The practicality of reinforcement learning algorithms has been limited due to poor scaling with respect to the problem size, as the sample complexity of learning an $\epsilon$-optimal policy is $\tilde{\Omega}\left(|S||A|H^3 / \epsilon^2\right)$ over worst case instances of an MDP with state space $S$, action space $A$, and horizon $H$. We consider a class of MDPs for which the associated optimal $Q^*$ function is low rank, where the latent features are unknown. While one would hope to achieve linear sample complexity in $|S|$ and $|A|$ due to the low rank structure, we show that without imposing further assumptions beyond low rank of $Q^*$, if one is constrained to estimate the $Q$ function using only observations from a subset of entries, there is a worst case instance in which one must incur a sample complexity exponential in the horizon $H$ to learn a near optimal policy. We subsequently show that under stronger low rank structural assumptions, given access to a generative model, Low Rank Monte Carlo Policy Iteration (LR-MCPI) and Low Rank Empirical Value Iteration (LR-EVI) achieve the desired sample complexity of $\tilde{O}\left((|S|+|A|)\mathrm{poly}(d,H)/\epsilon^2\right)$ for a rank $d$ setting, which is minimax optimal with respect to the scaling of $|S|, |A|$, and $\epsilon$. In contrast to literature on linear and low-rank MDPs, we do not require a known feature mapping, our algorithm is computationally simple, and our results hold for long time horizons. Our results provide insights on the minimal low-rank structural assumptions required on the MDP with respect to the transition kernel versus the optimal action-value function.
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We study a game between liquidity provider and liquidity taker agents interacting in an over-the-counter market, for which the typical example is foreign exchange. We show how a suitable design of parameterized families of reward functions coupled with shared policy learning constitutes an efficient solution to this problem. By playing against each other, our deep-reinforcement-learning-driven agents learn emergent behaviors relative to a wide spectrum of objectives encompassing profit-and-loss, optimal execution and market share. In particular, we find that liquidity providers naturally learn to balance hedging and skewing, where skewing refers to setting their buy and sell prices asymmetrically as a function of their inventory. We further introduce a novel RL-based calibration algorithm which we found performed well at imposing constraints on the game equilibrium. On the theoretical side, we are able to show convergence rates for our multi-agent policy gradient algorithm under a transitivity assumption, closely related to generalized ordinal potential games.
The Traveling Salesman Problem (TSP) is a well-known problem in combinatorial optimization with applications in various domains. However, existing TSP solvers face challenges in producing high-quality solutions with low latency. To address this issue, we propose NAR4TSP, which produces TSP solutions in a Non-Autoregressive (NAR) manner using a specially designed Graph Neural Network (GNN), achieving faster inference speed. Moreover, NAR4TSP is trained using an enhanced Reinforcement Learning (RL) strategy, eliminating the dependency on costly labels used to train conventional supervised learning-based NAR models. To the best of our knowledge, NAR4TSP is the first TSP solver that successfully combines RL and NAR decoding. The experimental results on both synthetic and real-world TSP instances demonstrate that NAR4TSP outperforms four state-of-the-art models in terms of solution quality, inference latency, and generalization ability. Lastly, we present visualizations of NAR4TSP's decoding process and its overall path planning to showcase the feasibility of implementing NAR4TSP in an end-to-end manner and its effectiveness, respectively.
Lifelong learning aims to create AI systems that continuously and incrementally learn during a lifetime, similar to biological learning. Attempts so far have met problems, including catastrophic forgetting, interference among tasks, and the inability to exploit previous knowledge. While considerable research has focused on learning multiple supervised classification tasks that involve changes in the input distribution, lifelong reinforcement learning (LRL) must deal with variations in the state and transition distributions, and in the reward functions. Modulating masks with a fixed backbone network, recently developed for classification, are particularly suitable to deal with such a large spectrum of task variations. In this paper, we adapted modulating masks to work with deep LRL, specifically PPO and IMPALA agents. The comparison with LRL baselines in both discrete and continuous RL tasks shows superior performance. We further investigated the use of a linear combination of previously learned masks to exploit previous knowledge when learning new tasks: not only is learning faster, the algorithm solves tasks that we could not otherwise solve from scratch due to extremely sparse rewards. The results suggest that RL with modulating masks is a promising approach to lifelong learning, to the composition of knowledge to learn increasingly complex tasks, and to knowledge reuse for efficient and faster learning.
