First-order methods play a central role in large-scale machine learning. Even though many variations exist, each suited to a particular problem, almost all such methods fundamentally rely on two types of algorithmic steps: gradient descent, which yields primal progress, and mirror descent, which yields dual progress. W…
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48 results for “coupled gradient descent”
Sharp pseudospectral bounds prevent transient amplification in coupled gradient descent.
problem Transient amplification in coupled gradient descent systems.
method Developed a sharp pseudospectral theory for block-triangular Jacobians, proving Kreiss constant bounds and matching minimax lower bounds.
result Obtained a finite-horizon iteration-complexity bound of for stochastic coupled descent.
Paper proposes a coupling-based diagnostic for SGD stepsize optimization.
problem Optimizing stepsize for SGD convergence.
method Coupling-based convergence diagnostic for monitoring stationarity.
result Proposed stepsize scheme achieves superior performance across convex and non-convex problems.
Paper studies asymmetric matrix sensing, proving gradient descent converges to low-rank solutions.
problem Reconstructing asymmetric low-rank matrices from linear measurements.
method Factorized gradient descent with coupling and regularization properties.
result Gradient descent from small random initialization converges to globally optimal and generalizing solutions.
Weak Convergence of Approximate reflection coupling and its Application to Non-convex Optimizationmath.PR
Paper proposes a weak approximation of reflection coupling for non-convex optimization.
problem Non-convex optimization problems with different drift terms.
method Proposes an approximate reflection coupling (ARC) for stochastic differential equations (SDEs).
result ARC converges weakly to the reflection coupling and can be applied to non-convex optimization.
A new method for sampling high-dimensional distributions overcomes overfitting.
problem Overfitting in energy-based models during gradient descent.
method Mean-field microcanonical gradient descent, which samples multiple data points simultaneously.
result The method reduces entropy loss while maintaining likelihood fit, improving overfitting issues.
Gradient descent forces neural network eigenvalues to a specific threshold.
problem Understanding why gradient descent drives eigenvalues to a specific threshold.
method Introduced edge coupling, a functional on consecutive iterate pairs, to explain the trajectory towards the eigenvalue threshold.
result Gradient descent forces the Hessian eigenvalue to the threshold from arbitrary initialization.
We present a convergence rate analysis for biased stochastic gradient descent (SGD), where individual gradient updates are corrupted by computation errors. We develop stochastic quadratic constraints to formulate a small linear matrix inequality (LMI) whose feasible points lead to convergence bounds of biased SGD. Base…
The paper provides statistical guarantees for SGD and ASGD in high-dimensional settings.
problem Theoretical understanding of SGD and ASGD in high-dimensional settings.
method Transfer of tools from high-dimensional time series to online learning, using coupling techniques.
result Established geometric-moment contraction and -th moment convergence of SGD and ASGD.
Paper proposes a mean-field gradient descent for zero-sum games, proving convergence to Nash equilibrium.
problem Finding mixed Nash equilibria in zero-sum games with multiple players.
method Mean-field gradient descent dynamics with time-averaging, incorporating exponentially discounted gradients.
result Exponential convergence rate to mixed Nash equilibrium with respect to total variation metric.
Natural gradient learning improves synaptic plasticity in spiking neurons.
problem Parametrization dependence leads to inconsistencies in classical synaptic plasticity theories.
method Proposes natural gradient descent in Riemannian geometry for spiking neurons.
result Derives a synaptic learning rule that explains biological phenomena.
A new method for SVGD reduces variance in high dimensions.
problem High-dimensional variance in SVGD.
method Grassmann Stein Variational Gradient Descent (GSVGD) projects onto arbitrary subspaces and uses coupled Grassmann-valued diffusion.
result GSVGD explores high-dimensional problems with intrinsic low-dimensional structure efficiently.
Paper develops unbiased gradient estimator for continuous-time models.
problem Estimating unbiased gradient of log-likelihood for continuous-time models.
method Doubly randomized scheme with coupled conditional particle filter (CCPF).
result Unbiased gradient estimate facilitates gradient-based algorithms.
New algorithm AG-OG optimizes separable convex-concave problems efficiently.
problem Efficiently solving separable convex-concave minimax optimization problems.
method Leverages Nesterov acceleration and optimistic gradient on component and coupling parts of the problem.
result Achieves optimal convergence rate for various settings including bilinearly coupled problems.
