Exact Recovery Guarantees for Parameterized Non-linear System Identification Problem under Adversarial Attacks

In this work, we study the system identification problem for parameterized non-linear systems using basis functions under adversarial attacks. Motivated by the LASSO-type estimators, we analyze the exact recovery property of a non-smooth estimator, which is generated by solving an embedded $\ell_1$-loss minimization problem. First, we derive necessary and sufficient conditions for the well-specifiedness of the estimator and the uniqueness of global solutions to the underlying optimization problem. Next, we provide exact recovery guarantees for the estimator under two different scenarios of boundedness and Lipschitz continuity of the basis functions. The non-asymptotic exact recovery is guaranteed with high probability, even when there are more severely corrupted data than clean data. Finally, we numerically illustrate the validity of our theory. This is the first study on the sample complexity analysis of a non-smooth estimator for the non-linear system identification problem.

A Black Swan Hypothesis in Markov Decision Process via Irrationality

Black swan events are statistically rare occurrences that carry extremely high risks. A typical view of defining black swan events is heavily assumed to originate from an unpredictable time-varying environments; however, the community lacks a comprehensive definition of black swan events. To this end, this paper challenges that the standard view is incomplete and claims that high-risk, statistically rare events can also occur in unchanging environments due to human misperception of their value and likelihood, which we call as spatial black swan event. We first carefully categorize black swan events, focusing on spatial black swan events, and mathematically formalize the definition of black swan events. We hope these definitions can pave the way for the development of algorithms to prevent such events by rationally correcting human perception

A CMDP-within-online framework for Meta-Safe Reinforcement Learning

Meta-reinforcement learning has widely been used as a learning-to-learn framework to solve unseen tasks with limited experience. However, the aspect of constraint violations has not been adequately addressed in the existing works, making their application restricted in real-world settings. In this paper, we study the problem of meta-safe reinforcement learning (Meta-SRL) through the CMDP-within-online framework to establish the first provable guarantees in this important setting. We obtain task-averaged regret bounds for the reward maximization (optimality gap) and constraint violations using gradient-based meta-learning and show that the task-averaged optimality gap and constraint satisfaction improve with task-similarity in a static environment or task-relatedness in a dynamic environment. Several technical challenges arise when making this framework practical. To this end, we propose a meta-algorithm that performs inexact online learning on the upper bounds of within-task optimality gap and constraint violations estimated by off-policy stationary distribution corrections. Furthermore, we enable the learning rates to be adapted for every task and extend our approach to settings with a competing dynamically changing oracle. Finally, experiments are conducted to demonstrate the effectiveness of our approach.

Pausing Policy Learning in Non-stationary Reinforcement Learning

Real-time inference is a challenge of real-world reinforcement learning due to temporal differences in time-varying environments: the system collects data from the past, updates the decision model in the present, and deploys it in the future. We tackle a common belief that continually updating the decision is optimal to minimize the temporal gap. We propose forecasting an online reinforcement learning framework and show that strategically pausing decision updates yields better overall performance by effectively managing aleatoric uncertainty. Theoretically, we compute an optimal ratio between policy update and hold duration, and show that a non-zero policy hold duration provides a sharper upper bound on the dynamic regret. Our experimental evaluations on three different environments also reveal that a non-zero policy hold duration yields higher rewards compared to continuous decision updates.

Absence of spurious solutions far from ground truth: A low-rank analysis with high-order losses

Matrix sensing problems exhibit pervasive non-convexity, plaguing optimization with a proliferation of suboptimal spurious solutions. Avoiding convergence to these critical points poses a major challenge. This work provides new theoretical insights that help demystify the intricacies of the non-convex landscape. In this work, we prove that under certain conditions, critical points sufficiently distant from the ground truth matrix exhibit favorable geometry by being strict saddle points rather than troublesome local minima. Moreover, we introduce the notion of higher-order losses for the matrix sensing problem and show that the incorporation of such losses into the objective function amplifies the negative curvature around those distant critical points. This implies that increasing the complexity of the objective function via high-order losses accelerates the escape from such critical points and acts as a desirable alternative to increasing the complexity of the optimization problem via over-parametrization. By elucidating key characteristics of the non-convex optimization landscape, this work makes progress towards a comprehensive framework for tackling broader machine learning objectives plagued by non-convexity.

Algorithmic Regularization in Tensor Optimization: Towards a Lifted Approach in Matrix Sensing

Gradient descent (GD) is crucial for generalization in machine learning models, as it induces implicit regularization, promoting compact representations. In this work, we examine the role of GD in inducing implicit regularization for tensor optimization, particularly within the context of the lifted matrix sensing framework. This framework has been recently proposed to address the non-convex matrix sensing problem by transforming spurious solutions into strict saddles when optimizing over symmetric, rank-1 tensors. We show that, with sufficiently small initialization scale, GD applied to this lifted problem results in approximate rank-1 tensors and critical points with escape directions. Our findings underscore the significance of the tensor parametrization of matrix sensing, in combination with first-order methods, in achieving global optimality in such problems.

