The minimisation of cost functions is crucial in various optimisation fields. However, identifying their global minimum remains challenging owing to the huge computational cost incurred. This work analytically expresses the computational cost to identify an approximate global minimum for a class of cost functions defined under a high-dimensional discrete state space. Then, we derive an optimal global search scheme that minimises the computational cost. Mathematical analyses demonstrate that a combination of the gradient descent algorithm and the selection and crossover algorithm--with a biased crossover weight--maximises the search efficiency. Remarkably, its computational cost is of the square root order in contrast to that of the conventional gradient descent algorithms, indicating a quadratic speedup of global search. We corroborate this proposition using numerical analyses of the travelling salesman problem. The simple computational architecture and minimal computational cost of the proposed scheme are highly desirable for biological organisms and neuromorphic hardware.
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Risk-averse problems receive far less attention than risk-neutral control problems in reinforcement learning, and existing risk-averse approaches are challenging to deploy to real-world applications. One primary reason is that such risk-averse algorithms often learn from consecutive trajectories with a certain length, which significantly increases the potential danger of causing dangerous failures in practice. This paper proposes Transition-based VOlatility-controlled Policy Search (TOPS), a novel algorithm that solves risk-averse problems by learning from (possibly non-consecutive) transitions instead of only consecutive trajectories. By using an actor-critic scheme with an overparameterized two-layer neural network, our algorithm finds a globally optimal policy at a sublinear rate with proximal policy optimization and natural policy gradient, with effectiveness comparable to the state-of-the-art convergence rate of risk-neutral policy-search methods. The algorithm is evaluated on challenging Mujoco robot simulation tasks under the mean-variance evaluation metric. Both theoretical analysis and experimental results demonstrate a state-of-the-art level of risk-averse policy search methods.
Learning optimal control policies directly on physical systems is challenging since even a single failure can lead to costly hardware damage. Most existing learning methods that guarantee safety, i.e., no failures, during exploration are limited to local optima. A notable exception is the GoSafe algorithm, which, unfortunately, cannot handle high-dimensional systems and hence cannot be applied to most real-world dynamical systems. This work proposes GoSafeOpt as the first algorithm that can safely discover globally optimal policies for complex systems while giving safety and optimality guarantees. Our experiments on a robot arm that would be prohibitive for GoSafe demonstrate that GoSafeOpt safely finds remarkably better policies than competing safe learning methods for high-dimensional domains.
The performance of spectral clustering heavily relies on the quality of affinity matrix. A variety of affinity-matrix-construction (AMC) methods have been proposed but they have hyperparameters to determine beforehand, which requires strong experience and lead to difficulty in real applications especially when the inter-cluster similarity is high or/and the dataset is large. In addition, we often need to choose different AMC methods for different datasets, which still depends on experience. To solve these two challenging problems, in this paper, we present a simple yet effective method for automated spectral clustering. The main idea is to find the most reliable affinity matrix among a set of candidates given by different AMC methods with different hyperparameters, where the reliability is quantified by the \textit{relative-eigen-gap} of graph Laplacian introduced in this paper. We also implement the method using Bayesian optimization.We extend the method to large-scale datasets such as MNIST, on which the time cost is less than 90s and the clustering accuracy is state-of-the-art. Extensive experiments of natural image clustering show that our method is more versatile, accurate, and efficient than baseline methods.
We propose a novel multibody dynamics simulation framework that can efficiently deal with large-dimensionality and complementarity multi-contact conditions. Typical contact simulation approaches perform contact impulse-level fixed-point iteration (IL-FPI), which has high time-complexity from large-size matrix inversion and multiplication, as well as susceptibility to ill-conditioned contact situations. To circumvent this, we propose a novel framework based on velocity-level fixed-point iteration (VL-FPI), which, by utilizing a certain surrogate dynamics and contact nodalization (with virtual nodes), can achieve not only inter-contact decoupling but also their inter-axes decoupling (i.e., contact diagonalization). This then enables us to one-shot/parallel-solve the contact problem during each VL-FPI iteration-loop, while the surrogate dynamics structure allows us to circumvent large-size/dense matrix inversion/multiplication, thereby, significantly speeding up the simulation time with improved convergence property. We theoretically show that the solution of our framework is consistent with that of the original problem and, further, elucidate mathematical conditions for the convergence of our proposed solver. Performance and properties of our proposed simulation framework are also demonstrated and experimentally-validated for various large-dimensional/multi-contact scenarios including deformable objects.
