The proximal policy optimization (PPO) algorithm stands as one of the most prosperous methods in the field of reinforcement learning (RL). Despite its success, the theoretical understanding of PPO remains deficient. Specifically, it is unclear whether PPO or its optimistic variants can effectively solve linear Markov decision processes (MDPs), which are arguably the simplest models in RL with function approximation. To bridge this gap, we propose an optimistic variant of PPO for episodic adversarial linear MDPs with full-information feedback, and establish a $\tilde{\mathcal{O}}(d^{3/4}H^2K^{3/4})$ regret for it. Here $d$ is the ambient dimension of linear MDPs, $H$ is the length of each episode, and $K$ is the number of episodes. Compared with existing policy-based algorithms, we achieve the state-of-the-art regret bound in both stochastic linear MDPs and adversarial linear MDPs with full information. Additionally, our algorithm design features a novel multi-batched updating mechanism and the theoretical analysis utilizes a new covering number argument of value and policy classes, which might be of independent interest.
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This work considers Bayesian inference under misspecification for complex statistical models comprised of simpler submodels, referred to as modules, that are coupled together. Such ``multi-modular" models often arise when combining information from different data sources, where there is a module for each data source. When some of the modules are misspecified, the challenges of Bayesian inference under misspecification can sometimes be addressed by using ``cutting feedback" methods, which modify conventional Bayesian inference by limiting the influence of unreliable modules. Here we investigate cutting feedback methods in the context of generalized posterior distributions, which are built from arbitrary loss functions, and present novel findings on their behaviour. We make three main contributions. First, we describe how cutting feedback methods can be defined in the generalized Bayes setting, and discuss the appropriate scaling of the loss functions for different modules to each other and the prior. Second, we derive a novel result about the large sample behaviour of the posterior for a given module's parameters conditional on the parameters of other modules. This formally justifies the use of conditional Laplace approximations, which provide better approximations of conditional posterior distributions compared to conditional distributions from a Laplace approximation of the joint posterior. Our final contribution leverages the large sample approximations of our second contribution to provide convenient diagnostics for understanding the sensitivity of inference to the coupling of the modules, and to implement a new semi-modular posterior approach for conducting robust Bayesian modular inference. The usefulness of the methodology is illustrated in several benchmark examples from the literature on cut model inference.
The main goal of this paper is to investigate distributed dynamic programming (DP) to solve networked multi-agent Markov decision problems (MDPs). We consider a distributed multi-agent case, where each agent does not have an access to the rewards of other agents except for its own reward. Moreover, each agent can share their parameters with its neighbors over a communication network represented by a graph. We propose a distributed DP in the continuous-time domain, and prove its convergence through control theoretic viewpoints. The proposed analysis can be viewed as a preliminary ordinary differential equation (ODE) analysis of a distributed temporal difference learning algorithm, whose convergence can be proved using Borkar-Meyn theorem and the single time-scale approach.
Preserving the topology from being inferred by external adversaries has become a paramount security issue for network systems (NSs), and adding random noises to the nodal states provides a promising way. Nevertheless, recent works have revealed that the topology cannot be preserved under i.i.d. noises in the asymptotic sense. How to effectively characterize the non-asymptotic preservation performance still remains an open issue. Inspired by the deviation quantification of concentration inequalities, this paper proposes a novel metric named trace-based variance-expectation ratio. This metric effectively captures the decaying rate of the topology inference error, where a slower rate indicates better non-asymptotic preservation performance. We prove that the inference error will always decay to zero asymptotically, as long as the added noises are non-increasing and independent (milder than the i.i.d. condition). Then, the optimal noise design that produces the slowest decaying rate for the error is obtained. More importantly, we amend the noise design by introducing one-lag time dependence, achieving the zero state deviation and the non-zero topology inference error in the asymptotic sense simultaneously. Extensions to a general class of noises with multi-lag time dependence are provided. Comprehensive simulations verify the theoretical findings.
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 prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $\Sigma,\Gamma$ and $\Phi$ are alphabets of constant size, and $\mu$ is a pairwise connected distribution over $\Sigma\times\Gamma\times\Phi$ with no $(\mathbb{Z},+)$ embeddings in which the probability of each atom is $\Omega(1)$, then the following holds. Any triplets of $1$-bounded functions $f\colon \Sigma^n\to\mathbb{C}$, $g\colon \Gamma^n\to\mathbb{C}$, $h\colon \Phi^n\to\mathbb{C}$ satisfying \[ \left|\mathbb{E}_{(x,y,z)\sim \mu^{\otimes n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon \] must arise from an Abelian group associated with the distribution $\mu$. More specifically, we show that there is an Abelian group $(H,+)$ of constant size such that for any such $f,g$ and $h$, the function $f$ (and similarly $g$ and $h$) is correlated with a function of the form $\tilde{f}(x) = \chi(\sigma(x_1),\ldots,\sigma(x_n)) L (x)$, where $\sigma\colon \Sigma \to H$ is some map, $\chi\in \hat{H}^{\otimes n}$ is a character, and $L\colon \Sigma^n\to\mathbb{C}$ is a low-degree function with bounded $2$-norm. En route we prove a few additional results that may be of independent interest, such as an improved direct product theorem, as well as a result we refer to as a ``restriction inverse theorem'' about the structure of functions that, under random restrictions, with noticeable probability have significant correlation with a product function. In companion papers, we show applications of our results to the fields of Probabilistically Checkable Proofs, as well as various areas in discrete mathematics such as extremal combinatorics and additive combinatorics.
