Deep linear ResNets converge globally with certain transformations.
problem Global convergence of training deep linear ResNets.
method Gradient descent and stochastic gradient descent for training L-hidden-layer linear ResNets. result GD and SGD can converge to global minimum for deep linear ResNets with specific transformations.
Gradient descent converges globally in deep linear residual networks with ZAS initialization.
problem Optimizing deep linear residual networks for convergence.
method Zero-asymmetric (ZAS) initialization for gradient descent.
result Gradient descent converges to an ε-optimal point in O(L^3 log(1/ε)) iterations.
This paper studies GAIL's global convergence for general MDP and nonlinear rewards.
problem Understanding when GAIL algorithms achieve global convergence for general MDP and nonlinear rewards.
method Characterization of global convergence for various policy gradient algorithms applied to GAIL.
result First systematic theoretical study of GAIL for global convergence.
Policy gradient converges to globally optimal policy in nearly linear-quadratic systems.
problem Finding optimal policies in nonlinear control systems with partial information.
method Policy gradient algorithm designed for nearly linear-quadratic regulators with small Lipschitz nonlinear components.
result Policy gradient algorithm converges to globally optimal policy with linear rate.
Global convergence proved for multi-agent LQRs with hierarchical actor-critic.
problem Challenges in understanding multi-agent reinforcement learning algorithms.
method Developed a hierarchical actor-critic algorithm for partially exchangeable agents.
result Global linear convergence to optimal policy proved.
We study the global convergence of generative adversarial imitation learning for linear quadratic regulators, which is posed as minimax optimization. To address the challenges arising from non-convex-concave geometry, we analyze the alternating gradient algorithm and establish its Q-linear rate of convergence to a uniq…
The Expectation-Maximization algorithm is perhaps the most broadly used algorithm for inference of latent variable problems. A theoretical understanding of its performance, however, largely remains lacking. Recent results established that EM enjoys global convergence for Gaussian Mixture Models. For Mixed Linear Regres…
We show that Newton's method converges globally at a linear rate for objective functions whose Hessians are stable. This class of problems includes many functions which are not strongly convex, such as logistic regression. Our linear convergence result is (i) affine-invariant, and holds even if an (ii) approximate Hess…
AdaLoss optimizes adaptive learning rates for efficient convergence in various models.
problem Efficiently optimizing adaptive learning rates for gradient descent methods.
method AdaLoss uses loss function information to dynamically adjust step sizes.
result AdaLoss achieves linear convergence in linear regression and robust global convergence in neural networks.
Wide neural networks converge linearly to zero loss with feature learning.
problem Optimizing wide neural networks with feature learning guarantees.
method Gradient flow analysis for wide shallow and multi-layer NNs.
result Training loss converges linearly to zero for wide NNs under GF, demonstrating feature learning and better generalization.
Paper proposes a quasi-Newton method for nonlinear equations with global convergence guarantees.
problem Solving smooth and monotone nonlinear equations efficiently and globally.
method Hybrid proximal extragradient framework combined with online learning for Jacobian approximation.
result First global convergence results showing quasi-Newton method's advantage over extragradient method.
Gradient descent achieves exact linear convergence rate for symmetric matrix completion.
problem Low-rank symmetric matrix completion using gradient descent.
method Local analysis of gradient descent for symmetric matrices without additional assumptions.
result Closed-form expression of exact linear convergence rate matches practice.
New method for accurately predicting linear dynamical systems.
problem Forecasting and estimating system matrices of linear dynamical systems.
method Non-convex polynomial optimization approach with global convergence guarantee.
result Global convergence of numerical solutions to a least-squares estimator.
Single-timescale actor-critic finds globally optimal policy.
problem Finding globally optimal policy in reinforcement learning.
method Simultaneous actor and critic updates with linear or deep neural network approximations.
result Actor sequence converges to globally optimal policy at O(K−1/2) rate. AGGLIO optimizes non-convex functions with local convexity guarantees.
problem Optimizing non-convex functions with local convexity.
method Stage-wise, graduated optimization technique for locally convex functions.
result Global convergence to the global optimum for non-convex and locally convex objectives.
Paper addresses global convergence of MLR estimation under weak data conditions.
problem Learning mixed linear regression models with general data conditions.
method Two-step recursive identification algorithm using least squares and EM principles.
result Global convergence and optimal clustering performance established under general data conditions.
