Every graph with maximum degree $\Delta$ can be colored with $(\Delta+1)$ colors using a simple greedy algorithm. Remarkably, recent work has shown that one can find such a coloring even in the semi-streaming model. But, in reality, one almost never needs $(\Delta+1)$ colors to properly color a graph. Indeed, the celebrated \Brooks' theorem states that every (connected) graph beside cliques and odd cycles can be colored with $\Delta$ colors. Can we find a $\Delta$-coloring in the semi-streaming model as well? We settle this key question in the affirmative by designing a randomized semi-streaming algorithm that given any graph, with high probability, either correctly declares that the graph is not $\Delta$-colorable or outputs a $\Delta$-coloring of the graph. The proof of this result starts with a detour. We first (provably) identify the extent to which the previous approaches for streaming coloring fail for $\Delta$-coloring: for instance, all these approaches can handle streams with repeated edges and they can run in $o(n^2)$ time -- we prove that neither of these tasks is possible for $\Delta$-coloring. These impossibility results however pinpoint exactly what is missing from prior approaches when it comes to $\Delta$-coloring. We then build on these insights to design a semi-streaming algorithm that uses $(i)$ a novel sparse-recovery approach based on sparse-dense decompositions to (partially) recover the "problematic" subgraphs of the input -- the ones that form the basis of our impossibility results -- and $(ii)$ a new coloring approach for these subgraphs that allows for recoloring of other vertices in a controlled way without relying on local explorations or finding "augmenting paths" that are generally impossible for semi-streaming algorithms. We believe both these techniques can be of independent interest.
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We define an extension of the posterior predictive $p$-value for multiple test statistics and establish a bound on its frequency under the assumption of model correctness. We argue that the conservativity of the posterior predictive $p$-value increases with model dimension, and we demonstrate the ability of the joint $p$-value to overcome this problem in many cases. We also compare the joint $p$-values to other alternative $p$-values designed to have higher power and show that the joint $p$-value can achieve similar performance for model rejection while maintaining more favorable computational and interpretive properties.
Stochastic Gradient Descent (SGD), a widely used optimization algorithm in deep learning, is often limited to converging to local optima due to the non-convex nature of the problem. Leveraging these local optima to improve model performance remains a challenging task. Given the inherent complexity of neural networks, the simple arithmetic averaging of the obtained local optima models in undesirable results. This paper proposes a {\em soft merging} method that facilitates rapid merging of multiple models, simplifies the merging of specific parts of neural networks, and enhances robustness against malicious models with extreme values. This is achieved by learning gate parameters through a surrogate of the $l_0$ norm using hard concrete distribution without modifying the model weights of the given local optima models. This merging process not only enhances the model performance by converging to a better local optimum, but also minimizes computational costs, offering an efficient and explicit learning process integrated with stochastic gradient descent. Thorough experiments underscore the effectiveness and superior performance of the merged neural networks.
The problem of non-monotone $k$-submodular maximization under a knapsack constraint ($\kSMK$) over the ground set size $n$ has been raised in many applications in machine learning, such as data summarization, information propagation, etc. However, existing algorithms for the problem are facing questioning of how to overcome the non-monotone case and how to fast return a good solution in case of the big size of data. This paper introduces two deterministic approximation algorithms for the problem that competitively improve the query complexity of existing algorithms. Our first algorithm, $\LAA$, returns an approximation ratio of $1/19$ within $O(nk)$ query complexity. The second one, $\RLA$, improves the approximation ratio to $1/5-\epsilon$ in $O(nk)$ queries, where $\epsilon$ is an input parameter. Our algorithms are the first ones that provide constant approximation ratios within only $O(nk)$ query complexity for the non-monotone objective. They, therefore, need fewer the number of queries than state-of-the-the-art ones by a factor of $\Omega(\log n)$. Besides the theoretical analysis, we have evaluated our proposed ones with several experiments in some instances: Influence Maximization and Sensor Placement for the problem. The results confirm that our algorithms ensure theoretical quality as the cutting-edge techniques and significantly reduce the number of queries.
We study the existence of optimal and p-optimal proof systems for classes in the Boolean hierarchy over $\mathrm{NP}$. Our main results concern $\mathrm{DP}$, i.e., the second level of this hierarchy: If all sets in $\mathrm{DP}$ have p-optimal proof systems, then all sets in $\mathrm{coDP}$ have p-optimal proof systems. The analogous implication for optimal proof systems fails relative to an oracle. As a consequence, we clarify such implications for all classes $\mathcal{C}$ and $\mathcal{D}$ in the Boolean hierarchy over $\mathrm{NP}$: either we can prove the implication or show that it fails relative to an oracle. Furthermore, we show that the sets $\mathrm{SAT}$ and $\mathrm{TAUT}$ have p-optimal proof systems, if and only if all sets in the Boolean hierarchy over $\mathrm{NP}$ have p-optimal proof systems which is a new characterization of a conjecture studied by Pudl\'ak.
We present C$\cdot$ASE, an efficient and effective framework that learns conditional Adversarial Skill Embeddings for physics-based characters. Our physically simulated character can learn a diverse repertoire of skills while providing controllability in the form of direct manipulation of the skills to be performed. C$\cdot$ASE divides the heterogeneous skill motions into distinct subsets containing homogeneous samples for training a low-level conditional model to learn conditional behavior distribution. The skill-conditioned imitation learning naturally offers explicit control over the character's skills after training. The training course incorporates the focal skill sampling, skeletal residual forces, and element-wise feature masking to balance diverse skills of varying complexities, mitigate dynamics mismatch to master agile motions and capture more general behavior characteristics, respectively. Once trained, the conditional model can produce highly diverse and realistic skills, outperforming state-of-the-art models, and can be repurposed in various downstream tasks. In particular, the explicit skill control handle allows a high-level policy or user to direct the character with desired skill specifications, which we demonstrate is advantageous for interactive character animation.
