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Federated Reinforcement Learning (FRL) allows multiple agents to collaboratively build a decision making policy without sharing raw trajectories. However, if a small fraction of these agents are adversarial, it can lead to catastrophic results. We propose a policy gradient based approach that is robust to adversarial agents which can send arbitrary values to the server. Under this setting, our results form the first global convergence guarantees with general parametrization. These results demonstrate resilience with adversaries, while achieving sample complexity of order $\tilde{\mathcal{O}}\left( \frac{1}{\epsilon^2} \left( \frac{1}{N-f} + \frac{f^2}{(N-f)^2}\right)\right)$, where $N$ is the total number of agents and $f$ is the number of adversarial agents.

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We tackle the problem of estimating the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR) of the infinite-horizon discounted cost within a Markov cost process. First, we derive a minimax lower bound of $\Omega(1/\sqrt{n})$ that holds both in an expected and in a probabilistic sense. Then, using a finite-horizon truncation scheme, we derive an upper bound for the error in CVaR estimation, which matches our lower bound up to constant factors. Finally, we discuss an extension of our estimation scheme that covers more general risk measures satisfying a certain continuity criterion, e.g., spectral risk measures, utility-based shortfall risk. To the best of our knowledge, our work is the first to provide lower and upper bounds on the estimation error for any risk measure within Markovian settings. We remark that our lower bounds also extend to the infinite-horizon discounted costs' mean. Even in that case, our result $\Omega(1/\sqrt{n}) $ improves upon the existing result $\Omega(1/n)$[13].

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We consider the measurement model $Y = AX,$ where $X$ and, hence, $Y$ are random variables and $A$ is an a priori known tall matrix. At each time instance, a sample of one of $Y$'s coordinates is available, and the goal is to estimate $\mu := \mathbb{E}[X]$ via these samples. However, the challenge is that a small but unknown subset of $Y$'s coordinates are controlled by adversaries with infinite power: they can return any real number each time they are queried for a sample. For such an adversarial setting, we propose the first asynchronous online algorithm that converges to $\mu$ almost surely. We prove this result using a novel differential inclusion based two-timescale analysis. Two key highlights of our proof include: (a) the use of a novel Lyapunov function for showing that $\mu$ is the unique global attractor for our algorithm's limiting dynamics, and (b) the use of martingale and stopping time theory to show that our algorithm's iterates are almost surely bounded.

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Despite the popularity of policy gradient methods, they are known to suffer from large variance and high sample complexity. To mitigate this, we introduce SoftTreeMax -- a generalization of softmax that takes planning into account. In SoftTreeMax, we extend the traditional logits with the multi-step discounted cumulative reward, topped with the logits of future states. We consider two variants of SoftTreeMax, one for cumulative reward and one for exponentiated reward. For both, we analyze the gradient variance and reveal for the first time the role of a tree expansion policy in mitigating this variance. We prove that the resulting variance decays exponentially with the planning horizon as a function of the expansion policy. Specifically, we show that the closer the resulting state transitions are to uniform, the faster the decay. In a practical implementation, we utilize a parallelized GPU-based simulator for fast and efficient tree search. Our differentiable tree-based policy leverages all gradients at the tree leaves in each environment step instead of the traditional single-sample-based gradient. We then show in simulation how the variance of the gradient is reduced by three orders of magnitude, leading to better sample complexity compared to the standard policy gradient. On Atari, SoftTreeMax demonstrates up to 5x better performance in a faster run time compared to distributed PPO. Lastly, we demonstrate that high reward correlates with lower variance.

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Evolution Strategy (ES) is a powerful black-box optimization technique based on the idea of natural evolution. In each of its iterations, a key step entails ranking candidate solutions based on some fitness score. For an ES method in Reinforcement Learning (RL), this ranking step requires evaluating multiple policies. This is presently done via on-policy approaches: each policy's score is estimated by interacting several times with the environment using that policy. This leads to a lot of wasteful interactions since, once the ranking is done, only the data associated with the top-ranked policies is used for subsequent learning. To improve sample efficiency, we propose a novel off-policy alternative for ranking, based on a local approximation for the fitness function. We demonstrate our idea in the context of a state-of-the-art ES method called the Augmented Random Search (ARS). Simulations in MuJoCo tasks show that, compared to the original ARS, our off-policy variant has similar running times for reaching reward thresholds but needs only around 70% as much data. It also outperforms the recent Trust Region ES. We believe our ideas should be extendable to other ES methods as well.

