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We present the Fast Chebyshev Transform (FCT), a fast, randomized algorithm to compute a Chebyshev approximation of functions in high-dimensions from the knowledge of the location of its nonzero Chebyshev coefficients. Rather than sampling a full-resolution Chebyshev grid in each dimension, we randomly sample several grids with varied resolutions and solve a least-squares problem in coefficient space in order to compute a polynomial approximating the function of interest across all grids simultaneously. We theoretically and empirically show that the FCT exhibits quasi-linear scaling and high numerical accuracy on challenging and complex high-dimensional problems. We demonstrate the effectiveness of our approach compared to alternative Chebyshev approximation schemes. In particular, we highlight our algorithm's effectiveness in high dimensions, demonstrating significant speedups over commonly-used alternative techniques.

The new field of Explainable Planning (XAIP) has produced a variety of approaches to explain and describe the behavior of autonomous agents to human observers. Many summarize agent behavior in terms of the constraints, or ''rules,'' which the agent adheres to during its trajectories. In this work, we narrow the focus from summary to specific moments in individual trajectories, offering a ''pointwise-in-time'' view. Our novel framework, which we define on Linear Temporal Logic (LTL) rules, assigns an intuitive status to any rule in order to describe the trajectory progress at individual time steps; here, a rule is classified as active, satisfied, inactive, or violated. Given a trajectory, a user may query for status of specific LTL rules at individual trajectory time steps. In this paper, we present this novel framework, named Rule Status Assessment (RSA), and provide an example of its implementation. We find that pointwise-in-time status assessment is useful as a post-hoc diagnostic, enabling a user to systematically track the agent's behavior with respect to a set of rules.

We present Direct Reward Fine-Tuning (DRaFT), a simple and effective method for fine-tuning diffusion models to maximize differentiable reward functions, such as scores from human preference models. We first show that it is possible to backpropagate the reward function gradient through the full sampling procedure, and that doing so achieves strong performance on a variety of rewards, outperforming reinforcement learning-based approaches. We then propose more efficient variants of DRaFT: DRaFT-K, which truncates backpropagation to only the last K steps of sampling, and DRaFT-LV, which obtains lower-variance gradient estimates for the case when K=1. We show that our methods work well for a variety of reward functions and can be used to substantially improve the aesthetic quality of images generated by Stable Diffusion 1.4. Finally, we draw connections between our approach and prior work, providing a unifying perspective on the design space of gradient-based fine-tuning algorithms.

Contrastive learning often relies on comparing positive anchor samples with multiple negative samples to perform Self-Supervised Learning (SSL). However, non-contrastive approaches like BYOL, SimSiam, and Barlow Twins achieve SSL without explicit negative samples. In this paper, we introduce a unified matrix information-theoretic framework that explains many contrastive and non-contrastive learning methods. We then propose a novel method Matrix-SSL based on matrix information theory. Experimental results reveal that Matrix-SSL significantly outperforms state-of-the-art methods on the ImageNet dataset under linear evaluation settings and on MS-COCO for transfer learning tasks. Specifically, when performing 100 epochs pre-training, our method outperforms SimCLR by 4.6%, and when performing transfer learning tasks on MS-COCO, our method outperforms previous SOTA methods such as MoCo v2 and BYOL up to 3.3% with only 400 epochs compared to 800 epochs pre-training. Code available at //github.com/yifanzhang-pro/Matrix-SSL.

In this study, we aim to enhance the arithmetic reasoning ability of Large Language Models (LLMs) through zero-shot prompt optimization. We identify a previously overlooked objective of query dependency in such optimization and elucidate two ensuing challenges that impede the successful and economical design of prompt optimization techniques. One primary issue is the absence of an effective method to evaluate prompts during inference when the golden answer is unavailable. Concurrently, learning via interactions with the LLMs to navigate the expansive natural language prompting space proves to be resource-intensive. To address this, we introduce Prompt-OIRL, which harnesses offline inverse reinforcement learning to draw insights from offline prompting demonstration data. Such data exists as by-products when diverse prompts are benchmarked on open-accessible datasets. With Prompt-OIRL, the query-dependent prompt optimization objective is achieved by first learning an offline reward model. This model can evaluate any query-prompt pairs without accessing LLMs. Subsequently, a best-of-N strategy is deployed to recommend the optimal prompt. Our experimental evaluations across various LLM scales and arithmetic reasoning datasets underscore both the efficacy and economic viability of the proposed approach.

