New shuffling methods improve convergence without Lipschitz smoothness.
problem Lack of convergence guarantees for shuffling methods under non-Lipschitz conditions.
method Revisit shuffling methods, prove convergence under general bounded variance condition.
result Matched current best-known convergence rates without Lipschitz smoothness.
New method improves optimization algorithms without Lipschitz smoothness.
problem Improving optimization algorithms in the absence of Lipschitz smoothness.
method Dual kernel conditioning (DKC) to provide dual Lipschitz continuity.
result First complexity bounds and iterate convergence for random reshuffling mirror descent.
RYU framework constructs safe regions for optimization problems.
problem Optimization problems with specific component functions.
method RYU framework for constructing safe regions.
result RYU framework improves upon state-of-the-art methods.
This paper analyzes convergence of large-scale Transformers with weight decay.
problem Understanding optimization guarantees in large-scale Transformer training.
method Construct mean-field limit, show gradient flow convergence to PDE, demonstrate global minimum consistency.
result Gradient flow reaches global minimum in large-scale Transformers with small weight decay.
The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothn…
We consider the stochastic composition optimization problem proposed in \cite{wang2017stochastic}, which has applications ranging from estimation to statistical and machine learning. We propose the first ADMM-based algorithm named com-SVR-ADMM, and show that com-SVR-ADMM converges linearly for strongly convex and Lipsc…
Sparse Polyak improves high-dimensional statistical estimation.
problem High-dimensional statistical estimation problems with growing problem dimension.
method Sparse Polyak modifies Polyak's adaptive step size to estimate restricted Lipschitz smoothness.
result Sparse Polyak achieves optimal statistical precision with fewer iterations.
Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.
problem Smooth metrics on manifolds with curvature and injectivity constraints.
method Bi-Lipschitz smoothing with controlled smoothing and volume lower bounds.
result Proves existence of smooth metrics with curvature bounds and injectivity radius constraints.
New method balances multivariate model fitting for mixed likelihoods.
problem Multivariate models often fit only a subset of observed variables.
method Lipschitz standardization for data preprocessing.
result Lipschitz standardization leads to more accurate multivariate models.
New method samples from non-log-concave distributions with weak dissipativity.
problem Sampling from distributions that are not log-concave and weakly dissipative.
method Taming scheme tailored to growth and decay properties of the target distribution.
result Explicit non-asymptotic guarantees for KL, TV, and Wasserstein distances.
Black-box variational inference tries to approximate a complex target distribution though a gradient-based optimization of the parameters of a simpler distribution. Provable convergence guarantees require structural properties of the objective. This paper shows that for location-scale family approximations, if the targ…
New privacy bounds for DP-SGD's last iterate, even with cyclic sampling.
problem Privacy of the last iterate in DP-SGD with cyclic sampling.
method Established new RDP upper bounds for the last iterate under realistic assumptions.
result Privacy bounds for DP-SGD's last iterate with cyclic sampling and clipping, even for nonconvex losses.
The study connects polyhedral manifolds to Riemannian ones with geometric bounds.
problem Connecting polyhedral manifolds to Riemannian manifolds with geometric constraints.
method Using a theorem by C. Lange and B. Bowditch, the study bounds the curvature and injectivity radius of Riemannian manifolds.
result Polyhedral manifolds with bounded geometry are bi-Lipschitz homeomorphic to Riemannian manifolds with controlled curvature and injectivity radius.
Study on privacy leakage in noisy gradient descent algorithms.
problem Information leakage of iterative randomized learning algorithms about training data.
method Analyzes the dynamics of Rényi differential privacy loss in noisy gradient descent algorithms.
result Privacy loss converges exponentially fast for smooth and strongly convex loss functions.
Efficient binary sampling method for global optimization of univariate functions with low regret.
problem Global optimization of univariate loss functions.
method Binary sampling approach to circumvent hard-to-determine query points in traditional methods.
result At most L log ( 3 T ) L\log (3T) L log ( 3 T ) and 2.25 H 2.25H 2.25 H regret for L L L -Lipschitz continuous and H H H -Lipschitz smooth functions respectively. Untuned SGD converges but with an exponential dependence on smoothness, adaptive methods prevent this.
problem The exponential dependence on smoothness in untuned SGD's convergence rate.
method Untuned SGD with arbitrary stepsize η, adaptive methods like NSGD, AMSGrad, and AdaGrad.
result Adaptive methods prevent the exponential dependence on smoothness in SGD.
Adaptive gradient methods such as AdaGrad and its variants update the stepsize in stochastic gradient descent on the fly according to the gradients received along the way; such methods have gained widespread use in large-scale optimization for their ability to converge robustly, without the need to fine-tune the stepsi…
AGD converges in polynomial iterations to optimal matrix factorization.
problem Matrix factorization optimization with alternating gradient descent.
method Alternating gradient descent with fixed step size, proving convergence in polynomial iterations.
result AGD reaches ε-optimal factorization in T iterations with high probability.