Introduces a new stochastic optimization method for deep learning.
problem Minimizing loss functions in deep neural networks.
method Introduces a second-order stochastic Runge-Kutta method and an adaptive SGD-G2.
result The method yields consistent minimization of loss functions and automatically adjusts learning rates.
Gradient descent with small initialization solves matrix completion without regularization.
problem Symmetric matrix completion from observed entries.
method Vanilla gradient descent with small initialization.
result GD converges to the ground truth matrix without regularization in over-parameterized scenario.
New method for online inference using SGD with random scaling.
problem Efficient online inference for SGD parameters.
method Asymptotic pivotal statistics via random scaling.
result Robust and efficient online inference without resampling.
This paper solves tensor robust principal component analysis via scaled gradient descent.
problem Extracting useful information from tensor data robust to corruptions and ill-conditioning.
method Directly recovers low-rank tensor factors via scaled gradient descent with adaptive thresholding.
result The proposed algorithm converges linearly to the true low-rank tensor at a constant rate independent of the condition number.
New algorithms help machines forget old data efficiently.
problem Machine learning models can retain old data, hindering new learning.
method Developed TV-stable algorithms based on noisy SGD for convex and non-convex functions.
result Achieved efficient unlearning with upper and lower bounds on risk.
New method speeds up change-point detection in data sequences.
problem Efficiently detecting change-points in long data sequences.
method Sequential Gradient Descent and Quasi-Newton's Method.
result New method can be orders of magnitude faster than existing methods.
Gradient descent reshapes the function space of neural networks.
problem Understanding how feature learning affects the function space of neural networks.
method Characterized the evolution of the feature space during training using a two-layer neural network.
result Gradient descent induces a data-adaptive deformation that selectively enhances signal-aligned directions.
Quantum field theory connects deep neural networks to criticality.
problem Understanding the criticality and training dynamics of deep neural networks.
method Constructing quantum field theory for deep neural networks, computing corrections to correlation functions.
result Found precise analogy with vector model, providing corrections to correlation length.
Mini-batch stochastic gradient descent and variants thereof have become standard for large-scale empirical risk minimization like the training of neural networks. These methods are usually used with a constant batch size chosen by simple empirical inspection. The batch size significantly influences the behavior of the …
Gradient descent in deep networks tends to find flat minima, which are nearly balanced.
problem Understanding the effect of gradient descent on the structure of minima in deep neural networks.
method Characterized flat minima in linear neural networks trained with a quadratic loss.
result Flat minima correspond to nearly balanced networks where the gain from input to intermediate representations is nearly constant.
Unified framework for coupled tensor completion improves recovery accuracy.
problem Improving recovery accuracy in coupled tensor completion.
method Unified framework using tensor ring (TR) decomposition with shared latent factors and novel optimization model.
result The proposed method achieves superior recovery accuracy on real-world data compared to state-of-the-art methods.
A new method for distributed optimization reduces communication rounds without minibatches.
problem Efficient training in distributed machine learning with different data distributions.
method A primal-dual method (GA-MSGD) applied to the Lagrangian of distributed optimization.
result Achieves linear convergence in communication rounds for strongly convex objectives.
A new method estimates protein evolutionary fields and couplings from alignments.
problem Estimating evolutionary fields and couplings from protein sequence alignments.
method Boltzmann machine with parallel, persistent Markov chain Monte Carlo method.
result Improved precision in predicting contact residue pairs.
Adapts SGD to noise and problem specifics for faster convergence.
problem Minimizing smooth, strongly-convex functions with varying noise and problem constants.
method Adaptive SGD with exponentially decreasing step-sizes, Nesterov acceleration, and stochastic line-search.
result Achieves near-optimal convergence rates without knowing noise or problem specifics.
New -step policy gradient method avoids local optima in restricted policy classes.
problem Suboptimal local optima in policy gradient methods for restricted policy classes.
method Proposes a -step policy gradient method to escape myopic local optima.
result The method converges to near optimal solutions exponentially close to the optimal deterministic policy.
Blind Descent avoids gradient issues, using a different learning approach.
problem Gradient issues like exploding and vanishing gradients.
method Does not use gradients to guide learning; instead, it is a more fundamental learning process.
result Gradient descent is a specific case of Blind Descent.
Data repetition improves SGD's learning of high-dimensional functions.
problem Learning pertinent features in multi-index models with high-dimensional noisy data.
method Investigation of two-layer shallow neural networks trained with gradient-based algorithms, focusing on data repetition.
result Data repetition significantly improves the computational efficiency of SGD, learning all directions with at most steps.
