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Recurrent Neural Networks (RNNs) have achieved great success in the prediction of sequential data. However, their theoretical studies are still lagging behind because of their complex interconnected structures. In this paper, we establish a new generalization error bound for vanilla RNNs, and provide a unified framework to calculate the Rademacher complexity that can be applied to a variety of loss functions. When the ramp loss is used, we show that our bound is tighter than the existing bounds based on the same assumptions on the Frobenius and spectral norms of the weight matrices and a few mild conditions. Our numerical results show that our new generalization bound is the tightest among all existing bounds in three public datasets. Our bound improves the second tightest one by an average percentage of 13.80% and 3.01% when the $\tanh$ and ReLU activation functions are used, respectively. Moreover, we derive a sharp estimation error bound for RNN-based estimators obtained through empirical risk minimization (ERM) in multi-class classification problems when the loss function satisfies a Bernstein condition.

Sampling a random permutation with restricted positions, or equivalently approximating the permanent of a 0-1 matrix, is a fundamental problem in computer science, with several notable results attained through the years. In this paper, we first improves the running time of the algorithms for a single permutation. We propose a fast approximation algorithm for the permanent of $\gamma$-dense 0-1 matrix, with an expected running time of $\tilde{O}\left(n^{2+(1-\gamma)/(2\gamma - 1)}\right)$. Our result removes the $n^4$ term from the previous best runtime and provides an improvement for $\gamma \geq 0.6$. When $\gamma = o(1)$, our runtime is $\tilde{\Theta}(n^2)$, which is nearly optimal for this problem. The core of our proof is to demonstrate that the Sinkhorn algorithm, a fundamental tool in matrix scaling, can achieve maximum accuracy of $1/\text{poly}(n)$ for dense matrices in $O(\log n)$ iterations. We further introduce a general model called permutations with disjunctive constraints (PDC) for handling multiple constrained permutations. We propose a novel Markov chain-based algorithm for sampling nearly uniform solutions of PDC within a Lov${\'a}$sz Local Lemma (LLL)-like regime by a novel sampling framework called correlated factorization. For uniform PDC formulas, where all constraints are of the same length and all permutations are of equal size, our algorithm runs in nearly linear time with respect to the number of variables.

In statistics on manifolds, the notion of the mean of a probability distribution becomes more involved than in a linear space. Several location statistics have been proposed, which reduce to the ordinary mean in Euclidean space. A relatively new family of contenders in this field are Diffusion Means, which are a one parameter family of location statistics modeled as initial points of isotropic diffusion with the diffusion time as parameter. It is natural to consider limit cases of the diffusion time parameter and it turns out that for short times the diffusion mean set approaches the intrinsic mean set. For long diffusion times, the limit is less obvious but for spheres of arbitrary dimension the diffusion mean set has been shown to converge to the extrinsic mean set. Here, we extend this result to the real projective spaces in their unique smooth isometric embedding into a linear space. We conjecture that the long time limit is always given by the extrinsic mean in the isometric embedding for connected compact symmetric spaces with unique isometric embedding.

We study the problem of fairly and efficiently allocating indivisible goods among agents with additive valuation functions. Envy-freeness up to one good (EF1) is a well-studied fairness notion for indivisible goods, while Pareto optimality (PO) and its stronger variant, fractional Pareto optimality (fPO), are widely recognized efficiency criteria. Although each property is straightforward to achieve individually, simultaneously ensuring both fairness and efficiency is challenging. Caragiannis et al.~\cite{caragiannis2019unreasonable} established the surprising result that maximizing Nash social welfare yields an allocation that is both EF1 and PO; however, since maximizing Nash social welfare is NP-hard, this approach does not provide an efficient algorithm. To overcome this barrier, Barman, Krishnamurthy, and Vaish~\cite{barman2018finding} designed a pseudo-polynomial time algorithm to compute an EF1 and PO allocation, and showed the existence of EF1 and fPO allocations. Nevertheless, the latter existence proof relies on a non-constructive convergence argument and does not directly yield an efficient algorithm for finding EF1 and fPO allocations. Whether a polynomial-time algorithm exists for finding an EF1 and PO (or fPO) allocation remains an important open problem. In this paper, we propose a polynomial-time algorithm to compute an allocation that achieves both EF1 and fPO under additive valuation functions when the number of agents is fixed. Our primary idea is to avoid processing the entire instance at once; instead, we sequentially add agents to the instance and construct an allocation that satisfies EF1 and fPO at each step.

