Unbalanced optimal transport extends classical optimal transport to measures with different total masses, but statistical guarantees for estimation of the transport map and the growth map remain limited. We study unbalanced optimal transport with quadratic cost and Kullback-Leibler marginal penalties and argue that the natural target of estimation is a transport-growth pair. Consequently, we develop two estimators for the transport-growth pair under two setups: a nearest-neighbor estimator for transporting between general densities, and a kernel-based estimator for transporting between smooth densities. We prove that the kernel-based estimator achieves the minimax optimal rate up to a logarithmic factor by deriving a matching lower bound on the minimax risk. Our main technical contribution is a reduction of estimation to a stability bound on the unbalanced optimal transport value. This bound controls the transport map and growth map risks through the duality gap of the true dual potentials for the estimated measures. These results provide a statistical foundation for estimation of the transport-growth pair in unbalanced optimal transport.
We prove early-stopped, target-to-output KL guarantees for Brownian interpolant Euler sampling from the true source under arbitrary Borel couplings, even with nonintegrable endpoints. At fixed stopping time and noise level, a fourth displacement moment yields an $O(N^{-1})$ discretization term. A second moment suffices for fixed-law convergence with grid refinement and vanishing drift error. For one fixed coupling with no displacement moment above two, the original and comparison paths have KL of orders one and $1/\log N$ relative to the exact-drift Euler path. On the same geometric grid, terminal KL is at most $C/N^2$. The path decomposition identifies the information about the source discarded by Markov projection. When the original bridge is Markov, current-state Euler sampling has an $O((M_2+d)/N)$ discretization term. We prove expected stopped-output KL consistency for a learner trained on iid endpoint pairs with fresh true-source initialization. In a spatial image control, iterative sampling has worse fair energy scores than direct prediction with resampled residuals. Uniform terminal rates for general second-moment couplings remain open.
Stationary tail computation for high-dimensional reflected Brownian motions (RBMs) remains computationally challenging. Building on NN4MGF's shared-coordinate basic-adjoint-relationship (BAR) framework, we develop Holomorphic Cauchy Networks (HCNs). Native complex-valued rational Cauchy subnetworks enforce holomorphicity on pole-free domains, removing sampled Cauchy-Riemann penalties and their input-derivative graphs. Exact origin anchoring and boundary-coordinate exclusion enforce structural identities, while anchored interactions represent dependent transforms. Under holomorphic-log assumptions, we establish simultaneous value and first-derivative approximation for the constrained architectures and a mixed-derivative lower bound for coordinate-only representations under nonzero cross-covariance. Closed-form real-axis derivatives and sequential backpropagation further control memory. On the same NVIDIA H100 NVL, the tested 30D HCN configuration uses approximately tenfold less peak allocated memory and has 9.7-fold lower mean update time than NN4MGF; extrapolation to the respective scheduled update counts gives a projected full-schedule update-time ratio of 58.2. With 64 million sampled base inputs per high-dimensional run, HCN reaches 50 dimensions on a laptop GPU with 6.367% mean relative error over thresholds 1 through 8. Degree-matched comparisons yield lower reported mean relative errors over the primary evaluation ranges, including 1.422% versus 6.684% on the 2D Harrison non-product benchmark. On a 10D correlated RBM benchmark, HCN achieves 3.829% mean relative error over thresholds 1 through 8 against a simulation reference, compared with 19.386% for an independent-marginal baseline.
The running-time distribution of a randomized algorithm is computed under independent randomness, but independence is rarely what a source guarantees. We study sources $\nu$ with $\nu[w]\le DP[w]^s$ for every finite prefix $w$ of the tape, where $P$ is an independent reference law, $0<s\le1$, and $D\ge1$; individual choices may be predictable, even determined by the past. The worst-case probability of any closed tape event $E$ is exactly $\min\{1,DC_s(E)\}$, where $C_s(E)$ is the infimum cost of covering $E$ by prefixes of varying depth. This tree max-flow/min-cut identity makes worst-case probabilities exact and optimizable. Three applications follow. In scheduling, $n$ tasks on one machine are each retried until the next random symbol lies in its success set; every schedule has the same makespan distribution under independent randomness, yet under our constraint two fixed orders miss a linear deadline with worst-case probabilities $1$ and $\exp[-\Theta(n)]$, and a static priority rule is optimal at every deadline. For the standard local walk on a $k$-CNF with an assignment satisfying a majority of each clause, we find the sharp source threshold $s_*$: below it, on some instances one admissible source forces nontermination for every clause selection; at it, with fixed $D$, the expected number of flips is at most cubic in the initial distance, against linear under independence, with matching instances at distances up to a constant fraction of the number of variables; a fixed margin above it restores linear dependence. For noisy predecessor search with prefix-bounded comparison errors, verified backtracking attains any prescribed error probability at or above a source threshold whose sharpness we leave open, with the same cubic-to-linear query bounds. Restarting on the shared tape restores exponential tails exactly when the nontermination event has a prefix cover of cost below one.
