Linear Mode Connectivity (LMC) is a notable phenomenon in the loss landscapes of neural networks, wherein independently trained models have been observed to be connected—up to permutation symmetries—by linear paths in parameter space along which the loss remains consistently low. This observation challenges classical views of non-convex optimization and has implications for model ensembling, generalization, and our understanding of neural loss geometry. Inspired by recent studies on LMC in standard neural networks, we systematically investigate this phenomenon within Mixture-of-Experts (MoE) architectures—a class of models known for their scalability and computational efficiency, which combine traditional neural networks—referred to as experts—through a learnable gating mechanism. We begin by conducting a comprehensive analysis of both dense and sparse gating regimes, demonstrating that the symmetries inherent to MoE architectures are fully characterized by permutations acting on both the expert components and the gating function. Building on these foundational findings, we propose a matching algorithm that enables alignment between independently trained MoEs, thereby facilitating the discovery of LMC. Finally, we empirically validate the presence of LMC using our proposed algorithm across diverse MoE configurations—including dense, sparse, and shared-expert variants—under a wide range of model settings and datasets of varying scales and modalities. Our results confirm the existence of LMC in MoE architectures and offer fundamental insights into the functional landscape and optimization dynamics of deep learning models.
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We study the online calibration of multi-dimensional forecasts over an arbitrary convex set $\mathcal{P} \subset \mathbb{R}^d$ relative to an arbitrary norm $\Vert\cdot\Vert$. We connect this with the problem of external regret minimization for online linear optimization, showing that if it is possible to guarantee $O(\sqrt{\rho T})$ worst-case regret after $T$ rounds when actions are drawn from $\mathcal{P}$ and losses are drawn from the dual $\Vert \cdot \Vert_*$ unit norm ball, then it is also possible to obtain $\epsilon$-calibrated forecasts after $T = \exp(O(\rho /\epsilon^2))$ rounds. When $\mathcal{P}$ is the $d$-dimensional simplex and $\Vert \cdot \Vert$ is the $\ell_1$-norm, the existence of $O(\sqrt{T\log d})$ algorithms for learning with experts implies that it is possible to obtain $\epsilon$-calibrated forecasts after $T = \exp(O(\log{d}/\epsilon^2)) = d^{O(1/\epsilon^2)}$ rounds, recovering a recent result of Peng 2025. Interestingly, our algorithm obtains this guarantee without requiring access to any online linear optimization subroutine or knowledge of the optimal rate $\rho$ -- in fact, our algorithm is identical for every setting of $\mathcal{P}$ and $\Vert \cdot \Vert$. Instead, we show that the optimal regularizer for the above OLO problem can be used to upper bound the above calibration error by a swap regret, which we then minimize by running the recent TreeSwap algorithm with Follow-The-Leader as a subroutine. The resulting algorithm is highly efficient and plays a distribution over simple averages of past observations in each round. Finally, we prove that any online calibration algorithm that guarantees $\epsilon T$ $\ell_1$-calibration error over the $d$-dimensional simplex requires $T \geq \exp(\mathrm{poly}(1/\epsilon))$ (assuming $d \geq \mathrm{poly}(1/\epsilon)$). This strengthens the corresponding $d^{\Omega(\log{1/\epsilon})}$ lower bound of Peng 2025, and shows that an exponential dependence on $1/\epsilon$ is necessary.
Theory · Learning Theory
Optimal transport (OT) provides a powerful framework for comparing and transforming probability distributions, with wide applications in generative modeling, AI4Science and statistical inference. However, existing estimation theory typically requires stringent smoothness conditions on the underlying Brenier potentials and assumes bounded distribution supports, limiting practical applicability. In this paper, we introduce a unified theoretical framework for semi-dual OT map estimation that relaxes both of these restrictions. Building on sieved convex conjugate, our framework has two key contributions: (i) a new map stability bounds that holds without any second-order regularity assumptions on the true Brenier potentials, and (ii) an oracle inequality that cleanly decomposes the estimation error into statistical error, sieved bias, and approximation error. Specifically, our approximation error is measured in the $L^\infty$ norm rather than Sobolev norm in the existing results, aligning more naturally with classical approximation theory. Leveraging these tools, we provide statistical error of semi-dual estimators with mild and verifiable conditions on the true OT map. Moreover, we establish the first theoretical guarantee for deep neural network OT map estimator between general distributions, with Tanh network function class as an example.
