We address the problem of estimating the mixing time $t_{\mathsf{mix}}$ of an arbitrary ergodic finite Markov chain from a single trajectory of length $m$. The reversible case was addressed by Hsu et al. [2017], who left the general case as an open problem. In the reversible case, the analysis is greatly facilitated by the fact that the Markov operator is self-adjoint, and Weyl's inequality allows for a dimension-free perturbation analysis of the empirical eigenvalues. As Hsu et al. point out, in the absence of reversibility (and hence, the non-symmetry of the pair probabilities matrix), the existing perturbation analysis has a worst-case exponential dependence on the number of states $d$. Furthermore, even if an eigenvalue perturbation analysis with better dependence on $d$ were available, in the non-reversible case the connection between the spectral gap and the mixing time is not nearly as straightforward as in the reversible case. Our key insight is to estimate the pseudo-spectral gap instead, which allows us to overcome the loss of self-adjointness and to achieve a polynomial dependence on $d$ and the minimal stationary probability $\pi_\star$. Additionally, in the reversible case, we obtain simultaneous nearly (up to logarithmic factors) minimax rates in $t_{\mathsf{mix}}$ and precision $\varepsilon$, closing a gap in Hsu et al., who treated $\varepsilon$ as constant in the lower bounds. Finally, we construct fully empirical confidence intervals for the pseudo-spectral gap, which shrink to zero at a rate of roughly $1/\sqrt m$, and improve the state of the art in even the reversible case.

We consider the problem of designing an artificial agent capable of interacting with humans in collaborative dialogue to produce creative, engaging narratives. In this task, the goal is to establish universe details, and to collaborate on an interesting story in that universe, through a series of natural dialogue exchanges. Our model can augment any probabilistic conversational agent by allowing it to reason about universe information established and what potential next utterances might reveal. Ideally, with each utterance, agents would reveal just enough information to add specificity and reduce ambiguity without limiting the conversation. We empirically show that our model allows control over the rate at which the agent reveals information and that doing so significantly improves accuracy in predicting the next line of dialogues from movies. We close with a case-study with four professional theatre performers, who preferred interactions with our model-augmented agent over an unaugmented agent.

We exhibit an efficient procedure for testing, based on a single long state sequence, whether an unknown Markov chain is identical to or $\varepsilon$-far from a given reference chain. We obtain nearly matching (up to logarithmic factors) upper and lower sample complexity bounds for our notion of distance, which is based on total variation. Perhaps surprisingly, we discover that the sample complexity depends solely on the properties of the known reference chain and does not involve the unknown chain at all, which is not even assumed to be ergodic.

We report the first, to the best of our knowledge, hand-in-hand collaboration between human rights activists and machine learners, leveraging crowd-sourcing to study online abuse against women on Twitter. On a technical front, we carefully curate an unbiased yet low-variance dataset of labeled tweets, analyze it to account for the variability of abuse perception, and establish baselines, preparing it for release to community research efforts. On a social impact front, this study provides the technical backbone for a media campaign aimed at raising public and deciders' awareness and elevating the standards expected from social media companies.

Most model-free reinforcement learning methods leverage state representations (embeddings) for generalization, but either ignore structure in the space of actions or assume the structure is provided a priori. We show how a policy can be decomposed into a component that acts in a low-dimensional space of action representations and a component that transforms these representations into actual actions. These representations improve generalization over large, finite action sets by allowing the agent to infer the outcomes of actions similar to actions already taken. We provide an algorithm to both learn and use action representations and provide conditions for its convergence. The efficacy of the proposed method is demonstrated on large-scale real-world problems.

A key learning scenario in large-scale applications is that of federated learning, where a centralized model is trained based on data originating from a large number of clients. We argue that, with the existing training and inference, federated models can be biased towards different clients. Instead, we propose a new framework of agnostic federated learning, where the centralized model is optimized for any target distribution formed by a mixture of the client distributions. We further show that this framework naturally yields a notion of fairness. We present data-dependent Rademacher complexity guarantees for learning with this objective, which guide the definition of an algorithm for agnostic federated learning. We also give a fast stochastic optimization algorithm for solving the corresponding optimization problem, for which we prove convergence bounds, assuming a convex loss function and hypothesis set. We further empirically demonstrate the benefits of our approach in several datasets. Beyond federated learning, our framework and algorithm can be of interest to other learning scenarios such as cloud computing, domain adaptation, drifting, and other contexts where the training and test distributions do not coincide.

