We introduce a framework to study the transformation of problems with manifold constraints into unconstrained problems through parametrizations in terms of a Euclidean space.

The paradigm of differentiable programming has significantly enhanced the scope of machine learning via the judicious use of gradient-based optimization. However, standard differentiable programming methods (such as autodiff) typically require that the machine learning models be differentiable, limiting their applicability. Our goal in this paper is to use a new, principled approach to extend gradient-based optimization to functions well modeled by splines, which encompass a large family of piecewise polynomial models. We derive the form of the (weak) Jacobian of such functions and show that it exhibits a block-sparse structure that can be computed implicitly and efficiently. Overall, we show that leveraging this redesigned Jacobian in the form of a differentiable layer'' in predictive models leads to improved performance in diverse applications such as image segmentation, 3D point cloud reconstruction, and finite element analysis.

We propose a novel method for gradient-based optimization of black-box simulators using differentiable local surrogate models. In fields such as physics and engineering, many processes are modeled with non-differentiable simulators with intractable likelihoods. Optimization of these forward models is particularly challenging, especially when the simulator is stochastic. To address such cases, we introduce the use of deep generative models to iteratively approximate the simulator in local neighborhoods of the parameter space. We demonstrate that these local surrogates can be used to approximate the gradient of the simulator, and thus enable gradient-based optimization of simulator parameters. In cases where the dependence of the simulator on the parameter space is constrained to a low dimensional submanifold, we observe that our method attains minima faster than baseline methods, including Bayesian optimization, numerical optimization and approaches using score function gradient estimators.

Boosting is a highly successful ML-born optimization setting in which one is required to computationally efficiently learn arbitrarily good models based on the access to a weak learner oracle, providing classifiers performing at least slightly differently from random guessing. A key difference with gradient-based optimization is that boosting's original model does not requires access to first order information about a loss, yet the decades long history of boosting has quickly evolved it into a first order optimization setting -- sometimes even wrongfully *defining* it as such. Owing to recent progress extending gradient-based optimization to use only a loss' zeroth ($0^{th}$) order information to learn, this begs the question: what loss functions be efficiently optimized with boosting and what is the information really needed for boosting to meet the *original* boosting blueprint's requirements?We provide a constructive formal answer essentially showing that *any* loss function can be optimized with boosting and thus boosting can achieve a feat not yet known to be possible in the classical $0^{th}$ order setting, since loss functions are not required to be be convex, nor differentiable or Lipschitz -- and in fact not required to be continuous either. Some tools we use are rooted in quantum calculus, the mathematical field -- not to be confounded with quantum computation -- that studies calculus without passing to the limit, and thus without using first order information.

Artificial intelligence methods are increasingly being explored for managing wildfires and other natural hazards. In particular, reinforcement learning (RL) is a promising path towards improving outcomes in such uncertain decision-making scenarios and moving beyond reactive strategies. However, training RL agents requires many environment interactions, and the speed of existing wildfire simulators is a severely limiting factor. We introduce $\texttt{JaxWildfire}$, a simulator underpinned by a principled probabilistic fire spread model based on cellular automata. It is implemented in JAX and enables vectorized simulations using $\texttt{vmap}$, allowing high throughput of simulations on GPUs. We demonstrate that $\texttt{JaxWildfire}$ achieves 6-35x speedup over existing software and enables gradient-based optimization of simulator parameters. Furthermore, we show that $\texttt{JaxWildfire}$ can be used to train RL agents to learn wildfire suppression policies. Our work is an important step towards enabling the advancement of RL techniques for managing natural hazards.

CLAX is a JAX-based library that implements classic click models using modern gradient-based optimization. While neural click models have emerged over the past decade, complex click models based on probabilistic graphical models (PGMs) have not systematically adopted gradient-based optimization, preventing practitioners from leveraging modern deep learning frameworks while preserving the interpretability of classic models. CLAX addresses this gap by replacing EM-based optimization with direct gradient-based optimization in a numerically stable manner. The framework's modular design enables the integration of any component, from embeddings and deep networks to custom modules, into classic click models for end-to-end optimization. We demonstrate CLAX's efficiency by running experiments on the full Baidu-ULTR dataset comprising over a billion user sessions in $\approx$ 2 hours on a single GPU, orders of magnitude faster than traditional EM approaches. CLAX implements ten classic click models, serving both industry practitioners seeking to understand user behavior and improve ranking performance at scale and researchers developing new click models. CLAX is available at: https://github.com/philipphager/clax

We wish to thank the reviewers for their time and thorough reviews. Our goal is to tackle high-dimensional problems (e.g. Ant has an observation space of 111 and action space of 8, we'll All of them suffered from the leaking patch problem. How this keeps planning from exploiting the inaccuracies of DAE is easiest to see in gradient-based optimization. V AE) has a spurious maximum.