Average-case Complexity of Teaching Convex Polytopes via Halfspace Queries

We examine the task of locating a target region among those induced by intersections of $n$ halfspaces in $\mathbb{R}^d$. This generic task connects to fundamental machine learning problems, such as training a perceptron and learning a $\phi$-separable dichotomy. We investigate the average teaching complexity of the task, i.e., the minimal number of samples (halfspace queries) required by a teacher to help a version-space learner in locating a randomly selected target. As our main result, we show that the average-case teaching complexity is $\Theta(d)$, which is in sharp contrast to the worst-case teaching complexity of $\Theta(n)$. If instead, we consider the average-case learning complexity, the bounds have a dependency on $n$ as $\Theta(n)$ for i.i.d. queries and $\Theta(d \log(n))$ for actively chosen queries by the learner. Our proof techniques are based on novel insights from computational geometry, which allow us to count the number of convex polytopes and faces in a Euclidean space depending on the arrangement of halfspaces. Our insights allow us to establish a tight bound on the average-case complexity for $\phi$-separable dichotomies, which generalizes the known $\mathcal{O}(d)$ bound on the average number of "extreme patterns" in the classical computational geometry literature (Cover, 1965).

Breaking the Sample Size Barrier in Model-Based Reinforcement Learning with a Generative Model

We investigate the sample efficiency of reinforcement learning in a $\gamma$-discounted infinite-horizon Markov decision process (MDP) with state space $\mathcal{S}$ and action space $\mathcal{A}$, assuming access to a generative model. Despite a number of prior work tackling this problem, a complete picture of the trade-offs between sample complexity and statistical accuracy is yet to be determined. In particular, prior results suffer from a sample size barrier, in the sense that their claimed statistical guarantees hold only when the sample size exceeds at least $\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma)^2}$ (up to some log factor). The current paper overcomes this barrier by certifying the minimax optimality of model-based reinforcement learning as soon as the sample size exceeds the order of $\frac{|\mathcal{S}||\mathcal{A}|}{1-\gamma}$ (modulo some log factor). More specifically, a perturbed model-based planning algorithm provably finds an $\varepsilon$-optimal policy with an order of $\frac{|\mathcal{S}||\mathcal{A}| }{(1-\gamma)^3\varepsilon^2}\log\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma)\varepsilon}$ samples for any $\varepsilon \in (0, \frac{1}{1-\gamma}]$. Along the way, we derive improved (instance-dependent) guarantees for model-based policy evaluation. To the best of our knowledge, this work provides the first minimax-optimal guarantee in a generative model that accommodates the entire range of sample sizes (beyond which finding a meaningful policy is information theoretically impossible).

Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality

We study the distribution and uncertainty of nonconvex optimization for noisy tensor completion -- the problem of estimating a low-rank tensor given incomplete and corrupted observations of its entries. Focusing on a two-stage estimation algorithm proposed by Cai et al. (2019), we characterize the distribution of this nonconvex estimator down to fine scales. This distributional theory in turn allows one to construct valid and short confidence intervals for both the unseen tensor entries and the unknown tensor factors. The proposed inferential procedure enjoys several important features: (1) it is fully adaptive to noise heteroscedasticity, and (2) it is data-driven and automatically adapts to unknown noise distributions. Furthermore, our findings unveil the statistical optimality of nonconvex tensor completion: it attains un-improvable $\ell_{2}$ accuracy -- including both the rates and the pre-constants -- when estimating both the unknown tensor and the underlying tensor factors.

Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction

Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP), based on a single trajectory of Markovian samples induced by a behavior policy. Focusing on a $\gamma$-discounted MDP with state space $\mathcal{S}$ and action space $\mathcal{A}$, we demonstrate that the $\ell_{\infty}$-based sample complexity of classical asynchronous Q-learning -- namely, the number of samples needed to yield an entrywise $\varepsilon$-accurate estimate of the Q-function -- is at most on the order of \begin{equation*} \frac{1}{\mu_{\mathsf{min}}(1-\gamma)^5\varepsilon^2}+ \frac{t_{\mathsf{mix}}}{\mu_{\mathsf{min}}(1-\gamma)} \end{equation*} up to some logarithmic factor, provided that a proper constant learning rate is adopted. Here, $t_{\mathsf{mix}}$ and $\mu_{\mathsf{min}}$ denote respectively the mixing time and the minimum state-action occupancy probability of the sample trajectory. The first term of this bound matches the complexity in the case with independent samples drawn from the stationary distribution of the trajectory. The second term reflects the expense taken for the empirical distribution of the Markovian trajectory to reach a steady state, which is incurred at the very beginning and becomes amortized as the algorithm runs. Encouragingly, the above bound improves upon the state-of-the-art result by a factor of at least $|\mathcal{S}||\mathcal{A}|$. Further, the scaling on the discount complexity can be improved by means of variance reduction.

