In this paper we investigate the notion of legibility in sequential decision tasks under uncertainty. Previous works that extend legibility to scenarios beyond robot motion either focus on deterministic settings or are computationally too expensive. Our proposed approach, dubbed PoL-MDP, is able to handle uncertainty while remaining computationally tractable. We establish the advantages of our approach against state-of-the-art approaches in several simulated scenarios of different complexity. We also showcase the use of our legible policies as demonstrations for an inverse reinforcement learning agent, establishing their superiority against the commonly used demonstrations based on the optimal policy. Finally, we assess the legibility of our computed policies through a user study where people are asked to infer the goal of a mobile robot following a legible policy by observing its actions.
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In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs $-$ which can be discretized to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox-Ingersoll-Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh-Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of $O(h^{3/2})$ and outperform schemes previously proposed in the literature.
In this paper we present a new gap-creating randomized self-reduction for parameterized Maximum Likelihood Decoding problem over $\mathbb{F}_p$ ($k$-MLD$_p$). The reduction takes a $k$-MLD$_p$ instance with $k\cdot n$ vectors as input, runs in time $f(k)n^{O(1)}$ for some computable function $f$, outputs a $(3/2-\varepsilon)$-Gap-$k'$-MLD$_p$ instance for any $\varepsilon>0$, where $k'=O(k^2\log k)$. Using this reduction, we show that assuming the randomized Exponential Time Hypothesis (ETH), no algorithms can approximate $k$-MLD$_p$ (and therefore its dual problem $k$-NCP$_p$) within factor $(3/2-\varepsilon)$ in $f(k)\cdot n^{o(\sqrt{k/\log k})}$ time for any $\varepsilon>0$. We then use reduction by Bhattacharyya, Ghoshal, Karthik and Manurangsi (ICALP 2018) to amplify the $(3/2-\varepsilon)$-gap to any constant. As a result, we show that assuming ETH, no algorithms can approximate $k$-NCP$_p$ and $k$-MDP$_p$ within $\gamma$-factor in $f(k)n^{o(k^{\varepsilon_\gamma})}$ time for some constant $\varepsilon_\gamma>0$. Combining with the gap-preserving reduction by Bennett, Cheraghchi, Guruswami and Ribeiro (STOC 2023), we also obtain similar lower bounds for $k$-MDP$_p$, $k$-CVP$_p$ and $k$-SVP$_p$. These results improve upon the previous $f(k)n^{\Omega(\mathsf{poly} \log k)}$ lower bounds for these problems under ETH using reductions by Bhattacharyya et al. (J.ACM 2021) and Bennett et al. (STOC 2023).
In this paper, we introduce a method for estimating blind spots for sensor setups of autonomous or automated vehicles and/or robotics applications. In comparison to previous methods that rely on geometric approximations, our presented approach provides more realistic coverage estimates by utilizing accurate and detailed 3D simulation environments. Our method leverages point clouds from LiDAR sensors or camera depth images from high-fidelity simulations of target scenarios to provide accurate and actionable visibility estimates. A Monte Carlo-based reference sensor simulation enables us to accurately estimate blind spot size as a metric of coverage, as well as detection probabilities of objects at arbitrary positions.
In this paper, we present new high-probability PAC-Bayes bounds for different types of losses. Firstly, for losses with a bounded range, we recover a strengthened version of Catoni's bound that holds uniformly for all parameter values. This leads to new fast rate and mixed rate bounds that are interpretable and tighter than previous bounds in the literature. In particular, the fast rate bound is equivalent to the Seeger--Langford bound. Secondly, for losses with more general tail behaviors, we introduce two new parameter-free bounds: a PAC-Bayes Chernoff analogue when the loss' cumulative generating function is bounded, and a bound when the loss' second moment is bounded. These two bounds are obtained using a new technique based on a discretization of the space of possible events for the "in probability" parameter optimization problem. This technique is both simpler and more general than previous approaches optimizing over a grid on the parameters' space. Finally, we extend all previous results to anytime-valid bounds using a simple technique applicable to any existing bound.
