A simple and effective method for the alignment of generative models is the best-of-$n$ policy, where $n$ samples are drawn from a base policy, and ranked based on a reward function, and the highest ranking one is selected. A commonly used analytical expression in the literature claims that the KL divergence between the best-of-$n$ policy and the base policy is equal to $\log (n) - (n-1)/n.$ We disprove the validity of this claim, and show that it is an upper bound on the actual KL divergence. We also explore the tightness of this upper bound in different regimes. Finally, we propose a new estimator for the KL divergence and empirically show that it provides a tight approximation through a few examples.
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Local variable selection aims to discover localized effects by assessing the impact of covariates on outcomes within specific regions defined by other covariates. We outline some challenges of local variable selection in the presence of non-linear relationships and model misspecification. Specifically, we highlight a potential drawback of common semi-parametric methods: even slight model misspecification can result in a high rate of false positives. To address these shortcomings, we propose a methodology based on orthogonal cut splines that achieves consistent local variable selection in high-dimensional scenarios. Our approach offers simplicity, handles both continuous and discrete covariates, and provides theory for high-dimensional covariates and model misspecification. We discuss settings with either independent or dependent data. Our proposal allows including adjustment covariates that do not undergo selection, enhancing flexibility in modeling complex scenarios. We illustrate its application in simulation studies with both independent and functional data, as well as with two real datasets. One dataset evaluates salary gaps associated with discrimination factors at different ages, while the other examines the effects of covariates on brain activation over time. The approach is implemented in the R package mombf.
Deep learning algorithms have recently shown to be a successful tool in estimating parameters of statistical models for which simulation is easy, but likelihood computation is challenging. But the success of these approaches depends on simulating parameters that sufficiently reproduce the observed data, and, at present, there is a lack of efficient methods to produce these simulations. We develop new black-box procedures to estimate parameters of statistical models based only on weak parameter structure assumptions. For well-structured likelihoods with frequent occurrences, such as in time series, this is achieved by pre-training a deep neural network on an extensive simulated database that covers a wide range of data sizes. For other types of complex dependencies, an iterative algorithm guides simulations to the correct parameter region in multiple rounds. These approaches can successfully estimate and quantify the uncertainty of parameters from non-Gaussian models with complex spatial and temporal dependencies. The success of our methods is a first step towards a fully flexible automatic black-box estimation framework.
In a regression model with multiple response variables and multiple explanatory variables, if the difference of the mean vectors of the response variables for different values of explanatory variables is always in the direction of the first principal eigenvector of the covariance matrix of the response variables, then it is called a multivariate allometric regression model. This paper studies the estimation of the first principal eigenvector in the multivariate allometric regression model. A class of estimators that includes conventional estimators is proposed based on weighted sum-of-squares matrices of regression sum-of-squares matrix and residual sum-of-squares matrix. We establish an upper bound of the mean squared error of the estimators contained in this class, and the weight value minimizing the upper bound is derived. Sufficient conditions for the consistency of the estimators are discussed in weak identifiability regimes under which the difference of the largest and second largest eigenvalues of the covariance matrix decays asymptotically and in ``large $p$, large $n$" regimes, where $p$ is the number of response variables and $n$ is the sample size. Several numerical results are also presented.
The presence of faulty or underactuated manipulators can disrupt the end-effector formation keeping of a team of manipulators. Based on two-link planar manipulators, we investigate this end-effector formation keeping problem for mixed fully- and under-actuated manipulators with flexible joints. In this case, the underactuated manipulators can comprise of active-passive (AP) manipulators, passive-active (PA) manipulators, or a combination thereof. We propose distributed control laws for the different types of manipulators to achieve and maintain the desired formation shape of the end-effectors. It is achieved by assigning virtual springs to the end-effectors for the fully-actuated ones and to the virtual end-effectors for the under-actuated ones. We study further the set of all desired and reachable shapes for the networked manipulators' end-effectors. Finally, we validate our analysis via numerical simulations.
We develop a new `subspace layered least squares' interior point method (IPM) for solving linear programs. Applied to an $n$-variable linear program in standard form, the iteration complexity of our IPM is up to an $O(n^{1.5} \log n)$ factor upper bounded by the \emph{straight line complexity} (SLC) of the linear program. This term refers to the minimum number of segments of any piecewise linear curve that traverses the \emph{wide neighborhood} of the central path, a lower bound on the iteration complexity of any IPM that follows a piecewise linear trajectory along a path induced by a self-concordant barrier. In particular, our algorithm matches the number of iterations of any such IPM up to the same factor $O(n^{1.5}\log n)$. As our second contribution, we show that the SLC of any linear program is upper bounded by $2^{n + o(1)}$, which implies that our IPM's iteration complexity is at most exponential. This in contrast to existing iteration complexity bounds that depend on either bit-complexity or condition measures; these can be unbounded in the problem dimension. We achieve our upper bound by showing that the central path is well-approximated by a combinatorial proxy we call the \emph{max central path}, which consists of $2n$ shadow vertex simplex paths. Our upper bound complements the lower bounds of Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018), and Allamigeon, Gaubert, and Vandame (STOC 2022), who constructed linear programs with exponential SLC. Finally, we show that each iteration of our IPM can be implemented in strongly polynomial time. Along the way, we develop a deterministic algorithm that approximates the singular value decomposition of a matrix in strongly polynomial time to high accuracy, which may be of independent interest.