Optimization · Stochastic & Convex Optimization
We study nonlinearly preconditioned gradient methods for smooth nonconvex optimization problems, focusing on sigmoid preconditioners that inherently perform a form of gradient clipping akin to the widely used gradient clipping technique. Building upon this idea, we introduce a novel heavy ball-type algorithm and provide convergence guarantees under a generalized smoothness condition that is less restrictive than traditional Lipschitz smoothness, thus covering a broader class of functions. Additionally, we develop a stochastic variant of the base method and study its convergence properties under different noise assumptions. We compare the proposed algorithms with baseline methods on diverse tasks from machine learning including neural network training.
General Machine Learning · Everything Else
Motivated by the recently shown connection between self-attention and (kernel) principal component analysis (PCA), we revisit the fundamentals of PCA. Using the difference-of-convex (DC) framework, we present several novel formulations and provide new theoretical insights. In particular, we show the kernelizability and out-of-sample applicability for a PCA-like family of problems. Moreover, we uncover that simultaneous iteration, which is connected to the classical QR algorithm, is an instance of the difference-of-convex algorithm (DCA), offering an optimization perspective on this longstanding method. Further, we describe new algorithms for PCA and empirically compare them with state-of-the-art methods. Lastly, we introduce a kernelizable dual formulation for a robust variant of PCA that minimizes the $l_1$-deviation of the reconstruction errors.
Optimization · Stochastic & Convex Optimization
We consider the problem of minimizing a $d$-dimensional Lipschitz convex function using a stochastic gradient oracle. We introduce and motivate a setting where the noise of the stochastic gradient is isotropic in that it is bounded in every direction with high probability. We then develop an algorithm for this setting which improves upon prior results by a factor of $d$ in certain regimes, and as a corollary, achieves a new state-of-the-art complexity for sub-exponential noise. We give matching lower bounds (up to polylogarithmic factors) for both results. Additionally, we develop an efficient quantum isotropifier, a quantum algorithm which converts a variance-bounded quantum sampling oracle into one that outputs an unbiased estimate with isotropic error. Combining our results, we obtain improved dimension-dependent rates for quantum stochastic convex optimization.
Applications · Robotics
Diffusion models have made significant strides in recent years, exhibiting strong generalization capabilities in planning and control tasks. However, most diffusion-based policies remain focused on reward maximization or cost minimization, often overlooking critical aspects of safety and stability. In this work, we propose Safe and Stable Diffusion ($S^2$Diff), a model-based framework that explores how diffusion models can ensure safety and stability from a Lyapunov perspective. We demonstrate that $S^2$Diff eliminates the reliance on both complex gradient-based solvers (e.g., quadratic programming, non-convex solvers) and control-affine structures, leading to globally valid control policies driven by the learned certificate functions. Additionally, we uncover intrinsic connections between diffusion sampling and almost Lyapunov theory, enabling the use of trajectory-level control policies to learn better certificate functions for safety and stability guarantees. To validate our approach, we conduct experiments on a wide variety of dynamical control systems, where $S^2$Diff consistently outperforms both certificate-based controllers and model-based diffusion baselines in terms of safety, stability, and overall control performance.
Optimization · Stochastic & Convex Optimization
The present article studies the minimization of convex, $L$-smooth functions defined on a separable real Hilbert space. We analyze regularized stochastic gradient descent (reg-SGD), a variant of stochastic gradient descent that uses a Tikhonov regularization with time-dependent, vanishing regularization parameter. We prove strong convergence of reg-SGD to the minimum-norm solution of the original problem without additional boundedness assumptions. Moreover, we quantify the rate of convergence and optimize the interplay between step-sizes and regularization decay. Our analysis reveals how vanishing Tikhonov regularization controls the flow of SGD and yields stable learning dynamics, offering new insights into the design of iterative algorithms for convex problems, including those that arise in ill-posed inverse problems. We validate our theoretical findings through numerical experiments on image reconstruction and ODE-based inverse problems.