We study the sample complexity of learning in average-reward weakly-coupled Markov decision processes (WCMDPs) and Restless Bandits (RBs) under a generative model. Naive reduction to a tabular MDP leads to high complexity bounds as the state-action space is exponentially large in the number of arms $N$. By exploiting the weakly coupled structure, we show that near-optimal policies can be learned with sample and computational complexities that are polynomial in $N$. Specifically, we analyze the plug-in approach, which applies an efficient planning algorithm to an empirical model estimated from data. For fully heterogeneous WCMDPs, we establish the first finite-sample PAC guarantee with polynomial complexity and an $O(1/\sqrt{N})$ optimality gap. For homogeneous RBs, we further prove that a smaller optimality gap is achievable under mild structural assumptions. A primary technical contribution is a novel Lyapunov-based analysis framework. Unlike classical approaches that rely on the difficult-to-control bias function, our framework uses an explicitly constructed Lyapunov function along with a drift transfer technique between the true and empirical models. A key step of independent interest in our framework is a fine-grained perturbation analysis for the underlying linear programming (LP) relaxation, which provides a general tool for analyzing LP-based policies and weakly-coupled systems.
Mode separation, namely how sharply a distribution fragments into barrier-separated clusters, is a fundamental geometric property of densities, difficult to quantify in high dimensions. Modes are often the meaningful alternatives in the data, such as the conformations of a molecule or the distinct semantic types among a generative model's outputs. Mode separation is distinct from dispersion, which is what standard label-free tools capture: differential entropy rises with spread regardless of fragmentation, and PCA orders directions by variance regardless of barriers. Mutual information does capture it, but only given the latent component labels, which are usually unavailable. We measure mode separation through a single stochastic process intrinsic to the density, namely the unique reversible diffusion with $f$ as its stationary distribution and a constant scalar diffusion coefficient. From its autocovariance matrix we extract two readouts. The first, SSA (Sum of Squared Autocorrelations), is a scalar barrier-sensitive measure. The second, DA (Dominant Autocorrelation directions), gives linear projections ordered by metastability rather than variance. Under an isotropic-Gaussian null, we derive the limiting spectrum of the empirical autocovariance in closed form, generalizing the Marchenko--Pastur law. Its upper edge gives a noise floor that separates metastable directions from sampling noise. Both readouts use only samples and a score function, scaling to high dimensions through pretrained score-based generative models via Tweedie's identity. We apply our framework to three settings: (i) synthetic Gaussian mixtures, where SSA tracks mutual information; (ii) SDXL text-to-image generations, where SSA and DA capture structure that entropy and PCA miss; and (iii) molecular dynamics of alanine dipeptide, where DA recovers the known slow backbone dihedrals from static samples alone.
In this note we prove that the Langevin Monte Carlo (LMC) algorithm can minimize the population risk of appropriately regularized depth-2 nets with no restrictions on the size of the net and for any choice of training data --- and we give non-asymptotic convergence rates for it. We achieve this via showing that in q-Renyi divergence, the iterates of LMC converge to the Gibbs distribution of certain Frobenius norm regularized losses for these nets --- in both classification and regression settings. The key technicality of the work is two-fold (a) to keep the amount of regularization needed be independent of the size of the net and (b) to obtain convergence rates that scale with just the Poincare constant of the corresponding Gibbs measure. This work synthesizes several recent observations about isoperimetry conditions under which LMC converges and that two-layer neural loss functions can always be regularized by a constant amount such that they satisfy the Villani conditions, and thus their Gibbs measures satisfy a Poincare inequality.
