Suppose h ( )=e x p ( ) . M -matrix (see [ 50 ]).

In standard reinforcement learning (RL), a learning agent seeks to optimize the overall reward. However, many key aspects of a desired behavior are more naturally expressed as constraints. For instance, the designer may want to limit the use of unsafe actions, increase the diversity of trajectories to enable exploration, or approximate expert trajectories when rewards are sparse. In this paper, we propose an algorithmic scheme that can handle a wide class of constraints in RL tasks: specifically, any constraints that require expected values of some vector measurements (such as the use of an action) to lie in a convex set. This captures previously studied constraints (such as safety and proximity to an expert), but also enables new classes of constraints (such as diversity). Our approach comes with rigorous theoretical guarantees and only relies on the ability to approximately solve standard RL tasks. As a result, it can be easily adapted to work with any model-free or model-based RL. In our experiments, we show that it matches previous algorithms that enforce safety via constraints, but can also enforce new properties that these algorithms do not incorporate, such as diversity.

In this paper, we consider an online optimization problem in which the reward functions are DR-submodular, and in addition to maximizing the total reward, the sequence of decisions must satisfy some convex constraints on average. Specifically, at each round $t\in\{1,\dots,T\}$, upon committing to an action $x_t$, a DR-submodular utility function $f_t(\cdot)$ and a convex constraint function $g_t(\cdot)$ are revealed, and the goal is to maximize the overall utility while ensuring the average of the constraint functions $\frac{1}{T}\sum_{t=1}^T g_t(x_t)$ is non-positive. Such cumulative constraints arise naturally in applications where the average resource consumption is required to remain below a prespecified threshold. We study this problem under an adversarial model and a stochastic model for the convex constraints, where the functions $g_t$ can vary arbitrarily or according to an i.i.d.

Pseudo-games are a natural and well-known generalization of normal-form games, in which the actions taken by each player affect not only the other players' payoffs, as in games, but also the other players' strategy sets. The solution concept par excellence for pseudo-games is the generalized Nash equilibrium (GNE), i.e., a strategy profile at which each player's strategy is feasible and no player can improve their payoffs by unilaterally deviating to another strategy in the strategy set determined by the other players' strategies. The computation of GNE in pseudo-games has long been a problem of interest, due to applications in a wide variety of fields, from environmental protection to logistics to telecommunications. Although computing GNE is PPAD-hard in general, it is still of interest to try to compute them in restricted classes of pseudo-games. One approach is to search for a strategy profile that minimizes exploitability, i.e., the sum of the regrets across all players. As exploitability is nondifferentiable in general, developing efficient first-order methods that minimize it might not seem possible at first glance. We observe, however, that the exploitability-minimization problem can be recast as a min-max optimization problem, and thereby obtain polynomial-time first-order methods to compute a refinement of GNE, namely the variational equilibria (VE), in convex-concave cumulative regret pseudo-games with jointly convex constraints. More generally, we also show that our methods find the stationary points of the exploitability in polynomial time in Lipschitz-smooth pseudo-games with jointly convex constraints. Finally, we demonstrate in experiments that our methods not only outperform known algorithms, but that even in pseudo-games where they are not guaranteed to converge to a GNE, they may do so nonetheless, with proper initialization.

In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient ascent on a continuous relaxation, directly connecting the discrete and continuous domains.