Task-oriented grasping (TOG) refers to the problem of predicting grasps on an object that enable subsequent manipulation tasks. To model the complex relationships between objects, tasks, and grasps, existing methods incorporate semantic knowledge as priors into TOG pipelines. However, the existing semantic knowledge is typically constructed based on closed-world concept sets, restraining the generalization to novel concepts out of the pre-defined sets. To address this issue, we propose GraspGPT, a large language model (LLM) based TOG framework that leverages the open-end semantic knowledge from an LLM to achieve zero-shot generalization to novel concepts. We conduct experiments on Language Augmented TaskGrasp (LA-TaskGrasp) dataset and demonstrate that GraspGPT outperforms existing TOG methods on different held-out settings when generalizing to novel concepts out of the training set. The effectiveness of GraspGPT is further validated in real-robot experiments. Our code, data, appendix, and video are publicly available at //sites.google.com/view/graspgpt/.
We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial $\chi_G(x)$ (in $\mathbb{C}$) lie inside the disc of radius $5.94 \Delta$ centered at $0$. This improves on the previously best known bound of approximately $6.91\Delta$. We also obtain improved bounds for graphs of high girth. We prove that for every $g$ there is a constant $K_g$ such that for any graph $G$ of maximum degree at most $\Delta$ and girth at least $g$, the zeros of its chromatic polynomial $\chi_G(x)$ lie inside the disc of radius $K_g \Delta$ centered at $0$, where $K_g$ is the solution to a certain optimization problem. In particular, $K_g < 5$ when $g \geq 5$ and $K_g < 4$ when $g \geq 25$ and $K_g$ tends to approximately $3.86$ as $g \to \infty$. Key to the proof is a classical theorem of Whitney which allows us to relate the chromatic polynomial of a graph $G$ to the generating function of so-called broken-circuit-free forests in $G$. We also establish a zero-free disc for the generating function of all forests in $G$ (aka the partition function of the arboreal gas) which may be of independent interest.
Graphs are a widely used data structure for collecting and analyzing relational data. However, when the graph structure is distributed across several parties, its analysis is particularly challenging. In particular, due to the sensitivity of the data each party might want to keep their partial knowledge of the graph private, while still willing to collaborate with the other parties for tasks of mutual benefit, such as data curation or the removal of poisoned data. To address this challenge, we propose Crypto'Graph, an efficient protocol for privacy-preserving link prediction on distributed graphs. More precisely, it allows parties partially sharing a graph with distributed links to infer the likelihood of formation of new links in the future. Through the use of cryptographic primitives, Crypto'Graph is able to compute the likelihood of these new links on the joint network without revealing the structure of the private individual graph of each party, even though they know the number of nodes they have, since they share the same graph but not the same links. Crypto'Graph improves on previous works by enabling the computation of a certain number of similarity metrics without any additional cost. The use of Crypto'Graph is illustrated for defense against graph poisoning attacks, in which it is possible to identify potential adversarial links without compromising the privacy of the graphs of individual parties. The effectiveness of Crypto'Graph in mitigating graph poisoning attacks and achieving high prediction accuracy on a graph neural network node classification task is demonstrated through extensive experimentation on a real-world dataset.
In multi-turn dialog, utterances do not always take the full form of sentences \cite{Carbonell1983DiscoursePA}, which naturally makes understanding the dialog context more difficult. However, it is essential to fully grasp the dialog context to generate a reasonable response. Hence, in this paper, we propose to improve the response generation performance by examining the model's ability to answer a reading comprehension question, where the question is focused on the omitted information in the dialog. Enlightened by the multi-task learning scheme, we propose a joint framework that unifies these two tasks, sharing the same encoder to extract the common and task-invariant features with different decoders to learn task-specific features. To better fusing information from the question and the dialog history in the encoding part, we propose to augment the Transformer architecture with a memory updater, which is designed to selectively store and update the history dialog information so as to support downstream tasks. For the experiment, we employ human annotators to write and examine a large-scale dialog reading comprehension dataset. Extensive experiments are conducted on this dataset, and the results show that the proposed model brings substantial improvements over several strong baselines on both tasks. In this way, we demonstrate that reasoning can indeed help better response generation and vice versa. We release our large-scale dataset for further research.
Most existing event extraction (EE) methods merely extract event arguments within the sentence scope. However, such sentence-level EE methods struggle to handle soaring amounts of documents from emerging applications, such as finance, legislation, health, etc., where event arguments always scatter across different sentences, and even multiple such event mentions frequently co-exist in the same document. To address these challenges, we propose a novel end-to-end model, Doc2EDAG, which can generate an entity-based directed acyclic graph to fulfill the document-level EE (DEE) effectively. Moreover, we reformalize a DEE task with the no-trigger-words design to ease the document-level event labeling. To demonstrate the effectiveness of Doc2EDAG, we build a large-scale real-world dataset consisting of Chinese financial announcements with the challenges mentioned above. Extensive experiments with comprehensive analyses illustrate the superiority of Doc2EDAG over state-of-the-art methods. Data and codes can be found at //github.com/dolphin-zs/Doc2EDAG.
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