Papers

Time series foundation models (TSFMs) are increasingly pre-trained on synthetically generated time series trajectories, where the data generating process is known. Current pre-training recipes are based on loss objectives which compare TSFM outputs to realized future values of each trajectory. We instead propose loss objectives which compare TSFM outputs to the conditional forecast distribution of each trajectory, a procedure we call synthetic data distillation (SDD). SDD corresponds to a Rao-Blackwellization of the training objective, in that it leaves the expectation of stochastic gradients unchanged while provably reducing the covariance of the stochastic gradient under the Loewner partial ordering. We empirically validate SDD on a TSFM model family of sizes from $4$M to $2.5$B parameters, and observe faster convergence of validation loss at every model size: on Gaussian Process data, SDD attains or improves upon the Status Quo loss whilst requiring $10\%-40\%$ less training iterations.

Synthetic data has become a promising way to scale model training beyond limited human-generated data but it may also induce strong model collapse (Dohmatob et al., 2024), where any fixed fraction of synthetic data prevents model performance from improving under data scaling, leaving a non-vanishing excess risk floor. In this paper, we study how synthetic data affects the generalization of one-pass SGD in high-dimensional linear regression with model shift. We establish finite-sample risk bounds for mixed and two-stage training, separating standard bias and variance from source-mismatch effects, namely fluctuation and persistent drift under mixing and filtered initialization bias under two-stage. These bounds reveal a sharp contrast: mixed training induces strong model collapse, while two-stage training avoids the floor by using synthetic data only in the first stage, showing that collapse is not inevitable under a simple data curriculum. Under a random sketch model, we further obtain scaling laws for both protocols, with tight results for mixed training in the optimization-saturated regime. These laws show that larger models may amplify synthetic-induced degradation under mixing, and quantify how high-quality synthetic pretraining may reduce bias in two-stage training. Finally, we establish an exact finite-sample necessary-and-sufficient condition for two-stage training to strictly outperform real-only training under the same real-data budget and identical real-stage updates. Overall, our results highlight that synthetic data is neither inherently harmful nor beneficial; its effect depends critically on both its quality and the training protocol used to incorporate it.

Neural networks can leverage feature superposition to encode more concepts than dimensions, but cross-feature interference constrains the linear accessibility of simultaneously active features. By framing linear accessibility as a compressed sensing problem, we derive high-probability bounds for fixed supports under subgaussian noise, proving the sufficient dimension scales linearly ($d=O_{\varepsilon}(k \log m)$) rather than prior worst-case quadratic limits. We characterize the asymmetry between active and inactive interference and the trade-off between interference and observation-noise budgets. We then validate these bounds across system parameters through Gaussian-tail approximations. We also introduce IHT-SAE, which uses learned iterative refinement to improve feature recovery beyond the limits of linear availability. These results quantify the geometric constraints of the linear representation hypothesis, providing a framework for evaluating sparse autoencoders, compositional generalization, and neural interpretability.

How much risk does a small reweighted training support retain? For finite weighted least squares with the minimum-norm learner, we prove the exact law $\Gamma_d(n)=3-n/d$ throughout $\lceil3d/2\rceil\leq n\leq2d-1$. The guarantee covers every observed feature rank and uses selections that preserve the full feature span. Balanced simplex anchors reduce dimension; positive-weight lifting and independent-line compression close the risk bound. Shifted coordinate pairs attain the matching lower bound. The complete dataset-level upper bound and sharpness construction are verified in Lean 4. At the smaller budget $(d,n)=(5,6)$, we also prove $\Gamma_5(6)=11/5$, matching the simplex-block prediction from $5=3+2$ over arbitrary interacting configurations. Circuit covers, comparison second moments, and circuit-plane probabilities give the sharp excess $6/5$, while polar-face geometry resolves shared rank-three circuits. The general simplex-block frontier connects these laws within the intermediate-budget selection problem.