In the rapidly advancing field of multi-agent systems, ensuring robustness in unfamiliar and adversarial settings is crucial. Notwithstanding their outstanding performance in familiar environments, these systems often falter in new situations due to overfitting during the training phase. This is especially pronounced in settings where both cooperative and competitive behaviours are present, encapsulating a dual nature of overfitting and generalisation challenges. To address this issue, we present Multi-Agent Diagnostics for Robustness via Illuminated Diversity (MADRID), a novel approach for generating diverse adversarial scenarios that expose strategic vulnerabilities in pre-trained multi-agent policies. Leveraging the concepts from open-ended learning, MADRID navigates the vast space of adversarial settings, employing a target policy's regret to gauge the vulnerabilities of these settings. We evaluate the effectiveness of MADRID on the 11vs11 version of Google Research Football, one of the most complex environments for multi-agent reinforcement learning. Specifically, we employ MADRID for generating a diverse array of adversarial settings for TiZero, the state-of-the-art approach which "masters" the game through 45 days of training on a large-scale distributed infrastructure. We expose key shortcomings in TiZero's tactical decision-making, underlining the crucial importance of rigorous evaluation in multi-agent systems.

Investigating the marginal causal effect of an intervention on an outcome from complex data remains challenging due to the inflexibility of employed models and the lack of complexity in causal benchmark datasets, which often fail to reproduce intricate real-world data patterns. In this paper we introduce Frugal Flows, a novel likelihood-based machine learning model that uses normalising flows to flexibly learn the data-generating process, while also directly inferring the marginal causal quantities from observational data. We propose that these models are exceptionally well suited for generating synthetic data to validate causal methods. They can create synthetic datasets that closely resemble the empirical dataset, while automatically and exactly satisfying a user-defined average treatment effect. To our knowledge, Frugal Flows are the first generative model to both learn flexible data representations and also exactly parameterise quantities such as the average treatment effect and the degree of unobserved confounding. We demonstrate the above with experiments on both simulated and real-world datasets.

Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs) due to complex geometries, interactions between physical variables, and the limited amounts of high-resolution training data. To address these issues, we propose Codomain Attention Neural Operator (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to function spaces. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations, fluid-structure interactions, and Rayleigh-B\'enard convection, we found CoDA-NO to outperform existing methods by over 36%.

Sequential recommendation as an emerging topic has attracted increasing attention due to its important practical significance. Models based on deep learning and attention mechanism have achieved good performance in sequential recommendation. Recently, the generative models based on Variational Autoencoder (VAE) have shown the unique advantage in collaborative filtering. In particular, the sequential VAE model as a recurrent version of VAE can effectively capture temporal dependencies among items in user sequence and perform sequential recommendation. However, VAE-based models suffer from a common limitation that the representational ability of the obtained approximate posterior distribution is limited, resulting in lower quality of generated samples. This is especially true for generating sequences. To solve the above problem, in this work, we propose a novel method called Adversarial and Contrastive Variational Autoencoder (ACVAE) for sequential recommendation. Specifically, we first introduce the adversarial training for sequence generation under the Adversarial Variational Bayes (AVB) framework, which enables our model to generate high-quality latent variables. Then, we employ the contrastive loss. The latent variables will be able to learn more personalized and salient characteristics by minimizing the contrastive loss. Besides, when encoding the sequence, we apply a recurrent and convolutional structure to capture global and local relationships in the sequence. Finally, we conduct extensive experiments on four real-world datasets. The experimental results show that our proposed ACVAE model outperforms other state-of-the-art methods.

Deep neural models in recent years have been successful in almost every field, including extremely complex problem statements. However, these models are huge in size, with millions (and even billions) of parameters, thus demanding more heavy computation power and failing to be deployed on edge devices. Besides, the performance boost is highly dependent on redundant labeled data. To achieve faster speeds and to handle the problems caused by the lack of data, knowledge distillation (KD) has been proposed to transfer information learned from one model to another. KD is often characterized by the so-called `Student-Teacher' (S-T) learning framework and has been broadly applied in model compression and knowledge transfer. This paper is about KD and S-T learning, which are being actively studied in recent years. First, we aim to provide explanations of what KD is and how/why it works. Then, we provide a comprehensive survey on the recent progress of KD methods together with S-T frameworks typically for vision tasks. In general, we consider some fundamental questions that have been driving this research area and thoroughly generalize the research progress and technical details. Additionally, we systematically analyze the research status of KD in vision applications. Finally, we discuss the potentials and open challenges of existing methods and prospect the future directions of KD and S-T learning.

Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis.