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.
New convergence results for NGVI with various step sizes and sample sizes.
problem Understanding convergence of stochastic NGVI for various schedules.
method Projected stochastic NGVI for exponential family variational distributions.
result Geometric convergence and $\mathcal{O}\left(\frac{1}{T^ρ}
ight)$ rates for different schedules.
Proposes an exponentially increasing step-size for faster parameter estimation in statistical models.
problem Slow convergence of gradient descent in locally convex loss functions.
method Exponentially increasing step-size in gradient descent algorithm.
result Converges linearly to optimal solution under homogeneous 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.
This manuscript shows that AdaBoost and its immediate variants can produce approximate maximum margin classifiers simply by scaling step size choices with a fixed small constant. In this way, when the unscaled step size is an optimal choice, these results provide guarantees for Friedman's empirically successful "shrink…
We highlight a pitfall when applying stochastic variational inference to general Bayesian networks. For global random variables approximated by an exponential family distribution, natural gradient steps, commonly starting from a unit length step size, are averaged to convergence. This useful insight into the scaling of…
We consider the least-squares regression problem and provide a detailed asymptotic analysis of the performance of averaged constant-step-size stochastic gradient descent (a.k.a. least-mean-squares). In the strongly-convex case, we provide an asymptotic expansion up to explicit exponentially decaying terms. Our analysis…
We provide a detailed study on the implicit bias of gradient descent when optimizing loss functions with strictly monotone tails, such as the logistic loss, over separable datasets. We look at two basic questions: (a) what are the conditions on the tail of the loss function under which gradient descent converges in the…
MSTGD optimizes gradient descent with stratified sampling for faster convergence.
problem Fluctuation in gradient expectation and variance between iterations.
method Memory Stochastic Stratified Gradient Descent (MSTGD) with stratified sampling and variance reduction.
result MSTGD achieves an exponential convergence rate independent of dataset size and batch size.
Stochastic gradient methods enable learning probabilistic models from large amounts of data. While large step-sizes (learning rates) have shown to be best for least-squares (e.g., Gaussian noise) once combined with parameter averaging, these are not leading to convergent algorithms in general. In this paper, we conside…
We provide non-asymptotic bounds for the well-known temporal difference learning algorithm TD(0) with linear function approximators. These include high-probability bounds as well as bounds in expectation. Our analysis suggests that a step-size inversely proportional to the number of iterations cannot guarantee optimal …
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.
SGD converges to an invariant distribution with sub-Gaussian or sub-exponential properties.
problem Optimizing smooth and strongly convex objectives using SGD.
method Analysis through Markov chains, focusing on convergence and concentration properties.
result SGD iterates and their invariant limit distribution inherit sub-Gaussian or sub-exponential concentration properties.
Stochastic approximation algorithms show exponential progress bounds.
problem Analyzing the convergence of stochastic approximation algorithms.
method Developed geometric ergodicity proofs to establish exponential concentration bounds.
result Proved faster convergence rates for specific algorithms.
Novel bounds for SGLD show generalization error decreases with more samples.
problem Understanding the generalization error of SGLD in non-convex optimization.
method Information-theoretic approach focusing on Kullback-Leibler divergence and sub-exponential loss function.
result Time-independent generalization bounds for SGLD, independent of step size and number of iterations.
Stochastic (sub)gradient methods require step size schedule tuning to perform well in practice. Classical tuning strategies decay the step size polynomially and lead to optimal sublinear rates on (strongly) convex problems. An alternative schedule, popular in nonconvex optimization, is called \emph{geometric step decay…
High-dimensional unimodal distributions can cause MCMC methods to fail.
problem Failure of MCMC methods in high-dimensional unimodal distributions.
method Examples and theoretical analysis of MCMC methods, including Metropolis-Hastings adjusted methods.
result MCMC methods can take an exponential run-time for high-dimensional unimodal distributions.
