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Many physical phenomena are described by Hamiltonian mechanics using an energy function (Hamiltonian). Recently, the Hamiltonian neural network, which approximates the Hamiltonian by a neural network, and its extensions have attracted much attention. This is a very powerful method, but theoretical studies are limited. In this study, by combining the statistical learning theory and KAM theory, we provide a theoretical analysis of the behavior of Hamiltonian neural networks when the learning error is not completely zero. A Hamiltonian neural network with non-zero errors can be considered as a perturbation from the true dynamics, and the perturbation theory of the Hamilton equation is widely known as KAM theory. To apply KAM theory, we provide a generalization error bound for Hamiltonian neural networks by deriving an estimate of the covering number of the gradient of the multi-layer perceptron, which is the key ingredient of the model. This error bound gives a sup-norm bound on the Hamiltonian that is required in the application of KAM theory.

AAAI 2022

KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-Zero Training Loss

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CAP Virtual Congress 2021

technical paper

The purpose of the AAAI conference is to promote research in artificial intelligence (AI) and scientific exchange among AI researchers, practitioners, scientists, and engineers in affiliated disciplines. 

Controllers for autonomous systems that operate in safety-critical settings must account for stochastic disturbances. Such disturbances are often modeled as process noise, and common assumptions are that the underlying distributions are known and/or Gaussian. In practice, however, these assumptions may be unrealistic and can lead to poor approximations of the true noise distribution. We present a novel planning method that does not rely on any explicit representation of the noise distributions. In particular, we address the problem of computing a controller that provides probabilistic guarantees on safely reaching a target. First, we abstract the continuous system into a discrete-state model that captures noise by probabilistic transitions between states. As a key contribution, we adapt tools from the scenario approach to compute probably approximately correct (PAC) bounds on these transition probabilities, based on a finite number of samples of the noise. We capture these bounds in the transition probability intervals of a so-called interval Markov decision process (iMDP). This iMDP is robust against uncertainty in the transition probabilities, and the tightness of the probability intervals can be controlled through the number of samples. We use state-of-the-art verification techniques to provide guarantees on the iMDP, and compute a controller for which these guarantees carry over to the autonomous system. Realistic benchmarks show the practical applicability of our method, even when the iMDP has millions of states or transitions.

Sampling-Based Robust Control of Autonomous Systems with Non-Gaussian Noise

The modeling and control of complex physical systems are essential in real-world problems. We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers. The procedure of the proposed framework is divided into two phases: solution operator learning for PDE constraints (Phase 1) and searching for optimal control (Phase 2). Once the surrogate model is trained in Phase 1, the optimal control can be inferred in Phase 2 without intensive computations. Our framework can be applied to both data-driven and data-free cases. We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation.

Solving PDE-Constrained Control Problems using Operator Learning

Many efforts have been made for revealing the decision-making process of black-box learning machines such as deep neural networks, resulting in useful local and global explanation methods. For local explanation, stochasticity is known to help: a simple method, called SmoothGrad, has improved the visual quality of gradient-based attribution by adding noise to the input space and averaging the explanations of the noisy inputs. In this paper, we extend this idea and propose NoiseGrad that enhances both local and global explanation methods. Specifically, NoiseGrad introduces stochasticity in the weight parameter space, such that the decision boundary is perturbed. NoiseGrad is expected to enhance the local explanation, similarly to SmoothGrad, due to the dual relationship between the input perturbation and the decision boundary perturbation. We evaluate NoiseGrad and its fusion with SmoothGrad - FusionGrad - qualitatively and quantitatively with several evaluation criteria, and show that our novel approach significantly outperforms the baseline methods. Both NoiseGrad and FusionGrad are method-agnostic and as handy as SmoothGrad using a simple heuristic for the choice of the hyperparameter setting without the need of fine-tuning.

NoiseGrad — Enhancing Explanationsby Introducing Stochasticity to Model Weights

We present a novel approach to hierarchical reinforcement learning for linearly-solvable Markov decision processes. Our approach assumes that the state space is partitioned, and defines subtasks for moving between the partitions. We represent value functions on several levels of abstraction, and use the compositionality of subtasks to estimate the optimal values of the states in each partition. The policy is implicitly defined on these optimal value estimates, rather than being decomposed among the subtasks. As a consequence, our approach can learn the globally optimal policy, and does not suffer from non-stationarities induced by high-level decisions. If several partitions have equivalent dynamics, the subtasks of those partitions can be shared. We show that our approach is significantly more sample efficient than that of a flat learner and similar hierarchical approaches when the set of boundary states is smaller than the entire state space.

