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Knowledge distillation aims to transfer useful information from a teacher network to a student network, with the primary goal of improving the student's performance for the task at hand. Over the years, there has a been a deluge of novel techniques and use cases of knowledge distillation. Yet, despite the various improvements, there seems to be a glaring gap in the community's fundamental understanding of the process. Specifically, what is the knowledge that gets distilled in knowledge distillation? In other words, in what ways does the student become similar to the teacher? Does it start to localize objects in the same way? Does it get fooled by the same adversarial samples? Does its data invariance properties become similar? Our work presents a comprehensive study to try to answer these questions. We show that existing methods can indeed indirectly distill these properties beyond improving task performance. We further study why knowledge distillation might work this way, and show that our findings have practical implications as well.

We extend classical methods of computational complexity to the setting of distributed computing, where they are sometimes more effective than in their original context. Our focus is on distributed decision in the LOCAL model, where multiple networked computers communicate via synchronous message-passing to collectively answer a question about their network topology. Rather unusually, we impose two orthogonal constraints on the running time of this model: the number of communication rounds is bounded by a constant, and the number of computation steps of each computer is polynomially bounded by the size of its local input and the messages it receives. By letting two players take turns assigning certificates to all computers in the network, we obtain a generalization of the polynomial hierarchy (and hence of the complexity classes $\mathbf{P}$ and $\mathbf{NP}$). We then extend some key results of complexity theory to this setting, in particular the Cook-Levin theorem (which identifies Boolean satisfiability as a complete problem for $\mathbf{NP}$), and Fagin's theorem (which characterizes $\mathbf{NP}$ as the problems expressible in existential second-order logic). The original results can be recovered as the special case where the network consists of a single computer. But perhaps more surprisingly, the task of separating complexity classes becomes easier in the general case: we can show that our hierarchy is infinite, while it remains notoriously open whether the same is true in the case of a single computer. (By contrast, a collapse of our hierarchy would have implied a collapse of the polynomial hierarchy.) As an application, we propose quantifier alternation as a new approach to measuring the locality of problems in distributed computing.

Reinforcement learning from Human Feedback (RLHF) learns from preference signals, while standard Reinforcement Learning (RL) directly learns from reward signals. Preferences arguably contain less information than rewards, which makes preference-based RL seemingly more difficult. This paper theoretically proves that, for a wide range of preference models, we can solve preference-based RL directly using existing algorithms and techniques for reward-based RL, with small or no extra costs. Specifically, (1) for preferences that are drawn from reward-based probabilistic models, we reduce the problem to robust reward-based RL that can tolerate small errors in rewards; (2) for general arbitrary preferences where the objective is to find the von Neumann winner, we reduce the problem to multiagent reward-based RL which finds Nash equilibria for factored Markov games with a restricted set of policies. The latter case can be further reduced to adversarial MDP when preferences only depend on the final state. We instantiate all reward-based RL subroutines by concrete provable algorithms, and apply our theory to a large class of models including tabular MDPs and MDPs with generic function approximation. We further provide guarantees when K-wise comparisons are available.

Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. This relies on another contribution of this paper, which eliminates the need to examine a factorial number of permutations of simplices with the same value. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude.

Recently, images are considered samples from a high-dimensional distribution, and deep learning has become almost synonymous with image generation. However, is a deep learning network truly necessary for image generation? In this paper, we investigate the possibility of image generation without using a deep learning network, motivated by validating the assumption that images follow a high-dimensional distribution. Since images are assumed to be samples from such a distribution, we utilize the Gaussian Mixture Model (GMM) to describe it. In particular, we employ a recent distribution learning technique named as Monte-Carlo Marginalization to capture the parameters of the GMM based on image samples. Moreover, we also use the Singular Value Decomposition (SVD) for dimensionality reduction to decrease computational complexity. During our evaluation experiment, we first attempt to model the distribution of image samples directly to verify the assumption that images truly follow a distribution. We then use the SVD for dimensionality reduction. The principal components, rather than raw image data, are used for distribution learning. Compared to methods relying on deep learning networks, our approach is more explainable, and its performance is promising. Experiments show that our images have a lower FID value compared to those generated by variational auto-encoders, demonstrating the feasibility of image generation without deep learning networks.

We analyze the structure of the market for foundation models, i.e., large AI models such as those that power ChatGPT and that are adaptable to downstream uses, and we examine the implications for competition policy and regulation. We observe that the most capable models will have a tendency towards natural monopoly and may have potentially vast markets. This calls for a two-pronged regulatory response: (i) Antitrust authorities need to ensure the contestability of the market by tackling strategic behavior, in particular by ensuring that monopolies do not propagate vertically to downstream uses, and (ii) given the diminished potential for market discipline, there is a role for regulators to ensure that the most capable models meet sufficient quality standards (including safety, privacy, non-discrimination, reliability and interoperability standards) to maximally contribute to social welfare. Regulators should also ensure a level regulatory playing field between AI and non-AI applications in all sectors of the economy. For models that are behind the frontier, we expect competition to be quite intense, implying a more limited role for competition policy, although a role for regulation remains.

In recent years, several authors have been investigating simplicial models, a model of epistemic logic based on higher-dimensional structures called simplicial complexes. In the original formulation, simplicial models were always assumed to be pure, meaning that all worlds have the same dimension. This is equivalent to the standard S5n semantics of epistemic logic, based on Kripke models. By removing the assumption that models must be pure, we can go beyond the usual Kripke semantics and study epistemic logics where the number of agents participating in a world can vary. This approach has been developed in a number of papers, with applications in fault-tolerant distributed computing where processes may crash during the execution of a system. A difficulty that arises is that subtle design choices in the definition of impure simplicial models can result in different axioms of the resulting logic. In this paper, we classify those design choices systematically, and axiomatize the corresponding logics. We illustrate them via distributed computing examples of synchronous systems where processes may crash.

Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving math problems and proving theorems has garnered significant interest in the fields of machine learning and natural language processing. For example, mathematics serves as a testbed for aspects of reasoning that are challenging for powerful deep learning models, driving new algorithmic and modeling advances. On the other hand, recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning. In this survey paper, we review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. We also evaluate existing benchmarks and methods, and discuss future research directions in this domain.

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.

Graph Neural Networks (GNNs) for representation learning of graphs broadly follow a neighborhood aggregation framework, where the representation vector of a node is computed by recursively aggregating and transforming feature vectors of its neighboring nodes. Many GNN variants have been proposed and have achieved state-of-the-art results on both node and graph classification tasks. However, despite GNNs revolutionizing graph representation learning, there is limited understanding of their representational properties and limitations. Here, we present a theoretical framework for analyzing the expressive power of GNNs in capturing different graph structures. Our results characterize the discriminative power of popular GNN variants, such as Graph Convolutional Networks and GraphSAGE, and show that they cannot learn to distinguish certain simple graph structures. We then develop a simple architecture that is provably the most expressive among the class of GNNs and is as powerful as the Weisfeiler-Lehman graph isomorphism test. We empirically validate our theoretical findings on a number of graph classification benchmarks, and demonstrate that our model achieves state-of-the-art performance.