Membership inference attacks (MIAs) have emerged as critical tools for auditing privacy risks in large language models (LLMs), aiming to determine whether a given text was included in a model's training corpus. However, most existing MIAs require access to per-token logits or probabilities, making them inapplicable in practice to proprietary LLMs that expose only textual continuations. To address this underexplored setting, we propose Word-level Probability MIA (WPMIA), a statistically principled MIA for strict black-box privacy auditing. WPMIA estimates word-level generation probabilities via Monte Carlo sampling with local kernel smoothing, then aggregates these estimates into a sequence-level likelihood estimator. Furthermore, WPMIA constructs the likelihood conditioned on different prefixes, thereby amplifying the distributional differences between members and non-members. We evaluate WPMIA across various open-source LLMs and find that it consistently outperforms existing black-box baselines. Importantly, we also evaluate WPMIA on modern proprietary LLMs, including GPT-5-Chat, Gemini-2.5-Flash, and Claude-4.5-Haiku, achieving an average TPR@5%FPR of 42.0 across these models. These results offer a sound foundation for future research on strict black-box membership inference. Code is available at \href{https://github.com/niusj03/WPMIA}{https://github.com/niusj03/WPMIA}.

We study the oracle complexity of computing a point with small fixed-point residual $\|T(x)-x\| \leq \epsilon$, for a general norm $\|\cdot\|$ and a self-map $T$ of a compact convex set. We study this problem in the setting where $T$ is nonexpansive with respect to the same norm $\|\cdot\|$ and accessed via an unbiased stochastic oracle with bounded variance $\sigma^2$. We provide an algorithm that solves such instances for any norm with a weak Rademacher type $q > 1$, with high probability. The algorithm is based on a recursive anchoring technique. For type-$2$ spaces, such as $\ell_p$-spaces for $p \in [2, \infty]$, our algorithm attains stochastic oracle complexity $\tilde O(\sigma^2 \epsilon^{-3} + \epsilon^{-1})$. We further prove a near-matching lower bound (i.e., matching up to poly-log factors) for such $\ell_{\infty}$-norm instances in high dimensions. Our lower bound holds against any randomized algorithm that succeeds with constant probability. It further extends to settings with “sparse” noise, where variance measured with respect to any $\ell_p$ norm is of the same order, ruling out the possibility of improving oracle complexity as a function of $\varepsilon$ by measuring variance in a non-matching $\ell_p$ norm.

We study the projected power method (PPM) for synchronizing \(n\) unknown permutations of \(m\) objects under a possibly sparse uniform corruption model. Each pair is observed with probability \(p\), and an observed measurement is uncorrupted with probability \(\pi_0\) and is otherwise an independent uniform permutation. Under \(\log m=o(np\pi_0^2)\), we prove exact one-step recovery (with high probability) of each prescribed block for an independent estimate with a fixed positive majority of correct blocks. When \(np\ge C_0\log n\) and \(m=o(np\pi_0^2)\), we prove that one high-probability event yields a block-error contraction simultaneously for every estimate whose optimally aligned error is at most \(0.5-\epsilon\). The contraction factor is \(O(m/(np\pi_0^2))\) and the error floor is \(O(e^{-cnp\pi_0}+e^{-cnp\pi_0^2}+{\log n}/{n})\). Consequently, one update maps every possibly data-dependent estimate in this basin to vanishing block error, and all subsequent iterates remain almost exact uniformly over the iteration index. The one-step and trajectory results extend to independent, non-identically distributed, permutation-valued corruptions with mean \(m^{-1} \mathbf{1}\mathbf{1}^{\top}\). Under the uniform model, a reference-block spectral initializer has aligned block error \(O_{\mathbb P}(m/(np\pi_0^2))\), yielding an end-to-end almost-exact recovery guarantee. Under a stronger all-block signal condition, PPM reaches exact recovery after finitely many iterations. The theory transfers exactly to partial permutations with common support; for varying supports, we establish deterministic and probabilistic co-visibility margins.

We establish nonasymptotic Gaussian approximation bounds in higher-order Wasserstein distance $W_p$, $p\geq2$, for deterministic matrix-weighted sums of multivariate Poisson martingale differences generated by uniformly ergodic Markov chains. Under an $L^{(2+\eta)p}$-moment condition, $\eta>0$, we obtain the explicit bound $$O\left(p^3\|A\|_4^2+pd^{1/4}\|A\|_2^{1/2}\|A\|_4^2\right),$$ where $A\in\mathbb{R}^n$ collects the moment sizes of the weighted increments. When these sizes are of order $n^{-1/2}$, the bound achieves the first optimal $O(n^{-1/2})$ rate for fixed $p$ and $d$. Consequently, we also obtain the first optimal $O(n^{-1/2})$ $W_p$ Gaussian approximation rate for multivariate additive functionals of uniformly ergodic Markov chains. Our analysis develops two techniques for addressing the interplay between higher-order Wasserstein distance and temporal dependence. First, building on the Ornstein–Uhlenbeck relative-score approach of Fang and Koike (2023), we formulate the bound in terms of antisymmetric Stein couplings while retaining the conditional tensor structure. Second, we develop a refresh-then-maximal coupling that combines an independent first-step resampling, which preserves the desired Stein identity, with a subsequent maximal coupling that provides effective control of the coupling increment. We apply these bounds to nonlinear Markovian stochastic approximation by decomposing normalized errors into matrix-weighted Poisson martingale sums and remainders. This yields the first sharp $W_p$ Gaussian approximation rates for both last iterates and Polyak–Ruppert averages. In particular, the optimized rates are $O(n^{-1/3})$ for polynomial-stepsize last iterates and $O(n^{-1/4})$ for Polyak–Ruppert averages, with an $O(n^{-1/2})$ rate for harmonic-stepsize last iterates under an additional spectral condition.

Normalizing flow Boltzmann generators retain a tractable pushforward density, but training with forward KL depends on target samples that may be biased or omit modes. As a result, a flow can miss target mass while its observed importance weights give a high effective sample size. We introduce the log-ratio variation $\X_\omega$, the mean absolute pairwise difference of the target-to-pushforward log-density ratio under a weighting measure $\omega$, and use it to define KLXX, a new loss function. Two log-ratio variations are added to the forward KL (denoted by the two X's): one weighted by the target to improve accuracy, the other by a mixture of quench and temper samples with pushforward samples to search candidate modes. We derive the Fisher–Rao gradient flow of KLXX, where both variations contribute nonpositive dissipation, and a fixed-surrogate error bound for KLXX. We use KLXX in an adaptive-staging Boltzmann generator, with importance reweighting at every stage. We bound the sampling error of its inference scheme when the stage weights are essentially bounded, and prove it asymptotically unbiased in the sample size. In the numerical tests, KLXX improves mode coverage over forward KL. It also improves the generator's per-stage diagnostics against the loss that built the schedule. The observables the generator recovers are close to independent references. The log-ratio variations thus supply information that the forward KL loss usually omits.

Minimum-norm confidence envelope strategies offer a nonparametric approach to constructing nonasymptotic, simultaneous confidence regions for band-limited functions, exploiting the theory of Reproducing Kernel Hilbert Spaces (RKHS). While the finite-sample coverage guarantees of these envelopes have been established, their consistency has not been analyzed so far. In this paper, we study this construction, here termed the Minimum-Norm Confidence Envelope (MiNCE) framework, and establish the strong uniform consistency of the resulting bands, both for noise-free and noisy observation models, under mild assumptions on the measurement noises. We further extend this formulation to the frequency domain, deriving nonasymptotic, simultaneous, strongly uniformly consistent confidence bands for the smoothed spectra. Numerical experiments in nonparametric regression and spectral estimation empirically confirm our theoretical results, illustrating the contraction of the confidence envelopes toward the target function as the sample size increases.

Weak Sparse Identification of Nonlinear Dynamics (WSINDy) provides a noise-robust approach for learning dynamical systems from data without requiring numerical differentiation. However, for high-dimensional systems, tensor-product libraries of candidate functions grow exponentially with the state dimension, making standard WSINDy expensive in both computation and memory. The Multidimensional Approximation of Nonlinear Dynamics (MANDy) addresses this scaling through a tensor-train (TT) representation of the candidate library, but does not provide a mechanism for sparse model selection. Here, we combine these approaches to develop TT-WSINDy, which performs the weak-form transformation, regression, and sparsification in TT format. We show that the TT formulation recovers the corresponding WSINDy regression problem and derive polynomial time and memory complexity bounds for the tensor-train sparsification procedure. Numerical experiments demonstrate robustness to measurement noise and computational savings for high-dimensional systems.

We present a method for recovering the moral graph of a causal DAG from a probability distribution over discrete variables, using fully connected tensor networks (FCTNs) with nuclear-norm-regularized bond corrections. Each bond matrix is parameterized as a baseline all-ones matrix plus a low-rank correction $C_{ij} = U_{ij}V_{ij}^\top$, and the nuclear norm of the correction implemented via the variational Frobenius norm penalty on the factors drives unnecessary bonds to zero. We prove that under faithfulness, positivity, and a no-implicit-rerouting assumption on the local tensor architecture, every optimal FCTN with zero reconstruction error $\varepsilon = 0$ has effective graph exactly equal to the moral graph. For the approximate regime ($\varepsilon > 0$), we provide explicit recovery bounds using the Fannes-Audenaert continuity of conditional mutual information, and derive a sufficient condition on the regularization parameter $\beta$. The effective graph is read directly from the optimized bond matrices.

Sensor-based AI systems are rarely operated under the conditions under which they were trained: devices, personnel and recording epochs change, and each change degrades performance in ways a random train-test split cannot reveal. We propose a staged, accountable evaluation protocol that treats the evaluation of a deployed model as a measurement with declared reference levels and a quantified uncertainty. Four cumulative generalisation stages hold out devices, subjects and time. Each stage is judged on quantiles of repeated trainings against chance references with the correct class count, an out-of-present-scope rate exposes silent misdirection towards classes that are no longer present in deployment relative to training, and an explicit decision rule ties roll-out decisions not to means but to 5% quantiles. We demonstrate the protocol on infrastructure-free geomagnetic localisation with smartphone-based recurrent classifiers in two real underground mines, including a replication of the scheme's training stages at the second site. Unchanged models are re-evaluated on data recorded 34 months after the training campaign, on a device generation unknown at training time and with a held-out surveyor. The 5% quantile of their present-conditioned precision there is 0.39 over 299 repeated trainings, 16.5 times the chance level; across the composition of the 42 reachable location classes the figure varies by +/-0.08, several times the spread between repeated runs. Repeated trainings of a single configuration show why means mislead: a bimodal configuration passes a mean-based test decisively while its 5% quantile lies more than an order of magnitude below chance.

Graph-data workloads such as diffusion estimation, ranking, semi-supervised learning, and network optimization often solve many Laplacian or symmetric diagonally dominant M-matrix (SDDM) systems with the same coefficient matrix. Approximate Cholesky preconditioners eliminate vertices one at a time and store the resulting sparse approximate factorization, the factor, whose construction cost is amortized across these solves. But eliminating a vertex, the pivot, creates a dense Schur-complement clique among its $d$ active neighbors. We introduce CAST (Canonical Approximate Schur Tree), which replaces this clique with a weighted random spanning tree sampled directly from it. Every realization is connected and contains exactly d-1 edges, while reweighting each selected edge by the reciprocal of its tree-inclusion probability makes the update unbiased. The distribution is independent of the ordering of the pivot neighbors, and we prove that its leverage-score marginals minimize the largest normalized reweighted-edge contribution among unbiased inverse-marginal one-tree estimators. We also introduce CAST-$\rho$, which replaces each pivot neighbor with $\rho$ copies, each carrying a $1/\rho$ share of that neighbor's incident weight, samples a weighted random spanning tree on the expanded clique, and contracts the copies back to the original neighborhood. The resulting update remains unbiased and connected, can be sampled exactly in $O(\rho d)$ time, and satisfies a $1/\rho$ bound on the second moment of the normalized local Schur error. Increasing $\rho$ therefore reduces certified local sampling variability, but may increase construction cost and downstream fill. Empirically, we observe that CAST-1 is the faster default, whereas CAST-2 is preferable when its additional edge contributions remain inexpensive.

