Weihao Kong

Abhimanyu Das, Weihao Kong, Biswajit Paria et al. · cmu

Probabilistic, hierarchically coherent forecasting is a key problem in many practical forecasting applications -- the goal is to obtain coherent probabilistic predictions for a large number of time series arranged in a pre-specified tree hierarchy. In this paper, we present an end-to-end deep probabilistic model for hierarchical forecasting that is motivated by a classical top-down strategy. It jointly learns the distribution of the root time series, and the (dirichlet) proportions according to which each parent time-series is split among its children at any point in time. The resulting forecasts are naturally coherent, and provide probabilistic predictions over all time series in the hierarchy. We experiment on several public datasets and demonstrate significant improvements of up to 26% on most datasets compared to state-of-the-art baselines. Finally, we also provide theoretical justification for the superiority of our top-down approach compared to the more traditional bottom-up modeling.

Jonathan N. Lee, Weihao Kong, Aldo Pacchiano et al.

In many bandit problems, the maximal reward achievable by a policy is often unknown in advance. We consider the problem of estimating the optimal policy value in the sublinear data regime before the optimal policy is even learnable. We refer to this as $V^*$ estimation. It was recently shown that fast $V^*$ estimation is possible but only in disjoint linear bandits with Gaussian covariates. Whether this is possible for more realistic context distributions has remained an open and important question for tasks such as model selection. In this paper, we first provide lower bounds showing that this general problem is hard. However, under stronger assumptions, we give an algorithm and analysis proving that $\widetilde{\mathcal{O}}(\sqrt{d})$ sublinear estimation of $V^*$ is indeed information-theoretically possible, where $d$ is the dimension. We then present a more practical, computationally efficient algorithm that estimates a problem-dependent upper bound on $V^*$ that holds for general distributions and is tight when the context distribution is Gaussian. We prove our algorithm requires only $\widetilde{\mathcal{O}}(\sqrt{d})$ samples to estimate the upper bound. We use this upper bound and the estimator to obtain novel and improved guarantees for several applications in bandit model selection and testing for treatment effects.

Jonathan N. Lee, Aldo Pacchiano, Vidya Muthukumar et al.

Deep reinforcement learning has achieved impressive successes yet often requires a very large amount of interaction data. This result is perhaps unsurprising, as using complicated function approximation often requires more data to fit, and early theoretical results on linear Markov decision processes provide regret bounds that scale with the dimension of the linear approximation. Ideally, we would like to automatically identify the minimal dimension of the approximation that is sufficient to encode an optimal policy. Towards this end, we consider the problem of model selection in RL with function approximation, given a set of candidate RL algorithms with known regret guarantees. The learner's goal is to adapt to the complexity of the optimal algorithm without knowing it \textit{a priori}. We present a meta-algorithm that successively rejects increasingly complex models using a simple statistical test. Given at least one candidate that satisfies realizability, we prove the meta-algorithm adapts to the optimal complexity with $\tilde{O}(L^{5/6} T^{2/3})$ regret compared to the optimal candidate's $\tilde{O}(\sqrt T)$ regret, where $T$ is the number of episodes and $L$ is the number of algorithms. The dimension and horizon dependencies remain optimal with respect to the best candidate, and our meta-algorithmic approach is flexible to incorporate multiple candidate algorithms and models. Finally, we show that the meta-algorithm automatically admits significantly improved instance-dependent regret bounds that depend on the gaps between the maximal values attainable by the candidates.

Weihao Kong, Gregory Valiant, Emma Brunskill

We study the problem of estimating the expected reward of the optimal policy in the stochastic disjoint linear bandit setting. We prove that for certain settings it is possible to obtain an accurate estimate of the optimal policy value even with a number of samples that is sublinear in the number that would be required to \emph{find} a policy that realizes a value close to this optima. We establish nearly matching information theoretic lower bounds, showing that our algorithm achieves near optimal estimation error. Finally, we demonstrate the effectiveness of our algorithm on joke recommendation and cancer inhibition dosage selection problems using real datasets.

Ilias Diakonikolas, Weihao Kong, Alistair Stewart

We study the problem of high-dimensional linear regression in a robust model where an $ε$-fraction of the samples can be adversarially corrupted. We focus on the fundamental setting where the covariates of the uncorrupted samples are drawn from a Gaussian distribution $\mathcal{N}(0, Σ)$ on $\mathbb{R}^d$. We give nearly tight upper bounds and computational lower bounds for this problem. Specifically, our main contributions are as follows: For the case that the covariance matrix is known to be the identity, we give a sample near-optimal and computationally efficient algorithm that outputs a candidate hypothesis vector $\widehatβ$ which approximates the unknown regression vector $β$ within $\ell_2$-norm $O(ε\log(1/ε) σ)$, where $σ$ is the standard deviation of the random observation noise. An error of $Ω(εσ)$ is information-theoretically necessary, even with infinite sample size. Prior work gave an algorithm for this problem with sample complexity $\tildeΩ(d^2/ε^2)$ whose error guarantee scales with the $\ell_2$-norm of $β$. For the case of unknown covariance, we show that we can efficiently achieve the same error guarantee as in the known covariance case using an additional $\tilde{O}(d^2/ε^2)$ unlabeled examples. On the other hand, an error of $O(εσ)$ can be information-theoretically attained with $O(d/ε^2)$ samples. We prove a Statistical Query (SQ) lower bound providing evidence that this quadratic tradeoff in the sample size is inherent. More specifically, we show that any polynomial time SQ learning algorithm for robust linear regression (in Huber's contamination model) with estimation complexity $O(d^{2-c})$, where $c>0$ is an arbitrarily small constant, must incur an error of $Ω(\sqrtε σ)$.

Weihao Kong, Gregory Valiant

We consider the problem of approximating the set of eigenvalues of the covariance matrix of a multivariate distribution (equivalently, the problem of approximating the "population spectrum"), given access to samples drawn from the distribution. The eigenvalues of the covariance of a distribution contain basic information about the distribution, including the presence or lack of structure in the distribution, the effective dimensionality of the distribution, and the applicability of higher-level machine learning and multivariate statistical tools. We consider this fundamental recovery problem in the regime where the number of samples is comparable, or even sublinear in the dimensionality of the distribution in question. First, we propose a theoretically optimal and computationally efficient algorithm for recovering the moments of the eigenvalues of the population covariance matrix. We then leverage this accurate moment recovery, via a Wasserstein distance argument, to show that the vector of eigenvalues can be accurately recovered. We provide finite--sample bounds on the expected error of the recovered eigenvalues, which imply that our estimator is asymptotically consistent as the dimensionality of the distribution and sample size tend towards infinity, even in the sublinear sample regime where the ratio of the sample size to the dimensionality tends to zero. In addition to our theoretical results, we show that our approach performs well in practice for a broad range of distributions and sample sizes.