This paper considers the problem of inference in cluster randomized experiments when cluster sizes are non-ignorable. Here, by a cluster randomized experiment, we mean one in which treatment is assigned at the cluster level. By non-ignorable cluster sizes, we refer to the possibility that the individual-level average treatment effects may depend non-trivially on the cluster sizes. We frame our analysis in a super-population framework in which cluster sizes are random. In this way, our analysis departs from earlier analyses of cluster randomized experiments in which cluster sizes are treated as non-random. We distinguish between two different parameters of interest: the equally-weighted cluster-level average treatment effect, and the size-weighted cluster-level average treatment effect. For each parameter, we provide methods for inference in an asymptotic framework where the number of clusters tends to infinity and treatment is assigned using a covariate-adaptive stratified randomization procedure. We additionally permit the experimenter to sample only a subset of the units within each cluster rather than the entire cluster and demonstrate the implications of such sampling for some commonly used estimators. A small simulation study and empirical demonstration show the practical relevance of our theoretical results.
In this paper, we propose a class of non-parametric classifiers, that learn arbitrary boundaries and generalize well. Our approach is based on a novel way to regularize 1NN classifiers using a greedy approach. We refer to this class of classifiers as Watershed Classifiers. 1NN classifiers are known to trivially over-fit but have very large VC dimension, hence do not generalize well. We show that watershed classifiers can find arbitrary boundaries on any dense enough dataset, and, at the same time, have very small VC dimension; hence a watershed classifier leads to good generalization. Traditional approaches to regularize 1NN classifiers are to consider $K$ nearest neighbours. Neighbourhood component analysis (NCA) proposes a way to learn representations consistent with ($n-1$) nearest neighbour classifier, where $n$ denotes the size of the dataset. In this article, we propose a loss function which can learn representations consistent with watershed classifiers, and show that it outperforms the NCA baseline.
In this paper, we analyze the discrete inf-sup condition and related error estimates for a modified Hilbert transformation as used in the space-time discretization of time-dependent partial differential equations. It turns out that the stability constant depends linearly on the finite element mesh parameter, but in most cases, we can show optimal convergence. We present a series of numerical experiments which illustrate the theoretical findings.
In this paper, we focus on simple bilevel optimization problems, where we minimize a convex smooth objective function over the optimal solution set of another convex smooth constrained optimization problem. We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem using a cutting plane approach and employs an accelerated gradient-based update to reduce the upper-level objective function over the approximated solution set. We measure the performance of our method in terms of suboptimality and infeasibility errors and provide non-asymptotic convergence guarantees for both error criteria. Specifically, when the feasible set is compact, we show that our method requires at most $\mathcal{O}(\max\{1/\sqrt{\epsilon_{f}}, 1/\epsilon_g\})$ iterations to find a solution that is $\epsilon_f$-suboptimal and $\epsilon_g$-infeasible. Moreover, under the additional assumption that the lower-level objective satisfies the $r$-th H\"olderian error bound, we show that our method achieves an iteration complexity of $\mathcal{O}(\max\{\epsilon_{f}^{-\frac{2r-1}{2r}},\epsilon_{g}^{-\frac{2r-1}{2r}}\})$, which matches the optimal complexity of single-level convex constrained optimization when $r=1$.
Random binning is a powerful and widely used tool in information theory. In this paper, considering the Tsallis measures, we examine the output statistics of random binning (OSRB). Using the OSRB framework, the achievable rate region of the wiretap channel with Tsallis divergence as a security measure is investigated.
Recent work pre-training Transformers with self-supervised objectives on large text corpora has shown great success when fine-tuned on downstream NLP tasks including text summarization. However, pre-training objectives tailored for abstractive text summarization have not been explored. Furthermore there is a lack of systematic evaluation across diverse domains. In this work, we propose pre-training large Transformer-based encoder-decoder models on massive text corpora with a new self-supervised objective. In PEGASUS, important sentences are removed/masked from an input document and are generated together as one output sequence from the remaining sentences, similar to an extractive summary. We evaluated our best PEGASUS model on 12 downstream summarization tasks spanning news, science, stories, instructions, emails, patents, and legislative bills. Experiments demonstrate it achieves state-of-the-art performance on all 12 downstream datasets measured by ROUGE scores. Our model also shows surprising performance on low-resource summarization, surpassing previous state-of-the-art results on 6 datasets with only 1000 examples. Finally we validated our results using human evaluation and show that our model summaries achieve human performance on multiple datasets.
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