Repeated-sampling evaluations increasingly extrapolate pass@k far beyond the number n of samples collected per problem. We show that, in the pooled/random-task conditional-Binomial model, fixed-n success counts identify only the n free moments of the latent per-task success distribution. Consequently, direct pass@k is identified for k <= n, but generic extrapolated pass@k, tail exponents, and tail constants are not identified for k > n, even with arbitrarily many exchangeable tasks at the same rollout budget. This is stronger than the observation that the usual estimator is undefined beyond n: it characterizes the information missing from the fixed-depth count-law experiment. We give exact count-law-preserving constructions with incompatible extrapolations, state the exceptional unique-extension case, and compute sharp population identified intervals through Hausdorff principal representations. On the public 10,000-rollout-per-problem release of Brown et al., counterfactual n = 16 evaluations leave failure at k = 1000 ambiguous by factors from 1.5 to over 2,600 across four MATH/GSM8K/CodeContests configurations. The calibration shows that intermediate-scale failure share alone does not determine width. Our result does not reject parametric inference-time scaling laws; it supplies the nonparametric baseline against which their assumptions can be evaluated. We give an exact, conservative one-coordinate finite-task confidence certificate and a reporting standard separating direct estimates, identified sets, and model-conditioned forecasts.

The edge-of-chaos condition preserves first-order input perturbations in wide randomly initialized networks, but physics-informed losses, score matching and derivative regularization depend on higher input derivatives. For smooth scalar-input fully connected networks, using a joint Gaussianity of the finite derivative jet that holds in the infinite-width limit at each fixed depth, we derive mean-field recursions through third order that are exact at the variance fixed point, with finite-depth corrections that decay geometrically. At criticality, the first-derivative variance is depth-invariant, whereas the second-derivative variance grows linearly whenever the activation has nonzero curvature. The resulting third-order system closes on mean-field susceptibilities. For residual networks with branch scale L^{-1/2}, we prove that every fixed finite derivative order has uniformly bounded variance under explicit regularity assumptions. Simulations verify the critical growth laws, the residual bound, and the closed recursion. The results concern initialization, not trained-network performance.

An input may activate few hidden units even when different inputs collectively use an entire network. We study the statistical complexity of this input-dependent sparsity in the one-hidden-layer ReLU model of Awasthi et al. (COLT 2024). For width $s$, at most $k$ active units per input, and effective weight and bias bounds $W,B$, every size-$m$ sample in the class's fixed radius-$R$ input domain satisfies $\mathcal{R}(S)\le CWR\min\{k,\sqrt{sk/m}\log^{3/2}(2m)\}+kB/\sqrt m$. A support-preserving cover and a single normalized chaining argument remove the previous explicit dimension factor, up to logarithms. Lower bounds on appropriate i.i.d. marginals match up to those logarithms, showing how changing active units across inputs retains a width dependence. The input domain matters: zero-bias networks sparse on the entire ball have at most $2k$ nonzero units and complexity $O(kWR/\sqrt m)$, whereas bias bounds comparable to $WR$ restore the worst-case rate on that same domain in only logarithmic dimension. A spherical-cap construction proves the latter claim without assuming sparsity merely on the sampling support. For a specified normalized bounded loss and biases comparable to $WR$, we also obtain agnostic minimax excess-risk bounds of order $\min\{1,\sqrt{s/(km)}\}$ up to logarithms.

Amari's contributions to information geometry and machine learning are well known. Here, we revisit Amari's work on Bayesian duality which has not received as much attention. We connect Amari's Bayesian duality to a convex duality of Bayes' rule. Using this connection, we present a generalization of Amari's Bayesian duality and discuss its relevance for modern artificial intelligence.