Improved privacy analysis for stochastic gradient descent.
problem Analyzing privacy leakage in noisy stochastic gradient descent.
method Modeling Rényi divergence dynamics with Langevin diffusions, proving exponential privacy loss convergence for smooth and strongly convex objectives.
result Privacy loss converges exponentially fast for smooth and strongly convex objectives under constant step size.
This paper shows that the implicit bias of gradient descent on linearly separable data is exactly characterized by the optimal solution of a dual optimization problem given by a smoothed margin, even for general losses. This is in contrast to prior results, which are often tailored to exponentially-tailed losses. For t…
New RL method MAC improves performance in sparse reward settings.
problem Slow mixing in large state spaces or sparse rewards.
method Multi-level Monte Carlo Actor-Critic (MAC) algorithm.
result Achieves convergence rate comparable to state-of-the-art AC algorithms.
A new algorithm reduces the time for ordinal embedding, making it faster and more scalable.
problem Efficiently learning representations from ordinal comparisons, especially for large datasets.
method SVRG-SBB: Stochastic variance reduced gradient with adaptive step size.
result Achieves $O( rac{1}{T})$ convergence rate and global linear convergence under certain assumptions.
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.
Paper analyzes SGD in kernel regression, showing it outperforms offline methods.
problem Performance of SGD in kernel regression compared to offline methods.
method Analyzes Stochastic Gradient Descent (SGD) in kernel regression under misspecified models.
result SGD achieves min-max optimal rates up to constants, avoiding saturation.
Reward-poisoning attacks can force RL agents to learn bad policies, and we categorize and quantify their feasibility.
problem Reward-poisoning attacks can manipulate RL agents to learn undesirable policies.
method Categorize attacks by infinity-norm constraint, provide thresholds for feasibility, and develop adaptive attack strategies.
result Adaptive reward-poisoning attacks can achieve the nefarious policy in polynomial steps, while non-adaptive attacks require exponential steps.
MaxVA improves Adam's step sizes by maximizing gradient variance.
problem Unstable or extreme adaptive learning rates in Adam.
method Maximizing the variance of gradient coordinates in Adam's running mean of squared gradients.
result Faster adaptation and more desirable convergence behaviors than Adam.
Information-Geometric Optimization (IGO) is a unified framework of stochastic algorithms for optimization problems. Given a family of probability distributions, IGO turns the original optimization problem into a new maximization problem on the parameter space of the probability distributions. IGO updates the parameter …
Implicit Q-learning and SARSA adjust step-sizes automatically, improving stability and performance.
problem Numerical instability and slow progress in Q-learning and SARSA due to step-size calibration.
method Reformulate iterative updates as fixed-point equations, scaling step-sizes inversely with feature norms.
result Implicit methods maintain stability over broader step-size ranges and achieve comparable convergence rates.
MSGD outperforms SGD in overparametrized settings with faster convergence rates.
problem Optimization of non-convex functions with momentum.
method Momentum Stochastic Gradient Descent (MSGD) with rigorous analysis.
result MSGD converges exponentially faster than SGD in overparametrized settings.
Exponentially smoothed RNNs improve industrial forecasting.
problem Complexity and non-stationarity in industrial time series data.
method Exponential smoothed recurrent neural networks (RNNs) for modeling non-linear dynamics.
result Exponentially smoothed RNNs outperform traditional models in multi-step forecasting.
Solves large-scale metric constrained problems using Project and Forget algorithm.
problem Finding consistent metric representations for large dissimilarity datasets.
method Active set algorithm with Bregman projections, converges to global optimal solution.
result Algorithm efficiently solves metric constrained problems with exponentially many constraints.
