526 research outputs found
Dynamic Tensor Clustering
Dynamic tensor data are becoming prevalent in numerous applications. Existing
tensor clustering methods either fail to account for the dynamic nature of the
data, or are inapplicable to a general-order tensor. Also there is often a gap
between statistical guarantee and computational efficiency for existing tensor
clustering solutions. In this article, we aim to bridge this gap by proposing a
new dynamic tensor clustering method, which takes into account both sparsity
and fusion structures, and enjoys strong statistical guarantees as well as high
computational efficiency. Our proposal is based upon a new structured tensor
factorization that encourages both sparsity and smoothness in parameters along
the specified tensor modes. Computationally, we develop a highly efficient
optimization algorithm that benefits from substantial dimension reduction. In
theory, we first establish a non-asymptotic error bound for the estimator from
the structured tensor factorization. Built upon this error bound, we then
derive the rate of convergence of the estimated cluster centers, and show that
the estimated clusters recover the true cluster structures with a high
probability. Moreover, our proposed method can be naturally extended to
co-clustering of multiple modes of the tensor data. The efficacy of our
approach is illustrated via simulations and a brain dynamic functional
connectivity analysis from an Autism spectrum disorder study.Comment: Accepted at Journal of the American Statistical Associatio
Provable Sparse Tensor Decomposition
We propose a novel sparse tensor decomposition method, namely Tensor
Truncated Power (TTP) method, that incorporates variable selection into the
estimation of decomposition components. The sparsity is achieved via an
efficient truncation step embedded in the tensor power iteration. Our method
applies to a broad family of high dimensional latent variable models, including
high dimensional Gaussian mixture and mixtures of sparse regressions. A
thorough theoretical investigation is further conducted. In particular, we show
that the final decomposition estimator is guaranteed to achieve a local
statistical rate, and further strengthen it to the global statistical rate by
introducing a proper initialization procedure. In high dimensional regimes, the
obtained statistical rate significantly improves those shown in the existing
non-sparse decomposition methods. The empirical advantages of TTP are confirmed
in extensive simulated results and two real applications of click-through rate
prediction and high-dimensional gene clustering.Comment: To Appear in JRSS-
Contextual Dynamic Pricing with Strategic Buyers
Personalized pricing, which involves tailoring prices based on individual
characteristics, is commonly used by firms to implement a consumer-specific
pricing policy. In this process, buyers can also strategically manipulate their
feature data to obtain a lower price, incurring certain manipulation costs.
Such strategic behavior can hinder firms from maximizing their profits. In this
paper, we study the contextual dynamic pricing problem with strategic buyers.
The seller does not observe the buyer's true feature, but a manipulated feature
according to buyers' strategic behavior. In addition, the seller does not
observe the buyers' valuation of the product, but only a binary response
indicating whether a sale happens or not. Recognizing these challenges, we
propose a strategic dynamic pricing policy that incorporates the buyers'
strategic behavior into the online learning to maximize the seller's cumulative
revenue. We first prove that existing non-strategic pricing policies that
neglect the buyers' strategic behavior result in a linear regret
with the total time horizon, indicating that these policies are not better
than a random pricing policy. We then establish that our proposed policy
achieves a sublinear regret upper bound of . Importantly, our
policy is not a mere amalgamation of existing dynamic pricing policies and
strategic behavior handling algorithms. Our policy can also accommodate the
scenario when the marginal cost of manipulation is unknown in advance. To
account for it, we simultaneously estimate the valuation parameter and the cost
parameter in the online pricing policy, which is shown to also achieve an
regret bound. Extensive experiments support our theoretical
developments and demonstrate the superior performance of our policy compared to
other pricing policies that are unaware of the strategic behaviors
Online Regularization for High-Dimensional Dynamic Pricing Algorithms
We propose a novel \textit{online regularization} scheme for
revenue-maximization in high-dimensional dynamic pricing algorithms. The online
regularization scheme equips the proposed optimistic online regularized maximum
likelihood pricing (\texttt{OORMLP}) algorithm with three major advantages:
encode market noise knowledge into pricing process optimism; empower online
statistical learning with always-validity over all decision points; envelop
prediction error process with time-uniform non-asymptotic oracle inequalities.
This type of non-asymptotic inference results allows us to design safer and
more robust dynamic pricing algorithms in practice. In theory, the proposed
\texttt{OORMLP} algorithm exploits the sparsity structure of high-dimensional
models and obtains a logarithmic regret in a decision horizon. These
theoretical advances are made possible by proposing an optimistic online LASSO
procedure that resolves dynamic pricing problems at the \textit{process} level,
based on a novel use of non-asymptotic martingale concentration. In
experiments, we evaluate \texttt{OORMLP} in different synthetic pricing problem
settings and observe that \texttt{OORMLP} performs better than \texttt{RMLP}
proposed in \cite{javanmard2019dynamic}
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