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We reexamine the the classical multidimensional scaling (MDS). We study some special cases, in particular, the exact solution for the sub-space formed by the 3 dimensional principal coordinates is derived. Also we give the extreme case when the points are collinear. Some insight into the effect on the MDS solution of the excluded eigenvalues (could be both positive as well as negative) of the doubly centered matrix is provided. As an illustration, we work through an example to understand the distortion in the MDS construction with positive and negative eigenvalues.

This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an integral form solution to Zubov's equation. For numerical computation, we develop two data-driven methods. One is based on the integration of an augmented system of differential equations; and the other one is based on deep learning. The former is effective for systems with a relatively low state space dimension and the latter is developed for high dimensional problems. The deep learning method is applied to a New England 10-generator power system model. We prove that a neural network approximation exists for the Lyapunov function of power systems such that the approximation error is a cubic polynomial of the number of generators. The error convergence rate as a function of n, the number of neurons, is proved.

We consider large-scale Markov decision processes with an unknown cost function and address the problem of learning a policy from a finite set of expert demonstrations. We assume that the learner is not allowed to interact with the expert and has no access to reinforcement signal of any kind. Existing inverse reinforcement learning methods come with strong theoretical guarantees, but are computationally expensive, while state-of-the-art policy optimization algorithms achieve significant empirical success, but are hampered by limited theoretical understanding. To bridge the gap between theory and practice, we introduce a novel bilinear saddle-point framework using Lagrangian duality. The proposed primal-dual viewpoint allows us to develop a model-free provably efficient algorithm through the lens of stochastic convex optimization. The method enjoys the advantages of simplicity of implementation, low memory requirements, and computational and sample complexities independent of the number of states. We further present an equivalent no-regret online-learning interpretation.

Learning a graph topology to reveal the underlying relationship between data entities plays an important role in various machine learning and data analysis tasks. Under the assumption that structured data vary smoothly over a graph, the problem can be formulated as a regularised convex optimisation over a positive semidefinite cone and solved by iterative algorithms. Classic methods require an explicit convex function to reflect generic topological priors, e.g. the $\ell_1$ penalty for enforcing sparsity, which limits the flexibility and expressiveness in learning rich topological structures. We propose to learn a mapping from node data to the graph structure based on the idea of learning to optimise (L2O). Specifically, our model first unrolls an iterative primal-dual splitting algorithm into a neural network. The key structural proximal projection is replaced with a variational autoencoder that refines the estimated graph with enhanced topological properties. The model is trained in an end-to-end fashion with pairs of node data and graph samples. Experiments on both synthetic and real-world data demonstrate that our model is more efficient than classic iterative algorithms in learning a graph with specific topological properties.

We study constrained reinforcement learning (CRL) from a novel perspective by setting constraints directly on state density functions, rather than the value functions considered by previous works. State density has a clear physical and mathematical interpretation, and is able to express a wide variety of constraints such as resource limits and safety requirements. Density constraints can also avoid the time-consuming process of designing and tuning cost functions required by value function-based constraints to encode system specifications. We leverage the duality between density functions and Q functions to develop an effective algorithm to solve the density constrained RL problem optimally and the constrains are guaranteed to be satisfied. We prove that the proposed algorithm converges to a near-optimal solution with a bounded error even when the policy update is imperfect. We use a set of comprehensive experiments to demonstrate the advantages of our approach over state-of-the-art CRL methods, with a wide range of density constrained tasks as well as standard CRL benchmarks such as Safety-Gym.

In real world settings, numerous constraints are present which are hard to specify mathematically. However, for the real world deployment of reinforcement learning (RL), it is critical that RL agents are aware of these constraints, so that they can act safely. In this work, we consider the problem of learning constraints from demonstrations of a constraint-abiding agent's behavior. We experimentally validate our approach and show that our framework can successfully learn the most likely constraints that the agent respects. We further show that these learned constraints are \textit{transferable} to new agents that may have different morphologies and/or reward functions. Previous works in this regard have either mainly been restricted to tabular (discrete) settings, specific types of constraints or assume the environment's transition dynamics. In contrast, our framework is able to learn arbitrary \textit{Markovian} constraints in high-dimensions in a completely model-free setting. The code can be found it: \url{//github.com/shehryar-malik/icrl}.

