Ali Asadi

Ali Asadi, Krishnendu Chatterjee, Ehsan Goharshady et al.

Markov decision processes (MDPs) are a fundamental model in sequential decision making. Robust MDPs (RMDPs) extend this framework by allowing uncertainty in transition probabilities and optimizing against the worst-case realization of that uncertainty. In particular, $(s, a)$-rectangular RMDPs with $L_\infty$ uncertainty sets form a fundamental and expressive model: they subsume classical MDPs and turn-based stochastic games. We consider this model with discounted payoffs. The existence of polynomial and strongly-polynomial time algorithms is a fundamental problem for these optimization models. For MDPs, linear programming yields polynomial-time algorithms for any arbitrary discount factor, and the seminal work of Ye established strongly--polynomial time for a fixed discount factor. The generalization of such results to RMDPs has remained an important open problem. In this work, we show that a robust policy iteration algorithm runs in strongly-polynomial time for $(s, a)$-rectangular $L_\infty$ RMDPs with a constant (fixed) discount factor, resolving an important algorithmic question.

Ali Asadi, Krishnendu Chatterjee, Alipasha Montaseri et al.

A basic model in sequential decision making is the Markov decision process (MDP), which is extended to Robust MDPs (RMDPs) by allowing uncertainty in transition probabilities and optimizing against the worst-case transition probabilities from the uncertainty sets. The class of $(s, a)$-rectangular RMDPs with $L_p$ uncertainty sets provides a flexible and expressive model for such problems. We study this class of RMDPs with a discounted-sum cost criterion and a constant discount factor. The existence of an efficient algorithm for this class is a fundamental theoretical question in optimization and sequential decision making. Previous results only establish a strongly polynomial-time algorithm for $L_\infty$ uncertainty sets. In this work, our main results are as follows: (a)~we show that for any compact uncertainty set, the policy iteration algorithm for RMDPs is strongly polynomial with oracle access to solutions of Robust Markov chains (RMCs); (b)~we present strongly polynomial-time bounds on the policy iteration algorithm for RMCs with $L_1$ and $L_\infty$ uncertainty sets; and (c)~we establish hardness results for RMCs with $L_p$ uncertainty sets for integer $p$ satisfying $1<p<\infty$. Finally, motivated by our theoretical bounds, we present experimental results showing how fast policy iteration converges for RMDPs with $L_1$ and $L_\infty$ uncertainty sets.

Ali Asadi, Krishnendu Chatterjee, Jakob de Raaij

Deterministic Markov Decision Processes (DMDPs) are a mathematical framework for decision-making where the outcomes and future possible actions are deterministically determined by the current action taken. DMDPs can be viewed as a finite directed weighted graph, where in each step, the controller chooses an outgoing edge. An objective is a measurable function on runs (or infinite trajectories) of the DMDP, and the value for an objective is the maximal cumulative reward (or weight) that the controller can guarantee. We consider the classical mean-payoff (aka limit-average) objective, which is a basic and fundamental objective. Howard's policy iteration algorithm is a popular method for solving DMDPs with mean-payoff objectives. Although Howard's algorithm performs well in practice, as experimental studies suggested, the best known upper bound is exponential and the current known lower bound is as follows: For the input size $I$, the algorithm requires $\tildeΩ(\sqrt{I})$ iterations, where $\tildeΩ$ hides the poly-logarithmic factors, i.e., the current lower bound on iterations is sub-linear with respect to the input size. Our main result is an improved lower bound for this fundamental algorithm where we show that for the input size $I$, the algorithm requires $\tildeΩ(I)$ iterations.

Ali Asadi, Krishnendu Chatterjee, Ehsan Kafshdar Goharshady et al.

Robust Markov Decision Processes (RMDPs) generalize classical MDPs that consider uncertainties in transition probabilities by defining a set of possible transition functions. An objective is a set of runs (or infinite trajectories) of the RMDP, and the value for an objective is the maximal probability that the agent can guarantee against the adversarial environment. We consider (a) reachability objectives, where given a target set of states, the goal is to eventually arrive at one of them; and (b) parity objectives, which are a canonical representation for $ω$-regular objectives. The qualitative analysis problem asks whether the objective can be ensured with probability 1. In this work, we study the qualitative problem for reachability and parity objectives on RMDPs without making any assumption over the structures of the RMDPs, e.g., unichain or aperiodic. Our contributions are twofold. We first present efficient algorithms with oracle access to uncertainty sets that solve qualitative problems of reachability and parity objectives. We then report experimental results demonstrating the effectiveness of our oracle-based approach on classical RMDP examples from the literature scaling up to thousands of states.

Ali Asadi, Krishnendu Chatterjee, Hongfei Fu et al.

In this work, we consider the fundamental problem of reachability analysis over imperative programs with real variables. The reachability property requires that a program can reach certain target states during its execution. Previous works that tackle reachability analysis are either unable to handle programs consisting of general loops (e.g. symbolic execution), or lack completeness guarantees (e.g. abstract interpretation), or are not automated (e.g. incorrectness logic/reverse Hoare logic). In contrast, we propose a novel approach for reachability analysis that can handle general programs, is (semi-)complete, and can be entirely automated for a wide family of programs. Our approach extends techniques from both invariant generation and ranking-function synthesis to reachability analysis through the notion of (Universal) Inductive Reachability Witnesses (IRWs/UIRWs). While traditional invariant generation uses over-approximations of reachable states, we consider the natural dual problem of under-approximating the set of program states that can reach a target state. We then apply an argument similar to ranking functions to ensure that all states in our under-approximation can indeed reach the target set in finitely many steps.

Ville Bergholm, Josh Izaac, Maria Schuld et al.

PennyLane is a Python 3 software framework for differentiable programming of quantum computers. The library provides a unified architecture for near-term quantum computing devices, supporting both qubit and continuous-variable paradigms. PennyLane's core feature is the ability to compute gradients of variational quantum circuits in a way that is compatible with classical techniques such as backpropagation. PennyLane thus extends the automatic differentiation algorithms common in optimization and machine learning to include quantum and hybrid computations. A plugin system makes the framework compatible with any gate-based quantum simulator or hardware. We provide plugins for hardware providers including the Xanadu Cloud, Amazon Braket, and IBM Quantum, allowing PennyLane optimizations to be run on publicly accessible quantum devices. On the classical front, PennyLane interfaces with accelerated machine learning libraries such as TensorFlow, PyTorch, JAX, and Autograd. PennyLane can be used for the optimization of variational quantum eigensolvers, quantum approximate optimization, quantum machine learning models, and many other applications.