A Scalable Walsh-Hadamard Regularizer to Overcome the Low-degree Spectral Bias of Neural Networks

Despite the capacity of neural nets to learn arbitrary functions, models trained through gradient descent often exhibit a bias towards ``simpler'' functions. Various notions of simplicity have been introduced to characterize this behavior. Here, we focus on the case of neural networks with discrete (zero-one) inputs through the lens of their Fourier (Walsh-Hadamard) transforms, where the notion of simplicity can be captured through the \emph{degree} of the Fourier coefficients. We empirically show that neural networks have a tendency to learn lower-degree frequencies. We show how this spectral bias towards simpler features can in fact \emph{hurt} the neural network's generalization on real-world datasets. To remedy this we propose a new scalable functional regularization scheme that aids the neural network to learn higher degree frequencies. Our regularizer also helps avoid erroneous identification of low-degree frequencies, which further improves generalization. We extensively evaluate our regularizer on synthetic datasets to gain insights into its behavior. Finally, we show significantly improved generalization on four different datasets compared to standard neural networks and other relevant baselines.

Safe Deep RL for Intraoperative Planning of Pedicle Screw Placement

Spinal fusion surgery requires highly accurate implantation of pedicle screw implants, which must be conducted in critical proximity to vital structures with a limited view of anatomy. Robotic surgery systems have been proposed to improve placement accuracy, however, state-of-the-art systems suffer from the limitations of open-loop approaches, as they follow traditional concepts of preoperative planning and intraoperative registration, without real-time recalculation of the surgical plan. In this paper, we propose an intraoperative planning approach for robotic spine surgery that leverages real-time observation for drill path planning based on Safe Deep Reinforcement Learning (DRL). The main contributions of our method are (1) the capability to guarantee safe actions by introducing an uncertainty-aware distance-based safety filter; and (2) the ability to compensate for incomplete intraoperative anatomical information, by encoding a-priori knowledge about anatomical structures with a network pre-trained on high-fidelity anatomical models. Planning quality was assessed by quantitative comparison with the gold standard (GS) drill planning. In experiments with 5 models derived from real magnetic resonance imaging (MRI) data, our approach was capable of achieving 90% bone penetration with respect to the GS while satisfying safety requirements, even under observation and motion uncertainty. To the best of our knowledge, our approach is the first safe DRL approach focusing on orthopedic surgeries.

Hallucinated Adversarial Control for Conservative Offline Policy Evaluation

We study the problem of conservative off-policy evaluation (COPE) where given an offline dataset of environment interactions, collected by other agents, we seek to obtain a (tight) lower bound on a policy's performance. This is crucial when deciding whether a given policy satisfies certain minimal performance/safety criteria before it can be deployed in the real world. To this end, we introduce HAMBO, which builds on an uncertainty-aware learned model of the transition dynamics. To form a conservative estimate of the policy's performance, HAMBO hallucinates worst-case trajectories that the policy may take, within the margin of the models' epistemic confidence regions. We prove that the resulting COPE estimates are valid lower bounds, and, under regularity conditions, show their convergence to the true expected return. Finally, we discuss scalable variants of our approach based on Bayesian Neural Networks and empirically demonstrate that they yield reliable and tight lower bounds in various continuous control environments.

Aligned Diffusion Schrödinger Bridges

Diffusion Schr\"odinger bridges (DSB) have recently emerged as a powerful framework for recovering stochastic dynamics via their marginal observations at different time points. Despite numerous successful applications, existing algorithms for solving DSBs have so far failed to utilize the structure of aligned data, which naturally arises in many biological phenomena. In this paper, we propose a novel algorithmic framework that, for the first time, solves DSBs while respecting the data alignment. Our approach hinges on a combination of two decades-old ideas: The classical Schr\"odinger bridge theory and Doob's $h$-transform. Compared to prior methods, our approach leads to a simpler training procedure with lower variance, which we further augment with principled regularization schemes. This ultimately leads to sizeable improvements across experiments on synthetic and real data, including the tasks of rigid protein docking and temporal evolution of cellular differentiation processes.

Linear Partial Monitoring for Sequential Decision-Making: Algorithms, Regret Bounds and Applications

Partial monitoring is an expressive framework for sequential decision-making with an abundance of applications, including graph-structured and dueling bandits, dynamic pricing and transductive feedback models. We survey and extend recent results on the linear formulation of partial monitoring that naturally generalizes the standard linear bandit setting. The main result is that a single algorithm, information-directed sampling (IDS), is (nearly) worst-case rate optimal in all finite-action games. We present a simple and unified analysis of stochastic partial monitoring, and further extend the model to the contextual and kernelized setting.