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Q-learning and SARSA(0) with linear function approximation, under $\epsilon$-greedy exploration, are leading methods to estimate the optimal policy in Reinforcement Learning (RL). It has been empirically known that the discontinuous nature of the greedy policies causes these algorithms to exhibit complex phenomena such as i.) instability, ii.) policy oscillation and chattering, iii.) multiple attractors, and iv.) worst policy convergence. However, the literature lacks a formal recipe to explain these behaviors and this has been a long-standing open problem (Sutton, 1999). Our work addresses this by building the necessary mathematical framework using stochastic recursive inclusions and Differential Inclusions (DIs). From this novel viewpoint, our main result states that these approximate algorithms asymptotically converge to suitable invariant sets of DIs instead of differential equations, as is common elsewhere in RL. Furthermore, the nature of these deterministic DIs completely governs the limiting behaviors of these algorithms.

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In Multi-Agent Reinforcement Learning (MARL), multiple agents interact with a common environment, as also with each other, for solving a shared problem in sequential decision-making. It has wide-ranging applications in gaming, robotics, finance, etc. In this work, we derive a novel law of iterated logarithm for a family of distributed nonlinear stochastic approximation schemes that is useful in MARL. In particular, our result describes the convergence rate on almost every sample path where the algorithm converges. This result is the first of its kind in the distributed setup and provides deeper insights than the existing ones, which only discuss convergence rates in the expected or the CLT sense. Importantly, our result holds under significantly weaker assumptions: neither the gossip matrix needs to be doubly stochastic nor the stepsizes square summable. As an application, we show that, for the stepsize $n^{-\gamma}$ with $\gamma \in (0, 1),$ the distributed TD(0) algorithm with linear function approximation has a convergence rate of $O(\sqrt{n^{-\gamma} \ln n })$ a.s.; for the $1/n$ type stepsize, the same is $O(\sqrt{n^{-1} \ln \ln n})$ a.s. These decay rates do not depend on the graph depicting the interactions among the different agents.

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Momentum methods are popularly used in accelerating stochastic iterative methods. Although a fair amount of literature is dedicated to momentum in stochastic optimisation, there are limited results that quantify the benefits of using heavy ball momentum in the specific case of stochastic approximation algorithms. We first show that the convergence rate with optimal step size does not improve when momentum is used (under some assumptions). Secondly, to quantify the behaviour in the initial phase we analyse the sample complexity of iterates with and without momentum. We show that the sample complexity bound for SA without momentum is $\tilde{\mathcal{O}}(\frac{1}{\alpha\lambda_{min}(A)})$ while for SA with momentum is $\tilde{\mathcal{O}}(\frac{1}{\sqrt{\alpha\lambda_{min}(A)}})$, where $\alpha$ is the step size and $\lambda_{min}(A)$ is the smallest eigenvalue of the driving matrix $A$. Although the sample complexity bound for SA with momentum is better for small enough $\alpha$, it turns out that for optimal choice of $\alpha$ in the two cases, the sample complexity bounds are of the same order.

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Overdetermined linear systems are common in reinforcement learning, e.g., in Q and value function estimation with function approximation. The standard least-squares criterion, however, leads to a solution that is unduly influenced by rows with large norms. This is a serious issue, especially when the matrices in these systems are beyond user control. To address this, we propose a scale-invariant criterion that we then use to develop two novel algorithms for value function estimation: Normalized Monte Carlo and Normalized TD(0). Separately, we also introduce a novel adaptive stepsize that may be useful beyond this work as well. We use simulations and theoretical guarantees to demonstrate the efficacy of our ideas.

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For providing quick and accurate search results, a search engine maintains a local snapshot of the entire web. And, to keep this local cache fresh, it employs a crawler for tracking changes across various web pages. It would have been ideal if the crawler managed to update the local snapshot as soon as a page changed on the web. However, finite bandwidth availability and server restrictions mean that there is a bound on how frequently the different pages can be crawled. This then brings forth the following optimisation problem: maximise the freshness of the local cache subject to the crawling frequency being within the prescribed bounds. Recently, tractable algorithms have been proposed to solve this optimisation problem under different cost criteria. However, these assume the knowledge of exact page change rates, which is unrealistic in practice. We address this issue here. Specifically, we provide three novel schemes for online estimation of page change rates. All these schemes only need partial information about the page change process, i.e., they only need to know if the page has changed or not since the last crawl instance. Our first scheme is based on the law of large numbers, the second on the theory of stochastic approximation, while the third is an extension of the second and involves an additional momentum term. For all of these schemes, we prove convergence and, also, provide their convergence rates. As far as we know, the results concerning the third estimator is quite novel. Specifically, this is the first convergence type result for a stochastic approximation algorithm with momentum. Finally, we provide some numerical experiments (on real as well as synthetic data) to compare the performance of our proposed estimators with the existing ones (e.g., MLE).

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