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Recently, Ivan Mihajlin and Alexander Smal proved a composition theorem of a universal relation and some function via so called xor composition, that is there exists some function $f:\{0,1\}^n \rightarrow \{0,1\}$ such that $\textsf{CC}(\text{U}_n \diamond \text{KW}_f) \geq 1.5n-o(n)$ where $\textsf{CC}$ denotes the communication complexity of the problem. In this paper, we significantly improve their result and present an asymptotically tight and much more general composition theorem of a universal relation and most functions, that is for most functions $f:\{0,1\}^n \rightarrow \{0,1\}$ we have $\textsf{CC}(\text{U}_m \diamond \text{KW}_f) \geq m+ n -O(\sqrt{m})$ when $m=\omega(\log^2 n),n =\omega(\sqrt{m})$. This is done by a direct proof of composition theorem of a universal relation and a multiplexor in the partially half-duplex model avoiding the xor composition. And the proof works even when the multiplexor only contains a few functions. One crucial ingredient in our proof involves a combinatorial problem of constructing a tree of many leaves and every leaf contains a non-overlapping set of functions. For each leaf, there is a set of inputs such that every function in the leaf takes the same value, that is all functions are restricted. We show how to choose a set of good inputs to effectively restrict these functions to force that the number of functions in each leaf is as small as possible while maintaining the total number of functions in all leaves. This results in a large number of leaves.

Conformal prediction is a statistical framework that generates prediction sets containing ground-truth labels with a desired coverage guarantee. The predicted probabilities produced by machine learning models are generally miscalibrated, leading to large prediction sets in conformal prediction. In this paper, we empirically and theoretically show that disregarding the probabilities' value will mitigate the undesirable effect of miscalibrated probability values. Then, we propose a novel algorithm named $\textit{Sorted Adaptive prediction sets}$ (SAPS), which discards all the probability values except for the maximum softmax probability. The key idea behind SAPS is to minimize the dependence of the non-conformity score on the probability values while retaining the uncertainty information. In this manner, SAPS can produce sets of small size and communicate instance-wise uncertainty. Theoretically, we provide a finite-sample coverage guarantee of SAPS and show that the expected value of set size from SAPS is always smaller than APS. Extensive experiments validate that SAPS not only lessens the prediction sets but also broadly enhances the conditional coverage rate and adaptation of prediction sets.

Gaussian processes scale prohibitively with the size of the dataset. In response, many approximation methods have been developed, which inevitably introduce approximation error. This additional source of uncertainty, due to limited computation, is entirely ignored when using the approximate posterior. Therefore in practice, GP models are often as much about the approximation method as they are about the data. Here, we develop a new class of methods that provides consistent estimation of the combined uncertainty arising from both the finite number of data observed and the finite amount of computation expended. The most common GP approximations map to an instance in this class, such as methods based on the Cholesky factorization, conjugate gradients, and inducing points. For any method in this class, we prove (i) convergence of its posterior mean in the associated RKHS, (ii) decomposability of its combined posterior covariance into mathematical and computational covariances, and (iii) that the combined variance is a tight worst-case bound for the squared error between the method's posterior mean and the latent function. Finally, we empirically demonstrate the consequences of ignoring computational uncertainty and show how implicitly modeling it improves generalization performance on benchmark datasets.

We develop a vector space semantics for Lambek Calculus with Soft Subexponentials, apply the calculus to construct compositional vector interpretations for parasitic gap noun phrases and discourse units with anaphora and ellipsis, and experiment with the constructions in a distributional sentence similarity task. As opposed to previous work, which used Lambek Calculus with a Relevant Modality the calculus used in this paper uses a bounded version of the modality and is decidable. The vector space semantics of this new modality allows us to meaningfully define contraction as projection and provide a linear theory behind what we could previously only achieve via nonlinear maps.