Optimization · Stochastic & Convex Optimization
Adding entropic regularization to Optimal Transport (OT) problems has become a standard approach for designing efficient and scalable solvers. However, regularization introduces a bias from the true solution. To mitigate this bias while still benefiting from the acceleration provided by regularization, a natural solver would adaptively decrease the regularization as it approaches the solution. Although some algorithms heuristically implement this idea, their theoretical guarantees and the extent of their acceleration compared to using a fixed regularization remain largely open. In the setting of semi-discrete OT, where the source measure is continuous and the target is discrete, we prove that decreasing the regularization can indeed accelerate convergence. To this end, we introduce DRAG: Decreasing (entropic) Regularization Averaged Gradient, a stochastic gradient descent algorithm where the regularization decreases with the number of optimization steps. We provide a theoretical analysis showing that DRAG benefits from decreasing regularization compared to a fixed scheme, achieving an unbiased $\mathcal{O}(1/t)$ sample and iteration complexity for both the OT cost and the potential estimation, and a $\mathcal{O}(1/\sqrt{t})$ rate for the OT map. Our theoretical findings are supported by numerical experiments that validate the effectiveness of DRAG and highlight its practical advantages.
Reinforcement Learning · Everything Else
Multi-task reinforcement learning aims to quickly identify solutions for new tasks with minimal or no additional interaction with the environment. Generalized Policy Improvement (GPI) addresses this by combining a set of base policies to produce a new one that is at least as good—though not necessarily optimal—as any individual base policy. Optimality can be ensured, particularly in the linear-reward case, via techniques that compute a Convex Coverage Set (CCS). However, these are computationally expensive and do not scale to complex domains. The Option Keyboard (OK) improves upon GPI by producing policies that are at least as good—and often better. It achieves this through a learned meta-policy that dynamically combines base policies. However, its performance critically depends on the choice of base policies. This raises a key question: is there an optimal set of base policies—an optimal *behavior basis*—that enables zero-shot identification of optimal solutions for *any* linear tasks? We solve this open problem by introducing a novel method that efficiently constructs such an optimal behavior basis. We show that it significantly reduces the number of base policies needed to ensure optimality in new tasks. We also prove that it is strictly more expressive than a CCS, enabling particular classes of *non-linear* tasks to be solved optimally. We empirically evaluate our technique in challenging domains and show that it outperforms state-of-the-art approaches, increasingly so as task complexity increases.
Deep Learning · Theory
Schrodinger Bridges (SBs) are diffusion processes that steer, in finite time, a given initial distribution to another final one while minimizing a suitable cost functional. Although various methods for computing SBs have recently been proposed in the literature, most of these approaches require computationally expensive training schemes, even for solving low-dimensional problems. In this work, we propose an analytic parametrization of a set of feasible policies for steering the distribution of a dynamical system from one Gaussian Mixture Model (GMM) to another. Instead of relying on standard non-convex optimization techniques, the optimal policy within the set can be approximated as the solution of a low-dimensional linear program whose dimension scales linearly with the number of components in each mixture. The proposed method generalizes naturally to more general classes of dynamical systems, such as controllable linear time-varying systems, enabling efficient solutions to multi-marginal momentum SBs between GMMs, a challenging distribution interpolation problem. We showcase the potential of this approach in low-to-moderate dimensional problems such as image-to-image translation in the latent space of an autoencoder, learning of cellular dynamics using multi-marginal momentum SBs, and various other examples. The implementation is publicly available at https://github.com/georgeRapa/GMMflow.
Theory · Learning Theory
We study the problem of regression with interval targets, where only upper and lower bounds on target values are available in the form of intervals. This problem arises when the exact target label is expensive or impossible to obtain, due to inherent uncertainties. In the absence of exact targets, traditional regression loss functions cannot be used. First, we study the methodology of using a loss functions compatible with interval targets, for which we establish non-asymptotic generalization bounds based on smoothness of the hypothesis class that significantly relaxing prior assumptions of realizability and small ambiguity degree. Second, we propose a novel min-max learning formulation: minimize against the worst-case (maximized) target labels within the provided intervals. The maximization problem in the latter is non-convex, but we show that good performance can be achieved with the incorporation of smoothness constraints. Finally, we perform extensive experiments on real-world datasets and show that our methods achieve state-of-the-art performance.