Meta-reinforcement learning (meta-RL) aims to learn from multiple training tasks the ability to adapt efficiently to unseen test tasks. Despite the success, existing meta-RL algorithms are known to be sensitive to the task distribution shift. When the test task distribution is different from the training task distribution, the performance may degrade significantly. To address this issue, this paper proposes Model-based Adversarial Meta-Reinforcement Learning (AdMRL), where we aim to minimize the worst-case sub-optimality gap -- the difference between the optimal return and the return that the algorithm achieves after adaptation -- across all tasks in a family of tasks, with a model-based approach. We propose a minimax objective and optimize it by alternating between learning the dynamics model on a fixed task and finding the adversarial task for the current model -- the task for which the policy induced by the model is maximally suboptimal. Assuming the family of tasks is parameterized, we derive a formula for the gradient of the suboptimality with respect to the task parameters via the implicit function theorem, and show how the gradient estimator can be efficiently implemented by the conjugate gradient method and a novel use of the REINFORCE estimator. We evaluate our approach on several continuous control benchmarks and demonstrate its efficacy in the worst-case performance over all tasks, the generalization power to out-of-distribution tasks, and in training and test time sample efficiency, over existing state-of-the-art meta-RL algorithms.

Auto-regressive sequence-to-sequence models with attention mechanism have achieved state-of-the-art performance in many tasks such as machine translation and speech synthesis. These models can be difficult to train. The standard approach, teacher forcing, guides a model with reference output history during training. The problem is that the model is unlikely to recover from its mistakes during inference, where the reference output is replaced by generated output. Several approaches deal with this problem, largely by guiding the model with generated output history. To make training stable, these approaches often require a heuristic schedule or an auxiliary classifier. This paper introduces attention forcing, which guides the model with generated output history and reference attention. This approach can train the model to recover from its mistakes, in a stable fashion, without the need for a schedule or a classifier. In addition, it allows the model to generate output sequences aligned with the references, which can be important for cascaded systems like many speech synthesis systems. Experiments on speech synthesis show that attention forcing yields significant performance gain. Experiments on machine translation show that for tasks where various re-orderings of the output are valid, guiding the model with generated output history is challenging, while guiding the model with reference attention is beneficial.

Machine learning methods are powerful in distinguishing different phases of matter in an automated way and provide a new perspective on the study of physical phenomena. We train a Restricted Boltzmann Machine (RBM) on data constructed with spin configurations sampled from the Ising Hamiltonian at different values of temperature and external magnetic field using Monte Carlo methods. From the trained machine we obtain the flow of iterative reconstruction of spin state configurations to faithfully reproduce the observables of the physical system. We find that the flow of the trained RBM approaches the spin configurations of the maximal possible specific heat which resemble the near criticality region of the Ising model. In the special case of the vanishing magnetic field the trained RBM converges to the critical point of the Renormalization Group (RG) flow of the lattice model. Our results suggest an alternative explanation of how the machine identifies the physical phase transitions, by recognizing certain properties of the configuration like the maximization of the specific heat, instead of associating directly the recognition procedure with the RG flow and its fixed points. Then from the reconstructed data we deduce the critical exponent associated to the magnetization to find satisfactory agreement with the actual physical value. We assume no prior knowledge about the criticality of the system and its Hamiltonian.

This paper proposes a Reinforcement Learning (RL) algorithm to synthesize policies for a Markov Decision Process (MDP), such that a linear time property is satisfied. We convert the property into a Limit Deterministic Buchi Automaton (LDBA), then construct a product MDP between the automaton and the original MDP. A reward function is then assigned to the states of the product automaton, according to accepting conditions of the LDBA. With this reward function, our algorithm synthesizes a policy that satisfies the linear time property: as such, the policy synthesis procedure is "constrained" by the given specification. Additionally, we show that the RL procedure sets up an online value iteration method to calculate the maximum probability of satisfying the given property, at any given state of the MDP - a convergence proof for the procedure is provided. Finally, the performance of the algorithm is evaluated via a set of numerical examples. We observe an improvement of one order of magnitude in the number of iterations required for the synthesis compared to existing approaches.