Tempo Adaption in Non-stationary Reinforcement Learning

We first raise and tackle ``time synchronization'' issue between the agent and the environment in non-stationary reinforcement learning (RL), a crucial factor hindering its real-world applications. In reality, environmental changes occur over wall-clock time ($\mathfrak{t}$) rather than episode progress ($k$), where wall-clock time signifies the actual elapsed time within the fixed duration $\mathfrak{t} \in [0, T]$. In existing works, at episode $k$, the agent rollouts a trajectory and trains a policy before transitioning to episode $k+1$. In the context of the time-desynchronized environment, however, the agent at time $\mathfrak{t}_k$ allocates $\Delta \mathfrak{t}$ for trajectory generation and training, subsequently moves to the next episode at $\mathfrak{t}_{k+1}=\mathfrak{t}_{k}+\Delta \mathfrak{t}$. Despite a fixed total episode ($K$), the agent accumulates different trajectories influenced by the choice of \textit{interaction times} ($\mathfrak{t}_1,\mathfrak{t}_2,...,\mathfrak{t}_K$), significantly impacting the sub-optimality gap of policy. We propose a Proactively Synchronizing Tempo (ProST) framework that computes optimal $\{ \mathfrak{t}_1,\mathfrak{t}_2,...,\mathfrak{t}_K \} (= \{ \mathfrak{t} \}_{1:K})$. Our main contribution is that we show optimal $\{ \mathfrak{t} \}_{1:K}$ trades-off between the policy training time (agent tempo) and how fast the environment changes (environment tempo). Theoretically, this work establishes an optimal $\{ \mathfrak{t} \}_{1:K}$ as a function of the degree of the environment's non-stationarity while also achieving a sublinear dynamic regret. Our experimental evaluation on various high dimensional non-stationary environments shows that the ProST framework achieves a higher online return at optimal $\{ \mathfrak{t} \}_{1:K}$ than the existing methods.

Scalable Primal-Dual Actor-Critic Method for Safe Multi-Agent RL with General Utilities

We investigate safe multi-agent reinforcement learning, where agents seek to collectively maximize an aggregate sum of local objectives while satisfying their own safety constraints. The objective and constraints are described by {\it general utilities}, i.e., nonlinear functions of the long-term state-action occupancy measure, which encompass broader decision-making goals such as risk, exploration, or imitations. The exponential growth of the state-action space size with the number of agents presents challenges for global observability, further exacerbated by the global coupling arising from agents' safety constraints. To tackle this issue, we propose a primal-dual method utilizing shadow reward and $\kappa$-hop neighbor truncation under a form of correlation decay property, where $\kappa$ is the communication radius. In the exact setting, our algorithm converges to a first-order stationary point (FOSP) at the rate of $\mathcal{O}\left(T^{-2/3}\right)$. In the sample-based setting, we demonstrate that, with high probability, our algorithm requires $\widetilde{\mathcal{O}}\left(\epsilon^{-3.5}\right)$ samples to achieve an $\epsilon$-FOSP with an approximation error of $\mathcal{O}(\phi_0^{2\kappa})$, where $\phi_0\in (0,1)$. Finally, we demonstrate the effectiveness of our model through extensive numerical experiments.

Exact Recovery for System Identification with More Corrupt Data than Clean Data

In this paper, we study the system identification problem for linear discrete-time systems under adversaries and analyze two lasso-type estimators. We study both asymptotic and non-asymptotic properties of these estimators in two separate scenarios, corresponding to deterministic and stochastic models for the attack times. Since the samples collected from the system are correlated, the existing results on lasso are not applicable. We show that when the system is stable and the attacks are injected periodically, the sample complexity for the exact recovery of the system dynamics is O(n), where n is the dimension of the states. When the adversarial attacks occur at each time instance with probability p, the required sample complexity for the exact recovery scales as O(\log(n)p/(1-p)^2). This result implies the almost sure convergence to the true system dynamics under the asymptotic regime. As a by-product, even when more than half of the data is compromised, our estimators still learn the system correctly. This paper provides the first mathematical guarantee in the literature on learning from correlated data for dynamical systems in the case when there is less clean data than corrupt data.

Over-parametrization via Lifting for Low-rank Matrix Sensing: Conversion of Spurious Solutions to Strict Saddle Points

This paper studies the role of over-parametrization in solving non-convex optimization problems. The focus is on the important class of low-rank matrix sensing, where we propose an infinite hierarchy of non-convex problems via the lifting technique and the Burer-Monteiro factorization. This contrasts with the existing over-parametrization technique where the search rank is limited by the dimension of the matrix and it does not allow a rich over-parametrization of an arbitrary degree. We show that although the spurious solutions of the problem remain stationary points through the hierarchy, they will be transformed into strict saddle points (under some technical conditions) and can be escaped via local search methods. This is the first result in the literature showing that over-parametrization creates a negative curvature for escaping spurious solutions. We also derive a bound on how much over-parametrization is requited to enable the elimination of spurious solutions.