We propose a robust Q-learning algorithm for Markov decision processes under model uncertainty when each state--action pair is associated with a finite ambiguity set of candidate transition kernels. This finite-measure framework enables user-designed uncertainty models, including competing fitted models and stress scenarios. {The learner observes successor states and their deterministic rewards from every candidate simulator, without requiring explicit knowledge of the transition probabilities or the reward function.} We establish almost sure convergence of the learned Q-function to the robust optimum, and derive a time-uniform high-probability error certificate that separates an observable empirical Bellman residual from transition-kernel estimation error. We also give quantitative finite-ambiguity approximation results for Wasserstein and parametric uncertainty models on finite state spaces. Numerical examples illustrate the effect of scenario selection and compare the tabular algorithm with robust value iteration.
We study two variants of the mirror descent-ascent (MDA) algorithm for solving min-max problems on the space of measures: simultaneous and alternating. We work under assumptions of convexity-concavity and relative smoothness of the payoff function with respect to a suitable Bregman divergence, defined on the space of measures via flat derivatives. We establish non-asymptotic convergence rates to mixed Nash equilibria, measured in the Nikaid\^{o}-Isoda error, proving an $\mathcal{O}(N^{-1/2})$ rate for simultaneous MDA and an improved $\mathcal{O}(N^{-2/3})$ rate for alternating MDA. The main technical contribution is an infinite-dimensional dual space analysis that relates Bregman divergences on measures to dual Bregman divergences on spaces of bounded continuous functions, allowing us to control asymmetric commutator terms created by alternating updates. The results substantially generalize prior analyses restricted to bilinear objectives and also apply to nonlinear convex-concave problems on measure spaces, thereby providing a unified theoretical foundation for MDA in mean-field min-max optimization.
We study causal inference under outcome interference for sequential, observational settings. Specifically, we consider settings where the binary outcomes over N units are Markovian across T time steps. At each time step, the outcomes of N units have dependencies captured through an Ising model; each outcome is also impacted through an external field capturing the effects of its treatment as well as latent confounders. Similar to panel data literature, these latent confounders are modeled to have a low-rank factor structure. Our data is a single sample from this high-dimensional distribution. To estimate causal quantities of interest, we provide a computationally efficient method based on Maximum Pseudo-Likelihood Estimation (MPLE) for learning the model parameters. Under mild assumptions, we establish non-asymptotic consistency for parameter estimation and show this translates to faithful estimation of causal quantities of interest after sampling from the learned model. We demonstrate the efficacy of the method through synthetic experiments as well as a real-world case-study investigating causal effects of vaccine rates on COVID-19 death rates within US counties nationwide.
The Bethe-Hessian matrix, introduced by Saade, Krzakala, and Zdeborov\'a (2014), is a Hermitian matrix designed for applying spectral clustering algorithms to sparse networks. Rather than employing a non-symmetric and high-dimensional non-backtracking operator, a spectral method based on the Bethe-Hessian matrix is conjectured to also reach the Kesten-Stigum detection threshold in the sparse stochastic block model (SBM). We provide the first rigorous analysis of the Bethe-Hessian spectral method in the SBM under both the bounded expected degree and the growing degree regimes. Specifically, we demonstrate that: (i) When the expected degree $d\geq 2$, the number of negative outliers of the Bethe-Hessian matrix can consistently estimate the number of blocks above the Kesten-Stigum threshold, thus confirming a conjecture from Saade, Krzakala, and Zdeborov\'a (2014) for $d\geq 2$. (ii) For sufficiently large $d$, its eigenvectors can be used to achieve weak recovery. (iii) As $d\to\infty$, we establish the concentration of the locations of its negative outlier eigenvalues, and weak consistency can be achieved via a spectral method based on the Bethe-Hessian matrix.
We study the long-context limit of softmax self-attention with a fixed query and a random context of $n$ i.i.d. keys on the sphere with the intrinsic dimension of the key distribution held fixed, viewing the inverse temperature $\beta_n$ as the scaling parameter that decides whether attention degenerates into uniform averaging or collapses onto the single closest key. We show that the critical scale at which selectivity emerges is determined by the local exponent of the distance-to-query distribution near zero rather than by global features of the context, and scales like $\beta_n^\ast \asymp n^{2/(d-1)}$ for uniform keys on $\mathbb{S}^{d-1}$. Furthermore, we characterize the limiting laws of the ordered attention weights and of the attention output across all regimes of $\beta_n$: a subcritical regime in which the output reduces to a local average around $q$ with explicit deterministic bias and Gaussian fluctuations; a critical regime in which a finite collection of nearest keys retains macroscopic mass without single-key collapse; and a supercritical regime in which all mass concentrates on the closest key. Of notable interest is the subcritical case with identity value matrix where the attention map approximately implements a backward heat equation.