In recent years, there has been a growing interest in understanding complex microstructures and their effect on macroscopic properties. In general, it is difficult to derive an effective constitutive law for such microstructures with reasonable accuracy and meaningful parameters. One numerical approach to bridge the scales is computational homogenization, in which a microscopic problem is solved at every macroscopic point, essentially replacing the effective constitutive model. Such approaches are, however, computationally expensive and typically infeasible in multi-query contexts such as optimization and material design. To render these analyses tractable, surrogate models that can accurately approximate and accelerate the microscopic problem over a large design space of shapes, material and loading parameters are required. In previous works, such models were constructed in a data-driven manner using methods such as Neural Networks (NN) or Gaussian Process Regression (GPR). However, these approaches currently suffer from issues, such as need for large amounts of training data, lack of physics, and considerable extrapolation errors. In this work, we develop a reduced order model based on Proper Orthogonal Decomposition (POD), Empirical Cubature Method (ECM) and a geometrical transformation method with the following key features: (i) large shape variations of the microstructure are captured, (ii) only relatively small amounts of training data are necessary, and (iii) highly non-linear history-dependent behaviors are treated. The proposed framework is tested and examined in two numerical examples, involving two scales and large geometrical variations. In both cases, high speed-ups and accuracies are achieved while observing good extrapolation behavior.
Deep neural networks (DNNs) trained with the logistic loss (i.e., the cross entropy loss) have made impressive advancements in various binary classification tasks. However, generalization analysis for binary classification with DNNs and logistic loss remains scarce. The unboundedness of the target function for the logistic loss is the main obstacle to deriving satisfying generalization bounds. In this paper, we aim to fill this gap by establishing a novel and elegant oracle-type inequality, which enables us to deal with the boundedness restriction of the target function, and using it to derive sharp convergence rates for fully connected ReLU DNN classifiers trained with logistic loss. In particular, we obtain optimal convergence rates (up to log factors) only requiring the H\"older smoothness of the conditional class probability $\eta$ of data. Moreover, we consider a compositional assumption that requires $\eta$ to be the composition of several vector-valued functions of which each component function is either a maximum value function or a H\"older smooth function only depending on a small number of its input variables. Under this assumption, we derive optimal convergence rates (up to log factors) which are independent of the input dimension of data. This result explains why DNN classifiers can perform well in practical high-dimensional classification problems. Besides the novel oracle-type inequality, the sharp convergence rates given in our paper also owe to a tight error bound for approximating the natural logarithm function near zero (where it is unbounded) by ReLU DNNs. In addition, we justify our claims for the optimality of rates by proving corresponding minimax lower bounds. All these results are new in the literature and will deepen our theoretical understanding of classification with DNNs.
The success of a multi-kilometre drive by a solar-powered rover at the lunar south pole depends upon careful planning in space and time due to highly dynamic solar illumination conditions. An additional challenge is that real-world robots may be subject to random faults that can temporarily delay long-range traverses. The majority of existing global spatiotemporal planners assume a deterministic rover-environment model and do not account for random faults. In this paper, we consider a random fault profile with a known, average spatial fault rate. We introduce a methodology to compute recovery policies that maximize the probability of survival of a solar-powered rover from different start states. A recovery policy defines a set of recourse actions to reach a location with sufficient battery energy remaining, given the local solar illumination conditions. We solve a stochastic reach-avoid problem using dynamic programming to find such optimal recovery policies. Our focus, in part, is on the implications of state space discretization, which is often required in practical implementations. We propose a modified dynamic programming algorithm that conservatively accounts for approximation errors. To demonstrate the benefits of our approach, we compare against existing methods in scenarios where a solar-powered rover seeks to safely exit from permanently shadowed regions in the Cabeus area at the lunar south pole. We also highlight the relevance of our methodology for mission formulation and trade safety analysis by empirically comparing different rover mobility models in simulated recovery drives from the LCROSS crash region.