Learning To Dive In Branch And Bound

Primal heuristics are important for solving mixed integer linear programs, because they find feasible solutions that facilitate branch and bound search. A prominent group of primal heuristics are diving heuristics. They iteratively modify and resolve linear programs to conduct a depth-first search from any node in the search tree. Existing divers rely on generic decision rules that fail to exploit structural commonality between similar problem instances that often arise in practice. Therefore, we propose L2Dive to learn specific diving heuristics with graph neural networks: We train generative models to predict variable assignments and leverage the duality of linear programs to make diving decisions based on the model's predictions. L2Dive is fully integrated into the open-source solver SCIP. We find that L2Dive outperforms standard divers to find better feasible solutions on a range of combinatorial optimization problems. For real-world applications from server load balancing and neural network verification, L2Dive improves the primal-dual integral by up to 7% (35%) on average over a tuned (default) solver baseline and reduces average solving time by 20% (29%).

Near-optimal Policy Identification in Active Reinforcement Learning

Many real-world reinforcement learning tasks require control of complex dynamical systems that involve both costly data acquisition processes and large state spaces. In cases where the transition dynamics can be readily evaluated at specified states (e.g., via a simulator), agents can operate in what is often referred to as planning with a \emph{generative model}. We propose the AE-LSVI algorithm for best-policy identification, a novel variant of the kernelized least-squares value iteration (LSVI) algorithm that combines optimism with pessimism for active exploration (AE). AE-LSVI provably identifies a near-optimal policy \emph{uniformly} over an entire state space and achieves polynomial sample complexity guarantees that are independent of the number of states. When specialized to the recently introduced offline contextual Bayesian optimization setting, our algorithm achieves improved sample complexity bounds. Experimentally, we demonstrate that AE-LSVI outperforms other RL algorithms in a variety of environments when robustness to the initial state is required.

Model-based Causal Bayesian Optimization

How should we intervene on an unknown structural causal model to maximize a downstream variable of interest? This optimization of the output of a system of interconnected variables, also known as causal Bayesian optimization (CBO), has important applications in medicine, ecology, and manufacturing. Standard Bayesian optimization algorithms fail to effectively leverage the underlying causal structure. Existing CBO approaches assume noiseless measurements and do not come with guarantees. We propose model-based causal Bayesian optimization (MCBO), an algorithm that learns a full system model instead of only modeling intervention-reward pairs. MCBO propagates epistemic uncertainty about the causal mechanisms through the graph and trades off exploration and exploitation via the optimism principle. We bound its cumulative regret, and obtain the first non-asymptotic bounds for CBO. Unlike in standard Bayesian optimization, our acquisition function cannot be evaluated in closed form, so we show how the reparameterization trick can be used to apply gradient-based optimizers. Empirically we find that MCBO compares favorably with existing state-of-the-art approaches.

PAC-Bayesian Meta-Learning: From Theory to Practice

Meta-Learning aims to accelerate the learning on new tasks by acquiring useful inductive biases from related data sources. In practice, the number of tasks available for meta-learning is often small. Yet, most of the existing approaches rely on an abundance of meta-training tasks, making them prone to overfitting. How to regularize the meta-learner to ensure generalization to unseen tasks, is a central question in the literature. We provide a theoretical analysis using the PAC-Bayesian framework and derive the first bound for meta-learners with unbounded loss functions. Crucially, our bounds allow us to derive the PAC-optimal hyper-posterior (PACOH) - the closed-form-solution of the PAC-Bayesian meta-learning problem, thereby avoiding the reliance on nested optimization, giving rise to an optimization problem amenable to standard variational methods that scale well. Our experiments show that, when instantiating the PACOH with Gaussian processes and Bayesian Neural Networks as base learners, the resulting methods are more scalable, and yield state-of-the-art performance, both in terms of predictive accuracy and the quality of uncertainty estimates. Finally, thanks to the principled treatment of uncertainty, our meta-learners can also be successfully employed for sequential decision problems.

Instance-Dependent Generalization Bounds via Optimal Transport

Existing generalization bounds fail to explain crucial factors that drive generalization of modern neural networks. Since such bounds often hold uniformly over all parameters, they suffer from over-parametrization, and fail to account for the strong inductive bias of initialization and stochastic gradient descent. As an alternative, we propose a novel optimal transport interpretation of the generalization problem. This allows us to derive instance-dependent generalization bounds that depend on the local Lipschitz regularity of the earned prediction function in the data space. Therefore, our bounds are agnostic to the parametrization of the model and work well when the number of training samples is much smaller than the number of parameters. With small modifications, our approach yields accelerated rates for data on low-dimensional manifolds, and guarantees under distribution shifts. We empirically analyze our generalization bounds for neural networks, showing that the bound values are meaningful and capture the effect of popular regularization methods during training.