This report examines the effectiveness of Chain-of-Thought (CoT) prompting in improving the multi-step reasoning abilities of large language models (LLMs). Inspired by previous studies \cite{Min2022RethinkingWork}, we analyze the impact of three types of CoT prompt perturbations, namely CoT order, CoT values, and CoT operators on the performance of GPT-3 on various tasks. Our findings show that incorrect CoT prompting leads to poor performance on accuracy metrics. Correct values in the CoT is crucial for predicting correct answers. Moreover, incorrect demonstrations, where the CoT operators or the CoT order are wrong, do not affect the performance as drastically when compared to the value based perturbations. This research deepens our understanding of CoT prompting and opens some new questions regarding the capability of LLMs to learn reasoning in context.

We prove a fundamental limitation on the efficiency of a wide class of Reinforcement Learning (RL) algorithms. This limitation applies to model-free RL methods as well as a broad range of model-based methods, such as planning with tree search. Under an abstract definition of this class, we provide a family of RL problems for which these methods suffer a lower bound exponential in the horizon for their interactions with the environment to find an optimal behavior. However, there exists a method, not tailored to this specific family of problems, which can efficiently solve the problems in the family. In contrast, our limitation does not apply to several types of methods proposed in the literature, for instance, goal-conditioned methods or other algorithms that construct an inverse dynamics model.

The Bayesian Cram\'er-Rao bound (CRB) provides a lower bound on the error of any Bayesian estimator under mild regularity conditions. It can be used to benchmark the performance of estimators, and provides a principled design metric for guiding system design and optimization. However, the Bayesian CRB depends on the prior distribution, which is often unknown for many problems of interest. This work develops a new data-driven estimator for the Bayesian CRB using score matching, a statistical estimation technique, to model the prior distribution. The performance of the estimator is analyzed in both the classical parametric modeling regime and the neural network modeling regime. In both settings, we develop novel non-asymptotic bounds on the score matching error and our Bayesian CRB estimator. Our proofs build on results from empirical process theory, including classical bounds and recently introduced techniques for characterizing neural networks, to address the challenges of bounding the score matching error. The performance of the estimator is illustrated empirically on a denoising problem example with a Gaussian mixture prior.

The problem of Multiple Object Tracking (MOT) consists in following the trajectory of different objects in a sequence, usually a video. In recent years, with the rise of Deep Learning, the algorithms that provide a solution to this problem have benefited from the representational power of deep models. This paper provides a comprehensive survey on works that employ Deep Learning models to solve the task of MOT on single-camera videos. Four main steps in MOT algorithms are identified, and an in-depth review of how Deep Learning was employed in each one of these stages is presented. A complete experimental comparison of the presented works on the three MOTChallenge datasets is also provided, identifying a number of similarities among the top-performing methods and presenting some possible future research directions.

Recurrent neural nets (RNN) and convolutional neural nets (CNN) are widely used on NLP tasks to capture the long-term and local dependencies, respectively. Attention mechanisms have recently attracted enormous interest due to their highly parallelizable computation, significantly less training time, and flexibility in modeling dependencies. We propose a novel attention mechanism in which the attention between elements from input sequence(s) is directional and multi-dimensional (i.e., feature-wise). A light-weight neural net, "Directional Self-Attention Network (DiSAN)", is then proposed to learn sentence embedding, based solely on the proposed attention without any RNN/CNN structure. DiSAN is only composed of a directional self-attention with temporal order encoded, followed by a multi-dimensional attention that compresses the sequence into a vector representation. Despite its simple form, DiSAN outperforms complicated RNN models on both prediction quality and time efficiency. It achieves the best test accuracy among all sentence encoding methods and improves the most recent best result by 1.02% on the Stanford Natural Language Inference (SNLI) dataset, and shows state-of-the-art test accuracy on the Stanford Sentiment Treebank (SST), Multi-Genre natural language inference (MultiNLI), Sentences Involving Compositional Knowledge (SICK), Customer Review, MPQA, TREC question-type classification and Subjectivity (SUBJ) datasets.