We introduce Triangular-Reference Schrödinger Bridges for Time Series (TR-SBTS), a conservative extension of the SBTS framework in which the Brownian reference is replaced by an intervalwise frozen, possibly degenerate diffusion reference, triangular across a hierarchy of latent volatility levels. The construction is a single entropy projection on the augmented state space, with the variational constraint imposed jointly across time and the latent levels and unfolded hierarchically by the disintegration of relative entropy. The variational core of SBTS is preserved: the entropy minimiser is the h-transform of the reference, and on each frozen interval the optimal dynamics admit a logarithmic-gradient drift formula on the affine leaves of the active covariance directions, valid even when the frozen covariance is rank-deficient. We establish stability of the frozen approximation and convergence of the corresponding regularised kernel estimators. The construction is realised through a finite-dimensional conditioning map assembled from three complementary reductions of the past -- a block PCR summary, a reference-aware Mahalanobis kernel on past increments induced by the runtime frozen covariance cumulants, and a past-window WLS drift regressor under the same reference metric -- together with a coupled state-covariance bridge step in which each latent level produces a dynamic reference for the level above, summarised by a covariance descriptor; the construction is evaluated on numerical experiments.

Robustness of neural networks is commonly quantified via local or global Lipschitz constants. However, Lipschitz continuity can be overly coarse or overly restrictive as global robustness measure, failing to capture nuanced, data-dependent behavior. We propose a data-driven, architecture-agnostic framework based on the discrete modulus of continuity (DMOC), a non linear generalization of Lipschitz continuity that provides a finer notion of robustness. Unlike many existing approaches, DMOC does not require access to model internals and instead evaluates regularity relative to the data distribution. This shifts the focus from the model to the data, which provide a data-driven baseline of regularity against which the network's robustness is assessed. We establish convergence results for DMOC-induced seminorms with explicit data-driven rates in terms of the separation distance, and introduce a scalable minibatch algorithm that reduces the quadratic cost of exact computation, enabling application to large-scale data sets such as ImageNet. Empirically, DMOC serves as an architecture independent diagnostic: it distinguishes trained from untrained networks, reveals underfitting and overfitting regimes, and yields, as a special case, tight Lipschitz estimates comparable to state-of-the-art method such as ECLipsE and ECLipsE-fast.

Probabilistic conditioning is concerned with the identification of a distribution of a random variable $X$ given a random variable $Y$. It is a cornerstone of scientific and engineering applications where modeling uncertainty is key. This problem has traditionally been addressed in machine learning by directly learning the conditional distribution of a fixed joint distribution. This paper introduces a novel perspective: we propose to solve the conditioning problem by identifying a single operator that maps any joint density to its conditional, thus amortizing over joint-conditional pairs. We establish that the conditioning operator can be approximated to arbitrary accuracy by neural operators. Our proof relies on new results establishing continuity of the conditioning operator over suitable classes of densities. Finally, we learn the conditioning map for a class of Gaussian mixtures using neural operators, illustrating the promise of our framework. This work provides the theoretical underpinnings for general-purpose, amortized methods for probabilistic conditioning, such as foundation models for Bayesian inference.

Algorithms in machine learning and AI do critically depend on at least three key components: (i) the risk function, which is the expectation of the loss function, (ii) the function space, which is often called the hypothesis space, and (iii) the set of probability measures, which are allowed for the specified algorithm. This paper gives a survey of a certain class of loss functions, which we call ratio-based. In supervised learning, margin-based loss functions for classification tasks depending on the product of the output values $y_i$ and the predictions $f(x_i)$ as well as distance-based loss functions depending on the difference of $y_i$ and $f(x_i)$ for regression are common. Distance-based loss functions are in particular useful, if an additive model assumption seems plausible, i.e. the common signal plus noise assumption. However, in the literature, several loss functions proposed for regression purposes have a multiplicative error structure in mind and pay attention to relative errors, i.e. to the ratio of $y_i$ and $f(x_i)$. In this survey article, we systematically investigate such ratio-based loss functions and propose a few new losses, which may be interesting for future research. We concentrate on investigating general properties of ratio-based loss functions like continuity, Lipschitz-continuity, convexity, and differentiability, because these properties play a central role in most machine learning algorithms. Therefore, we do not focus on some specific machine learning algorithm to derive universal consistency, learning rates, or stability results. Instead, we want to enable future research in this direction.

Although application examples of multilevel optimization have already been discussed since the 1990s, the development of solution methods was almost limited to bilevel cases due to the difficulty of the problem. In recent years, in machine learning, Franceschi et al. have proposed a method for solving bilevel optimization problems by replacing their lower-level problems with the T steepest descent update equations with some prechosen iteration number T. In this paper, we have developed a gradient-based algorithm for multilevel optimization with n levels based on their idea and proved that our reformulation asymptotically converges to the original multilevel problem. As far as we know, this is one of the first algorithms with some theoretical guarantee for multilevel optimization. Numerical experiments show that a trilevel hyperparameter learning model considering data poisoning produces more stable prediction results than an existing bilevel hyperparameter learning model in noisy data settings.

Policy gradient lies at the core of deep reinforcement learning (RL) in continuous domains. Despite much success, it is often observed in practice that RL training with policy gradient can fail for many reasons, even on standard control problems with known solutions. We propose a framework for understanding one inherent limitation of the policy gradient approach: the optimization landscape in the policy space can be extremely non-smooth or fractal for certain classes of MDPs, such that there does not exist gradient to be estimated in the first place. We draw on techniques from chaos theory and non-smooth analysis, and analyze the maximal Lyapunov exponents and Hölder exponents of the policy optimization objectives. Moreover, we develop a practical method that can estimate the local smoothness of objective function from samples to identify when the training process has encountered fractal landscapes. We show experiments to illustrate how some failure cases of policy optimization can be explained by such fractal landscapes.

Policy gradient lies at the core of deep reinforcement learning (RL) in continuous domains. Despite much success, it is often observed in practice that RL training with policy gradient can fail for many reasons, even on standard control problems with known solutions. We propose a framework for understanding one inherent limitation of the policy gradient approach: the optimization landscape in the policy space can be extremely non-smooth or fractal for certain classes of MDPs, such that there does not exist gradient to be estimated in the first place. We draw on techniques from chaos theory and non-smooth analysis, and analyze the maximal Lyapunov exponents and Hölder exponents of the policy optimization objectives. Moreover, we develop a practical method that can estimate the local smoothness of objective function from samples to identify when the training process has encountered fractal landscapes. We show experiments to illustrate how some failure cases of policy optimization can be explained by such fractal landscapes.

We extend the traditional worst-case, minimax analysis of stochastic convex optimization by introducing a localized form of minimax complexity for individual functions. Our main result gives function-specific lower and upper bounds on the number of stochastic subgradient evaluations needed to optimize either the function or its "hardest local alternative" to a given numerical precision. The bounds are expressed in terms of a localized and computational analogue of the modulus of continuity that is central to statistical minimax analysis. We show how the computational modulus of continuity can be explicitly calculated in concrete cases, and relates to the curvature of the function at the optimum. We also prove a superefficiency result that demonstrates it is a meaningful benchmark, acting as a computational analogue of the Fisher information in statistical estimation. The nature and practical implications of the results are demonstrated in simulations.