Ather Gattami

Hamid Reza Feyzmahdavian, Ather Gattami, Mikael Johansson

This paper develops a controller synthesis method for distributed LQG control problems under output-feedback. We consider a system consisting of three interconnected linear subsystems with a delayed information sharing structure. While the state-feedback case has previously been solved, the extension to output-feedback is nontrivial as the classical separation principle fails. To find the optimal solution, the controller is decomposed into two independent components: a centralized LQG-optimal controller under delayed state observations, and a sum of correction terms based on additional local information available to decision makers. Explicit discrete-time equations are derived whose solutions are the gains of the optimal controller.

Johan Östman, Ather Gattami, Daniel Gillblad

We consider a decentralized multiplayer game, played over $T$ rounds, with a leader-follower hierarchy described by a directed acyclic graph. For each round, the graph structure dictates the order of the players and how players observe the actions of one another. By the end of each round, all players receive a joint bandit-reward based on their joint action that is used to update the player strategies towards the goal of minimizing the joint pseudo-regret. We present a learning algorithm inspired by the single-player multi-armed bandit problem and show that it achieves sub-linear joint pseudo-regret in the number of rounds for both adversarial and stochastic bandit rewards. Furthermore, we quantify the cost incurred due to the decentralized nature of our problem compared to the centralized setting.

Ather Gattami

In this paper, we consider reinforcement learning of nonlinear systems with continuous state and action spaces. We present an episodic learning algorithm, where we for each episode use convex optimization to find a two-layer neural network approximation of the optimal $Q$-function. The convex optimization approach guarantees that the weights calculated at each episode are optimal, with respect to the given sampled states and actions of the current episode. For stable nonlinear systems, we show that the algorithm converges and that the converging parameters of the trained neural network can be made arbitrarily close to the optimal neural network parameters. In particular, if the regularization parameter in the training phase is given by $ρ$, then the parameters of the trained neural network converge to $w$, where the distance between $w$ and the optimal parameters $w^\star$ is bounded by $\mathcal{O}(ρ)$. That is, when the number of episodes goes to infinity, there exists a constant $C$ such that \[ \|w-w^\star\| \le Cρ. \] In particular, our algorithm converges arbitrarily close to the optimal neural network parameters as the regularization parameter goes to zero. As a consequence, our algorithm converges fast due to the polynomial-time convergence of convex optimization algorithms.

Qinbo Bai, Vaneet Aggarwal, Ather Gattami

In the optimization of dynamical systems, the variables typically have constraints. Such problems can be modeled as a constrained Markov Decision Process (CMDP). This paper considers a model-free approach to the problem, where the transition probabilities are not known. In the presence of long-term (or average) constraints, the agent has to choose a policy that maximizes the long-term average reward as well as satisfy the average constraints in each episode. The key challenge with the long-term constraints is that the optimal policy is not deterministic in general, and thus standard Q-learning approaches cannot be directly used. This paper uses concepts from constrained optimization and Q-learning to propose an algorithm for CMDP with long-term constraints. For any $γ\in(0,\frac{1}{2})$, the proposed algorithm is shown to achieve $O(T^{1/2+γ})$ regret bound for the obtained reward and $O(T^{1-γ/2})$ regret bound for the constraint violation, where $T$ is the total number of steps. We note that these are the first results on regret analysis for MDP with long-term constraints, where the transition probabilities are not known apriori.