A growing body of work establishes that large language models are not mere statistical memorizers, but are capable of in-context learning: performing inference at test time using only examples provided in the prompt, without any parameter updates. Prior theoretical work has shown that this capability extends to supervised learning tasks such as linear regression. We prove that in-context learning extends further to data generation: frozen transformers can simulate iterative generative samplers from in-context samples. We first show that transformers can realize closed-form and smoothed closed-form diffusion samplers. The construction identifies a concrete generative role for softmax attention: it computes responsibility weights and weighted empirical averages, while feedforward layers implement Euler updates. To empirically relate these constructions to pretrained language models, we study semantic-topic sampling: prompts consisting of words drawn from a common semantic category, such as animals, foods, or cities. Across transformer layers, the normalized hidden states exhibit a two-stage geometry: they move toward a uniform spherical reference in intermediate layers and then return to structured, topic-dependent representations near the output. We further measure an interacting-particle energy on these hidden-state clouds and observe the same U-shape pattern. We then prove that transformers can approximate an energy-based sampler, constructing the same U-shape energy across the layers.

Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $\mathcal{X}_k^d:=\{\mathbf{x}\in\{\pm 1\}^d:|\{i:\mathbf{x}_i=1\}|=k\}$, in high-dimensional regimes where $k\ll d$ (i.e., where $\mathcal{X}_k^d$ is highly magnetized). We use our frameworks to design improved samplers for canonical problems in the study of Ising models and Bayesian sparse linear regression. Our first main result considers the Sherrington–Kirkpatrick (SK) model restricted to fixed-magnetization slices $\mathcal{X}_k^d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $\beta>0$, under arbitrary external fields, provided that $k\le c_\beta d$ for an appropriate constant $c_\beta$. By combining this result with an annealing strategy for estimating normalizing constants, we obtain polynomial-time samplers for the SK model at arbitrarily low temperatures under a sufficiently strong external field of strength $h$. In the large-$\beta$ limit, our framework permits sampling at field strengths within constant factors of the Almeida–Thouless line delineating the replica-symmetric and replica-symmetry-breaking regions ([dAT78]), improving polynomially over the field strength $h(\beta)$ required by the recent work of [BAR26]. Our second main result concerns the measurement complexity of polynomial-time Bayesian sparse linear regression. Recent work by [KSTZ25] shows how to sample from the canonical Gaussian spike-and-slab posterior with expected sparsity $k$, at any signal-to-noise ratio, given $n\gtrsim k^3\log^3 d$ Gaussian measurements. We improve this requirement to $n\gtrsim k^{3/2}\log^2 d+k\log^3 d$, using a common sparsity-aware framework underlying both our results.

We study continuous-time relational event data, where time-stamped dyadic interactions reflect both individual node propensities and evolving relational proximity. We propose a dynamic latent space model for inhomogeneous Poisson processes, where event intensities depend on node-specific activity parameters and time-varying latent distances modeled via flexible B-splines. We prove model identifiability by decoupling baseline activity from latent position, ensuring high interaction volumes do not warp the spatial map. For scalability, we develop a minibatch stochastic gradient algorithm with stable initialization and geometric anchoring, alongside an effective-degrees-of-freedom BIC for tuning model complexity. Simulations confirm accurate parameter recovery and out-of-sample prediction. Applied to cooperative diplomatic events among 60 major economies (1995–2022), the model uncovers shifting patterns of international cooperation and isolates mobile geopolitical actors from stationary institutional anchors.

In this paper, we consider the scalar-on-function linear regression model under a realistic sampling scheme in which the functional covariates are observed on a regular grid and contaminated by additive noise. We propose a two-step estimation procedure: first, the underlying curves are reconstructed from the discrete noisy observations using a Fourier-based projection method; second, the slope function is estimated by a penalized least-squares criterion over finite-dimensional trigonometric spaces, with data-driven selection of the model dimension. We establish oracle-type inequalities for the prediction error, both with respect to the reconstructed curves and to the true latent curves. Under regularity assumptions on the slope function and polynomial decay of the eigenvalues of the covariate, we derive convergence rates for the prediction error and show that our estimator attains the minimax rate when the number of grid points is sufficiently large. Finally, the proposed method is illustrated on simulated data and on a real meteorological dataset.

