Gaussian process (GP) regression provides a strategy for accelerating saddle point searches on high-dimensional energy surfaces by reducing the number of times the energy and its derivatives with respect to atomic coordinates need to be evaluated. The computational overhead in the hyperparameter optimization can, however, be large and make the approach inefficient. Failures can also occur if the search ventures too far into regions that are not represented well enough by the GP model. Here, these challenges are resolved by using geometry-aware optimal transport measures and an active pruning strategy using a summation over Wasserstein-1 distances for each atom-type in farthest-point sampling, selecting a fixed-size subset of geometrically diverse configurations to avoid rapidly increasing cost of GP updates as more observations are made. Stability is enhanced by permutation-invariant metric that provides a reliable trust radius for early-stopping and a logarithmic barrier penalty for the growth of the signal variance. These physically motivated algorithmic changes prove their efficacy by reducing to less than a half the mean computational time on a set of 238 challenging configurations from a previously published data set of chemical reactions. With these improvements, the GP approach is established as, a robust and scalable algorithm for accelerating saddle point searches when the evaluation of the energy and atomic forces requires significant computational effort.

While the phenomenon of grokking, i.e., delayed generalization, has been studied extensively, it remains an open problem whether there is a mathematical framework that characterizes what kind of features will emerge, how and in which conditions it happens, and is closely related to the gradient dynamics of the training, for complex structured inputs. We propose a novel framework, named $\mathbf{Li}_2$, that captures three key stages for the grokking behavior of 2-layer nonlinear networks: (I) Lazy learning, (II) independent feature learning and (III) interactive feature learning. At the lazy learning stage, top layer overfits to random hidden representation and the model appears to memorize, and at the same time, the backpropagated gradient $G_F$ from the top layer now carries information about the target label, with a specific structure that enables each hidden node to learn their representation independently. Interestingly, the independent dynamics follows exactly the gradient ascent of an energy function $E$, and its local maxima are precisely the emerging features. We study whether these local-optima induced features are generalizable, their representation power, and how they change on sample size, in group arithmetic tasks. When hidden nodes start to interact in the later stage of learning, we provably show how $G_F$ changes to focus on missing features that need to be learned. Our study sheds lights on roles played by key hyperparameters such as weight decay, learning rate and sample sizes in grokking, leads to provable scaling laws of feature emergence, memorization and generalization, and reveals why recent optimizers such as Muon can be effective, from the first principles of gradient dynamics. Our analysis can be extended to multi-layers. The code is available at https://github.com/yuandong-tian/understanding/tree/main/ssl/real-dataset/cogo.

Uncertainty in optimization is often represented as stochastic parameters in the optimization model. In Predict-Then-Optimize approaches, predictions of a machine learning model are used as values for such parameters, effectively transforming the stochastic optimization problem into a deterministic one. This two-stage framework is built on the assumption that more accurate predictions result in solutions that are closer to the actual optimal solution. However, providing evidence for this assumption in the context of complex, constrained optimization problems is challenging and often overlooked in the literature. Simulating predictions of machine learning models offers a way to (experimentally) analyze how prediction error impacts solution quality without the need to train real models. Complementing an algorithm from the literature for simulating binary classification, we introduce a new algorithm for simulating predictions of multiclass classifiers. We conduct a computational study to evaluate the performance of these algorithms, and show that classifier performance can be simulated with reasonable accuracy, although some variability is observed. Additionally, we apply these algorithms to assess the performance of a Predict-Then-Optimize algorithm for a machine scheduling problem. The experiments demonstrate that the relationship between prediction error and how close solutions are to the actual optimum is non-trivial, highlighting important considerations for the design and evaluation of decision-making systems based on machine learning predictions.

The performance of quantum simulations heavily depends on the efficiency of noise mitigation techniques and error correction algorithms. Reinforcement has emerged as a powerful strategy to enhance the efficiency of learning and optimization algorithms. In this study, we demonstrate that a reinforced quantum dynamics can exhibit significant robustness against interactions with a noisy environment. We study a quantum annealing process where, through reinforcement, the system is encouraged to maintain its current state or follow a noise-free evolution. A learning algorithm is employed to derive a concise approximation of this reinforced dynamics, reducing the total evolution time and, consequently, the system's exposure to noisy interactions. This also avoids the complexities associated with implementing quantum feedback in such reinforcement algorithms. The efficacy of our method is demonstrated through numerical simulations of reinforced quantum annealing with one- and two-qubit systems under Pauli noise.

Constrained decision-making is essential for designing safe policies in real-world control systems, yet simulated environments often fail to capture real-world adversities. We consider the problem of learning a policy that will maximize the cumulative reward while satisfying a constraint, even when there is a mismatch between the real model and an accessible simulator/nominal model. In particular, we consider the robust constrained Markov decision problem (RCMDP) where an agent needs to maximize the reward and satisfy the constraint against the worst possible stochastic model under the uncertainty set centered around an unknown nominal model. Primal-dual methods, effective for standard constrained MDP (CMDP), are not applicable here because of the lack of the strong duality property. Further, one cannot apply the standard robust value-iteration based approach on the composite value function either as the worst case models may be different for the reward value function and the constraint value function. We propose a novel technique that effectively minimizes the constraint value function--to satisfy the constraints; on the other hand, when all the constraints are satisfied, it can simply maximize the robust reward value function. We prove that such an algorithm finds a policy with at most $ε$ sub-optimality and feasible policy after $O(ε^{-2})$ iterations. In contrast to the state-of-the-art method, we do not need to employ a binary search, thus, we reduce the computation time by at least 4x for smaller value of discount factor ($γ$) and by at least 6x for larger value of $γ$.

