Classical mixture models (MMs) are widely used tractable proposals for approximate inference settings such as variational inference (VI) and importance sampling (IS). Recently, mixture models with negative coefficients, called subtractive mixture models (SMMs), have been proposed as a potentially more expressive alternative. However, how to effectively use SMMs for VI and IS is still an open question as they do not provide latent variable semantics and therefore cannot use sampling schemes for classical MMs. In this work, we study how to circumvent this issue by designing several expectation estimators for IS and learning schemes for VI with SMMs, and we empirically evaluate them for distribution approximation. Finally, we discuss the additional challenges in estimation stability and learning efficiency that they carry and propose ways to overcome them. Code is available at: https://github.com/april-tools/delta-vi.

Uncertainty quantification is crucial in safety-critical systems, where decisions must be made under uncertainty. In particular, we consider the problem of online uncertainty quantification, where data points arrive sequentially. Online conformal prediction is a principled online uncertainty quantification method that dynamically constructs a prediction set at each time step. While existing methods for online conformal prediction provide long-run coverage guarantees without any distributional assumptions, they typically assume a full feedback setting in which the true label is always observed. In this paper, we propose a novel learning method for online conformal prediction with partial feedback from an adaptive adversary-a more challenging setup where the true label is revealed only when it lies inside the constructed prediction set. Specifically, we formulate online conformal prediction as an adversarial bandit problem by treating each candidate prediction set as an arm. Building on an existing algorithm for adversarial bandits, our method achieves a long-run coverage guarantee by explicitly establishing its connection to the regret of the learner. Finally, we empirically demonstrate the effectiveness of our method in both independent and identically distributed (i.i.d.) and non-i.i.d. settings, showing that it successfully controls the miscoverage rate while maintaining a reasonable size of the prediction set.

Converting betting odds into accurate outcome probabilities is a fundamental challenge in order to use betting odds as a benchmark for sports forecasting and market efficiency analysis. In this study, we propose two methods to overcome the limitations of existing conversion methods. Firstly, we propose an odds-only method to convert betting odds to probabilities without using historical data for model fitting. While existing odds-only methods, such as Multiplicative, Shin, and Power exist, they do not adjust for biases or relationships we found in our betting odds dataset, which consists of 90014 football matches across five different bookmakers. To overcome these limitations, our proposed Odds-Only-Equal-Profitability-Confidence (OO-EPC) method aligns with the bookmakers' pricing objectives of having equal confidence in profitability for each outcome. We provide empirical evidence from our betting odds dataset that, for the majority of bookmakers, our proposed OO-EPC method outperforms the existing odds-only methods. Beyond controlled experiments, we applied the OO-EPC method under real-world uncertainty by using it for six iterations of an annual basketball outcome forecasting competition. Secondly, we propose a generalised linear model that utilises historical data for model fitting and then converts betting odds to probabilities. Existing generalised linear models attempt to capture relationships that the Efficient Market Hypothesis already captures. To overcome this shortcoming, our proposed Favourite-Longshot-Bias-Adjusted Generalised Linear Model (FL-GLM) fits just one parameter to capture the favourite-longshot bias, providing a more interpretable alternative. We provide empirical evidence from historical football matches where, for all bookmakers, our proposed FL-GLM outperforms the existing multinomial and logistic generalised linear models.

Empirical studies of trained models often report a transient regime in which signal is detectable in a finite gradient descent time window before overfitting dominates. We provide an analytically tractable random-matrix model that reproduces this phenomenon for gradient flow in a linear teacher--student setting. In this framework, learning occurs when an isolated eigenvalue separates from a noisy bulk, before eventually disappearing in the overfitting regime. The key ingredient is anisotropy in the input covariance, which induces fast and slow directions in the learning dynamics. In a two-block covariance model, we derive the full time-dependent bulk spectrum of the symmetrized weight matrix through a $2\times 2$ Dyson equation, and we obtain an explicit outlier condition for a rank-one teacher via a rank-two determinant formula. This yields a transient Baik-Ben Arous-Péché (BBP) transition: depending on signal strength and covariance anisotropy, the teacher spike may never emerge, emerge and persist, or emerge only during an intermediate time interval before being reabsorbed into the bulk. We map the corresponding phase diagrams and validate the theory against finite-size simulations. Our results provide a minimal solvable mechanism for early stopping as a transient spectral effect driven by anisotropy and noise.