Social Aspects · Privacy-preserving Statistics and Machine Learning
Private Evolution (PE) is a promising training-free method for differentially private (DP) synthetic data generation. While it achieves strong performance in some domains (e.g., images and text), its behavior in others (e.g., tabular data) is less consistent. To date, the only theoretical analysis of the convergence of PE depends on unrealistic assumptions about both the algorithm’s behavior and the structure of the sensitive dataset. In this work, we develop a new theoretical framework to understand PE’s practical behavior and identify sufficient conditions for its convergence. For $d$-dimensional sensitive datasets with $n$ data points from a convex and compact domain, we prove that under the right hyperparameter settings and given access to the Gaussian variation API proposed in \cite{PE23}, PE produces an $(\varepsilon, \delta)$-DP synthetic dataset with expected 1-Wasserstein distance $\tilde{O}(d(n\varepsilon)^{-1/d})$ from the original; this establishes worst-case convergence of the algorithm as $n \to \infty$. Our analysis extends to general Banach spaces as well. We also connect PE to the Private Signed Measure Mechanism, a method for DP synthetic data generation that has thus far not seen much practical adoption. We demonstrate the practical relevance of our theoretical findings in experiments.
Deep Learning · Algorithms
Many machine learning applications require outputs that satisfy complex, dynamic constraints. This task is particularly challenging in Graph Neural Network models due to the variable output sizes of graph-structured data. In this paper, we introduce ProjNet, a Graph Neural Network framework which satisfies input-dependant constraints. ProjNet combines a sparse vector clipping method with the Component-Averaged Dykstra (CAD) algorithm, an iterative scheme for solving the best-approximation problem. We establish a convergence result for CAD and develop a GPU-accelerated implementation capable of handling large-scale inputs efficiently. To enable end-to-end training, we introduce a surrogate gradient for CAD that is both computationally efficient and better suited for optimization than the exact gradient. We validate ProjNet on four classes of constrained optimisation problems: linear programming, two classes of non-convex quadratic programs, and radio transmit power optimization, demonstrating its effectiveness across diverse problem settings.
Optimization · Optimization and Learning under Uncertainty
We study Online Convex Optimization with adversarial constraints (COCO). At each round a learner selects an action from a convex decision set and then an adversary reveals a convex cost and a convex constraint function. The goal of the learner is to select a sequence of actions to minimize both regret and the cumulative constraint violation (CCV) over a horizon of length $T$. The best-known policy for this problem achieves $O(\sqrt{T})$ regret and $\tilde{O}(\sqrt{T})$ CCV. In this paper, we improve this by trading off regret to achieve substantially smaller CCV. This trade-off is especially important in safety-critical applications, where satisfying the safety constraints is non-negotiable. Specifically, for any bounded convex cost and constraint functions, we propose an online policy that achieves $\tilde{O}(\sqrt{dT}+ T^\beta)$ regret and $\tilde{O}(dT^{1-\beta})$ CCV, where $d$ is the dimension of the decision set and $\beta \in [0,1]$ is a tunable parameter. We begin with a special case, called the $\textsf{Constrained Expert}$ problem, where the decision set is a probability simplex and the cost and constraint functions are linear. Leveraging a new adaptive small-loss regret bound, we propose a computationally efficient policy for the $\textsf{Constrained Expert}$ problem, that attains $O(\sqrt{T\ln N}+T^{\beta})$ regret and $\tilde{O}(T^{1-\beta} \ln N)$ CCV for $N$ number of experts. The original problem is then reduced to the $\textsf{Constrained Expert}$ problem via a covering argument. Finally, with an additional $M$-smoothness assumption, we propose a computationally efficient first-order policy attaining $O(\sqrt{MT}+T^{\beta})$ regret and $\tilde{O}(MT^{1-\beta})$ CCV.
Probabilistic Methods · Bayesian Models and Methods
We introduce a highly expressive yet distinctly tractable family for black-box variational inference (BBVI). Each member of this family is a weighted product of experts (PoE), and each weighted expert in the product is proportional to a multivariate $t$-distribution. These products of experts can model distributions with skew, heavy tails, and multiple modes, but to use them for BBVI, we must be able to sample from their densities. We show how to do this by reformulating these products of experts as latent variable models with auxiliary Dirichlet random variables. These Dirichlet variables emerge from a Feynman identity, originally developed for loop integrals in quantum field theory, that expresses the product of multiple fractions (or in our case, $t$-distributions) as an integral over the simplex. We leverage this simplicial latent space to draw weighted samples from these products of experts---samples which BBVI then uses to find the PoE that best approximates a target density. Given a collection of experts, we derive an iterative procedure to optimize the exponents that determine their geometric weighting in the PoE. At each iteration, this procedure minimizes a regularized Fisher divergence to match the scores of the variational and target densities at a batch of samples drawn from the current approximation. This minimization reduces to a convex quadratic program, and we prove under general conditions that these updates converge exponentially fast to a near-optimal weighting of experts. We conclude by evaluating this approach on a variety of synthetic and real-world target distributions.
