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This paper presents fault-tolerant asynchronous Stochastic Gradient Descent (SGD) algorithms. SGD is widely used for approximating the minimum of a cost function $Q$, as a core part of optimization and learning algorithms. Our algorithms are designed for the cluster-based model, which combines message-passing and shared-memory communication layers. Processes may fail by crashing, and the algorithm inside each cluster is wait-free, using only reads and writes. For a strongly convex function $Q$, our algorithm tolerates any number of failures, and provides convergence rate that yields the maximal distributed acceleration over the optimal convergence rate of sequential SGD. For arbitrary functions, the convergence rate has an additional term that depends on the maximal difference between the parameters at the same iteration. (This holds under standard assumptions on $Q$.) In this case, the algorithm obtains the same convergence rate as sequential SGD, up to a logarithmic factor. This is achieved by using, at each iteration, a multidimensional approximate agreement algorithm, tailored for the cluster-based model. The algorithm for arbitrary functions requires that at least a majority of the clusters contain at least one nonfaulty process. We prove that this condition is necessary when optimizing some non-convex functions.

Polynomial kernel regression is one of the standard and state-of-the-art learning strategies. However, as is well known, the choices of the degree of polynomial kernel and the regularization parameter are still open in the realm of model selection. The first aim of this paper is to develop a strategy to select these parameters. On one hand, based on the worst-case learning rate analysis, we show that the regularization term in polynomial kernel regression is not necessary. In other words, the regularization parameter can decrease arbitrarily fast when the degree of the polynomial kernel is suitable tuned. On the other hand,taking account of the implementation of the algorithm, the regularization term is required. Summarily, the effect of the regularization term in polynomial kernel regression is only to circumvent the " ill-condition" of the kernel matrix. Based on this, the second purpose of this paper is to propose a new model selection strategy, and then design an efficient learning algorithm. Both theoretical and experimental analysis show that the new strategy outperforms the previous one. Theoretically, we prove that the new learning strategy is almost optimal if the regression function is smooth. Experimentally, it is shown that the new strategy can significantly reduce the computational burden without loss of generalization capability.

Multi-agent reinforcement learning (MARL) addresses sequential decision-making problems with multiple agents, where each agent optimizes its own objective. In many real-world instances, the agents may not only want to optimize their objectives, but also ensure safe behavior. For example, in traffic routing, each car (agent) aims to reach its destination quickly (objective) while avoiding collisions (safety). Constrained Markov Games (CMGs) are a natural formalism for safe MARL problems, though generally intractable. In this work, we introduce and study Constrained Markov Potential Games (CMPGs), an important class of CMGs. We first show that a Nash policy for CMPGs can be found via constrained optimization. One tempting approach is to solve it by Lagrangian-based primal-dual methods. As we show, in contrast to the single-agent setting, however, CMPGs do not satisfy strong duality, rendering such approaches inapplicable and potentially unsafe. To solve the CMPG problem, we propose our algorithm Coordinate-Ascent for CMPGs (CA-CMPG), which provably converges to a Nash policy in tabular, finite-horizon CMPGs. Furthermore, we provide the first sample complexity bounds for learning Nash policies in unknown CMPGs, and, which under additional assumptions, guarantee safe exploration.

The subset cover problem for $k \geq 1$ hash functions, which can be seen as an extension of the collision problem, was introduced in 2002 by Reyzin and Reyzin to analyse the security of their hash-function based signature scheme HORS. The security of many hash-based signature schemes relies on this problem or a variant of this problem (e.g. HORS, SPHINCS, SPHINCS+, $\dots$). Recently, Yuan, Tibouchi and Abe (2022) introduced a variant to the subset cover problem, called restricted subset cover, and proposed a quantum algorithm for this problem. In this work, we prove that any quantum algorithm needs to make $\Omega\left((k+1)^{-\frac{2^{k}}{2^{k+1}-1}}\cdot N^{\frac{2^{k}-1}{2^{k+1}-1}}\right)$ queries to the underlying hash functions with codomain size $N$ to solve the restricted subset cover problem, which essentially matches the query complexity of the algorithm proposed by Yuan, Tibouchi and Abe. We also analyze the security of the general $(r,k)$-subset cover problem, which is the underlying problem that implies the unforgeability of HORS under a $r$-chosen message attack (for $r \geq 1$). We prove that a generic quantum algorithm needs to make $\Omega\left(N^{k/5}\right)$ queries to the underlying hash functions to find a $(1,k)$-subset cover. We also propose a quantum algorithm that finds a $(r,k)$-subset cover making $O\left(N^{k/(2+2r)}\right)$ queries to the $k$ hash functions.