Let $(M,d,\mu)$ be a nice geodesic probability space (such as a manifold) and $X=\{x_i\}_{i\in [N]} \subset M$ a finite $\mu$-i.i.d. sample. Suppose we observe noisy distance random variables $d'(x_i,x_j)$. When can the true distances be recovered to small additive error? This question has attracted interest in two disparate fields: latent-space graph models and boundary-value inverse problems. We first show that, taking $N \asymp \epsilon^{-2d-2}|\log\epsilon|$ and using runtime $O(N^{2.5+o_d (1)})$, we can recover all distances in $X$ to additive error $O(\epsilon|\log\epsilon|)$ with high probability, assuming $\mathbb{E} d'(x_i, x_j) = F(d(x_i,x_j))$ for an unknown, increasing bilipschitz $F$ and with the $d'(x_i,x_j)$ allowed to be arbitrarily heterogeneous in their distribution as long as they are bounded in subgaussian norm. This generalizes existing work, which required $F$ to be known and made strict assumptions on the distribution of $d'(x_i, x_j)$. Achieving this generalization required a novel ``ruler-building'' procedure to learn $F$. We extend our results to a missing data case where we randomly lose access to noisy distances. Our second result uses ideas from the first to tackle a partial observability case: $X = Y \cup Z$ with $Z$ sampled from $M$ and $Y$ sampled from an observer subset satisfying visibility conditions that model the assumptions made on the boundary in boundary-value inverse problems. In this setting, we recover all pairwise distances in $Z$ to additive error $O(\epsilon)$ with $|Z| \asymp \epsilon^{-2d-2}|\log\epsilon|$, $|Y| \asymp \epsilon^{-\max\{d_\nu,4\}}$ with $d_\nu$ determined by the volume-growth properties of the probability measure on the observer set, and runtime $O(|Z|^3)$. This is, to our knowledge, the first quantitative guarantee for this observer/observed split under noise, which complements existing stability results for boundary data.
It is known that randomly initialized neural networks have Gaussian behaviour in the $n\to \infty$ limit. Unfortunately, this approximation can be poor for finite $n$ and gets worse as one goes deeper into the network. In this work, we develop a more accurate approximation that is \emph{slightly} non-Gaussian, which we call the para-Gaussian distribution and is constructed from the Hermite polynomials. We show how this para-Gaussian approximation can lead to a practical recursive method to approximate the deeper layers of the network. Finally, we apply this theory to linear neural networks (where things can be solved exactly) and to ReLU neural networks (where things can be approximated).
For a known finite-dimensional model subspace, classical Gaussian regression theory guarantees that least squares eventually attains the parametric rate, with prediction error scaling as the subspace dimension divided by the sample size. Yet subspaces with the same number of unknown coefficients can require very different sample sizes to reach this regime. For subspaces of polynomials on the Boolean cube, a fundamental question is how restrictions on degree and ambient dimension jointly determine this threshold. Our main result sharply characterizes the worst-subspace minimax sample threshold over a substantial parameter range, revealing an exponential dependence on degree and precisely quantifying the joint effects of degree, ambient dimension, and subspace dimension. A small modification of our lower-bound construction reveals a strong separation: a single subspace can admit noiseless recovery with optimal sample complexity independent of degree, yet require essentially the worst-case sample size, exponential in degree, to attain the noisy parametric rate. To establish the sampling lower bound, we sharpen the Polyanskiy-Samorodnitsky uncertainty principle (J. Funct. Anal., '19) to uniformly optimal precision and extend its concentration guarantee to large subspaces in which every function concentrates most of its energy on one shared Hamming ball. The sampling upper bound follows from the sharp moment bound of Kirshner and Samorodnitsky (IEEE Trans. Inf. Theory, '21) combined with empirical-process bounds.
This monograph provides an introduction to mean field reinforcement learning through the lens of Markov decision processes arising from large-population stochastic control with mean field interactions and common noise. Starting from the connection between multi-agent reinforcement learning and mean field control, it develops the probabilistic, mathematical, and control-theoretic framework needed to formulate representative-agent learning problems, analyze their relationship with finite-population systems, and study both general and linear-quadratic models. The presentation includes dynamic programming principles, propagation-of-chaos limits, and theoretical analyses of tabular Q-learning and policy-gradient methods. It also discusses numerical implementations, including tabular schemes and deep reinforcement learning methods such as deep deterministic policy gradient. The goal is to give readers a coherent bridge between mean field control theory and reinforcement learning methodology, emphasizing the mathematical structure of the problems and the design of tractable learning approaches for large stochastic populations.
