Score matching is an approach to learning probability distributions parametrized up to a constant of proportionality (e.g. Energy-Based Models). The idea is to fit the score of the distribution, rather than the likelihood, thus avoiding the need to evaluate the constant of proportionality. While there's a clear algorithmic benefit, the statistical "cost'' can be steep: recent work by Koehler et al. 2022 showed that for distributions that have poor isoperimetric properties (a large Poincar\'e or log-Sobolev constant), score matching is substantially statistically less efficient than maximum likelihood. However, many natural realistic distributions, e.g. multimodal distributions as simple as a mixture of two Gaussians in one dimension -- have a poor Poincar\'e constant. In this paper, we show a close connection between the mixing time of a broad class of Markov processes with generator $\mathcal{L}$ and an appropriately chosen generalized score matching loss that tries to fit $\frac{\mathcal{O} p}{p}$. This allows us to adapt techniques to speed up Markov chains to construct better score-matching losses. In particular, ``preconditioning'' the diffusion can be translated to an appropriate ``preconditioning'' of the score loss. Lifting the chain by adding a temperature like in simulated tempering can be shown to result in a Gaussian-convolution annealed score matching loss, similar to Song and Ermon, 2019. Moreover, we show that if the distribution being learned is a finite mixture of Gaussians in $d$ dimensions with a shared covariance, the sample complexity of annealed score matching is polynomial in the ambient dimension, the diameter of the means, and the smallest and largest eigenvalues of the covariance -- obviating the Poincar\'e constant-based lower bounds of the basic score matching loss shown in Koehler et al. 2022.

This dissertation will combine new tools and methodologies to answer pressing questions regarding inundation area and hurricane events in complex, heterogeneous changing environments. In addition to remote sensing approaches, citizen science and machine learning are both emerging fields that harness advancing technology to answer environmental management and disaster response questions. Freshwater lakes supply a large amount of inland water resources to sustain local and regional developments. However, some lake systems depend upon great fluctuation in water surface area.

In this paper, we propose a novel personalized decision support system that combines Theory of Mind (ToM) modeling and explainable Reinforcement Learning (XRL) to provide effective and interpretable interventions. Our method leverages DRL to provide expert action recommendations while incorporating ToM modeling to understand users' mental states and predict their future actions, enabling appropriate timing for intervention. To explain interventions, we use counterfactual explanations based on RL's feature importance and users' ToM model structure. Our proposed system generates accurate and personalized interventions that are easily interpretable by end-users. We demonstrate the effectiveness of our approach through a series of crowd-sourcing experiments in a simulated team decision-making task, where our system outperforms control baselines in terms of task performance. Our proposed approach is agnostic to task environment and RL model structure, therefore has the potential to be generalized to a wide range of applications.

Deep learning for Earth imagery plays an increasingly important role in geoscience applications such as agriculture, ecology, and natural disaster management. Still, progress is often hindered by the limited training labels. Given Earth imagery with limited training labels, a base deep neural network model, and a spatial knowledge base with label constraints, our problem is to infer the full labels while training the neural network. The problem is challenging due to the sparse and noisy input labels, spatial uncertainty within the label inference process, and high computational costs associated with a large number of sample locations. Existing works on neuro-symbolic models focus on integrating symbolic logic into neural networks (e.g., loss function, model architecture, and training label augmentation), but these methods do not fully address the challenges of spatial data (e.g., spatial uncertainty, the trade-off between spatial granularity and computational costs). To bridge this gap, we propose a novel Spatial Knowledge-Infused Hierarchical Learning (SKI-HL) framework that iteratively infers sample labels within a multi-resolution hierarchy. Our framework consists of a module to selectively infer labels in different resolutions based on spatial uncertainty and a module to train neural network parameters with uncertainty-aware multi-instance learning. Extensive experiments on real-world flood mapping datasets show that the proposed model outperforms several baseline methods. The code is available at \url{https://github.com/ZelinXu2000/SKI-HL}.

Transferring latent structure from one environment or problem to another is a mechanism by which humans and animals generalize with very little data. Inspired by cognitive and neurobiological insights, we propose graph schemas as a mechanism of abstraction for transfer learning. Graph schemas start with latent graph learning where perceptually aliased observations are disambiguated in the latent space using contextual information. Latent graph learning is also emerging as a new computational model of the hippocampus to explain map learning and transitive inference. Our insight is that a latent graph can be treated as a flexible template -- a schema -- that models concepts and behaviors, with slots that bind groups of latent nodes to the specific observations or groundings. By treating learned latent graphs (schemas) as prior knowledge, new environments can be quickly learned as compositions of schemas and their newly learned bindings. We evaluate graph schemas on two previously published challenging tasks: the memory & planning game and one-shot StreetLearn, which are designed to test rapid task solving in novel environments. Graph schemas can be learned in far fewer episodes than previous baselines, and can model and plan in a few steps in novel variations of these tasks. We also demonstrate learning, matching, and reusing graph schemas in more challenging 2D and 3D environments with extensive perceptual aliasing and size variations, and show how different schemas can be composed to model larger and more complex environments. To summarize, our main contribution is a unified system, inspired and grounded in cognitive science, that facilitates rapid transfer learning of new environments using schemas via map-induction and composition that handles perceptual aliasing.