Recent research has demonstrated the potential of reinforcement learning (RL) in enabling effective multi-robot collaboration, particularly in social dilemmas where robots face a trade-off between self-interests and collective benefits. However, environmental factors such as miscommunication and adversarial robots can impact cooperation, making it crucial to explore how multi-robot communication can be manipulated to achieve different outcomes. This paper presents a novel approach, namely PIMbot, to manipulating the reward function in multi-robot collaboration through two distinct forms of manipulation: policy and incentive manipulation. Our work introduces a new angle for manipulation in recent multi-agent RL social dilemmas that utilize a unique reward function for incentivization. By utilizing our proposed PIMbot mechanisms, a robot is able to manipulate the social dilemma environment effectively. PIMbot has the potential for both positive and negative impacts on the task outcome, where positive impacts lead to faster convergence to the global optimum and maximized rewards for any chosen robot. Conversely, negative impacts can have a detrimental effect on the overall task performance. We present comprehensive experimental results that demonstrate the effectiveness of our proposed methods in the Gazebo-simulated multi-robot environment. Our work provides insights into how inter-robot communication can be manipulated and has implications for various robotic applications. %, including robotics, transportation, and manufacturing.
In many real-world multi-agent cooperative tasks, due to high cost and risk, agents cannot continuously interact with the environment and collect experiences during learning, but have to learn from offline datasets. However, the transition dynamics in the dataset of each agent can be much different from the ones induced by the learned policies of other agents in execution, creating large errors in value estimates. Consequently, agents learn uncoordinated low-performing policies. In this paper, we propose a framework for offline decentralized multi-agent reinforcement learning, which exploits value deviation and transition normalization to deliberately modify the transition probabilities. Value deviation optimistically increases the transition probabilities of high-value next states, and transition normalization normalizes the transition probabilities of next states. They together enable agents to learn high-performing and coordinated policies. Theoretically, we prove the convergence of Q-learning under the altered non-stationary transition dynamics. Empirically, we show that the framework can be easily built on many existing offline reinforcement learning algorithms and achieve substantial improvement in a variety of multi-agent tasks.
In this paper, we consider decentralized optimization problems where agents have individual cost functions to minimize subject to subspace constraints that require the minimizers across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus or single-task optimization as special cases, and allows for more general task relatedness models such as multitask smoothness and coupled optimization. In order to cope with communication constraints, we propose and study an adaptive decentralized strategy where the agents employ differential randomized quantizers to compress their estimates before communicating with their neighbors. The analysis shows that, under some general conditions on the quantization noise, and for sufficiently small step-sizes $\mu$, the strategy is stable both in terms of mean-square error and average bit rate: by reducing $\mu$, it is possible to keep the estimation errors small (on the order of $\mu$) without increasing indefinitely the bit rate as $\mu\rightarrow 0$. Simulations illustrate the theoretical findings and the effectiveness of the proposed approach, revealing that decentralized learning is achievable at the expense of only a few bits.
In this manuscript we derive the optimal out-of-sample causal predictor for a linear system that has been observed in $k+1$ within-sample environments. In this model we consider $k$ shifted environments and one observational environment. Each environment corresponds to a linear structural equation model (SEM) with its own shift and noise vector, both in $L^2$. The strength of the shifts can be put in a certain order, and we may therefore speak of all shifts that are less or equally strong than a given shift. We consider the space of all shifts are $\gamma$ times less or equally strong than any weighted average of the observed shift vectors with weights on the unit sphere. For each $\beta\in\mathbb{R}^p$ we show that the supremum of the risk functions $R_{\tilde{A}}(\beta)$ over $\tilde{A}\in C^\gamma$ has a worst-risk decomposition into a (positive) linear combination of risk functions, depending on $\gamma$. We then define the causal regularizer, $\beta_\gamma$, as the argument $\beta$ that minimizes this risk. The main result of the paper is that this regularizer can be consistently estimated with a plug-in estimator outside a set of zero Lebesgue measure in the parameter space. A practical obstacle for such estimation is that it involves the solution of a general degree polynomial which cannot be done explicitly. Therefore we also prove that an approximate plug-in estimator using the bisection method is also consistent. An interesting by-product of the proof of the main result is that the plug-in estimation of the argmin of the maxima of a finite set of quadratic risk functions is consistent outside a set of zero Lebesgue measure in the parameter space.
Modelling in biology must adapt to increasingly complex and massive data. The efficiency of the inference algorithms used to estimate model parameters is therefore questioned. Many of these are based on stochastic optimization processes which waste a significant part of the computation time due to their rejection sampling approaches. We introduce the Fixed Landscape Inference MethOd (flimo), a new likelihood-free inference method for continuous state-space stochastic models. It applies deterministic gradient-based optimization algorithms to obtain a point estimate of the parameters, minimizing the difference between the data and some simulations according to some prescribed summary statistics. In this sense, it is analogous to Approximate Bayesian Computation (ABC). Like ABC, it can also provide an approximation of the distribution of the parameters. Three applications are proposed: a usual theoretical example, namely the inference of the parameters of g-and-k distributions; a population genetics problem, not so simple as it seems, namely the inference of a selective value from time series in a Wright-Fisher model; and simulations from a Ricker model, representing chaotic population dynamics. In the two first applications, the results show a drastic reduction of the computational time needed for the inference phase compared to the other methods, despite an equivalent accuracy. Even when likelihood-based methods are applicable, the simplicity and efficiency of flimo make it a compelling alternative. Implementations in Julia and in R are available on //metabarcoding.org/flimo. To run flimo, the user must simply be able to simulate data according to the chosen model.
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