Sanjit Dandapanthula

Sanjit Dandapanthula, Nicholas M. Boffi

Reward guidance algorithms steer a learned generative process toward the reward-tilted measure at inference time. While empirically powerful, these methods are prone to reward hacking: the guided model over-optimizes the reward at the cost of fidelity to the learned distribution. Prior work has attributed this to the complexity of neural reward functions or implicit biases in diffusion training, but its fundamental origins remain poorly understood. We show that reward hacking arises from an approximation made in most practical implementations of reward-guided diffusion -- finite-particle plug-in estimation of the Doob h-function -- even in the simplest non-trivial settings of Gaussian and Gaussian mixture targets with quadratic rewards. In closed form, we isolate two distinct failure modes of the plug-in estimator: it leads to reward hacking within each mode and it cannot select high-reward modes. We propose a closed-form reward damping schedule that corrects the within-mode bias with no additional compute, and clarify the role of best-of-n sampling in compensating for the mode selection failure. Experiments on Gaussian mixture targets, a 2D checkerboard, and FLUX.1 text-to-image generation confirm that our theoretical insights carry over to practical settings.

Sanjit Dandapanthula, Aleksandr Podkopaev, Shiva Prasad Kasiviswanathan et al.

Optimal transport (OT) and Gromov-Wasserstein (GW) alignment provide interpretable geometric frameworks for comparing, transforming, and aggregating heterogeneous datasets -- tasks ubiquitous in data science and machine learning. Because these frameworks are computationally expensive, large-scale applications often rely on closed-form solutions for Gaussian distributions under quadratic cost. This work provides a comprehensive treatment of Gaussian, quadratic cost OT and inner product GW (IGW) alignment, closing several gaps in the literature to broaden applicability. First, we treat the open problem of IGW alignment between uncentered Gaussians on separable Hilbert spaces by giving a closed-form expression up to a quadratic optimization over unitary operators, for which we derive tight analytic upper and lower bounds. If at least one Gaussian measure is centered, the solution reduces to a fully closed-form expression, which we further extend to an analytic solution for the IGW barycenter between centered Gaussians. We also present a reduction of Gaussian multimarginal OT with pairwise quadratic costs to a tractable optimization problem and provide an efficient algorithm to solve it using a rank-deficiency constraint. To demonstrate utility, we apply our results to knowledge distillation and heterogeneous clustering on synthetic and real-world datasets.

Sanjit Dandapanthula, Margaret Johnson, Madeleine Pascolini-Campbell et al.

Accurate and high-resolution estimation of land surface temperature (LST) is crucial in estimating evapotranspiration, a measure of plant water use and a central quantity in agricultural applications. In this work, we develop a novel statistical method for downscaling LST data obtained from NASA's ECOSTRESS mission, using high-resolution data from the Landsat 8 mission as a proxy for modeling agricultural field structure. Using the Landsat data, we identify the boundaries of agricultural fields through edge detection techniques, allowing us to capture the inherent block structure present in the spatial domain. We propose a block-diagonal Gaussian process (BDGP) model that captures the spatial structure of the agricultural fields, leverages independence of LST across fields for computational tractability, and accounts for the change of support present in ECOSTRESS observations. We use the resulting BDGP model to perform Gaussian process regression and obtain high-resolution estimates of LST from ECOSTRESS data, along with uncertainty quantification. Our results demonstrate the practicality of the proposed method in producing reliable high-resolution LST estimates, with potential applications in agriculture, urban planning, and climate studies.

Sanjit Dandapanthula, Aaditya Ramdas

Deep equilibrium models (DEQs) have recently emerged as a powerful paradigm for training infinitely deep weight-tied neural networks that achieve state of the art performance across many modern machine learning tasks. Despite their practical success, theoretically understanding the gradient descent dynamics for training DEQs remains an area of active research. In this work, we rigorously study the gradient descent dynamics for DEQs in the simple setting of linear models and single-index models, filling several gaps in the literature. We prove a conservation law for linear DEQs which implies that the parameters remain trapped on spheres during training and use this property to show that gradient flow remains well-conditioned for all time. We then prove linear convergence of gradient descent to a global minimizer for linear DEQs and deep equilibrium single-index models under appropriate initialization and with a sufficiently small step size. Finally, we validate our theoretical findings through experiments.