SGD and stochastic gradient descent converge at optimal rates for certain non-convex functions.
problem Optimal convergence rates for non-convex functions under gradient noise.
method Geometric interpretation of the PL-condition to analyze convergence rates.
result Convergence rates of SGD and stochastic gradient descent match those of strongly convex quadratics.
Large over-parametrized models learned via stochastic gradient descent (SGD) methods have become a key element in modern machine learning. Although SGD methods are very effective in practice, most theoretical analyses of SGD suggest slower convergence than what is empirically observed. In our recent work [8] we analyze…
This work proposes a mathematical framework for loss landscapes and optimization in deep neural networks.
problem The effectiveness of gradient-based optimization in over-parameterized neural networks.
method A modern view and mathematical framework of loss landscapes and efficient optimization in over-parameterized machine learning models.
result Wide neural networks satisfy the PL∗ condition, explaining (S)GD convergence to a global minimum. Paper tackles fast convergence for non-convex strongly-concave min-max problems.
problem Non-convex strongly-concave min-max problems in deep learning.
method Proximal stage-based method with PL condition for faster convergence.
result Established fast convergence in primal objective gap and duality gap.
Paper proposes algorithms for solving nonconvex-nonconcave problems with complexity guarantees.
problem Nonconvex-nonconcave minimax problems with PL condition.
method Zeroth-order AGDA and VRAGDA algorithms.
result Iteration complexities for obtaining ε-stationary points.
Stochastic AUC maximization has garnered an increasing interest due to better fit to imbalanced data classification. However, existing works are limited to stochastic AUC maximization with a linear predictive model, which restricts its predictive power when dealing with extremely complex data. In this paper, we conside…
Improved SGD bounds for machine learning models with Markovian noise.
problem Uniform high-probability bounds for SGD under PL condition with Markovian noise.
method Combining Poisson equation for Markovian noise and probabilistic induction for almost-sure bounds.
result Matching 1/k decay rate for expected suboptimality. Stagewise training strategy is widely used for learning neural networks, which runs a stochastic algorithm (e.g., SGD) starting with a relatively large step size (aka learning rate) and geometrically decreasing the step size after a number of iterations. It has been observed that the stagewise SGD has much faster conve…
Improved convergence analysis for decentralized non-convex optimization.
problem Minimizing a sum of smooth non-convex functions over a network.
method Gradient tracking in decentralized stochastic gradient descent (GT-DSGD).
result GT-DSGD achieves network-independent performances matching centralized SGD under certain conditions.
Unified analysis of SGD variants for nonconvex federated optimization.
problem Performance of stochastic gradient methods in nonconvex optimization.
method Proposed a unified assumption for modeling stochastic gradient second moment, leading to a single convergence analysis for various methods.
result Unified convergence analysis for a wide range of SGD variants and distributed methods.
This work provides formal guarantees for heuristic optimization methods in machine learning.
problem Lack of theoretical understanding of heuristic optimization methods in machine learning.
method Analysis and formal guarantees for AdaGrad, SGD with exponential and cosine step sizes, and momentum methods.
result First formal guarantees for AdaGrad and SGD variants, including convergence and adaptivity to noise.
SAIL-RevKL improves SAIL's convergence by regularizing the objective function.
problem Convergence of self-improving online LLM alignment algorithms.
method Proposed SAIL-RevKL, a regularized objective function to improve optimization landscape.
result Proved SAIL-RevKL satisfies the Polyak-Lojasiewicz (PL) condition with near-linear sample complexity.
SLEDGE algorithm reduces gradient computation errors in optimization.
problem Accumulated errors in gradient estimation methods for large-scale optimization.
method Single-loop method for finite-sum nonconvex optimization without periodic gradient refresh.
result Achieves nearly optimal gradient complexity and second-order optimality.
New algorithms improve convergence of minimax optimization.
problem Minimax optimization convergence issues in nonconvex problems.
method Established new convergence results for two single-loop algorithms.
result Improved convergence rates for minimax optimization.
New method tackles nonconvex-nonconcave problems with local KL condition.
problem Nonconvex-nonconcave minimax problems under varying KL conditions.
method Inexact proximal gradient method for KL-structured subproblems.
result Complexity guarantees for approximate stationary points.
New algorithm reduces communication in federated learning.
problem Communication inefficiency in distributed federated learning.
method Distributionally Robust Federated Averaging (DRFA) with snapshotting scheme.
result DRFA achieves efficient convergence in various settings.
Recent applications that arise in machine learning have surged significant interest in solving min-max saddle point games. This problem has been extensively studied in the convex-concave regime for which a global equilibrium solution can be computed efficiently. In this paper, we study the problem in the non-convex reg…
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.
Unified analysis improves SAM for non-convex optimization.
problem Improving generalization in machine learning models.
method Sharpness-aware minimization (SAM) and Unified SAM.
result Unified SAM provides convergence guarantees under relaxed assumptions.
Develops a generalized version of Chung's Lemma for stochastic optimization methods.
problem Establishing asymptotic convergence rates for stochastic optimization methods under various step size rules.
method Generalized version of Chung's Lemma for a broader family of step size rules.
result Demonstrates tight non-asymptotic convergence rates for various stochastic methods.
In federated distributed learning, the goal is to optimize a global training objective defined over distributed devices, where the data shard at each device is sampled from a possibly different distribution (a.k.a., heterogeneous or non i.i.d. data samples). In this paper, we generalize the local stochastic and full gr…
Local convergence theory for mildly over-parameterized neural nets.
problem Understanding why over-parameterization works in neural networks.
method Developed a local convergence theory for two-layer neural nets, showing neuron convergence under certain conditions.
result All student neurons converge to one of teacher neurons when the loss is below a threshold.
We analyze stochastic gradient algorithms for optimizing nonconvex, nonsmooth finite-sum problems. In particular, the objective function is given by the summation of a differentiable (possibly nonconvex) component, together with a possibly non-differentiable but convex component. We propose a proximal stochastic gradie…
The paper analyzes and validates two step size schedules for SGD: exponential and cosine, proving their adaptivity and performance.
problem The variability of SGD performance due to step size choice.
method Analysis and empirical evaluation of exponential and cosine step sizes.
result Exponential and cosine step sizes are adaptive to noise and achieve optimal performance without tuning hyperparameters.
PAGE is a simple gradient estimator for nonconvex optimization problems.
problem Nonconvex optimization problems in machine learning.
method PAGE is a probabilistic gradient estimator that uses vanilla SGD with probability and a small adjustment with probability 1-p.
result PAGE achieves optimal convergence rates for nonconvex finite-sum and online problems.
Proposes an online method for solving non-convex DRO with KL regularization.
problem Solving distributionally robust optimization with non-convex objectives.
method Practical online stochastic methods for DRO with KL regularization, avoiding high-dimensional dual variables and online learning issues.
result Empirical studies show significant speedup and efficiency in training deep learning models.
Paper proves SHB convergence with biased gradients and approximate step sizes.
problem Establishing convergence of SHB with biased gradients and approximate step sizes.
method Generalizes SHB convergence conditions for biased gradients, approximate step sizes, and block updating.
result Proves convergence of SHB with new conditions for biased gradients and approximate step sizes.
Paper analyzes convergence rates of SGD for non-convex functions under various assumptions.
problem Analyzing convergence rates of SGD for non-convex functions.
method Studied convergence properties of Stochastic Gradient Descent (SGD) for invex functions under weaker and stronger hypotheses.
result Derives estimates on the rate of convergence of $J(oldsymbolθ_t)$ to its limit for functions satisfying the Polyak-Lojasiewicz (PL) condition.
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.