Normalizing flow is a class of deep generative models for efficient sampling and likelihood estimation, which achieves attractive performance, particularly in high dimensions. The flow is often implemented using a sequence of invertible residual blocks. Existing works adopt special network architectures and regularization of flow trajectories. In this paper, we develop a neural ODE flow network called JKO-iFlow, inspired by the Jordan-Kinderleherer-Otto (JKO) scheme, which unfolds the discrete-time dynamic of the Wasserstein gradient flow. The proposed method stacks residual blocks one after another, allowing efficient block-wise training of the residual blocks, avoiding sampling SDE trajectories and score matching or variational learning, thus reducing the memory load and difficulty in end-to-end training. We also develop adaptive time reparameterization of the flow network with a progressive refinement of the induced trajectory in probability space to improve the model accuracy further. Experiments with synthetic and real data show that the proposed JKO-iFlow network achieves competitive performance compared with existing flow and diffusion models at a significantly reduced computational and memory cost.
Explaining generative diffusion models via visual analysis for interpretable decision-making process
Diffusion models have demonstrated remarkable performance in generation tasks. Nevertheless, explaining the diffusion process remains challenging due to it being a sequence of denoising noisy images that are difficult for experts to interpret. To address this issue, we propose the three research questions to interpret the diffusion process from the perspective of the visual concepts generated by the model and the region where the model attends in each time step. We devise tools for visualizing the diffusion process and answering the aforementioned research questions to render the diffusion process human-understandable. We show how the output is progressively generated in the diffusion process by explaining the level of denoising and highlighting relationships to foundational visual concepts at each time step through the results of experiments with various visual analyses using the tools. Throughout the training of the diffusion model, the model learns diverse visual concepts corresponding to each time-step, enabling the model to predict varying levels of visual concepts at different stages. We substantiate our tools using Area Under Cover (AUC) score, correlation quantification, and cross-attention mapping. Our findings provide insights into the diffusion process and pave the way for further research into explainable diffusion mechanisms.
We introduce a text-to-speech (TTS) model called BASE TTS, which stands for $\textbf{B}$ig $\textbf{A}$daptive $\textbf{S}$treamable TTS with $\textbf{E}$mergent abilities. BASE TTS is the largest TTS model to-date, trained on 100K hours of public domain speech data, achieving a new state-of-the-art in speech naturalness. It deploys a 1-billion-parameter autoregressive Transformer that converts raw texts into discrete codes ("speechcodes") followed by a convolution-based decoder which converts these speechcodes into waveforms in an incremental, streamable manner. Further, our speechcodes are built using a novel speech tokenization technique that features speaker ID disentanglement and compression with byte-pair encoding. Echoing the widely-reported "emergent abilities" of large language models when trained on increasing volume of data, we show that BASE TTS variants built with 10K+ hours and 500M+ parameters begin to demonstrate natural prosody on textually complex sentences. We design and share a specialized dataset to measure these emergent abilities for text-to-speech. We showcase state-of-the-art naturalness of BASE TTS by evaluating against baselines that include publicly available large-scale text-to-speech systems: YourTTS, Bark and TortoiseTTS. Audio samples generated by the model can be heard at //amazon-ltts-paper.com/.
Under a generalised estimating equation analysis approach, approximate design theory is used to determine Bayesian D-optimal designs. For two examples, considering simple exchangeable and exponential decay correlation structures, we compare the efficiency of identified optimal designs to balanced stepped-wedge designs and corresponding stepped-wedge designs determined by optimising using a normal approximation approach. The dependence of the Bayesian D-optimal designs on the assumed correlation structure is explored; for the considered settings, smaller decay in the correlation between outcomes across time periods, along with larger values of the intra-cluster correlation, leads to designs closer to a balanced design being optimal. Unlike for normal data, it is shown that the optimal design need not be centro-symmetric in the binary outcome case. The efficiency of the Bayesian D-optimal design relative to a balanced design can be large, but situations are demonstrated in which the advantages are small. Similarly, the optimal design from a normal approximation approach is often not much less efficient than the Bayesian D-optimal design. Bayesian D-optimal designs can be readily identified for stepped-wedge cluster randomised trials with binary outcome data. In certain circumstances, principally ones with strong time period effects, they will indicate that a design unlikely to have been identified by previous methods may be substantially more efficient. However, they require a larger number of assumptions than existing optimal designs, and in many situations existing theory under a normal approximation will provide an easier means of identifying an efficient design for binary outcome data.
We analyze a Discontinuous Galerkin method for a problem with linear advection-reaction and $p$-type diffusion, with Sobolev indices $p\in (1, \infty)$. The discretization of the diffusion term is based on the full gradient including jump liftings and interior-penalty stabilization while, for the advective contribution, we consider a strengthened version of the classical upwind scheme. The developed error estimates track the dependence of the local contributions to the error on local P\'eclet numbers. A set of numerical tests supports the theoretical derivations.
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