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In the field of Learning from Demonstration (LfD), Dynamical Systems (DSs) have gained significant attention due to their ability to generate real-time motions and reach predefined targets. However, the conventional convergence-centric behavior exhibited by DSs may fall short in safety-critical tasks, specifically, those requiring precise replication of demonstrated trajectories or strict adherence to constrained regions even in the presence of perturbations or human intervention. Moreover, existing DS research often assumes demonstrations solely in Euclidean space, overlooking the crucial aspect of orientation in various applications. To alleviate these shortcomings, we present an innovative approach geared toward ensuring the safe execution of learned orientation skills within constrained regions surrounding a reference trajectory. This involves learning a stable DS on SO(3), extracting time-varying conic constraints from the variability observed in expert demonstrations, and bounding the evolution of the DS with Conic Control Barrier Function (CCBF) to fulfill the constraints. We validated our approach through extensive evaluation in simulation and showcased its effectiveness for a cutting skill in the context of assisted teleoperation.

We consider the differentially private (DP) facility location problem in the so called super-set output setting proposed by Gupta et al. [SODA 2010]. The current best known expected approximation ratio for an $\epsilon$-DP algorithm is $O\left(\frac{\log n}{\sqrt{\epsilon}}\right)$ due to Cohen-Addad et al. [AISTATS 2022] where $n$ denote the size of the metric space, meanwhile the best known lower bound is $\Omega(1/\sqrt{\epsilon})$ [NeurIPS 2019]. In this short note, we give a lower bound of $\tilde{\Omega}\left(\min\left\{\log n, \sqrt{\frac{\log n}{\epsilon}}\right\}\right)$ on the expected approximation ratio of any $\epsilon$-DP algorithm, which is the first evidence that the approximation ratio has to grow with the size of the metric space.

Large language models (LLMs) are notorious for hallucinating, i.e., producing erroneous claims in their output. Such hallucinations can be dangerous, as occasional factual inaccuracies in the generated text might be obscured by the rest of the output being generally factual, making it extremely hard for the users to spot them. Current services that leverage LLMs usually do not provide any means for detecting unreliable generations. Here, we aim to bridge this gap. In particular, we propose a novel fact-checking and hallucination detection pipeline based on token-level uncertainty quantification. Uncertainty scores leverage information encapsulated in the output of a neural network or its layers to detect unreliable predictions, and we show that they can be used to fact-check the atomic claims in the LLM output. Moreover, we present a novel token-level uncertainty quantification method that removes the impact of uncertainty about what claim to generate on the current step and what surface form to use. Our method Claim Conditioned Probability (CCP) measures only the uncertainty of particular claim value expressed by the model. Experiments on the task of biography generation demonstrate strong improvements for CCP compared to the baselines for six different LLMs and three languages. Human evaluation reveals that the fact-checking pipeline based on uncertainty quantification is competitive with a fact-checking tool that leverages external knowledge.

Reinforcement Learning from Human Feedback (\textbf{RLHF}) has emerged as a dominant approach for aligning LLM outputs with human preferences. Inspired by the success of RLHF, we study the performance of multiple algorithms that learn from feedback (Expert Iteration, Proximal Policy Optimization (\textbf{PPO}), Return-Conditioned RL) on improving LLM reasoning capabilities. We investigate both sparse and dense rewards provided to the LLM both heuristically and via a learned reward model. We additionally start from multiple model sizes and initializations both with and without supervised fine-tuning (\textbf{SFT}) data. Overall, we find all algorithms perform comparably, with Expert Iteration performing best in most cases. Surprisingly, we find the sample complexity of Expert Iteration is similar to that of PPO, requiring at most on the order of $10^6$ samples to converge from a pretrained checkpoint. We investigate why this is the case, concluding that during RL training models fail to explore significantly beyond solutions already produced by SFT models. Additionally, we discuss a trade off between maj@1 and pass@96 metric performance during SFT training and how conversely RL training improves both simultaneously. We then conclude by discussing the implications of our findings for RLHF and the future role of RL in LLM fine-tuning.

