Hypercomplex Phase Retrieval

Hypercomplex signal processing (HSP) offers powerful tools for analyzing and processing multidimensional signals by explicitly exploiting inter-dimensional correlations through Clifford algebra. In recent years, hypercomplex formulations of the phase retrieval (PR) problem, wheren a complex-valued signal is recovered from intensity-only measurements, have attracted growing interest. Hypercomplex phase retrieval (HPR) naturally arises in a range of optical imaging and computational sensing applications, where signals are often modeled using quaternion- or octonion-valued representations. Similar to classical PR, HPR problems may involve measurements obtained via complex, hypercomplex, Fourier, or other structured sensing operators. These formulations open new avenues for the development of advanced HSP-based algorithms and theoretical frameworks. This chapter surveys emerging methodologies and applications of HPR, with particular emphasis on optical imaging systems.

Cyber Security of Sensor Systems for State Sequence Estimation: an AI Approach

Sensor systems are extremely popular today and vulnerable to sensor data attacks. Due to possible devastating consequences, counteracting sensor data attacks is an extremely important topic, which has not seen sufficient study. This paper develops the first methods that accurately identify/eliminate only the problematic attacked sensor data presented to a sequence estimation/regression algorithm under a powerful attack model constructed based on known/observed attacks. The approach does not assume a known form for the statistical model of the sensor data, allowing data-driven and machine learning sequence estimation/regression algorithms to be protected. A simple protection approach for attackers not endowed with knowledge of the details of our protection approach is first developed, followed by additional processing for attacks based on protection system knowledge. In the cases tested for which it was designed, experimental results show that the simple approach achieves performance indistinguishable, to two decimal places, from that for an approach which knows which sensors are attacked. For cases where the attacker has knowledge of the protection approach, experimental results indicate the additional processing can be configured so that the worst-case degradation under the additional processing and a large number of sensors attacked can be made significantly smaller than the worst-case degradation of the simple approach, and close to an approach which knows which sensors are attacked, for the same number of attacked sensors with just a slight degradation under no attacks. Mathematical descriptions of the worst-case attacks are used to demonstrate the additional processing will provide similar advantages for cases for which we do not have numerical results. All the data-driven processing used in our approaches employ only unattacked training data.

Value of Information-based Deceptive Path Planning Under Adversarial Interventions

Existing methods for deceptive path planning (DPP) address the problem of designing paths that conceal their true goal from a passive, external observer. Such methods do not apply to problems where the observer has the ability to perform adversarial interventions to impede the path planning agent. In this paper, we propose a novel Markov decision process (MDP)-based model for the DPP problem under adversarial interventions and develop new value of information (VoI) objectives to guide the design of DPP policies. Using the VoI objectives we propose, path planning agents deceive the adversarial observer into choosing suboptimal interventions by selecting trajectories that are of low informational value to the observer. Leveraging connections to the linear programming theory for MDPs, we derive computationally efficient solution methods for synthesizing policies for performing DPP under adversarial interventions. In our experiments, we illustrate the effectiveness of the proposed solution method in achieving deceptiveness under adversarial interventions and demonstrate the superior performance of our approach to both existing DPP methods and conservative path planning approaches on illustrative gridworld problems.

WaveMax: Radar Waveform Design via Convex Maximization of FrFT Phase Retrieval

The ambiguity function (AF) is a critical tool in radar waveform design, representing the two-dimensional correlation between a transmitted signal and its time-delayed, frequency-shifted version. Obtaining a radar signal to match a specified AF magnitude is a bi-variate variant of the well-known phase retrieval problem. Prior approaches to this problem were either limited to a few classes of waveforms or lacked a computable procedure to estimate the signal. Our recent work provided a framework for solving this problem for both band- and time-limited signals using non-convex optimization. In this paper, we introduce a novel approach WaveMax that formulates waveform recovery as a convex optimization problem by relying on the fractional Fourier transform (FrFT)-based AF. We exploit the fact that AF of the FrFT of the original signal is equivalent to a rotation of the original AF. In particular, we reconstruct the radar signal by solving a low-rank minimization problem, which approximates the waveform using the leading eigenvector of a matrix derived from the AF. Our theoretical analysis shows that unique waveform reconstruction is achievable with a sample size no more than three times the signal frequencies or time samples. Numerical experiments validate the efficacy of WaveMax in recovering signals from noiseless and noisy AF, including scenarios with randomly and uniformly sampled sparse data.

Hierarchical Preference Optimization: Learning to achieve goals via feasible subgoals prediction

This work introduces Hierarchical Preference Optimization (HPO), a novel approach to hierarchical reinforcement learning (HRL) that addresses non-stationarity and infeasible subgoal generation issues when solving complex robotic control tasks. HPO leverages maximum entropy reinforcement learning combined with token-level Direct Preference Optimization (DPO), eliminating the need for pre-trained reference policies that are typically unavailable in challenging robotic scenarios. Mathematically, we formulate HRL as a bi-level optimization problem and transform it into a primitive-regularized DPO formulation, ensuring feasible subgoal generation and avoiding degenerate solutions. Extensive experiments on challenging robotic navigation and manipulation tasks demonstrate impressive performance of HPO, where it shows an improvement of up to 35% over the baselines. Furthermore, ablation studies validate our design choices, and quantitative analyses confirm the ability of HPO to mitigate non-stationarity and infeasible subgoal generation issues in HRL.