We analyze reinforcement learning algorithms using a distributional approach.
problem Theoretical analysis of reinforcement learning algorithms for constant step-sizes.
method Distributional approach to theoretical analyses of reinforcement learning algorithms.
result TD(λ) and Q-Learning have contractive update rules in the space of distributions of functions, leading to exponentially fast convergence. We design a randomised parallel version of Adaboost based on previous studies on parallel coordinate descent. The algorithm uses the fact that the logarithm of the exponential loss is a function with coordinate-wise Lipschitz continuous gradient, in order to define the step lengths. We provide the proof of convergence …
The paper analyzes fixed step-size SA schemes on Riemannian manifolds.
problem Developing efficient algorithms for optimization on curved spaces.
method Fixed step-size stochastic approximation schemes in a Riemannian framework.
result The schemes converge to the solution as the step-size approaches zero.
A random walk wn on a separable, geodesic hyperbolic metric space X converges to the boundary ∂X with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when …
Meta-strategy learns tuning parameters for online learning methods.
problem Difficulty in setting tuning parameters for online learning methods.
method Meta-learning approach to learn parameters from past tasks.
result Meta-strategy improves on learning each task in isolation.
Community detection in hypergraphs is explored. Under a generative hypergraph model called "d-wise hypergraph stochastic block model" (d-hSBM) which naturally extends the Stochastic Block Model from graphs to d-uniform hypergraphs, the asymptotic minimax mismatch ratio is characterized. For proving the achievability, w…
The CSA-ES is an Evolution Strategy with Cumulative Step size Adaptation, where the step size is adapted measuring the length of a so-called cumulative path. The cumulative path is a combination of the previous steps realized by the algorithm, where the importance of each step decreases with time. This article studies …
In this paper, we introduce a method for adapting the step-sizes of temporal difference (TD) learning. The performance of TD methods often depends on well chosen step-sizes, yet few algorithms have been developed for setting the step-size automatically for TD learning. An important limitation of current methods is that…
New algorithm improves stability of optimization algorithms by adapting step-size.
problem Optimization algorithms' effectiveness is sensitive to step-size hyperparameters.
method Adapts NGN step-size method with momentum to enhance stability.
result Achieves convergence rate of O(1/√K) without restrictive assumptions.
AutoStep MCMC adapts step size locally for better sampling efficiency.
problem Challenging step size selection for complex, multiscale targets.
method AutoStep MCMC uses a locally adaptive step size for involutive proposals.
result AutoStep MCMC is π-invariant, irreducible, and aperiodic.
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…
ELU algorithm improves on EM for over-specified Gaussian mixtures.
problem Slow convergence of EM in over-specified Gaussian mixtures.
method Developed ELU algorithm for two-component mixtures, combining exponential location update and gradient descent.
result ELU converges to final statistical radius after logarithmic iterations, resolving open question.
Improved averaging method for noisy observations converges strongly.
problem Noisy observations from random dynamical systems require stable estimates.
method Introduced p-EMA, a modified exponential moving average with subharmonic weight decay. result Stochastic convergence guarantees for p-EMA under mild assumptions. Polyak step size GD reaches final radius of convergence after log iterations.
problem Statistical and computational complexities of Polyak step size GD.
method Generalized smoothness and Lojasiewicz conditions, stability of gradients.
result Polyak step size GD reaches final statistical radius of convergence after logarithmic number of iterations.
Warm starts improve variational quantum algorithms by avoiding barren plateaus.
problem Barren plateaus in variational quantum algorithms limit scaling.
method Exploring warm starts in iterative variational methods for quantum circuits.
result Warm starts can lead to substantial gradients in small regions, suggesting trainability.
Accelerated gradient method's stability deteriorates exponentially with steps.
problem Algorithmic stability of Nesterov's accelerated gradient method.
method Analysis of two notions of algorithmic stability for Nesterov's accelerated gradient method.
result Stability of Nesterov's accelerated method deteriorates exponentially with the number of gradient steps.
We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small ste…
Diagonal Frog: High-order positivity-preserving FD schemes for anisotropic Fokker-Planck equations
problem Positivity-preserving discretizations for anisotropic Fokker-Planck equations
method Diagonal Frog discretization
result Second-order accuracy and mass conservation