Understanding the Power and Limitations of Teaching with Imperfect Knowledge

Machine teaching studies the interaction between a teacher and a student/learner where the teacher selects training examples for the learner to learn a specific task. The typical assumption is that the teacher has perfect knowledge of the task---this knowledge comprises knowing the desired learning target, having the exact task representation used by the learner, and knowing the parameters capturing the learning dynamics of the learner. Inspired by real-world applications of machine teaching in education, we consider the setting where teacher's knowledge is limited and noisy, and the key research question we study is the following: When does a teacher succeed or fail in effectively teaching a learner using its imperfect knowledge? We answer this question by showing connections to how imperfect knowledge affects the teacher's solution of the corresponding machine teaching problem when constructing optimal teaching sets. Our results have important implications for designing robust teaching algorithms for real-world applications.

An Online Learning Framework for Energy-Efficient Navigation of Electric Vehicles

Energy-efficient navigation constitutes an important challenge in electric vehicles, due to their limited battery capacity. We employ a Bayesian approach to model energy consumption at road-segments for efficient navigation. In order to learn the model parameters, we develop an online learning framework and investigate several exploration strategies such as Thompson Sampling and Upper Confidence Bound. We then extend our online learning framework to multi-agent setting, where multiple vehicles adaptively navigate and learn the parameters of the energy model. We analyze Thompson Sampling and establish rigorous regret bounds on its performance. Finally, we demonstrate the performance of our methods via several real-world experiments on Luxembourg SUMO Traffic dataset.

A Financial Service Chatbot based on Deep Bidirectional Transformers

We develop a chatbot using Deep Bidirectional Transformer models (BERT) to handle client questions in financial investment customer service. The bot can recognize 381 intents, and decides when to say "I don't know" and escalates irrelevant/uncertain questions to human operators. Our main novel contribution is the discussion about uncertainty measure for BERT, where three different approaches are systematically compared on real problems. We investigated two uncertainty metrics, information entropy and variance of dropout sampling in BERT, followed by mixed-integer programming to optimize decision thresholds. Another novel contribution is the usage of BERT as a language model in automatic spelling correction. Inputs with accidental spelling errors can significantly decrease intent classification performance. The proposed approach combines probabilities from masked language model and word edit distances to find the best corrections for misspelled words. The chatbot and the entire conversational AI system are developed using open-source tools, and deployed within our company's intranet. The proposed approach can be useful for industries seeking similar in-house solutions in their specific business domains. We share all our code and a sample chatbot built on a public dataset on Github.

Adaptive Teaching of Temporal Logic Formulas to Learners with Preferences

Machine teaching is an algorithmic framework for teaching a target hypothesis via a sequence of examples or demonstrations. We investigate machine teaching for temporal logic formulas -- a novel and expressive hypothesis class amenable to time-related task specifications. In the context of teaching temporal logic formulas, an exhaustive search even for a myopic solution takes exponential time (with respect to the time span of the task). We propose an efficient approach for teaching parametric linear temporal logic formulas. Concretely, we derive a necessary condition for the minimal time length of a demonstration to eliminate a set of hypotheses. Utilizing this condition, we propose a myopic teaching algorithm by solving a sequence of integer programming problems. We further show that, under two notions of teaching complexity, the proposed algorithm has near-optimal performance. The results strictly generalize the previous results on teaching preference-based version space learners. We evaluate our algorithm extensively under a variety of learner types (i.e., learners with different preference models) and interactive protocols (e.g., batched and adaptive). The results show that the proposed algorithms can efficiently teach a given target temporal logic formula under various settings, and that there are significant gains of teaching efficacy when the teacher adapts to the learner's current hypotheses or uses oracles.

Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Data

This paper delivers improved theoretical guarantees for the convex programming approach in low-rank matrix estimation, in the presence of (1) random noise, (2) gross sparse outliers, and (3) missing data. This problem, often dubbed as robust principal component analysis (robust PCA), finds applications in various domains. Despite the wide applicability of convex relaxation, the available statistical support (particularly the stability analysis vis-a-vis random noise) remains highly suboptimal, which we strengthen in this paper. When the unknown matrix is well-conditioned, incoherent, and of constant rank, we demonstrate that a principled convex program achieves near-optimal statistical accuracy, in terms of both the Euclidean loss and the $\ell_{\infty}$ loss. All of this happens even when nearly a constant fraction of observations are corrupted by outliers with arbitrary magnitudes. The key analysis idea lies in bridging the convex program in use and an auxiliary nonconvex optimization algorithm, and hence the title of this paper.