We show that marginals of subchains of length $t$ of any finitely correlated translation invariant state on a chain can be learned, in trace distance, with $O(t^2)$ copies -- with an explicit dependence on local dimension, memory dimension and spectral properties of a certain map constructed from the state -- and computational complexity polynomial in $t$. The algorithm requires only the estimation of a marginal of a controlled size, in the worst case bounded by a multiple of the minimum bond dimension, from which it reconstructs a translation invariant matrix product operator. In the analysis, a central role is played by the theory of operator systems. A refined error bound can be proven for $C^*$-finitely correlated states, which have an operational interpretation in terms of sequential quantum channels applied to the memory system. We can also obtain an analogous error bound for a class of matrix product density operators reconstructible by local marginals. In this case, a linear number of marginals must be estimated, obtaining a sample complexity of $\tilde{O}(t^3)$. The learning algorithm also works for states that are only close to a finitely correlated state, with the potential of providing competitive algorithms for other interesting families of states.

Given a pre-trained classifier and multiple human experts, we investigate the task of online classification where model predictions are provided for free but querying humans incurs a cost. In this practical but under-explored setting, oracle ground truth is not available. Instead, the prediction target is defined as the consensus vote of all experts. Given that querying full consensus can be costly, we propose a general framework for online Bayesian consensus estimation, leveraging properties of the multivariate hypergeometric distribution. Based on this framework, we propose a family of methods that dynamically estimate expert consensus from partial feedback by producing a posterior over expert and model beliefs. Analyzing this posterior induces an interpretable trade-off between querying cost and classification performance. We demonstrate the efficacy of our framework against a variety of baselines on CIFAR-10H and ImageNet-16H, two large-scale crowdsourced datasets.

The Multi-Reference Alignment (MRA) problem aims at the recovery of an unknown signal from repeated observations under the latent action of a group of cyclic isometries, in the presence of additive noise of high intensity $\sigma$. It is a more tractable version of the celebrated cryo EM model. In the crucial high noise regime, it is known that its sample complexity scales as $\sigma^6$. Recent investigations have shown that for the practically significant setting of sparse signals, the sample complexity of the maximum likelihood estimator asymptotically scales with the noise level as $\sigma^4$. In this work, we investigate minimax optimality for signal estimation under the MRA model for so-called collision-free signals. In particular, this signal class covers the setting of generic signals of dilute sparsity (wherein the support size $s=O(L^{1/3})$, where $L$ is the ambient dimension. We demonstrate that the minimax optimal rate of estimation in for the sparse MRA problem in this setting is $\sigma^2/\sqrt{n}$, where $n$ is the sample size. In particular, this widely generalizes the sample complexity asymptotics for the restricted MLE in this setting, establishing it as the statistically optimal estimator. Finally, we demonstrate a concentration inequality for the restricted MLE on its deviations from the ground truth.

When the data used for reinforcement learning (RL) are collected by multiple agents in a distributed manner, federated versions of RL algorithms allow collaborative learning without the need for agents to share their local data. In this paper, we consider federated Q-learning, which aims to learn an optimal Q-function by periodically aggregating local Q-estimates trained on local data alone. Focusing on infinite-horizon tabular Markov decision processes, we provide sample complexity guarantees for both the synchronous and asynchronous variants of federated Q-learning. In both cases, our bounds exhibit a linear speedup with respect to the number of agents and near-optimal dependencies on other salient problem parameters. In the asynchronous setting, existing analyses of federated Q-learning, which adopt an equally weighted averaging of local Q-estimates, require that every agent covers the entire state-action space. In contrast, our improved sample complexity scales inverse proportionally to the minimum entry of the average stationary state-action occupancy distribution of all agents, thus only requiring the agents to collectively cover the entire state-action space, unveiling the blessing of heterogeneity in enabling collaborative learning by relaxing the coverage requirement of the single-agent case. However, its sample complexity still suffers when the local trajectories are highly heterogeneous. In response, we propose a novel federated Q-learning algorithm with importance averaging, giving larger weights to more frequently visited state-action pairs, which achieves a robust linear speedup as if all trajectories are centrally processed, regardless of the heterogeneity of local behavior policies.

Sequential planning in large state space and action space quickly becomes intractable due to combinatorial explosion of the search space. Heuristic methods, like monte-carlo tree search, though effective for large state space, but struggle if action space is large. Pure reinforcement learning methods, relying only on reward signals, needs prohibitively large interactions with the environment to device a viable plan. If the state space, observations and actions can be represented in natural language then Large Language models (LLM) can be used to generate action plans. Recently several such goal-directed agents like Reflexion, CLIN, SayCan were able to surpass the performance of other state-of-the-art methods with minimum or no task specific training. But they still struggle with exploration and get stuck in local optima. Their planning capabilities are limited by the limited reasoning capability of the foundational LLMs on text data. We propose a hybrid agent "neoplanner", that synergizes both state space search with queries to foundational LLM to get the best action plan. The reward signals are quantitatively used to drive the search. A balance of exploration and exploitation is maintained by maximizing upper confidence bounds of values of states. In places where random exploration is needed, the LLM is queried to generate an action plan. Learnings from each trial are stored as entity relationships in text format. Those are used in future queries to the LLM for continual improvement. Experiments in the Scienceworld environment reveals a 124% improvement from the current best method in terms of average reward gained across multiple tasks.