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We provide a framework for accelerating reinforcement learning (RL) algorithms by heuristics constructed from domain knowledge or offline data. Tabula rasa RL algorithms require environment interactions or computation that scales with the horizon of the sequential decision-making task. Using our framework, we show how heuristic-guided RL induces a much shorter-horizon subproblem that provably solves the original task. Our framework can be viewed as a horizon-based regularization for controlling bias and variance in RL under a finite interaction budget. On the theoretical side, we characterize properties of a good heuristic and its impact on RL acceleration. In particular, we introduce the novel concept of an improvable heuristic, a heuristic that allows an RL agent to extrapolate beyond its prior knowledge. On the empirical side, we instantiate our framework to accelerate several state-of-the-art algorithms in simulated robotic control tasks and procedurally generated games. Our framework complements the rich literature on warm-starting RL with expert demonstrations or exploratory datasets, and introduces a principled method for injecting prior knowledge into RL.

We analyze the complexity of learning directed acyclic graphical models from observational data in general settings without specific distributional assumptions. Our approach is information-theoretic and uses a local Markov boundary search procedure in order to recursively construct ancestral sets in the underlying graphical model. Perhaps surprisingly, we show that for certain graph ensembles, a simple forward greedy search algorithm (i.e. without a backward pruning phase) suffices to learn the Markov boundary of each node. This substantially improves the sample complexity, which we show is at most polynomial in the number of nodes. This is then applied to learn the entire graph under a novel identifiability condition that generalizes existing conditions from the literature. As a matter of independent interest, we establish finite-sample guarantees for the problem of recovering Markov boundaries from data. Moreover, we apply our results to the special case of polytrees, for which the assumptions simplify, and provide explicit conditions under which polytrees are identifiable and learnable in polynomial time. We further illustrate the performance of the algorithm, which is easy to implement, in a simulation study. Our approach is general, works for discrete or continuous distributions without distributional assumptions, and as such sheds light on the minimal assumptions required to efficiently learn the structure of directed graphical models from data.

In Bayesian analysis, the selection of a prior distribution is typically done by considering each parameter in the model. While this can be convenient, in many scenarios it may be desirable to place a prior on a summary measure of the model instead. In this work, we propose a prior on the model fit, as measured by a Bayesian coefficient of determination (R2), which then induces a prior on the individual parameters. We achieve this by placing a beta prior on R2 and then deriving the induced prior on the global variance parameter for generalized linear mixed models. We derive closed-form expressions in many scenarios and present several approximation strategies when an analytic form is not possible and/or to allow for easier computation. In these situations, we suggest to approximate the prior by using a generalized beta prime distribution that matches it closely. This approach is quite flexible and can be easily implemented in standard Bayesian software. Lastly, we demonstrate the performance of the method on simulated data where it particularly shines in high-dimensional examples as well as real-world data which shows its ability to model spatial correlation in the random effects.

Real-world robotic tasks require complex reward functions. When we define the problem the robot needs to solve, we pretend that a designer specifies this complex reward exactly, and it is set in stone from then on. In practice, however, reward design is an iterative process: the designer chooses a reward, eventually encounters an "edge-case" environment where the reward incentivizes the wrong behavior, revises the reward, and repeats. What would it mean to rethink robotics problems to formally account for this iterative nature of reward design? We propose that the robot not take the specified reward for granted, but rather have uncertainty about it, and account for the future design iterations as future evidence. We contribute an Assisted Reward Design method that speeds up the design process by anticipating and influencing this future evidence: rather than letting the designer eventually encounter failure cases and revise the reward then, the method actively exposes the designer to such environments during the development phase. We test this method in a simplified autonomous driving task and find that it more quickly improves the car's behavior in held-out environments by proposing environments that are "edge cases" for the current reward.

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules $X\xrightarrow{f} Y \xrightarrow{g} Z$ such that $M\cong \ker{g}/\mathrm{im}{f}$. It runs in time $O(|X|^3+|Y|^3+|Z|^3)$ and requires $O(|X|^2+|Y|^2+|Z|^2)$ memory, where $|\cdot |$ denotes the rank. Given the presentation computed by our algorithm, the bigraded Betti numbers of $M$ are readily computed. Our approach is based on a simple matrix reduction algorithm, slight variants of which compute kernels of morphisms between free modules, minimal generating sets, and Gr\"obner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

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.

Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.

Why deep neural networks (DNNs) capable of overfitting often generalize well in practice is a mystery in deep learning. Existing works indicate that this observation holds for both complicated real datasets and simple datasets of one-dimensional (1-d) functions. In this work, for natural images and low-frequency dominant 1-d functions, we empirically found that a DNN with common settings first quickly captures the dominant low-frequency components, and then relatively slowly captures high-frequency ones. We call this phenomenon Frequency Principle (F-Principle). F-Principle can be observed over various DNN setups of different activation functions, layer structures and training algorithms in our experiments. F-Principle can be used to understand (i) the behavior of DNN training in the information plane and (ii) why DNNs often generalize well albeit its ability of overfitting. This F-Principle potentially can provide insights into understanding the general principle underlying DNN optimization and generalization for real datasets.

Machine Learning models become increasingly proficient in complex tasks. However, even for experts in the field, it can be difficult to understand what the model learned. This hampers trust and acceptance, and it obstructs the possibility to correct the model. There is therefore a need for transparency of machine learning models. The development of transparent classification models has received much attention, but there are few developments for achieving transparent Reinforcement Learning (RL) models. In this study we propose a method that enables a RL agent to explain its behavior in terms of the expected consequences of state transitions and outcomes. First, we define a translation of states and actions to a description that is easier to understand for human users. Second, we developed a procedure that enables the agent to obtain the consequences of a single action, as well as its entire policy. The method calculates contrasts between the consequences of a policy derived from a user query, and of the learned policy of the agent. Third, a format for generating explanations was constructed. A pilot survey study was conducted to explore preferences of users for different explanation properties. Results indicate that human users tend to favor explanations about policy rather than about single actions.

This manuscript surveys reinforcement learning from the perspective of optimization and control with a focus on continuous control applications. It surveys the general formulation, terminology, and typical experimental implementations of reinforcement learning and reviews competing solution paradigms. In order to compare the relative merits of various techniques, this survey presents a case study of the Linear Quadratic Regulator (LQR) with unknown dynamics, perhaps the simplest and best studied problem in optimal control. The manuscript describes how merging techniques from learning theory and control can provide non-asymptotic characterizations of LQR performance and shows that these characterizations tend to match experimental behavior. In turn, when revisiting more complex applications, many of the observed phenomena in LQR persist. In particular, theory and experiment demonstrate the role and importance of models and the cost of generality in reinforcement learning algorithms. This survey concludes with a discussion of some of the challenges in designing learning systems that safely and reliably interact with complex and uncertain environments and how tools from reinforcement learning and controls might be combined to approach these challenges.