We consider the problem of forming prediction sets in an online setting where the distribution generating the data is allowed to vary over time. Previous approaches to this problem suffer from over-weighting historical data and thus may fail to quickly react to the underlying dynamics. Here we correct this issue and develop a novel procedure with provably small regret over all local time intervals of a given width. We achieve this by modifying the adaptive conformal inference (ACI) algorithm of Gibbs and Cand\`{e}s (2021) to contain an additional step in which the step-size parameter of ACI's gradient descent update is tuned over time. Crucially, this means that unlike ACI, which requires knowledge of the rate of change of the data-generating mechanism, our new procedure is adaptive to both the size and type of the distribution shift. Our methods are highly flexible and can be used in combination with any baseline predictive algorithm that produces point estimates or estimated quantiles of the target without the need for distributional assumptions. We test our techniques on two real-world datasets aimed at predicting stock market volatility and COVID-19 case counts and find that they are robust and adaptive to real-world distribution shifts.

Multi-object rearrangement is a crucial skill for service robots, and commonsense reasoning is frequently needed in this process. However, achieving commonsense arrangements requires knowledge about objects, which is hard to transfer to robots. Large language models (LLMs) are one potential source of this knowledge, but they do not naively capture information about plausible physical arrangements of the world. We propose LLM-GROP, which uses prompting to extract commonsense knowledge about semantically valid object configurations from an LLM and instantiates them with a task and motion planner in order to generalize to varying scene geometry. LLM-GROP allows us to go from natural-language commands to human-aligned object rearrangement in varied environments. Based on human evaluations, our approach achieves the highest rating while outperforming competitive baselines in terms of success rate while maintaining comparable cumulative action costs. Finally, we demonstrate a practical implementation of LLM-GROP on a mobile manipulator in real-world scenarios. Supplementary materials are available at: //sites.google.com/view/llm-grop

Video instance segmentation (VIS) is the task that requires simultaneously classifying, segmenting and tracking object instances of interest in video. Recent methods typically develop sophisticated pipelines to tackle this task. Here, we propose a new video instance segmentation framework built upon Transformers, termed VisTR, which views the VIS task as a direct end-to-end parallel sequence decoding/prediction problem. Given a video clip consisting of multiple image frames as input, VisTR outputs the sequence of masks for each instance in the video in order directly. At the core is a new, effective instance sequence matching and segmentation strategy, which supervises and segments instances at the sequence level as a whole. VisTR frames the instance segmentation and tracking in the same perspective of similarity learning, thus considerably simplifying the overall pipeline and is significantly different from existing approaches. Without bells and whistles, VisTR achieves the highest speed among all existing VIS models, and achieves the best result among methods using single model on the YouTube-VIS dataset. For the first time, we demonstrate a much simpler and faster video instance segmentation framework built upon Transformers, achieving competitive accuracy. We hope that VisTR can motivate future research for more video understanding tasks.

It is always well believed that modeling relationships between objects would be helpful for representing and eventually describing an image. Nevertheless, there has not been evidence in support of the idea on image description generation. In this paper, we introduce a new design to explore the connections between objects for image captioning under the umbrella of attention-based encoder-decoder framework. Specifically, we present Graph Convolutional Networks plus Long Short-Term Memory (dubbed as GCN-LSTM) architecture that novelly integrates both semantic and spatial object relationships into image encoder. Technically, we build graphs over the detected objects in an image based on their spatial and semantic connections. The representations of each region proposed on objects are then refined by leveraging graph structure through GCN. With the learnt region-level features, our GCN-LSTM capitalizes on LSTM-based captioning framework with attention mechanism for sentence generation. Extensive experiments are conducted on COCO image captioning dataset, and superior results are reported when comparing to state-of-the-art approaches. More remarkably, GCN-LSTM increases CIDEr-D performance from 120.1% to 128.7% on COCO testing set.

Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.

While existing machine learning models have achieved great success for sentiment classification, they typically do not explicitly capture sentiment-oriented word interaction, which can lead to poor results for fine-grained analysis at the snippet level (a phrase or sentence). Factorization Machine provides a possible approach to learning element-wise interaction for recommender systems, but they are not directly applicable to our task due to the inability to model contexts and word sequences. In this work, we develop two Position-aware Factorization Machines which consider word interaction, context and position information. Such information is jointly encoded in a set of sentiment-oriented word interaction vectors. Compared to traditional word embeddings, SWI vectors explicitly capture sentiment-oriented word interaction and simplify the parameter learning. Experimental results show that while they have comparable performance with state-of-the-art methods for document-level classification, they benefit the snippet/sentence-level sentiment analysis.