Text generation with Large Language Models (LLMs) is known to be memory bound due to the combination of their auto-regressive nature, huge parameter counts, and limited memory bandwidths, often resulting in low token rates. Speculative decoding has been proposed as a solution for LLM inference acceleration. However, since draft models are often unavailable in the modern open-source LLM families, e.g., for Llama 2 7B, training a high-quality draft model is required to enable inference acceleration via speculative decoding. In this paper, we propose a simple draft model training framework for direct alignment to chat-capable target models. With the proposed framework, we train Llama 2 Chat Drafter 115M, a draft model for Llama 2 Chat 7B or larger, with only 1.64\% of the original size. Our training framework only consists of pretraining, distillation dataset generation, and finetuning with knowledge distillation, with no additional alignment procedure. For the finetuning step, we use instruction-response pairs generated by target model for distillation in plausible data distribution, and propose a new Total Variation Distance++ (TVD++) loss that incorporates variance reduction techniques inspired from the policy gradient method in reinforcement learning. Our empirical results show that Llama 2 Chat Drafter 115M with speculative decoding achieves up to 2.3 block efficiency and 2.4$\times$ speed-up relative to autoregressive decoding on various tasks with no further task-specific fine-tuning.

An asymptotic theory is established for linear functionals of the predictive function given by kernel ridge regression, when the reproducing kernel Hilbert space is equivalent to a Sobolev space. The theory covers a wide variety of linear functionals, including point evaluations, evaluation of derivatives, $L_2$ inner products, etc. We establish the upper and lower bounds of the estimates and their asymptotic normality. It is shown that $\lambda\sim n^{-1}$ is the universal optimal order of magnitude for the smoothing parameter to balance the variance and the worst-case bias. The theory also implies that the optimal $L_\infty$ error of kernel ridge regression can be attained under the optimal smoothing parameter $\lambda\sim n^{-1}\log n$. These optimal rates for the smoothing parameter differ from the known optimal rate $\lambda\sim n^{-\frac{2m}{2m+d}}$ that minimizes the $L_2$ error of the kernel ridge regression.

We demonstrate the superior capabilities of the recently proposed Lorentz quantum computer (LQC) compared to conventional quantum computers. We introduce an associated computational complexity class, bounded-error Lorentz quantum polynomial-time (BLQP), and prove that the complexity class ${\text P}^{\sharp \text{P}}$ is contained within BLQP. We present LQC algorithms that solve in polynomial time the problem of maximum independent set and the problems in the classes of NP, co-NP, PH (polynomial hierarchy), PP (probabilistic polynomial-time), and ${\text P}^{\sharp \text{P}}$. We show that the quantum computing with postselection proposed by Aaronson can be simulated efficiently by LQC, but not vice versa.

The competition complexity of an auction setting is the number of additional bidders needed such that the simple mechanism of selling items separately (with additional bidders) achieves greater revenue than the optimal but complex (randomized, prior-dependent, Bayesian-truthful) optimal mechanism without the additional bidders. Our main result settles the competition complexity of $n$ bidders with additive values over $m < n$ independent items at $\Theta(\sqrt{nm})$. The $O(\sqrt{nm})$ upper bound is due to [BW19], and our main result improves the prior lower bound of $\Omega(\ln n)$ to $\Omega(\sqrt{nm})$. Our main result follows from an explicit construction of a Bayesian IC auction for $n$ bidders with additive values over $m<n$ independent items drawn from the Equal Revenue curve truncated at $\sqrt{nm}$ ($\mathcal{ER}_{\le \sqrt{nm}}$), which achieves revenue that exceeds $\text{SRev}_{n+\sqrt{nm}}(\mathcal{ER}_{\le \sqrt{nm}}^m)$. Along the way, we show that the competition complexity of $n$ bidders with additive values over $m$ independent items is exactly equal to the minimum $c$ such that $\text{SRev}_{n+c}(\mathcal{ER}_{\le p}^m) \geq \text{Rev}_n(\mathcal{ER}_{\le p}^m)$ for all $p$ (that is, some truncated Equal Revenue witnesses the worst-case competition complexity). Interestingly, we also show that the untruncated Equal Revenue curve does not witness the worst-case competition complexity when $n > m$: $\text{SRev}_n(\mathcal{ER}^m) = nm+O_m(\ln (n)) \leq \text{SRev}_{n+O_m(\ln (n))}(\mathcal{ER}^m)$, and therefore our result can only follow by considering all possible truncations.

Recently, Mutual Information (MI) has attracted attention in bounding the generalization error of Deep Neural Networks (DNNs). However, it is intractable to accurately estimate the MI in DNNs, thus most previous works have to relax the MI bound, which in turn weakens the information theoretic explanation for generalization. To address the limitation, this paper introduces a probabilistic representation of DNNs for accurately estimating the MI. Leveraging the proposed MI estimator, we validate the information theoretic explanation for generalization, and derive a tighter generalization bound than the state-of-the-art relaxations.

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.