We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as x↦WNWN−1⋯W1x) by minimizing the ℓ2 loss over whitened data. Convergence at a linear rate is guaranteed when the following hold: (i) dimensions of hidden layers are…
Gradient EM converges globally for over-parameterized Gaussian mixtures.
problem Global convergence of gradient EM for Gaussian mixtures with more than 2 components.
method Likelihood-based convergence analysis framework.
result Gradient EM converges globally with a sublinear rate of O(1/√t).
Gradient descent converges linearly for overparameterized linear networks.
problem Convergence of gradient descent for overparameterized neural networks.
method Local Polyak-Lojasiewicz and Descent Lemma for overparameterized linear models.
result Gradient descent achieves linear convergence for two-layer linear networks under relaxed assumptions.
Despite the empirical success of the actor-critic algorithm, its theoretical understanding lags behind. In a broader context, actor-critic can be viewed as an online alternating update algorithm for bilevel optimization, whose convergence is known to be fragile. To understand the instability of actor-critic, we focus o…
We consider stochastic second-order methods for minimizing smooth and strongly-convex functions under an interpolation condition satisfied by over-parameterized models. Under this condition, we show that the regularized subsampled Newton method (R-SSN) achieves global linear convergence with an adaptive step-size and a…
RWR converges to global optimum in certain settings.
problem Proving convergence of RWR to optimal policy.
method Iterative learning with return-weighted log-likelihood.
result RWR converges to global optimum under certain conditions.
Paper improves a method for fast global and local convergence in optimization.
problem Slow global convergence in optimization methods with noisy Hessian estimates.
method Stochastic Newton Proximal Extragradient method using HPE framework.
result Faster global linear rate and superlinear convergence in fewer iterations.
This paper extends the convergence rate of DEQs with ReLU to any general activation.
problem Proving global convergence rate for DEQs with general activations.
method Developed a novel population Gram matrix and new form of dual activation with Hermite polynomial expansion.
result Gradient descent converges to a globally optimal solution at a linear rate for DEQs with general activations.
Paper addresses online identification and clustering for mixed linear regression models.
problem Online identification and clustering of mixed linear regression models.
method Introduces two online identification algorithms based on the EM principle, proving global convergence without i.i.d. data assumptions.
result Global convergence of the proposed algorithms for mixed linear regression models.
New insights into continual learning for deep models, showing convergence issues but local linear solutions.
problem Challenges in continual learning for homogeneous deep models.
method Sequential projections onto task margin sets, leveraging nonconvex projection theory.
result Local linear convergence under certain conditions for homogeneous deep networks.
We study the global convergence of policy optimization for finding the Nash equilibria (NE) in zero-sum linear quadratic (LQ) games. To this end, we first investigate the landscape of LQ games, viewing it as a nonconvex-nonconcave saddle-point problem in the policy space. Specifically, we show that despite its nonconve…
Many statistical M-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-dimensional framewor…
This work shows linear convergence for two-layer neural networks in mean-field regime.
problem Optimizing two-layer neural networks in the mean-field regime.
method Mean-field analysis and continuous-time noisy gradient descent.
result Establishes linear convergence rate for two-layer neural networks.
A new framework for bilevel optimization tackles stochastic and global variance reduction.
problem Bilevel optimization challenges in large-scale empirical risk minimization.
method Introducing a novel framework where inner and main variables evolve simultaneously, leading to unbiased estimates and global variance reduction algorithms.
result SABA algorithm achieves $O( rac{1}{T})$ convergence rate and linear convergence under Polyak-Lojasciewicz assumption.
While convergence of the Alternating Direction Method of Multipliers (ADMM) on convex problems is well studied, convergence on nonconvex problems is only partially understood. In this paper, we consider the Gaussian phase retrieval problem, formulated as a linear constrained optimization problem with a biconvex objecti…
GBML with deep nets converges globally and generalizes well.
problem Theoretical guarantees for few-shot learning with deep nets.
method Proving global convergence and generalization bounds for GBML with over-parameterized DNNs.
result GBML with over-parameterized DNNs converges globally to the optimum at a linear rate and achieves good generalization.