The dynamic ranking, due to its increasing importance in many applications, is becoming crucial, especially with the collection of voluminous time-dependent data. One such application is sports statistics, where dynamic ranking aids in forecasting the performance of competitive teams, drawing on historical and current data. Despite its usefulness, predicting and inferring rankings pose challenges in environments necessitating time-dependent modeling. This paper introduces a spectral ranker called Kernel Rank Centrality, designed to rank items based on pairwise comparisons over time. The ranker operates via kernel smoothing in the Bradley-Terry model, utilizing a Markov chain model. Unlike the maximum likelihood approach, the spectral ranker is nonparametric, demands fewer model assumptions and computations, and allows for real-time ranking. We establish the asymptotic distribution of the ranker by applying an innovative group inverse technique, resulting in a uniform and precise entrywise expansion. This result allows us to devise a new inferential method for predictive inference, previously unavailable in existing approaches. Our numerical examples showcase the ranker's utility in predictive accuracy and constructing an uncertainty measure for prediction, leveraging data from the National Basketball Association (NBA). The results underscore our method's potential compared to the gold standard in sports, the Arpad Elo rating system.
Reinforcement learning is still struggling with solving long-horizon surgical robot tasks which involve multiple steps over an extended duration of time due to the policy exploration challenge. Recent methods try to tackle this problem by skill chaining, in which the long-horizon task is decomposed into multiple subtasks for easing the exploration burden and subtask policies are temporally connected to complete the whole long-horizon task. However, smoothly connecting all subtask policies is difficult for surgical robot scenarios. Not all states are equally suitable for connecting two adjacent subtasks. An undesired terminate state of the previous subtask would make the current subtask policy unstable and result in a failed execution. In this work, we introduce value-informed skill chaining (ViSkill), a novel reinforcement learning framework for long-horizon surgical robot tasks. The core idea is to distinguish which terminal state is suitable for starting all the following subtask policies. To achieve this target, we introduce a state value function that estimates the expected success probability of the entire task given a state. Based on this value function, a chaining policy is learned to instruct subtask policies to terminate at the state with the highest value so that all subsequent policies are more likely to be connected for accomplishing the task. We demonstrate the effectiveness of our method on three complex surgical robot tasks from SurRoL, a comprehensive surgical simulation platform, achieving high task success rates and execution efficiency. Code is available at $\href{//github.com/med-air/ViSkill}{\text{//github.com/med-air/ViSkill}}$.
We evaluate benchmark deep reinforcement learning (DRL) algorithms on the task of portfolio optimisation under a simulator. The simulator is based on correlated geometric Brownian motion (GBM) with the Bertsimas-Lo (BL) market impact model. Using the Kelly criterion (log utility) as the objective, we can analytically derive the optimal policy without market impact and use it as an upper bound to measure performance when including market impact. We found that the off-policy algorithms DDPG, TD3 and SAC were unable to learn the right Q function due to the noisy rewards and therefore perform poorly. The on-policy algorithms PPO and A2C, with the use of generalised advantage estimation (GAE), were able to deal with the noise and derive a close to optimal policy. The clipping variant of PPO was found to be important in preventing the policy from deviating from the optimal once converged. In a more challenging environment where we have regime changes in the GBM parameters, we found that PPO, combined with a hidden Markov model (HMM) to learn and predict the regime context, is able to learn different policies adapted to each regime. Overall, we find that the sample complexity of these algorithms is too high, requiring more than 2m steps to learn a good policy in the simplest setting, which is equivalent to almost 8,000 years of daily prices.
We consider estimation and inference with data collected from episodic reinforcement learning (RL) algorithms; i.e. adaptive experimentation algorithms that at each period (aka episode) interact multiple times in a sequential manner with a single treated unit. Our goal is to be able to evaluate counterfactual adaptive policies after data collection and to estimate structural parameters such as dynamic treatment effects, which can be used for credit assignment (e.g. what was the effect of the first period action on the final outcome). Such parameters of interest can be framed as solutions to moment equations, but not minimizers of a population loss function, leading to $Z$-estimation approaches in the case of static data. However, such estimators fail to be asymptotically normal in the case of adaptive data collection. We propose a re-weighted $Z$-estimation approach with carefully designed adaptive weights to stabilize the episode-varying estimation variance, which results from the nonstationary policy that typical episodic RL algorithms invoke. We identify proper weighting schemes to restore the consistency and asymptotic normality of the re-weighted Z-estimators for target parameters, which allows for hypothesis testing and constructing uniform confidence regions for target parameters of interest. Primary applications include dynamic treatment effect estimation and dynamic off-policy evaluation.
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