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A support vector machine (SVM) is an algorithm that finds a hyperplane which optimally separates labeled data points in $\mathbb{R}^n$ into positive and negative classes. The data points on the margin of this separating hyperplane are called support vectors. We connect the possible configurations of support vectors to Radon's theorem, which provides guarantees for when a set of points can be divided into two classes (positive and negative) whose convex hulls intersect. If the convex hulls of the positive and negative support vectors are projected onto a separating hyperplane, then the projections intersect if and only if the hyperplane is optimal. Further, with a particular type of general position, we show that (a) the projected convex hulls of the support vectors intersect in exactly one point, (b) the support vectors are stable under perturbation, (c) there are at most $n+1$ support vectors, and (d) every number of support vectors from 2 up to $n+1$ is possible. Finally, we perform computer simulations studying the expected number of support vectors, and their configurations, for randomly generated data. We observe that as the distance between classes of points increases for this type of randomly generated data, configurations with fewer support vectors become more likely.

Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. Here we consider a detailed and comprehensive analysis on an estimation of the error term obtained by truncating the asymptotic expansion for $p(n)$ at any positive integer $n$. This gives rise to an infinite family of inequalities for $p(n)$ which finally answers to a question proposed by Chen. Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation.

This paper is concerned with the sample efficiency of reinforcement learning, assuming access to a generative model (or simulator). We first consider $\gamma$-discounted infinite-horizon Markov decision processes (MDPs) with state space $\mathcal{S}$ and action space $\mathcal{A}$. Despite a number of prior works tackling this problem, a complete picture of the trade-offs between sample complexity and statistical accuracy is yet to be determined. In particular, all prior results suffer from a severe sample size barrier, in the sense that their claimed statistical guarantees hold only when the sample size exceeds at least $\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma)^2}$. The current paper overcomes this barrier by certifying the minimax optimality of two algorithms -- a perturbed model-based algorithm and a conservative model-based algorithm -- as soon as the sample size exceeds the order of $\frac{|\mathcal{S}||\mathcal{A}|}{1-\gamma}$ (modulo some log factor). Moving beyond infinite-horizon MDPs, we further study time-inhomogeneous finite-horizon MDPs, and prove that a plain model-based planning algorithm suffices to achieve minimax-optimal sample complexity given any target accuracy level. To the best of our knowledge, this work delivers the first minimax-optimal guarantees that accommodate the entire range of sample sizes (beyond which finding a meaningful policy is information theoretically infeasible).

In natural language, referencing objects at different levels of specificity is a fundamental pragmatic mechanism for efficient communication in context. We develop a novel communication game, the hierarchical reference game, to study the emergence of such reference systems in artificial agents. We consider a simplified world, in which concepts are abstractions over a set of primitive attributes (e.g., color, style, shape). Depending on how many attributes are combined, concepts are more general ("circle") or more specific ("red dotted circle"). Based on the context, the agents have to communicate at different levels of this hierarchy. Our results show that the agents learn to play the game successfully and can even generalize to novel concepts. To achieve abstraction, they use implicit (omitting irrelevant information) and explicit (indicating that attributes are irrelevant) strategies. In addition, the compositional structure underlying the concept hierarchy is reflected in the emergent protocols, indicating that the need to develop hierarchical reference systems supports the emergence of compositionality.

Markov decision processes (MDPs) are formal models commonly used in sequential decision-making. MDPs capture the stochasticity that may arise, for instance, from imprecise actuators via probabilities in the transition function. However, in data-driven applications, deriving precise probabilities from (limited) data introduces statistical errors that may lead to unexpected or undesirable outcomes. Uncertain MDPs (uMDPs) do not require precise probabilities but instead use so-called uncertainty sets in the transitions, accounting for such limited data. Tools from the formal verification community efficiently compute robust policies that provably adhere to formal specifications, like safety constraints, under the worst-case instance in the uncertainty set. We continuously learn the transition probabilities of an MDP in a robust anytime-learning approach that combines a dedicated Bayesian inference scheme with the computation of robust policies. In particular, our method (1) approximates probabilities as intervals, (2) adapts to new data that may be inconsistent with an intermediate model, and (3) may be stopped at any time to compute a robust policy on the uMDP that faithfully captures the data so far. We show the effectiveness of our approach and compare it to robust policies computed on uMDPs learned by the UCRL2 reinforcement learning algorithm in an experimental evaluation on several benchmarks.

How can we train an assistive human-machine interface (e.g., an electromyography-based limb prosthesis) to translate a user's raw command signals into the actions of a robot or computer when there is no prior mapping, we cannot ask the user for supervision in the form of action labels or reward feedback, and we do not have prior knowledge of the tasks the user is trying to accomplish? The key idea in this paper is that, regardless of the task, when an interface is more intuitive, the user's commands are less noisy. We formalize this idea as a completely unsupervised objective for optimizing interfaces: the mutual information between the user's command signals and the induced state transitions in the environment. To evaluate whether this mutual information score can distinguish between effective and ineffective interfaces, we conduct an observational study on 540K examples of users operating various keyboard and eye gaze interfaces for typing, controlling simulated robots, and playing video games. The results show that our mutual information scores are predictive of the ground-truth task completion metrics in a variety of domains, with an average Spearman's rank correlation of 0.43. In addition to offline evaluation of existing interfaces, we use our unsupervised objective to learn an interface from scratch: we randomly initialize the interface, have the user attempt to perform their desired tasks using the interface, measure the mutual information score, and update the interface to maximize mutual information through reinforcement learning. We evaluate our method through a user study with 12 participants who perform a 2D cursor control task using a perturbed mouse, and an experiment with one user playing the Lunar Lander game using hand gestures. The results show that we can learn an interface from scratch, without any user supervision or prior knowledge of tasks, in under 30 minutes.

Human-in-the-loop aims to train an accurate prediction model with minimum cost by integrating human knowledge and experience. Humans can provide training data for machine learning applications and directly accomplish some tasks that are hard for computers in the pipeline with the help of machine-based approaches. In this paper, we survey existing works on human-in-the-loop from a data perspective and classify them into three categories with a progressive relationship: (1) the work of improving model performance from data processing, (2) the work of improving model performance through interventional model training, and (3) the design of the system independent human-in-the-loop. Using the above categorization, we summarize major approaches in the field, along with their technical strengths/ weaknesses, we have simple classification and discussion in natural language processing, computer vision, and others. Besides, we provide some open challenges and opportunities. This survey intends to provide a high-level summarization for human-in-the-loop and motivates interested readers to consider approaches for designing effective human-in-the-loop solutions.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.

Recent years have witnessed significant advances in technologies and services in modern network applications, including smart grid management, wireless communication, cybersecurity as well as multi-agent autonomous systems. Considering the heterogeneous nature of networked entities, emerging network applications call for game-theoretic models and learning-based approaches in order to create distributed network intelligence that responds to uncertainties and disruptions in a dynamic or an adversarial environment. This paper articulates the confluence of networks, games and learning, which establishes a theoretical underpinning for understanding multi-agent decision-making over networks. We provide an selective overview of game-theoretic learning algorithms within the framework of stochastic approximation theory, and associated applications in some representative contexts of modern network systems, such as the next generation wireless communication networks, the smart grid and distributed machine learning. In addition to existing research works on game-theoretic learning over networks, we highlight several new angles and research endeavors on learning in games that are related to recent developments in artificial intelligence. Some of the new angles extrapolate from our own research interests. The overall objective of the paper is to provide the reader a clear picture of the strengths and challenges of adopting game-theoretic learning methods within the context of network systems, and further to identify fruitful future research directions on both theoretical and applied studies.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.