New algorithm converges to optimal filter for predicting linear dynamical systems.
problem Direct policy search for optimal dynamic filters in partially observable systems.
method Regularizer enforcing informativity over filter states.
result Gradient descent converges to globally optimal solution at rate O(1/T).
This paper proves SGD converges to global minimum for over-parameterized ReLU networks.
problem Theoretical understanding of implicit neural networks is limited.
method Gradient flow analysis of ReLU activated implicit neural networks.
result Randomly initialized gradient descent converges to global minimum at a linear rate for square loss function in over-parameterized ReLU networks.
Global convergence for robust regression problems via IRLS with enhancements.
problem Global convergence for robust regression problems.
method Augmentations to IRLS to ensure global recovery and improved robustness.
result Global recovery guarantees for robust regression problems, outperforming state-of-the-art algorithms.
Gradient descent with preconditioning finds global optima in overparameterized nonconvex factorization.
problem Finding global optima in nonconvex Burer-Monteiro factorization.
method Preconditioned gradient descent for overparameterized nonconvex function minimization.
result Gradient descent with preconditioning achieves linear convergence in the overparameterized case.
New method solves subspace optimization problems efficiently.
problem Finding a k-dimensional subspace in high dimensions.
method Local linear convergence of gradient methods under strict complementarity.
result Gradient method converges linearly in high dimensions.
One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies this surprising phenomenon for two-layer fully connected ReLU activated neural n…
The paper analyzes PPM for nonconvex-nonconcave problems, identifying three regions with varying convergence guarantees.
problem Challenges in nonconvex-nonconcave minimax optimization.
method Classic proximal point method with insights from the Moreau envelope.
result Identification of three regions with varying convergence guarantees for PPM.
We show that there are no spurious local minima in the non-convex factorized parametrization of low-rank matrix recovery from incoherent linear measurements. With noisy measurements we show all local minima are very close to a global optimum. Together with a curvature bound at saddle points, this yields a polynomial ti…
Policy gradient methods converge for LQR problems with noisy state dynamics.
problem Finding optimal policies in noisy LQR problems over finite time horizons.
method Policy gradient methods with convergence guarantees for finite time and stochastic state dynamics.
result Global linear convergence for policy gradient methods in LQR problems with weak assumptions.
DEQs converge to optimal solutions with mild over-parameterization.
problem Training over-parameterized deep equilibrium models.
method Solves equilibrium point directly, uses gradient descent, and analyzes convergence via linear rate.
result Gradient descent converges to a globally optimal solution at a linear rate for quadratic loss.
GD converges faster to flatter minima than gradient flow in shallow networks.
problem Understanding the dynamics of gradient descent in shallow linear networks.
method Analyzing the convergence rate and solution of gradient descent in depth-2 linear neural networks.
result GD converges linearly to flatter minima than gradient flow, even with large step sizes.
Policy gradient methods find Nash equilibrium in noisy games.
problem Finding Nash equilibrium in noisy games.
method Policy gradient methods with noise added.
result Policy gradient methods converge to Nash equilibrium in noisy games.
Gradient descent proves global convergence for deep networks with a single wide layer.
problem Proving global convergence of gradient descent for deep ReLU networks.
method Simplified proof using a single wide layer, leveraging ReLU's Lipschitz property.
result Gradient descent converges globally for networks with a single wide layer.
SGD methods fail to converge to global minimizers in deep neural networks with ReLU activation.
problem Failure of SGD methods to converge to global minimizers in deep neural networks.
method Stochastic Gradient Descent (SGD) and its variants like Adam, RMSProp, etc.
result SGD methods fail to converge to global minimizers with high probability in deep neural networks with ReLU activation.
Gradient descent converges to a global minimum in nonlinear ReLU implicit networks with linear width.
problem Understanding convergence of gradient methods in nonlinear, infinitely deep ReLU networks.
method Introduced a scaling constant to ensure well-posedness of the equilibrium equation, proving convergence to a global minimum for linear width networks.
result Gradient descent converges to a global minimum at a linear rate for nonlinear ReLU implicit networks with linear width.
New neural approach for estimating SEMs with provable convergence.
problem Estimating structural parameters in SEMs.
method Formulated as a min-max game with neural networks, learned using stochastic gradient descent.
result Global convergence in overparametrized regime, improving state-of-the-art.