Momentum affects optimization differently at small vs large batch sizes near instability.
problem Understanding how momentum impacts optimization near the edge of stability.
method Demonstrated through batch-size dependent behavior of SGD with momentum.
result Momentum operates in two distinct regimes: amplifying stochastic fluctuations at small batch sizes and stabilizing at large batch sizes.
A new method improves Bayesian filtering in nonlinear systems.
problem Bayesian filtering in nonlinear dynamical systems with non-Gaussian posteriors.
method Transport maps with block-triangular structure and gradient flows for MMD minimization.
result Accurate approximation of non-Gaussian posteriors without particle collapse.
Bias correction improves language model training performance.
problem Stochastic update bias in preconditioned optimizers.
method Cross-fitted preconditioning and variance-corrected inversion.
result Reduces held-out pretraining loss by 0.15 nats.
PURE-CD algorithm proves complexity bounds for convex-concave problems.
problem Solving convex-concave min-max problems with bilinear coupling.
method Primal-dual algorithm with random extrapolation and coordinate descent (PURE-CD).
result Complexity bounds match or improve existing results for dense and sparse problems.
New analysis shows SNG's effectiveness in small samples.
problem Limited understanding of SNG in small data settings.
method Sketch-and-project analysis of SNG.
result Global convergence and rate characterization for SNG.
In this note, we observe the behavior of gradient flow and discrete and noisy gradient descent in some simple settings. It is commonly noted that addition of noise to gradient descent can affect the trajectory of gradient descent. Here, we run some computer experiments for gradient descent on some simple functions, and…
Stochastic gradient descent on manifolds improves low-rank approximation.
problem Efficiently approximate large matrices with lower rank.
method Stochastic gradient descent on a manifold.
result Algorithm outperforms Euclidean space methods on Netflix Prize data.
A new method improves stochastic gradient descent for faster and more efficient estimation.
problem Efficient and fast parametric estimation methods.
method Projected stochastic gradient descent corrected by Fisher scoring.
result The method is faster and more efficient than traditional methods.
We develop randomized (block) coordinate descent (CD) methods for linearly constrained convex optimization. Unlike most CD methods, we do not assume the constraints to be separable, but let them be coupled linearly. To our knowledge, ours is the first CD method that allows linear coupling constraints, without making th…
New algorithm for low-rank optimal transport with improved interpretability and efficiency.
problem Quadratic scaling of optimal transport coupling matrix for massive datasets.
method Factor Relaxation with Latent Coupling (FRLC) algorithm.
result Superior performance on diverse applications including graph clustering and spatial transcriptomics.
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.
In this sequel to arXiv:1510.03817, we apply our abstract Lojasiewicz-Simon gradient inequality to prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient …
Derives Mirror Descent from gradient flow on a Riemannian manifold.
problem No specific problem stated; focuses on derivation.
method Derives Mirror Descent from gradient flow on a Riemannian manifold with a natural discretization.
result Generalizes Mirror Descent to non-Hessian metrics.
We consider the behavior of gradient flow and of discrete and noisy gradient descent. It is commonly noted that the addition of noise to the process of discrete gradient descent can affect the trajectory of gradient descent. In previous work, we observed such effects. There, we considered the case where the minima had …
Stochasticity helps to navigate rough landscapes: comparing gradient-descent-based algorithms in the phase retrieval problemcond-mat.dis-nn
Gradient descent variants improve phase retrieval accuracy.
problem Phase retrieval problem in high-dimensional spaces.
method Gradient descent, stochastic gradient descent, Langevin algorithm, dynamical mean-field theory.
result Stochastic variants of gradient descent achieve better generalization in phase retrieval.
A new framework optimizes fMRI and behavioral data for better understanding of Autism.
problem Linking complex fMRI data to behavioral measures is challenging.
method Coupled manifold optimization framework projecting fMRI onto a shared manifold and mapping to behavioral measures.
result Framework outperforms traditional methods in predicting clinical severity of Autism.
Reparameterizes mirror descent as gradient descent for efficient sparse learning.
problem Efficiently training small sparse networks with mirror descent.
method Develops a framework to convert mirror descent updates into gradient descent updates on different parameters.
result Mirror descent can be reparameterized as gradient descent on modified parameters, facilitating standard backpropagation.