Qinbo Bai, Vaneet Aggarwal, Ather Gattami

In the optimization of dynamic systems, the variables typically have constraints. Such problems can be modeled as a Constrained Markov Decision Process (CMDP). This paper considers the peak Constrained Markov Decision Process (PCMDP), where the agent chooses the policy to maximize total reward in the finite horizon as well as satisfy constraints at each epoch with probability 1. We propose a model-free algorithm that converts PCMDP problem to an unconstrained problem and a Q-learning based approach is applied. We define the concept of probably approximately correct (PAC) to the proposed PCMDP problem. The proposed algorithm is proved to achieve an $(ε,p)$-PAC policy when the episode $K\geqΩ(\frac{I^2H^6SA\ell}{ε^2})$, where $S$ and $A$ are the number of states and actions, respectively. $H$ is the number of epochs per episode. $I$ is the number of constraint functions, and $\ell=\log(\frac{SAT}{p})$. We note that this is the first result on PAC kind of analysis for PCMDP with peak constraints, where the transition dynamics are not known apriori. We demonstrate the proposed algorithm on an energy harvesting problem and a single machine scheduling problem, where it performs close to the theoretical upper bound of the studied optimization problem.

Hanwei Wu, Ather Gattami, Markus Flierl

We present a representation learning framework for financial time series forecasting. One challenge of using deep learning models for finance forecasting is the shortage of available training data when using small datasets. Direct trend classification using deep neural networks trained on small datasets is susceptible to the overfitting problem. In this paper, we propose to first learn compact representations from time series data, then use the learned representations to train a simpler model for predicting time series movements. We consider a class-conditioned latent variable model. We train an encoder network to maximize the mutual information between the latent variables and the trend information conditioned on the encoded observed variables. We show that conditional mutual information maximization can be approximated by a contrastive loss. Then, the problem is transformed into a classification task of determining whether two encoded representations are sampled from the same class or not. This is equivalent to performing pairwise comparisons of the training datapoints, and thus, improves the generalization ability of the encoder network. We use deep autoregressive models as our encoder to capture long-term dependencies of the sequence data. Empirical experiments indicate that our proposed method has the potential to advance state-of-the-art performance.

Ather Gattami

In this paper, we consider reinforcement learning of Markov Decision Processes (MDP) with peak constraints, where an agent chooses a policy to optimize an objective and at the same time satisfy additional constraints. The agent has to take actions based on the observed states, reward outputs, and constraint-outputs, without any knowledge about the dynamics, reward functions, and/or the knowledge of the constraint-functions. We introduce a game theoretic approach to construct reinforcement learning algorithms where the agent maximizes an unconstrained objective that depends on the simulated action of the minimizing opponent which acts on a finite set of actions and the output data of the constraint functions (rewards). We show that the policies obtained from maximin Q-learning converge to the optimal policies. To the best of our knowledge, this is the first time learning algorithms guarantee convergence to optimal stationary policies for the MDP problem with peak constraints for both discounted and expected average rewards.

Seungil You, Ather Gattami, John C. Doyle

This paper reformulates and streamlines the core tools of robust stability and performance for LTI systems using now-standard methods in convex optimization. In particular, robustness analysis can be formulated directly as a primal convex (semidefinite program or SDP) optimization problem using sets of gramians whose closure is a semidefinite cone. This allows various constraints such as structured uncertainty to be included directly, and worst-case disturbances and perturbations constructed directly from the primal variables. Well known results such as the KYP lemma and various scaled small gain tests can also be obtained directly through standard SDP duality. To readers familiar with robustness and SDPs, the framework should appear obvious, if only in retrospect. But this is also part of its appeal and should enhance pedagogy, and we hope suggest new research. There is a key lemma proving closure of a grammian that is also obvious but our current proof appears unnecessarily cumbersome, and a final aim of this paper is to enlist the help of experts in robust control and convex optimization in finding simpler alternatives.

Seungil You, Ather Gattami

This paper proposes a direct, and simple approach to the H infinity norm calculation in more general settings. In contrast to the method based on the Kalman-Yakubovich-Popov lemma, our approach does not require a controllability assumption, and returns a sinusoidal input that achieves the H infinity norm of the system including its frequency. In addition, using a semidefinite programming duality, we present a new proof of the Kalman- Yakubovich-Popov lemma, and make a connection between strong duality and controllability. Finally, we generalize our approach towards the generalized Kalman-Yakubovich-Popov lemma, which considers input signals within a finite spectrum.