We consider binary classification restricted to a class of continuous piecewise linear functions whose decision boundaries are (possibly nonconvex) starshaped polyhedral sets, supported on a fixed polyhedral simplicial fan. We investigate the expressivity of these function classes and describe the combinatorial and geometric structure of the loss landscape, most prominently the sublevel sets, for two loss-functions: the 0/1-loss (discrete loss) and a log-likelihood loss function. In particular, we give explicit bounds on the VC dimension of this model, and concretely describe the sublevel sets of the discrete loss as chambers in a hyperplane arrangement. For the log-likelihood loss, we give sufficient conditions for the optimum to be unique, and describe the geometry of the optimum when varying the rate parameter of the underlying exponential probability distribution.

Safety-critical environments are inherently dynamic. Distribution shifts, emerging vulnerabilities, and evolving requirements demand continuous updates to machine learning models. Yet even benign parameter updates can have unintended consequences, such as catastrophic forgetting in classical models or alignment drift in foundation models. Existing heuristic approaches (e.g., regularization, parameter isolation) can mitigate these effects but cannot certify that updated models continue to satisfy required performance specifications. We address this problem by introducing a framework for provably safe model updates. Our approach first formalizes the problem as computing the largest locally invariant domain (LID): a connected region in parameter space where all points are certified to satisfy a given specification. While exact maximal LID computation is intractable, we show that relaxing the problem to parameterized abstract domains (orthotopes, zonotopes) yields a tractable primal-dual formulation. This enables efficient certification of updates - independent of the data or algorithm used - by projecting them onto the safe domain. Our formulation further allows computation of multiple approximately optimal LIDs, incorporation of regularization-inspired biases, and use of lookahead data buffers. Across continual learning and foundation model fine-tuning benchmarks, our method matches or exceeds heuristic baselines for avoiding forgetting while providing formal safety guarantees.

This paper introduces a comprehensive unified framework for constructing multi-view diffusion geometries through intertwined multi-view diffusion trajectories (MDTs), a class of inhomogeneous diffusion processes that iteratively combine the random walk operators of multiple data views. Each MDT defines a trajectory-dependent diffusion operator with a clear probabilistic and geometric interpretation, capturing over time the interplay between data views. Our formulation encompasses existing multi-view diffusion models, while providing new degrees of freedom for view interaction and fusion. We establish theoretical properties under mild assumptions, including ergodicity of both the point-wise operator and the process in itself. We also derive MDT-based diffusion distances, and associated embeddings via singular value decompositions. Finally, we propose various strategies for learning MDT operators within the defined operator space, guided by internal quality measures. Beyond enabling flexible model design, MDTs also offer a neutral baseline for evaluating diffusion-based approaches through comparison with randomly selected MDTs. Experiments show the practical impact of the MDT operators in a manifold learning and data clustering context.

Large language model (LLM) reinforcement learning has increasingly relied on group-based policy optimization frameworks, such as GRPO and GSPO, to achieve stable fine-tuning at scale. However, a fundamental trade-off persists between optimization granularity and training stability. While GSPO improves robustness via sequence-level optimization, its monolithic treatment of sequences introduces severe inefficiencies: its conservative clipping mechanism indiscriminately discards valid training samples-a phenomenon we term gradient underutilization-and its uniform credit assignment fails to capture the heterogeneous contributions of critical reasoning steps. In this work, we propose Entropy Importance Sampling Policy Optimization (ESPO), a novel framework that reconciles fine-grained control with training stability. ESPO decomposes sequences into groups based on predictive entropy, enabling (1) Entropy-driven Importance Sampling to capture intra-sequence heterogeneity, and (2) Entropy-adaptive Clipping to dynamically allocate trust regions based on model uncertainty. Extensive experiments on mathematical reasoning benchmarks demonstrate that ESPO not only accelerates convergence but also achieves state-of-the-art performance, notably improving accuracy on the challenging HMMT benchmark from 4.4% to 13.13%.

Many emerging applications - such as adversarial training, AI alignment, and robust optimization - can be framed as zero-sum games between neural nets, with von Neumann-Nash equilibria (NE) capturing the desirable system behavior. While such games often involve non-convex non-concave objectives, empirical evidence shows that simple gradient methods frequently converge, suggesting a hidden geometric structure. In this paper, we provide a theoretical framework that explains this phenomenon through the lens of hidden convexity and overparameterization. We identify sufficient conditions - spanning initialization, training dynamics, and network width - that guarantee global convergence to a NE in a broad class of non-convex min-max games. To our knowledge, this is the first such result for games that involve two-layer neural networks. Technically, our approach is twofold: (a) we derive a novel path-length bound for the alternating gradient descent-ascent scheme in min-max games; and (b) we show that the reduction from a hidden convex-concave geometry to two-sided Polyak-Łojasiewicz (PŁ) min-max condition hold with high probability under overparameterization, using tools from random matrix theory.