Test-time adaptation (TTA) enhances the zero-shot robustness under distribution shifts by leveraging unlabeled test data during inference. Despite notable advances, several challenges still limit its broader applicability. First, most methods rely on backpropagation or iterative optimization, which limits scalability and hinders real-time deployment. Second, they lack explicit modeling of class-conditional feature distributions. This modeling is crucial for producing reliable decision boundaries and calibrated predictions, but it remains underexplored due to the lack of both source data and supervision at test time. In this paper, we propose ADAPT, an Advanced Distribution-Aware and backPropagation-free Test-time adaptation method. We reframe TTA as a Gaussian probabilistic inference task by modeling class-conditional likelihoods using gradually updated class means and a shared covariance matrix. This enables closed-form, training-free inference. To correct potential likelihood bias, we introduce lightweight regularization guided by CLIP priors and a historical knowledge bank. ADAPT requires no source data, no gradient updates, and no full access to target data, supporting both online and transductive settings. Extensive experiments across diverse benchmarks demonstrate that our method achieves state-of-the-art performance under a wide range of distribution shifts with superior scalability and robustness.

Optimization of deep neural networks (DNNs) has been a driving force in the advancement of modern machine learning and artificial intelligence. With DNNs characterized by a prolonged sequence of nonlinear propagation, determining their optimal parameters given an objective naturally fits within the framework of Optimal Control Programming. Such an interpretation of DNNs as dynamical systems has proven crucial in offering a theoretical foundation for principled analysis from numerical equations to physics. In parallel to these theoretical pursuits, this paper focuses on an algorithmic perspective. Our motivated observation is the striking algorithmic resemblance between the Backpropagation algorithm for computing gradients in DNNs and the optimality conditions for dynamical systems, expressed through another backward process known as dynamic programming. Consolidating this connection, where Backpropagation admits a variational structure, solving an approximate dynamic programming up to the first-order expansion leads to a new class of optimization methods exploring higher-order expansions of the Bellman equation. The resulting optimizer, termed Optimal Control Theoretic Neural Optimizer (OCNOpt), enables rich algorithmic opportunities, including layer-wise feedback policies, game-theoretic applications, and higher-order training of continuous-time models such as Neural ODEs. Extensive experiments demonstrate that OCNOpt improves upon existing methods in robustness and efficiency while maintaining manageable computational complexity, paving new avenues for principled algorithmic design grounded in dynamical systems and optimal control theory.

Recent insights have revealed that rate-coding is a primary form of information representation captured by surrogate-gradient-based Backpropagation Through Time (BPTT) in training deep Spiking Neural Networks (SNNs).

Both of the techniques assume bitwise access to the oracle as a classical function.

Computing gradients through backpropagation is crucial to the success of modern deep neural networks.

Neural network approaches that parameterize value functions have succeeded in approximating high-dimensional optimal feedback controllers when the Hamiltonian admits explicit formulas. However, many practical problems, such as the space shuttle reentry problem and bicycle dynamics, among others, may involve implicit Hamiltonians that do not admit explicit formulas, limiting the applicability of existing methods. Rather than directly parameterizing controls, which does not leverage the Hamiltonian's underlying structure, we propose an end-to-end implicit deep learning approach that directly parameterizes the value function to learn optimal control laws. Our method enforces physical principles by ensuring trained networks adhere to the control laws by exploiting the fundamental relationship between the optimal control and the value function's gradient; this is a direct consequence of the connection between Pontryagin's Maximum Principle and dynamic programming. Using Jacobian-Free Backpropagation (JFB), we achieve efficient training despite temporal coupling in trajectory optimization. We show that JFB produces descent directions for the optimal control objective and experimentally demonstrate that our approach effectively learns high-dimensional feedback controllers across multiple scenarios involving implicit Hamiltonians, which existing methods cannot address.

Fine-tuning large language models (LLMs) with backpropagation\textemdash even for a subset of parameters such as LoRA\textemdash can be much more memory-consuming than inference and is often deemed impractical for resource-constrained mobile devices. Alternative methods, such as zeroth-order optimization (ZO), can greatly reduce the memory footprint but come at the cost of significantly slower model convergence (10$\times$ to 100$\times$ more steps than backpropagation). We propose a memory-efficient implementation of backpropagation (MeBP) on mobile devices that provides better trade-off between memory usage and compute time, while converging faster and achieving better performance than the ZO baseline. We verify the effectiveness of MeBP on an iPhone 15 Pro Max and show that various LLMs, ranging from 0.5B to 4B parameters, can be fine-tuned using less than 1GB of memory. We release an example of the MeBP implementation at https://github.com/apple/ml-mebp.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

We propose a second-order (Hessian or Hessian-free) based optimization method for variational inference inspired by Gaussian backpropagation, and argue that quasi-Newton optimization can be developed as well. This is accomplished by generalizing the gradient computation in stochastic backpropagation via a reparametrization trick with lower complexity. As an illustrative example, we apply this approach to the problems of Bayesian logistic regression and variational auto-encoder (V AE). Additionally, we compute bounds on the estimator variance of intractable expectations for the family of Lipschitz continuous function. Our method is practical, scalable and model free. We demonstrate our method on several real-world datasets and provide comparisons with other stochastic gradient methods to show substantial enhancement in convergence rates.