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Krause, Andreas ; Brunskill, Emma ; Cho, Kyunghyun ; Engelhardt, Barbara ; Sabato, Sivan ; Scarlett, Jonathan (Ed.)Transfer operators provide a rich framework for representing the dynamics of very general, nonlinear dynamical systems. When interacting with reproducing kernel Hilbert spaces (RKHS), descriptions of dynamics often incur prohibitive data storage requirements, motivating dataset sparsification as a precursory step to computation. Further, in practice, data is available in the form of trajectories, introducing correlation between samples. In this work, we present a method for sparse learning of transfer operators from $\beta$mixing stochastic processes, in both discrete and continuous time, and provide sample complexity analysis extending existing theoretical guarantees for learning from nonsparse, i.i.d. data. In addressing continuoustime settings, we develop precise descriptions using covariancetype operators for the infinitesimal generator that aids in the sample complexity analysis. We empirically illustrate the efficacy of our sparse embedding approach through deterministic and stochastic nonlinear system examples.more » « less

Krause, Andreas ; Brunskill, Emma ; Cho, Kyunghyun ; Engelhardt, Barbara ; Sabato, Sivan ; Scarlett, Jonathan (Ed.)Differentiable Search Index is a recently proposed paradigm for document retrieval, that encodes information about a corpus of documents within the parameters of a neural network and directly maps queries to corresponding documents. These models have achieved stateoftheart performances for document retrieval across many benchmarks. These kinds of models have a significant limitation: it is not easy to add new documents after a model is trained. We propose IncDSI, a method to add documents in real time (about 2050ms per document), without retraining the model on the entire dataset (or even parts thereof). Instead we formulate the addition of documents as a constrained optimization problem that makes minimal changes to the network parameters. Although orders of magnitude faster, our approach is competitive with retraining the model on the whole dataset and enables the development of document retrieval systems that can be updated with new information in realtime. Our code for IncDSI is available at \href{https://github.com/varshakishore/IncDSI}{https://github.com/varshakishore/IncDSI}.more » « less

Krause, Andreas ; Brunskill, Emma ; Cho, Kyunghyun ; Engelhardt, Barbara ; Sabato, Sivan ; Scarlett, Jonathan (Ed.)We consider the problem of estimating the optimal transport map between two probability distributions, P and Q in R^d, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution Q is a discrete measure supported on a finite number of points in R^d. We study a computationally efficient estimator initially proposed by Pooladian & NilesWeed (2021), based on entropic optimal transport, and show in the semidiscrete setting that it converges at the minimaxoptimal rate n^{−1/2}, independent of dimension. Other standard map estimation techniques both lack finitesample guarantees in this setting and provably suffer from the curse of dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.more » « less

Krause, Andreas ; Brunskill, Emma ; Cho, Kyunghyun ; Engelhardt, Barbara ; Sabato, Sivan ; Scarlett, Jonathan. (Ed.)The parameter space for any fixed architecture of feedforward ReLU neural networks serves as a proxy during training for the associated class of functions  but how faithful is this representation? It is known that many different parameter settings $\theta$ can determine the same function $f$. Moreover, the degree of this redundancy is inhomogeneous: for some networks, the only symmetries are permutation of neurons in a layer and positive scaling of parameters at a neuron, while other networks admit additional hidden symmetries. In this work, we prove that, for any network architecture where no layer is narrower than the input, there exist parameter settings with no hidden symmetries. We also describe a number of mechanisms through which hidden symmetries can arise, and empirically approximate the functional dimension of different network architectures at initialization. These experiments indicate that the probability that a network has no hidden symmetries decreases towards 0 as depth increases, while increasing towards 1 as width and input dimension increase.more » « less

Krause, Andreas ; Brunskill, Emma ; Cho, Kyunghyun ; Engelhardt ; Barbara ; Sabato, Sivan ; Scarlett, Jonathan (Ed.)Robust Markov decision processes (MDPs) address the challenge of model uncertainty by optimizing the worstcase performance over an uncertainty set of MDPs. In this paper, we focus on the robust averagereward MDPs under the modelfree setting. We first theoretically characterize the structure of solutions to the robust averagereward Bellman equation, which is essential for our later convergence analysis. We then design two modelfree algorithms, robust relative value iteration (RVI) TD and robust RVI Qlearning, and theoretically prove their convergence to the optimal solution. We provide several widely used uncertainty sets as examples, including those def ined by the contamination model, total variation, Chisquared divergence, KullbackLeibler (KL) divergence and Wasserstein distance.more » « less

Krause, Andreas ; Brunskill, Emma ; Cho, Kyunghyun ; Engelhardt, Barbara ; Sabato, Sivan ; Scarlett, Jonathan (Ed.)We consider a deep matrix factorization model of covariance matrices trained with the BuresWasserstein distance. While recent works have made advances in the study of the optimization problem for overparametrized lowrank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another type of loss and connects with the generative setting. We characterize the critical points and minimizers of the BuresWasserstein distance over the space of rankbounded matrices. The Hessian of this loss at lowrank matrices can theoretically blow up, which creates challenges to analyze convergence of gradient optimization methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss as well as convergence results for finite step size gradient descent under certain assumptions on the initial weights.more » « less