We investigate learning the equilibria in non-stationary multi-agent systems and address the challenges that differentiate multi-agent learning from single-agent learning. Specifically, we focus on games with bandit feedback, where testing an equilibrium can result in substantial regret even when the gap to be tested is small, and the existence of multiple optimal solutions (equilibria) in stationary games poses extra challenges. To overcome these obstacles, we propose a versatile black-box approach applicable to a broad spectrum of problems, such as general-sum games, potential games, and Markov games, when equipped with appropriate learning and testing oracles for stationary environments. Our algorithms can achieve $\widetilde{O}\left(\Delta^{1/4}T^{3/4}\right)$ regret when the degree of nonstationarity, as measured by total variation $\Delta$, is known, and $\widetilde{O}\left(\Delta^{1/5}T^{4/5}\right)$ regret when $\Delta$ is unknown, where $T$ is the number of rounds. Meanwhile, our algorithm inherits the favorable dependence on number of agents from the oracles. As a side contribution that may be independent of interest, we show how to test for various types of equilibria by a black-box reduction to single-agent learning, which includes Nash equilibria, correlated equilibria, and coarse correlated equilibria.

We propose a hybrid quantum-classical algorithm to compute approximate solutions of binary combinatorial problems. We employ a shallow-depth quantum circuit to implement a unitary and Hermitian operator that block-encodes the weighted maximum cut or the Ising Hamiltonian. Measuring the expectation of this operator on a variational quantum state yields the variational energy of the quantum system. The system is enforced to evolve towards the ground state of the problem Hamiltonian by optimizing a set of angles using normalized gradient descent. Experimentally, our algorithm outperforms the state-of-the-art quantum approximate optimization algorithm on random fully connected graphs and challenges D-Wave quantum annealers by producing better approximate solutions. Source code and data files are publicly available.

In this paper, we present a novel hybrid method for solving a Stokes interface problem in a regular domain with jump discontinuities on an interface. Our approach combines the expressive power of neural networks with the convergence of finite difference schemes to achieve efficient implementations and accurate results. The key concept of our method is to decompose the solution into two parts: the singular part and the regular part. We employ neural networks to approximate the singular part, which captures the jump discontinuities across the interface. We then utilize a finite difference scheme to approximate the regular part, which handles the smooth variations of the solution in that regular domain. To validate the effectiveness of our approach, we present two- and three-dimensional examples to demonstrate the accuracy and convergence of the proposed method, and show that our proposed hybrid method provides an innovative and reliable approach to tackle Stokes interface problems.

Pseudorandom quantum states (PRS) are efficiently constructible states that are computationally indistinguishable from being Haar-random, and have recently found cryptographic applications. We explore new definitions, new properties and applications of pseudorandom states, and present the following contributions: 1. New Definitions: We study variants of pseudorandom function-like state (PRFS) generators, introduced by Ananth, Qian, and Yuen (CRYPTO'22), where the pseudorandomness property holds even when the generator can be queried adaptively or in superposition. We show feasibility of these variants assuming the existence of post-quantum one-way functions. 2. Classical Communication: We show that PRS generators with logarithmic output length imply commitment and encryption schemes with classical communication. Previous constructions of such schemes from PRS generators required quantum communication. 3. Simplified Proof: We give a simpler proof of the Brakerski--Shmueli (TCC'19) result that polynomially-many copies of uniform superposition states with random binary phases are indistinguishable from Haar-random states. 4. Necessity of Computational Assumptions: We also show that a secure PRS with output length logarithmic, or larger, in the key length necessarily requires computational assumptions.

We consider the problems of testing and learning quantum $k$-junta channels, which are $n$-qubit to $n$-qubit quantum channels acting non-trivially on at most $k$ out of $n$ qubits and leaving the rest of qubits unchanged. We show the following. 1. An $\widetilde{O}\left(\sqrt{k}\right)$-query algorithm to distinguish whether the given channel is $k$-junta channel or is far from any $k$-junta channels, and a lower bound $\Omega\left(\sqrt{k}\right)$ on the number of queries; 2. An $\widetilde{O}\left(4^k\right)$-query algorithm to learn a $k$-junta channel, and a lower bound $\Omega\left(4^k/k\right)$ on the number of queries. This answers an open problem raised by Chen et al. (2023). In order to settle these problems, we develop a Fourier analysis framework over the space of superoperators and prove several fundamental properties, which extends the Fourier analysis over the space of operators introduced in Montanaro and Osborne (2010).

Game theory has by now found numerous applications in various fields, including economics, industry, jurisprudence, and artificial intelligence, where each player only cares about its own interest in a noncooperative or cooperative manner, but without obvious malice to other players. However, in many practical applications, such as poker, chess, evader pursuing, drug interdiction, coast guard, cyber-security, and national defense, players often have apparently adversarial stances, that is, selfish actions of each player inevitably or intentionally inflict loss or wreak havoc on other players. Along this line, this paper provides a systematic survey on three main game models widely employed in adversarial games, i.e., zero-sum normal-form and extensive-form games, Stackelberg (security) games, zero-sum differential games, from an array of perspectives, including basic knowledge of game models, (approximate) equilibrium concepts, problem classifications, research frontiers, (approximate) optimal strategy seeking techniques, prevailing algorithms, and practical applications. Finally, promising future research directions are also discussed for relevant adversarial games.