The Gauss-Newton method is analyzed for neural networks using Riemannian optimization techniques.
problem Training neural networks with smooth activations and convergence rates.
method Riemannian optimization perspective, analyzing the Gauss-Newton method in both underparameterized and overparameterized regimes.
result Geometric convergence rates independent of conditioning and eigenvalues, demonstrating accelerated convergence.
Anderson acceleration (or Anderson mixing) is an efficient acceleration method for fixed point iterations x t + 1 = G ( x t ) x_{t+1}=G(x_t) x t + 1 = G ( x t ) , e.g., gradient descent can be viewed as iteratively applying the operation G ( x ) ≜ x − α ∇ f ( x ) G(x) \triangleq x-α\nabla f(x) G ( x ) ≜ x − α ∇ f ( x ) . It is known that Anderson acceleration is quite efficient in practice and can be viewed…
New analysis for black-box learning without gradients, improving generalization bounds.
problem Generalization error analysis for derivative-free optimization.
method Zeroth-order Stochastic Search (ZoSS) algorithm for Lipschitz and smooth losses.
result Generalization bounds independent of model dimension, batch size, and number of perturbed evaluations.
This work closes the theory-practice gap for distributed optimization methods by introducing a new regularity condition.
problem Existing convergence conditions for distributed optimization methods are violated by nearly all kernels used in practice.
method Introduces Hessian relative uniform continuity (HRUC) to guarantee convergence under mild conditions.
result Derives convergence guarantees for mirror descent-based gradient tracking without restrictive assumptions.
Improved learning rates with new smoothness measure.
problem Learning with noisy data and unknown function class.
method Generalized Hölder smoothness to average smoothness, proving upper and lower bounds.
result Achieved nearly optimal learning rates in realizable and agnostic settings.
New framework optimizes decisions under uncertainty considering causal and continuous data.
problem Optimizing decisions under uncertain distributions with causal and continuous data structures.
method Developed a framework using Causal Sinkhorn DRO with Soft Regression Forest decision rules.
result Framework provides interpretable and tractable decision rules for optimizing under uncertainty.
We prove the following new characterization of C p C^p C p (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X X X has a C p C^p C p smooth (Lipschitz) bump function if and only if it has another C p C^p C p smooth (Lipschitz) bump function f f f such that f ′ ( x ) ≠ 0 f'(x)\neq 0 f ′ ( x ) = 0 for every point x x x in the interior of the …
New algorithm for private non-convex optimization with optimal rates.
problem Private optimization of non-convex functions under KL condition.
method Variance-reduced gradient descent and proximal point method.
result Achieves nearly optimal rates for excess empirical risk.
This work analyzes Adam's preconditioning effect on quadratic functions and quantifies its impact on condition number.
problem Understanding and quantifying the preconditioning effect of Adam to alleviate ill-conditioning in gradient descent.
method Detailed analysis of Adam's preconditioning effect for quadratic functions, including empirical evidence.
result Adam can mitigate the condition number but at a dimension-dependent cost, with specific bounds for different types of Hessians.
Optimizes differentially private kernel learning with random projection.
problem Privacy-preserving learning algorithms with optimal performance.
method Differentially private kernel ERM algorithm based on random projection in reproducing kernel Hilbert space.
result Achieves minimax-optimal excess risk rates for various loss functions.
On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator D B \mathrm{D}_{\mathcal B} D B in L 2 \mathrm{L}^{2} L 2 depends Riesz continuously on L ∞ \mathrm{L}^{\infty} L ∞ perturbations of local boundary conditions B {\mathcal B} B . The Lipschitz bound for the map ${…
Selecting more uniformly distributed data improves training efficiency and performance.
problem Improving data selection for training large language models (LLMs).
method Established a convergence framework for gradient descent beyond the NTK regime, proving that more uniform data leads to larger minimum pairwise distances and faster training.
result Selecting more uniformly distributed data accelerates training and achieves comparable or better performance in LLMs.
New method solves complex optimization problems with real-time learning.
problem Nonconvex nonsmooth conditional stochastic optimization problems.
method Single time-scale stochastic method with parametric model approximation.
result Method converges with probability one using differential inclusions and Lyapunov function.
BaNk-UCB tackles batched nonparametric bandits with k-NN regression and UCB.
problem Sequential decision-making with limited online feedback in domains like medicine and marketing.
method Combines k-NN regression with UCB principle for fully nonparametric, adaptive, and simple implementation.
result Near-optimal regret guarantees under Lipschitz smoothness and margin assumptions, with minimax-optimal rates.
Study efficient derivative computation for nondifferentiable maps in machine learning.
problem Efficiently compute derivatives of fixed-point of nondifferentiable contractions.
method Iterative Differentiation (ITD), Approximate Implicit Differentiation (AID), and New Stochastic Implicit Differentiation (NSID).
result Established convergence rates for ITD, AID, and NSID, matching or improving smooth setting rates.
