihood of its system breach. Our two-sided model additionally has an Attacker, assumed to be rational and also guided by economic considerations in its decision-making, to which the Defender responds. Our model is a simplified adaptation of a model proposed during the Cold War for weapons deployment in the US. Our model may also be viewed as a Stackelberg game and, from an analytic perspective, as a Max-Min problem, the analysis of which is known to have to contend with discontinuous behavior. The complexity of our simple model is rooted in its inherent nonlinearity and, more consequentially, non-convexity of the objective function in the optimization. The possibilities of the Attacker's actions add substantially to the risk to the Defender, and the Defender's rational, risk-neutral optimal investments in general substantially exceed the optimal investments predicted by the one-sided Gordon-Loeb model. We obtain a succinct set of three decision types that categorize all of the Defender's optimal investment decisions. Also, the Defender's optimal decisions exhibit discontinuous behavior as the initial vulnerability of its system is varied. The analysis is supplemented by extensive numerical illustrations. The results from our model open several major avenues for future work.

Resource-constrained classification tasks are common in real-world applications such as allocating tests for disease diagnosis, hiring decisions when filling a limited number of positions, and defect detection in manufacturing settings under a limited inspection budget. Typical classification algorithms treat the learning process and the resource constraints as two separate and sequential tasks. Here we design an adaptive learning approach that considers resource constraints and learning jointly by iteratively fine-tuning misclassification costs. Via a structured experimental study using a publicly available data set, we evaluate a decision tree classifier that utilizes the proposed approach. The adaptive learning approach performs significantly better than alternative approaches, especially for difficult classification problems in which the performance of common approaches may be unsatisfactory. We envision the adaptive learning approach as an important addition to the repertoire of techniques for handling resource-constrained classification problems.

This paper proposes two convergent adaptive mesh-refining algorithms for the hybrid high-order method in convex minimization problems with two-sided p-growth. Examples include the p-Laplacian, an optimal design problem in topology optimization, and the convexified double-well problem. The hybrid high-order method utilizes a gradient reconstruction in the space of piecewise Raviart-Thomas finite element functions without stabilization on triangulations into simplices or in the space of piecewise polynomials with stabilization on polytopal meshes. The main results imply the convergence of the energy and, under further convexity properties, of the approximations of the primal resp. dual variable. Numerical experiments illustrate an efficient approximation of singular minimizers and improved convergence rates for higher polynomial degrees. Computer simulations provide striking numerical evidence that an adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even with empirical higher convergence rates.

We introduce a practical method to enforce linear partial differential equation (PDE) constraints for functions defined by neural networks (NNs), up to a desired tolerance. By combining methods in differentiable physics and applications of the implicit function theorem to NN models, we develop a differentiable PDE-constrained NN layer. During training, our model learns a family of functions, each of which defines a mapping from PDE parameters to PDE solutions. At inference time, the model finds an optimal linear combination of the functions in the learned family by solving a PDE-constrained optimization problem. Our method provides continuous solutions over the domain of interest that exactly satisfy desired physical constraints. Our results show that incorporating hard constraints directly into the NN architecture achieves much lower test error, compared to training on an unconstrained objective.

Policy optimization, which finds the desired policy by maximizing value functions via optimization techniques, lies at the heart of reinforcement learning (RL). In addition to value maximization, other practical considerations arise as well, including the need of encouraging exploration, and that of ensuring certain structural properties of the learned policy due to safety, resource and operational constraints. These can often be accounted for via regularized RL, which augments the target value function with a structure-promoting regularizer. Focusing on discounted infinite-horizon Markov decision processes, we propose a generalized policy mirror descent (GPMD) algorithm for solving regularized RL. As a generalization of policy mirror descent (arXiv:2102.00135), our algorithm accommodates a general class of convex regularizers and promotes the use of Bregman divergence in cognizant of the regularizer in use. We demonstrate that our algorithm converges linearly to the global solution over an entire range of learning rates, in a dimension-free fashion, even when the regularizer lacks strong convexity and smoothness. In addition, this linear convergence feature is provably stable in the face of inexact policy evaluation and imperfect policy updates. Numerical experiments are provided to corroborate the appealing performance of GPMD.

Zeroth-order optimization methods are developed to overcome the practical hurdle of having knowledge of explicit derivatives. Instead, these schemes work with merely access to noisy functions evaluations. The predominant approach is to mimic first-order methods by means of some gradient estimator. The theoretical limitations are well-understood, yet, as most of these methods rely on finite-differencing for shrinking differences, numerical cancellation can be catastrophic. The numerical community developed an efficient method to overcome this by passing to the complex domain. This approach has been recently adopted by the optimization community and in this work we analyze the practically relevant setting of dealing with computational noise. To exemplify the possibilities we focus on the strongly-convex optimization setting and provide a variety of non-asymptotic results, corroborated by numerical experiments, and end with local non-convex optimization.

Graph-based procedural materials are ubiquitous in content production industries. Procedural models allow the creation of photorealistic materials with parametric control for flexible editing of appearance. However, designing a specific material is a time-consuming process in terms of building a model and fine-tuning parameters. Previous work [Hu et al. 2022; Shi et al. 2020] introduced material graph optimization frameworks for matching target material samples. However, these previous methods were limited to optimizing differentiable functions in the graphs. In this paper, we propose a fully differentiable framework which enables end-to-end gradient based optimization of material graphs, even if some functions of the graph are non-differentiable. We leverage the Differentiable Proxy, a differentiable approximator of a non-differentiable black-box function. We use our framework to match structure and appearance of an output material to a target material, through a multi-stage differentiable optimization. Differentiable Proxies offer a more general optimization solution to material appearance matching than previous work.

It is common to address the curse of dimensionality in Markov decision processes (MDPs) by exploiting low-rank representations. This motivates much of the recent theoretical study on linear MDPs. However, most approaches require a given representation under unrealistic assumptions about the normalization of the decomposition or introduce unresolved computational challenges in practice. Instead, we consider an alternative definition of linear MDPs that automatically ensures normalization while allowing efficient representation learning via contrastive estimation. The framework also admits confidence-adjusted index algorithms, enabling an efficient and principled approach to incorporating optimism or pessimism in the face of uncertainty. To the best of our knowledge, this provides the first practical representation learning method for linear MDPs that achieves both strong theoretical guarantees and empirical performance. Theoretically, we prove that the proposed algorithm is sample efficient in both the online and offline settings. Empirically, we demonstrate superior performance over existing state-of-the-art model-based and model-free algorithms on several benchmarks.

This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.