Optimization
In this paper, we propose a novel tube-wise local tensor compressed sensing (CS) model, where sensing operators are independently applied to each tube of a third-order tensor. To recover the low-rank ground truth tensor, we minimize a non-convex objective via Burer–Monteiro factorization and solve it using gradient descent with spectral initialization. We prove that this approach achieves exact recovery with a linear convergence rate. Notably, our method attains provably lower sample complexity than existing TCS methods. Our proof leverages the leave-one-out technique to show that gradient descent generates iterates implicitly biased towards solutions with bounded incoherence, which ensures contraction of optimization error in consecutive iterates. Empirical results validate the effectiveness of GD in solving the proposed local TCS model.
Probabilistic Methods · Monte Carlo and Sampling Methods
We consider the problem of sampling distributions stemming from non-convex potentials with Unadjusted Langevin Algorithm (ULA). We prove the stability of the discrete-time ULA to drift approximations under the assumption that the potential is strongly convex at infinity. In many context, e.g. imaging inverse problems, potentials are non-convex and non-smooth. Proximal Stochastic Gradient Langevin Algorithm (PSGLA) is a popular algorithm to handle such potentials. It combines the forward-backward optimization algorithm with a ULA step. Our main stability result combined with properties of the Moreau envelope allows us to derive the first proof of convergence of the PSGLA for non-convex potentials. We empirically validate our methodology on synthetic data and in the context of imaging inverse problems. In particular, we observe that PSGLA exhibits faster convergence rates than Stochastic Gradient Langevin Algorithm for posterior sampling while preserving its restoration properties.
Optimization · Neural Network Optimization
Learn to Optimize (L2O) trains deep neural network-based solvers for optimization, achieving success in accelerating convex problems and improving non-convex solutions. However, L2O lacks rigorous theoretical backing for its own training convergence, as existing analyses often use unrealistic assumptions-a gap this work highlights empirically. We bridge this gap by proving the training convergence of L2O models that learn Gradient Descent (GD) hyperparameters for quadratic programming, leveraging the Neural Tangent Kernel (NTK) theory. We propose a deterministic initialization strategy to support our theoretical results and promote stable training over extended optimization horizons by mitigating gradient explosion. Our L2O framework demonstrates over 50% better optimality than GD and superior robustness over state-of-the-art L2O methods on synthetic datasets. The code of our method can be found from https://github.com/NetX-lab/MathL2OProof-Official.
Optimization
In this paper, we study the problem of solving a simple bilevel optimization problem, where the upper-level objective is minimized over the solution set of the lower-level problem. We focus on the general setting in which both the upper- and lower-level objectives are smooth but potentially nonconvex. Due to the absence of additional structural assumptions for the lower-level objective—such as convexity or the Polyak–Łojasiewicz (PL) condition—guaranteeing global optimality is generally intractable. Instead, we introduce a suitable notion of stationarity for this class of problems and aim to design a first-order algorithm that finds such stationary points in polynomial time. Intuitively, stationarity in this setting means the upper-level objective cannot be substantially improved locally without causing a larger deterioration in the lower-level objective. To this end, we show that a simple and implementable variant of the dynamic barrier gradient descent (DBGD) framework can effectively solve the considered nonconvex simple bilevel problems up to stationarity. Specifically, to reach an $(\epsilon_f, \epsilon_g)$-stationary point—where $\epsilon_f$ and $\epsilon_g$ denote the target stationarity accuracies for the upper- and lower-level objectives, respectively—the considered method achieves a complexity of $\mathcal{O}(\max(\epsilon_f^{-\frac{3+p}{1+p}}, \epsilon_g^{-\frac{3+p}{2}}))$, where $p \geq 0$ is an arbitrary constant balancing the terms. To the best of our knowledge, this is the first complexity result for a discrete-time algorithm that guarantees joint stationarity for both levels in general nonconvex simple bilevel problems.