Can the classical Heavy-Ball method, with arbitrary horizon-dependent parameters chosen in advance, achieve Nesterov's $O(T^{-2})$ last-iterate rate on every smooth convex objective? We provide a negative answer. For every horizon $T\ge2$ and every predetermined schedule with nonnegative step sizes and momenta in $[0,1)$, there exists a convex $1$-smooth objective, with initialization distance at most one and zero initial velocity, for which the last iterate of the Heavy-Ball method satisfies \[ f(x_T)-f^\star=\Omega\!\left(\frac{1}{T^\alpha\log T}\right), \qquad \alpha=\frac{1+\sqrt5}{2}. \] Thus even fully nonstationary, horizon-dependent tuning cannot give the classical Heavy-Ball method a Nesterov-rate guarantee on the smooth convex class.

Formulaic alpha discovery is a pool-dependent symbolic search problem in which informative feedback is observed primarily when a complete expression is evaluated. This delayed feedback creates two coupled difficulties: the retained alpha pool does not preserve the full history of realized evaluation feedback, and the value of an intermediate construction action is uncertain because its consequence depends on the formula eventually completed. We introduce AlphaRJM, which addresses these difficulties through Reward-Jump Memory, an event-driven latent state that remains fixed during token construction and updates only at terminal evaluation events using the realized pool reward and evaluation outcome, and an action-conditioned SDE return critic that represents future discounted discovery returns with stochastic particles. The particles guide action selection through their mean and uncertainty and are learned using a distributional Bellman objective combining energy-distance matching, mean calibration, and jump regularization. Empirically, AlphaRJM delivers strong and stable gains across multiple equity universes, forecasting horizons, and random seeds, while ablations confirm the complementary roles of persistent evaluation history, stochastic return modeling, and distributional supervision.

Understanding patient heterogeneity is key to improving prognostic modeling in traumatic brain injury (TBI). Unsupervised clustering is widely used to explore patterns in patient characteristics that may define subgroups. However, it involves a multitude of decisions, including the choice of algorithm, the distance metric, and the method used to determine the "optimal" number of clusters. The aim of this study is to investigate how these choices influence the resulting clustering solution. We analyzed data from 4,509 patients enrolled in the Collaborative European NeuroTrauma Effectiveness Research in TBI (CENTER-TBI) study. K-medoids, agglomerative, and spectral clustering were applied in a complete 3 X 2 X 2 factorial design, in combination with Euclidean or Gower's distances, and silhouette score or gap statistic to choose the number of clusters. We investigated the agreement of clustering solutions with UpSet Plots and stability with the (adjusted) Rand index. Comparisons were made both across approaches using the original dataset and within approaches using bootstrap resampling. Clustering results varied substantially depending on the analysis choices. The number of suggested clusters varied widely, from one to twenty-five. Adjusted Rand indices confirmed low concordance between methods. Moreover, none of the clustering solutions demonstrated discriminatory performance comparable to a supervised logistic regression model in classifying patient recovery illustrating the limited usefulness of clustering for this purpose. The high instability in clustering results compromises interpretability and underscores that such solutions should not be blindly interpreted as underlying structure.