We propose a generic feature compression method for Approximate Nearest Neighbor Search (ANNS) problems, which speeds up existing ANNS methods in a plug-and-play manner. Specifically, we propose a new network structure called Compression Network with Transformer (CNT) to compress the feature into a low dimensional space, and an inhomogeneous neighborhood relationship preserving (INRP) loss that aims to maintain high search accuracy. In CNT, we use multiple compression projections to cast the feature into many low dimensional spaces, and then use transformer to globally optimize these projections such that the features are well compressed following the guidance from our loss function. The loss function is designed to assign high weights on point pairs that are close in original feature space, and keep their distances in projected space. Keeping these distances helps maintain the eventual top-k retrieval accuracy, and down weighting others creates room for feature compression. Experimental results show that our CNT can significantly improve the efficiency of popular ANNS methods while preserves or even improves search accuracy, suggesting its broad potential impact on real world applications.
We study the problem of estimating the fixed point of a contractive operator defined on a separable Banach space. Focusing on a stochastic query model that provides noisy evaluations of the operator, we analyze a variance-reduced stochastic approximation scheme, and establish non-asymptotic bounds for both the operator defect and the estimation error, measured in an arbitrary semi-norm. In contrast to worst-case guarantees, our bounds are instance-dependent, and achieve the local asymptotic minimax risk non-asymptotically. For linear operators, contractivity can be relaxed to multi-step contractivity, so that the theory can be applied to problems like average reward policy evaluation problem in reinforcement learning. We illustrate the theory via applications to stochastic shortest path problems, two-player zero-sum Markov games, as well as policy evaluation and $Q$-learning for tabular Markov decision processes.
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.
Imitation learning seeks to circumvent the difficulty in designing proper reward functions for training agents by utilizing expert behavior. With environments modeled as Markov Decision Processes (MDP), most of the existing imitation algorithms are contingent on the availability of expert demonstrations in the same MDP as the one in which a new imitation policy is to be learned. In this paper, we study the problem of how to imitate tasks when there exist discrepancies between the expert and agent MDP. These discrepancies across domains could include differing dynamics, viewpoint, or morphology; we present a novel framework to learn correspondences across such domains. Importantly, in contrast to prior works, we use unpaired and unaligned trajectories containing only states in the expert domain, to learn this correspondence. We utilize a cycle-consistency constraint on both the state space and a domain agnostic latent space to do this. In addition, we enforce consistency on the temporal position of states via a normalized position estimator function, to align the trajectories across the two domains. Once this correspondence is found, we can directly transfer the demonstrations on one domain to the other and use it for imitation. Experiments across a wide variety of challenging domains demonstrate the efficacy of our approach.
Non-convex optimization is ubiquitous in modern machine learning. Researchers devise non-convex objective functions and optimize them using off-the-shelf optimizers such as stochastic gradient descent and its variants, which leverage the local geometry and update iteratively. Even though solving non-convex functions is NP-hard in the worst case, the optimization quality in practice is often not an issue -- optimizers are largely believed to find approximate global minima. Researchers hypothesize a unified explanation for this intriguing phenomenon: most of the local minima of the practically-used objectives are approximately global minima. We rigorously formalize it for concrete instances of machine learning problems.
Distance metric learning based on triplet loss has been applied with success in a wide range of applications such as face recognition, image retrieval, speaker change detection and recently recommendation with the CML model. However, as we show in this article, CML requires large batches to work reasonably well because of a too simplistic uniform negative sampling strategy for selecting triplets. Due to memory limitations, this makes it difficult to scale in high-dimensional scenarios. To alleviate this problem, we propose here a 2-stage negative sampling strategy which finds triplets that are highly informative for learning. Our strategy allows CML to work effectively in terms of accuracy and popularity bias, even when the batch size is an order of magnitude smaller than what would be needed with the default uniform sampling. We demonstrate the suitability of the proposed strategy for recommendation and exhibit consistent positive results across various datasets.
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