Robust Stochastic Shortest-Path Planning via Risk-Sensitive Incremental Sampling

With the pervasiveness of Stochastic Shortest-Path (SSP) problems in high-risk industries, such as last-mile autonomous delivery and supply chain management, robust planning algorithms are crucial for ensuring successful task completion while mitigating hazardous outcomes. Mainstream chance-constrained incremental sampling techniques for solving SSP problems tend to be overly conservative and typically do not consider the likelihood of undesirable tail events. We propose an alternative risk-aware approach inspired by the asymptotically-optimal Rapidly-Exploring Random Trees (RRT*) planning algorithm, which selects nodes along path segments with minimal Conditional Value-at-Risk (CVaR). Our motivation rests on the step-wise coherence of the CVaR risk measure and the optimal substructure of the SSP problem. Thus, optimizing with respect to the CVaR at each sampling iteration necessarily leads to an optimal path in the limit of the sample size. We validate our approach via numerical path planning experiments in a two-dimensional grid world with obstacles and stochastic path-segment lengths. Our simulation results show that incorporating risk into the tree growth process yields paths with lengths that are significantly less sensitive to variations in the noise parameter, or equivalently, paths that are more robust to environmental uncertainty. Algorithmic analyses reveal similar query time and memory space complexity to the baseline RRT* procedure, with only a marginal increase in processing time. This increase is offset by significantly lower noise sensitivity and reduced planner failure rates.

DIPPER: Direct Preference Optimization to Accelerate Primitive-Enabled Hierarchical Reinforcement Learning

Learning control policies to perform complex robotics tasks from human preference data presents significant challenges. On the one hand, the complexity of such tasks typically requires learning policies to perform a variety of subtasks, then combining them to achieve the overall goal. At the same time, comprehensive, well-engineered reward functions are typically unavailable in such problems, while limited human preference data often is; making efficient use of such data to guide learning is therefore essential. Methods for learning to perform complex robotics tasks from human preference data must overcome both these challenges simultaneously. In this work, we introduce DIPPER: Direct Preference Optimization to Accelerate Primitive-Enabled Hierarchical Reinforcement Learning, an efficient hierarchical approach that leverages direct preference optimization to learn a higher-level policy and reinforcement learning to learn a lower-level policy. DIPPER enjoys improved computational efficiency due to its use of direct preference optimization instead of standard preference-based approaches such as reinforcement learning from human feedback, while it also mitigates the well-known hierarchical reinforcement learning issues of non-stationarity and infeasible subgoal generation due to our use of primitive-informed regularization inspired by a novel bi-level optimization formulation of the hierarchical reinforcement learning problem. To validate our approach, we perform extensive experimental analysis on a variety of challenging robotics tasks, demonstrating that DIPPER outperforms hierarchical and non-hierarchical baselines, while ameliorating the non-stationarity and infeasible subgoal generation issues of hierarchical reinforcement learning.

PIPER: Primitive-Informed Preference-based Hierarchical Reinforcement Learning via Hindsight Relabeling

In this work, we introduce PIPER: Primitive-Informed Preference-based Hierarchical reinforcement learning via Hindsight Relabeling, a novel approach that leverages preference-based learning to learn a reward model, and subsequently uses this reward model to relabel higher-level replay buffers. Since this reward is unaffected by lower primitive behavior, our relabeling-based approach is able to mitigate non-stationarity, which is common in existing hierarchical approaches, and demonstrates impressive performance across a range of challenging sparse-reward tasks. Since obtaining human feedback is typically impractical, we propose to replace the human-in-the-loop approach with our primitive-in-the-loop approach, which generates feedback using sparse rewards provided by the environment. Moreover, in order to prevent infeasible subgoal prediction and avoid degenerate solutions, we propose primitive-informed regularization that conditions higher-level policies to generate feasible subgoals for lower-level policies. We perform extensive experiments to show that PIPER mitigates non-stationarity in hierarchical reinforcement learning and achieves greater than 50$\%$ success rates in challenging, sparse-reward robotic environments, where most other baselines fail to achieve any significant progress.

Global Optimality without Mixing Time Oracles in Average-reward RL via Multi-level Actor-Critic

In the context of average-reward reinforcement learning, the requirement for oracle knowledge of the mixing time, a measure of the duration a Markov chain under a fixed policy needs to achieve its stationary distribution-poses a significant challenge for the global convergence of policy gradient methods. This requirement is particularly problematic due to the difficulty and expense of estimating mixing time in environments with large state spaces, leading to the necessity of impractically long trajectories for effective gradient estimation in practical applications. To address this limitation, we consider the Multi-level Actor-Critic (MAC) framework, which incorporates a Multi-level Monte Carlo (MLMC) gradient estimator. With our approach, we effectively alleviate the dependency on mixing time knowledge, a first for average-reward MDPs global convergence. Furthermore, our approach exhibits the tightest-available dependence of $\mathcal{O}\left( \sqrt{\tau_{mix}} \right)$ relative to prior work. With a 2D gridworld goal-reaching navigation experiment, we demonstrate that MAC achieves higher reward than a previous PG-based method for average reward, Parameterized Policy Gradient with Advantage Estimation (PPGAE), especially in cases with relatively small training sample budget restricting trajectory length.

Sampling-based Safe Reinforcement Learning for Nonlinear Dynamical Systems

We develop provably safe and convergent reinforcement learning (RL) algorithms for control of nonlinear dynamical systems, bridging the gap between the hard safety guarantees of control theory and the convergence guarantees of RL theory. Recent advances at the intersection of control and RL follow a two-stage, safety filter approach to enforcing hard safety constraints: model-free RL is used to learn a potentially unsafe controller, whose actions are projected onto safe sets prescribed, for example, by a control barrier function. Though safe, such approaches lose any convergence guarantees enjoyed by the underlying RL methods. In this paper, we develop a single-stage, sampling-based approach to hard constraint satisfaction that learns RL controllers enjoying classical convergence guarantees while satisfying hard safety constraints throughout training and deployment. We validate the efficacy of our approach in simulation, including safe control of a quadcopter in a challenging obstacle avoidance problem, and demonstrate that it outperforms existing benchmarks.