We study distributed one-dimensional mean estimation under a 1-bit communication constraint. Each agent observes one sample, drawn independently from an unknown distribution, and returns a single bit in response to a query $Q: \mathbb{R}\to\{0,1\}$ chosen by a central learner. The distribution has mean in $[-\lambda,\lambda]$ and $k$-th central moment at most $\sigma^k$, for a fixed $k>1$. The order-optimal two-stage protocol of Lau and Scarlett uses responses from the first batch to choose the second-batch queries, motivating the question of whether this single round of interaction is necessary. We answer this negatively: for every $k>1$, a non-adaptive protocol attains the adaptive 1-bit minimax rate (and concurrent works reached the same conclusion via different strategies). We further determine the minimax sample complexity among non-adaptive 1-bit estimators when every one-set $Q^{-1}(1)$ is restricted to a union of at most $s$ intervals. Relative to unrestricted non-adaptive 1-bit querying, this constraint adds a term of order $(\lambda\sigma/(s\varepsilon^2))\log(1/\delta)$, giving the full tradeoff between sample complexity and interval complexity to within $k$-dependent constant factors. As a corollary, we identify, order-wise, the minimum interval budget needed to retain the unrestricted 1-bit minimax sample rate.

We study the time-uniform convergence of the raw iterate of standard stochastic gradient descent (SGD) for unconstrained smooth convex objectives. We prove that, under standard noise assumptions, the time-uniform convergence rate gets arbitrarily close to $\sqrt{\log n / n}$ but never reaches it. More specifically, we prove that for every positive, eventually nondecreasing sequence $h$ satisfying $h(n) = o(\sqrt{n})$, a bound of order $h(n)/\sqrt{n}$, holding simultaneously for all $n$ with probability at least $1-\alpha$ and uniformly over the problem class, is achievable if and only if \[ \sum_{j = 1}^{\infty} \frac{1}{h(2^j)^2} < \infty. \] The constructive sufficiency result follows from a dyadic horizon-free schedule together with an additive conditional-restart inequality. The necessity counterpart applies to every deterministic nonnegative schedule and holds even for a one-dimensional analytic smooth convex objective with Gaussian noise.

Suppose we are given an ordered sequence of independent data whose distribution changes $K$ times at unknown locations, for some unknown $K \geq 0$. In this paper, we study the problem of performing distribution-free inference on $K$. First, we show an impossibility result: any distribution-free upper confidence bound on $K$ must be trivial and uninformative. Then, using conformal $p$-values, and under only the assumption that the data segments induced by the changepoints are exchangeable (within themselves) and mutually independent, we construct a finite-sample valid lower confidence bound on $K$, which we call the Conformal LOwer bound on Changepoint Count (CLOCC). We show that CLOCC is the only feasible way to provide a lower bound on $K$ under the stated assumptions, a property we refer to as its universality. We provide practical guidelines for choosing score functions that yield efficient and tight lower bounds. We evaluate CLOCC in several synthetic and real-data experiments, where it provides informative lower bounds on $K$, demonstrating its practical applicability.

Neural networks acquire internal representations through learning. In this work, we formulate stochastic gradient descent (SGD) as a Markovian stochastic process and derive a Fisher-information flow speed limit that bounds the rate at which trainable parameters can acquire information about latent variables in the data-generating process. The resulting inequality decomposes the information flow into drift and noise contributions, thereby quantifying the roles of deterministic learning forces and SGD-induced fluctuations from an information-theoretic perspective. We verify the bound in analytically tractable basis-function linear regression, where the information budget predicted by the bound reproduces the ordering and characteristic time scales with which different latent variables are encoded in the learned parameters. These results establish Fisher-information speed limits as a quantitative framework for diagnosing when and how different aspects of the data-generating mechanism are acquired during stochastic learning.

Slice sampling is a Markov chain Monte Carlo algorithm that draws its next state uniformly from a "slice"—a super-level set of the target density function—at each iteration, thereby providing automatic local adaptivity to the scale of the target. In practice the exact slice is not known, so general-purpose implementations use an approximate slice that is grown from a starting interval of length $w>0$, with a computational cost that depends on $w$. This work presents an analysis of the average per-iteration number of target density evaluations, as a function of $w$, of hybrid slice sampling with various slice-finding schemes for targets with contiguous slices. The paper uses the results of the analysis to develop automated, slice-adaptive tuning schemes along with suboptimality bounds and asymptotic convergence guarantees. Simulations demonstrate that the tuning schemes reliably yield near-optimal slice-adaptive tuning with essentially no dependence on the initial setting of $w$.