Globally Optimal Hierarchical Reinforcement Learning for Linearly-Solvable Markov Decision Processes

Reward-Weighted Regression (RWR) belongs to a family of widely known iterative Reinforcement Learning algorithms based on the Expectation-Maximization framework. In this family, learning at each iteration consists of sampling a batch of trajectories using the current policy and fitting a new policy to maximize a return-weighted log-likelihood of actions. Although RWR is known to yield monotonic improvement of the policy under certain circumstances, whether and under which conditions RWR converges to the optimal policy have remained open questions. In this paper, we provide for the first time a proof that RWR converges to a global optimum when no function approximation is used, in a general compact setting. Furthermore, for the simpler case with finite state and action spaces we prove R-linear convergence of the state-value function to the optimum.

Reward-Weighted Regression Converges to a Global Optimum

As neural network classifiers are deployed in real-world applications, it is crucial that their failures can be detected reliably. One practical solution is to assign confidence scores to each prediction, then use these scores to filter out possible misclassifications. However, existing confidence metrics are not yet sufficiently reliable for this role. This paper presents a new framework that produces a quantitative metric for detecting misclassification errors. This framework, RED, builds an error detector on top of the base classifier and estimates uncertainty of the detection scores using Gaussian Processes. Experimental comparisons with other error detection methods on 125 UCI datasets demonstrate that this approach is effective. Further implementations on two probabilistic base classifiers and two large deep learning architecture in vision tasks further confirm that the method is robust and scalable. Third, an empirical analysis of RED with out-of-distribution and adversarial samples shows that the method can be used not only to detect errors but also to understand where they come from. RED can thereby be used to improve trustworthiness of neural network classifiers more broadly in the future.

Detecting Misclassification Errors in Neural Networks with a Gaussian Process Model

Two-sided matching is an important research area that has had a major impact on the design of real-world matching markets. One consistent feature in many of the real-world applications is that they impose new feasibility constraints that lead to research challenges. We survey developments in the field of two-sided matching with various constraints, including those based on regions, diversity, multi-dimensional capacities, and matroids.

Matching Market Design with Constraints

Recent  breakthroughs in AI have shown the remarkable power of deep learning and deep reinforcement learning. These developments, however, have been tied to specific tasks, and progress  in out-of-distribution generalization has been limited.  While it is assumed that these limitations can be overcome by incorporating suitable inductive biases, the notion of inductive biases itself is often left vague and does not provide meaningful guidance. In the paper, I articulate a different learning approach where representations do not emerge from  biases in a neural architecture  but are learned over a given  target language with  a known  semantics. The basic ideas are  implicit in mainstream AI where representations have been encoded in languages ranging from fragments of first-order logic to probabilistic structural causal models. The challenge is to learn from data the representations that have traditionally been crafted by hand. Generalization is then a  result of the semantics  of the language. The goals of this paper  are to make these ideas explicit, to place them in a broader context where the design of the target language is crucial, and to illustrate them in the context of learning to act and plan. For this, after a general discussion, I consider  learning representations of actions, general policies, and subgoals ("intrinsic rewards"). In  these cases, learning is formulated  as a combinatorial  problem  but nothing prevents the use of deep learning techniques instead. Indeed, learning representations over languages with a known semantics  provides an account of what is to be learned, while learning representations with neural nets   provides  a complementary account of how representations can be learned. The challenge and the opportunity is to bring the two  together.

Target Languages (vs. Inductive Biases) for Learning to Act and Plan

Despite the surprising power of many modern AI systems that often learn their own representations, there is significant discontent about their inscrutability and the attendant problems in their ability to interact with humans. While alternatives such as  neuro-symbolic approaches have been proposed, there is a lack of consensus on what they are about. There are often two independent motivations (i) symbols as a lingua franca for human-AI interaction and (ii) symbols as (system-produced) abstractions use in its internal reasoning. The jury is still out on whether AI systems will need to use symbols in their internal reasoning to achieve general intelligence capabilities. Whatever the answer there is, the need for (human-understandable) symbols in human-AI interaction seems quite compelling. Symbols, like emotions, may well not be sine qua non for intelligence per se, but they will be crucial for AI systems to interact with us humans--as we can neither turn off our emotions not get by without our symbols. In particular, in many human-designed domains, humans would be interested in providing explicit (symbolic) knowledge and advice--and expect machine explanations in kind. This alone requires AI systems to at least do their I/O in symbolic terms. In this blue sky paper, we argue this point of view, and discuss research directions that need to be pursued to allow for this type of human-AI interaction.

Symbols as a Lingua Franca for Bridging Human-AI Chasm for Explainable and Advisable AI Systems

The rapid development of drone technology is leading to more and more use cases being proposed. In response, regulators are drawing up drone traffic management frameworks. However, to design solutions that are efficient, fair, simple, non-manipulable, and scalable, we need market design and AI expertise. To this end, we introduce the drone traffic management problem as a new research challenge to the market design and AI communities. We present five design desiderata that we have derived from our interviews with stakeholders from the regulatory side as well as from public and private enterprises. Finally, we provide an overview of the solution space to point out possible directions for future research.

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