We study a loss function that is the negative log-likelihood of the model. We analyse the number of local minima at a fixed distance from the signal/spike with the Kac-Rice formula, and locate trivialization of the landscape at large signal-to-noise ratios. We evaluate in a closed form the performance of a gradient flow algorithm using integro-differential PDEs as developed in physics of disordered systems for the Langevin dynamics. We analyze the performance of an approximate message passing algorithm estimating the maximum likelihood configuration via its state evolution. We conclude by comparing the above results: while we observe a drastic slow down of the gradient flow dynamics even in the region where the landscape is trivial, both the analyzed algorithms are shown to perform well even in the part of the region of parameters where spurious local minima are present.

Motivation: Ab initio protein docking represents a major challenge for optimizing a noisy and costly "black box"-like function in a high-dimensional space. Despite progress in this field, there is no docking method available for rigorous uncertainty quantification (UQ) of its solution quality (e.g. interface RMSD or iRMSD). Results: We introduce a novel algorithm, Bayesian Active Learning (BAL), for optimization and UQ of such black-box functions and flexible protein docking. BAL directly models the posterior distribution of the global optimum (or native structures for protein docking) with active sampling and posterior estimation iteratively feeding each other. Furthermore, we use complex normal modes to represent a homogeneous Euclidean conformation space suitable for high-dimension optimization and construct funnel-like energy models for encounter complexes. Over a protein docking benchmark set and a CAPRI set including homology docking, we establish that BAL significantly improve against both starting points by rigid docking and refinements by particle swarm optimization, providing for one third targets a top-3 near-native prediction. BAL also generates tight confidence intervals with half range around 25% of iRMSD and confidence level at 85%. Its estimated probability of a prediction being native or not achieves binary classification AUROC at 0.93 and AUPRC over 0.60 (compared to 0.14 by chance); and also found to help ranking predictions. To the best of our knowledge, this study represents the first uncertainty quantification solution for protein docking, with theoretical rigor and comprehensive assessment. Source codes are available at https://github.com/Shen-Lab/BAL.

Optimizing an expensive-to-query function is a common task in science and engineering, where it is beneficial to keep the number of queries to a minimum. A popular strategy is Bayesian optimization (BO), which leverages probabilistic models for this task. Most BO today uses Gaussian processes (GPs), or a few other surrogate models. However, there is a broad set of Bayesian modeling techniques that we may want to use to capture complex systems and reduce the number of queries. Probabilistic programs (PPs) are modern tools that allow for flexible model composition, incorporation of prior information, and automatic inference. In this paper, we develop ProBO, a framework for BO using only standard operations common to most PPs. This allows a user to drop in an arbitrary PP implementation and use it directly in BO. To do this, we describe black box versions of popular acquisition functions that can be used in our framework automatically, without model-specific derivation, and show how to optimize these functions. We also introduce a model, which we term the Bayesian Product of Experts, that integrates into ProBO and can be used to combine information from multiple models implemented with different PPs. We show empirical results using multiple PP implementations, and compare against standard BO methods.

Recently there has been an increasing interest in the multivariate Gaussian process (MGP) which extends the Gaussian process (GP) to deal with multiple outputs. One approach to construct the MGP and account for non-trivial commonalities amongst outputs employs a convolution process (CP). The CP is based on the idea of sharing latent functions across several convolutions. Despite the elegance of the CP construction, it provides new challenges that need yet to be tackled. First, even with a moderate number of outputs, model building is extremely prohibitive due to the huge increase in computational demands and number of parameters to be estimated. Second, the negative transfer of knowledge may occur when some outputs do not share commonalities. In this paper we address these issues. We propose a regularized pairwise modeling approach for the MGP established using CP. The key feature of our approach is to distribute the estimation of the full multivariate model into a group of bivariate GPs which are individually built. Interestingly pairwise modeling turns out to possess unique characteristics, which allows us to tackle the challenge of negative transfer through penalizing the latent function that facilitates information sharing in each bivariate model. Predictions are then made through combining predictions from the bivariate models within a Bayesian framework. The proposed method has excellent scalability when the number of outputs is large and minimizes the negative transfer of knowledge between uncorrelated outputs. Statistical guarantees for the proposed method are studied and its advantageous features are demonstrated through numerical studies.