Inductive bias refers to restrictions on the hypothesis class that enable a learning method to generalize effectively from limited data. A canonical example in control is linearity, which underpins low sample-complexity guarantees for stabilization and optimal control. For general nonlinear dynamics, by contrast, guarantees often rely on smoothness assumptions (e.g., Lipschitz continuity) which, when combined with covering arguments, can lead to data requirements that grow exponentially with the ambient dimension. In this paper we argue that data-efficient nonlinear control demands exploiting inductive bias embedded in nature itself, namely, structure imposed by physical laws. Focusing on Hamiltonian systems, we leverage symplectic geometry and intrinsic recurrence on energy level sets to solve target reachability problems. Our approach combines the recurrence property with a recently proposed class of policies, called chain policies, which composes locally certified trajectory segments extracted from demonstrations to achieve target reachability. We provide sufficient conditions for reachability under this construction and show that the resulting data requirements depend on explicit geometric and recurrence properties of the Hamiltonian rather than the state dimension.

Recursive architectures such as Tiny Recursive Models (TRMs) perform implicit reasoning through iterative latent computation, yet the geometric structure of these reasoning trajectories remains poorly understood. We investigate the activation manifold of TRMs during recursive unrolling and find that activations occupy an effectively linear, low-dimensional subspace whose principal directions can be tracked dynamically with cheap power iterations. This suggests that weight-sharing concentrates iterative computation along a small number of dominant eigendirections, and we find that this concentration varies sharply across computational sites. We exploit this structure through LASER (Low-Rank Activation SVD for Efficient Recursion), a dynamic compression framework that maintains an evolving low-rank basis via matrix-free subspace tracking with a fidelity-triggered reset mechanism, achieving ${\sim}60\%$ activation memory savings with no statistically significant accuracy degradation. Our analysis raises questions about how recursive architectures allocate representational capacity during implicit reasoning, and whether this concentration can be exploited to improve the efficiency and stability of latent computation.

Factor-based Structural Equation Modeling (SEM) relies on likelihood-based estimation assuming a nonsingular sample covariance matrix, which breaks down in small-sample settings with $p>n$. To address this, we propose a novel estimation principle that reformulates the covariance structure into self-covariance and cross-covariance components. The resulting framework defines a likelihood-based feasible set combined with a relative error constraint, enabling stable estimation in small-sample settings where $p>n$ for sign and direction. Experiments on synthetic and real-world data show improved stability, particularly in recovering the sign and direction of structural parameters. These results extend covariance-based SEM to small-sample settings and provide practically useful directional information for decision-making.

We derive explicit non-asymptotic PAC-Bayes generalization bounds for Gibbs posteriors, that is, data-dependent distributions over model parameters obtained by exponentially tilting a prior with the empirical risk. Unlike classical worst-case complexity bounds based on uniform laws of large numbers, which require explicit control of the model space in terms of metric entropy (integrals), our analysis yields posterior-averaged risk bounds that can be applied to overparameterized models and adapt to the data structure and the intrinsic model complexity. The bound involves a marginal-type integral over the parameter space, which we analyze using tools from singular learning theory to obtain explicit and practically meaningful characterizations of the posterior risk. Applications to low-rank matrix completion and ReLU neural network regression and classification show that the resulting bounds are analytically tractable and substantially tighter than classical complexity-based bounds. Our results highlight the potential of PAC-Bayes analysis for precise finite-sample generalization guarantees in modern overparameterized and singular models.

Variational inference (VI) is a central tool in modern machine learning, used to approximate an intractable target density by optimising over a tractable family of distributions. As the variational family cannot typically represent the target exactly, guarantees on the quality of the resulting approximation are crucial for understanding which of its properties VI can faithfully capture. Recent work has identified instances in which symmetries of the target and the variational family enable the recovery of certain statistics, even under model misspecification. However, these guarantees are inherently problem-specific and offer little insight into the fundamental mechanism by which symmetry forces statistic recovery. In this paper, we overcome this limitation by developing a general theory of symmetry-induced statistic recovery in variational inference. First, we characterise when variational minimisers inherit the symmetries of the target and establish conditions under which these pin down identifiable statistics. Second, we unify existing results by showing that previously known statistic recovery guarantees in location-scale families arise as special cases of our theory. Third, we apply our framework to distributions on the sphere to obtain novel guarantees for directional statistics in von Mises-Fisher families. Together, these results provide a modular blueprint for deriving new recovery guarantees for VI in a broad range of symmetry settings.

It's the third time Amazon has poured major money into the AI company. Amazon and Anthropic are strengthening their ties once again, with steep financial commitments made on both sides. Today, Amazon announced that it will invest $5 billion in the AI company, along with as much as $20 billion in additional payments if certain milestones are met. This news follows the initial $4 billion investment Amazon made in Anthropic in 2023 and a second $4 billion round from 2024 . On Anthropic's side, it has committed to continued use of Amazon's custom Trainium silicon for its AI models.