Develops a privacy-preserving algorithm for sparse robust regression.
problem Privacy-preserving machine learning for sparse robust regression.
method Develops FRAPPE algorithm for non-smooth loss under differential privacy.
result Achieves better privacy and statistical accuracy trade-off.
This paper recovers smooth functions from noisy modulo samples using a three-stage strategy.
problem Recovering Hölder smooth functions from noisy modulo samples.
method Three-stage strategy: denoising with local polynomial estimators, unwrapping, and spline-based quasi-interpolant.
result Uniform error rates for Hölder class functions with high probability.
Two algorithms learn Gaussian graphical models from Glauber dynamics trajectories.
problem Learning Gaussian graphical models from dependent data.
method Two complementary approaches: local edge-testing and burn-in/thinning reduction.
result Both approaches provide finite-sample recovery guarantees and empirical comparisons.
Improved KLMC for sampling under various conditions.
problem Stable simulation of kinetic Langevin dynamics under different parameters.
method Revisited synchronous Wasserstein coupling analysis with stochastic exponential Euler discretization.
result Exponential integrator can simulate kinetic Langevin dynamics in the overdamped regime with proper time acceleration.
This work establishes near-minimax optimal guarantees for ODE-based samplers under mild assumptions.
problem Develop rigorous statistical guarantees for ODE-based samplers in generative modeling.
method Proposes a smooth regularized score estimator and refined convergence analysis.
result Achieves minimax rate in total variation distance for ODE-based samplers under mild assumptions.
New method provides valid confidence intervals for spatial associations.
problem Limited insight into covariate-response relationships in spatial settings.
method Lipschitz-driven uncertainty quantification for spatial association.
result Valid frequentist confidence intervals for associations in spatial settings.
The paper introduces PD learning to improve deep learning theory.
problem Lack of theoretical understanding in deep learning model fitting and generalization.
method Proposes a PD learning framework to analyze optimization and generalization mechanisms of deep learning.
result Established theoretical guarantees on optimizability and derived generalization error bounds.
Just as an explicit parameterisation of system dynamics by state, i.e., a choice of coordinates, can impede the identification of general structure, so it is too with an explicit parameterisation of system dynamics by control. However, such explicit and fixed parameterisation by control is commonplace in control theory…
This paper tackles robust control of noisy systems with uncertain distributions.
problem Optimal control of sampled-data stochastic systems with multiplicative noise and distributional ambiguity.
method Develops a convex relaxation to handle the ``concave-max'' geometry and derives a probabilistic performance guarantee.
result Derives an explicit, non-asymptotic bound on the duality gap and proves robust viability conditions.
SNGP improves single-model deep uncertainty by enhancing distance-awareness.
problem Improving uncertainty estimation in deep learning models, especially for real-time applications.
method SNGP improves distance-awareness of DNNs through spectral normalization and Gaussian process layers.
result SNGP outperforms other single-model approaches in prediction, calibration, and out-of-domain detection.
We shrink confidence sets for equivalent discrete distributions using permutation equivalence.
problem Building high-probability confidence sets for equivalent discrete distributions.
method Exploiting permutation-equivalence to refine confidence sets.
result Confidence sets shrink at asymptotic rates of O ( 1 / ∑ k ∈ K n k ) O(1/\sqrt{\sum_{k\in \mathcal K} n_k}) O ( 1/ ∑ k ∈ K n k ) and O ( 1 / max k ∈ K n k ) O(1/\max_{k\in K} n_{k}) O ( 1/ max k ∈ K n k ) . Study shows gradient variance increases during deep learning training, contrary to common belief.
problem Understanding and minimizing gradient variance in deep learning models.
method Gradient Clustering method using stratified sampling to minimize gradient variance.
result Gradient variance increases during training, and smaller learning rates coincide with higher variance.
Researchers compare different gradient methods for ridge regression, finding conjugate gradients have similar performance.
problem Comparing statistical properties of different gradient methods in ridge regression.
method Explicit non-standard error decomposition to bound prediction error of conjugate gradient iterates.
result Conjugate gradient iterates share optimality properties with gradient flow and ridge regression up to a constant factor.
Gradient descent implicitly regularizes neural networks by penalizing large loss gradients.
problem How to optimize deep neural networks without explicit regularization.
method Backward error analysis to calculate implicit gradient regularization and demonstrate its effectiveness empirically.
result Implicit gradient regularization biases gradient descent toward flat minima, improving model robustness and test errors.
Improved sampling method using regularized Stein Variational Gradient Flow.
problem Improving the accuracy of sampling methods in machine learning.
method Proposed Regularized Stein Variational Gradient Flow to interpolate between SVGD and Wasserstein Gradient Flow.
result Established theoretical properties and provided preliminary numerical evidence of improved performance.