We develop a quantitative statistical theory of transformers in the large-context regime by adopting the abstraction of contextual flow maps (CFMs): dynamical systems that evolve a distinguished token in the presence of a contextual measure across a stack of attention blocks. Within this framework, the finite-context model approximates an idealized infinite-context system in which the contextual measure is replaced by its underlying population, so that the context length $n$ becomes a statistical resource. Exploiting the McKean--Vlasov structure of the dynamics and the classical machinery of propagation of chaos, we establish a forward bound controlling the deviation between the finite- and infinite-context CFMs uniformly along depth, and a backward bound controlling the deviation between the corresponding training trajectories uniformly across iterations of online gradient descent. Both bounds achieve the optimal Wasserstein rate $n^{-1/d}$ for general CFMs and parametric rate $n^{-1/2}$ for a restricted class of CFMs that includes transformers as a special case. The analysis rests on a new Eulerian adjoint formulation of the loss gradient and stability estimates for the resulting forward--adjoint system, both of which may be of independent interest.
We study stationary solutions of McKean-Vlasov equations on the circle. We establish an exact equivalence between solutions of the stationary McKean-Vlasov equation and an infinite-dimensional quadratic system of equations over Fourier coefficients, allowing explicit characterization of stationary states in a sequence space rather than a function space. This framework provides a transparent description of local bifurcations, their periodicity and resonance structures, while accommodating singular potentials. We derive analytic expressions for the emergence, form and shape (supercritical, critical, subcritical or transcritical) of bifurcations involving possibly multiple Fourier modes and connect them with discontinuous phase transitions. We also characterize, under suitable assumptions, the detailed structure of the stationary bifurcating solutions through leading small-amplitude coefficients for arbitrarily many fixed Fourier modes. At the global level, we establish regularity and concavity properties of the free energy landscape, proving existence, compactness, and coexistence of globally minimizing stationary measures, and identifying discontinuous phase transitions with points of non-differentiability of the minimum free energy map. We apply the theory to the Noisy Mean-Field Transformer model, where we show how changing the inverse temperature $\beta$ affects the geometry of the infinitely many bifurcations from the uniform measure. We also discuss how increasing $\beta$ motivates the study of approximate multi-mode stationary profiles as candidates for a separate metastability analysis; persistence times require additional dynamical estimates. Moreover, we determine the exact threshold $\beta_0\simeq2.447$ separating continuous and discontinuous phase transitions in this model. The same threshold marks the change of the first bifurcation from supercritical to subcritical.
Objective functions based on Hellinger distance yield robust and efficient estimators of model parameters. Motivated by privacy and regulatory requirements encountered in contemporary applications, we derive in this paper \emph{private minimum Hellinger distance estimators}. The estimators satisfy a new privacy constraint, namely, Hellinger differential privacy, while retaining the robustness and efficiency properties. We demonstrate that Hellinger differential privacy shares several features of standard differential privacy while allowing for sharper inference. Additionally, for computational purposes, we also develop Hellinger differentially private gradient descent and Newton-Raphson algorithms. We illustrate the behavior of our estimators in finite samples using numerical experiments and verify that they retain robustness properties under gross-error contamination.
We consider the infinite-horizon average-reward restless bandit problem. We propose a novel two-set policy that maintains two dynamic subsets of arms: one subset of arms has a nearly optimal state distribution and takes actions according to an LP-Priority routine; the other subset of arms is driven towards the optimal state distribution and gradually merged into the first subset. We show that our two-set policy is asymptotically optimal with an $O(\exp(-C N))$ optimality gap for an $N$-armed problem, under the assumptions of aperiodic-unichain, non-degeneracy, and local stability. Our policy is the first to achieve exponential asymptotic optimality under the above set of easy-to-verify assumptions, whereas prior work either requires a global attractor assumption or only achieves an $O(1/\sqrt{N})$ optimality gap. We further discuss obstacles in weakening the assumptions by demonstrating examples where exponential asymptotic optimality is not achievable when any of the three assumptions is violated. Notably, we prove a lower bound for a large class of locally unstable restless bandits, showing that local stability is particularly fundamental for exponential asymptotic optimality. Finally, we use simulations to demonstrate that the two-set policy outperforms previous policies on certain restless bandit problems and performs competitively overall.