This paper introduces a new metric, ULI, for RL that ensures both cumulative and instantaneous performance.
problem High-stakes applications require RL algorithms to avoid playing bad policies.
method Introduces uniform last-iterate (ULI) guarantee, a stronger metric capturing both cumulative and instantaneous performance.
result ULI directly implies near-optimal cumulative performance across various metrics, but not the other way around.
OMWU shows last iterate convergence in convex-concave games.
problem Optimizing in constrained min-max optimization landscapes.
method OMWU (Optimistic Multiplicative-Weights Update) in the no-regret online learning framework.
result OMWU exhibits last iterate convergence for convex-concave games, generalizing previous results.
In this paper we study the smooth convex-concave saddle point problem. Specifically, we analyze the last iterate convergence properties of the Extragradient (EG) algorithm. It is well known that the ergodic (averaged) iterates of EG converge at a rate of O ( 1 / T ) O(1/T) O ( 1/ T ) (Nemirovski, 2004). In this paper, we show that the last…
Last SGD iterate bounds for overparameterized linear regression.
problem Analyzing the last iterate risk bounds of SGD with decaying stepsize for overparameterized linear regression.
method Problem-dependent analysis of last iterate risk bounds of SGD with geometrically decaying stepsize.
result Proved nearly matching upper and lower bounds on the excess risk for last iterate SGD with geometrically decaying stepsize.
New convergence rates for shuffling gradient methods without strong convexity.
problem Theoretical gap between shuffling gradient methods' empirical success and established convergence rates.
method Proved last-iterate convergence rates for shuffling gradient methods using function value gap.
result First last-iterate convergence rates for shuffling gradient methods without strong convexity.
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.
While classic work in convex-concave min-max optimization relies on average-iterate convergence results, the emergence of nonconvex applications such as training Generative Adversarial Networks has led to renewed interest in last-iterate convergence guarantees. Proving last-iterate convergence is challenging because ma…
This work addresses the convergence of SGD's final iterate without restrictive assumptions.
problem Prove optimal convergence rate of SGD's final iterate without compact domains or bounded noise.
method Unified proof for general domains, composite objectives, non-Euclidean norms, etc.
result First unified convergence rates in expectation and high probability.
Many problems in machine learning and game theory can be formulated as saddle-point problems, for which various first-order methods have been developed and proven efficient in practice. Under the general convex-concave assumption, most first-order methods only guarantee an ergodic convergence rate, that is, the uniform…
Improved convergence rates for saddle-point optimization algorithms.
problem Understanding last-iterate convergence rates for saddle-point optimization algorithms in constrained settings.
method Expanding the understanding of last-iterate convergence for Optimistic Gradient Descent Ascent (OGDA) and Optimistic Multiplicative Weights Update (OMWU) in the constrained setting.
result Linear last-iterate convergence achieved with a universal constant learning rate for OMWU in bilinear games over the simplex.
Last-iterate guarantees for learning in co-coercive games under noisy feedback.
problem Learning in co-coercive games with noisy feedback.
method Vanilla stochastic gradient descent with a new noise model.
result Last-iterate bound of order O ( log ( t ) / t 1 / 3 ) O(\log(t)/t^{1/3}) O ( log ( t ) / t 1/3 ) for co-coercive games. Optimizes convergence rate of stochastic proximal algorithms for composite convex problems.
problem Solving composite convex optimization problems with composite regularizers.
method Analyzed proximal stochastic gradient method and randomized incremental proximal method under relaxed variance assumptions.
result Proves O ( 1 / T ) O(1/\sqrt{T}) O ( 1/ T ) convergence rate for last iterate of both algorithms under componentwise convexity and smoothness. Paper proves suboptimal convergence rate of last iterate for SGDM.
problem Proves suboptimal convergence rate of last iterate for SGDM.
method Focuses on convergence rate of last iterate of SGDM, introduces Follow-The-Regularized-Leader-based algorithms.
result Shows optimal convergence rate of last iterate for unconstrained convex stochastic optimization problems.
SGD achieves near optimal convergence rate in smooth interpolation regime.
problem Optimization of smooth convex objectives with zero noise at optimum.
method Stochastic Gradient Descent (SGD) with large stepsize analysis.
result Last iterate of SGD achieves expected excess risk of O(1/T + σ* / √T) with optimal stepsize.
Efficient algorithm converges to Nash equilibrium in bilinear problems with bandit feedback.
problem Learning dynamics in bilinear saddle-point problems with bandit feedback.
method Uncoupled learning algorithm combining experimental design and FTRL with a tailored regularizer.
result Last-iterate convergence rate of i l d e O ( T − 1 / 4 ) ilde{O}(T^{-1/4}) i l d e O ( T − 1/4 ) in high probability. Paper analyzes convergence rates for multi-agent learning in games.
problem Convergence rates for multi-agent learning in games.
method Characterizes finite-time convergence rates for joint OGD learning on λ λ λ -cocoercive games and develops adaptive algorithms. result Adaptive algorithms achieve same convergence rates as non-adaptive counterparts.
Improved shuffling gradient methods converge faster for nonsmooth convex optimization.
problem Improving convergence rates for nonsmooth convex optimization problems.
method Analysis of shuffling gradient methods, focusing on Random Reshuffle and Single Shuffle strategies.
result Shuffling gradient methods, particularly Random Reshuffle and Single Shuffle, converge faster than Proximal Gradient Descent for nonsmooth convex optimization.
New ODE models show saddle-point optimization methods converge differently, with last-iterate convergence for OGDA.
problem Analyzing convergence properties of saddle-point optimization methods.
method High-Resolution Differential Equations (HRDEs) to design differential equation models for saddle-point optimization methods.
result HRDEs reveal last-iterate convergence for Optimistic Gradient Descent Ascent (OGDA) in bilinear games.
Mutation improves FTRL convergence in zero-sum games.
problem Lack of last-iterate convergence in FTRL variants.
method Introduced mutation to perturb action probabilities in FTRL.
result M-FTRL converges to Nash equilibria under full-information feedback.
Two accelerated extragradient methods converge at O ( 1 / k ) O(1/k) O ( 1/ k ) rate for co-hypomonotone inclusions.
problem Solving co-hypomonotone inclusions with sum of Lipschitz and multivalued operators.
method Developed two Nesterov's accelerated extragradient methods for co-hypomonotone inclusions.
result Achieve O ( 1 / k ) \mathcal{O}(1/k) O ( 1/ k ) last-iterate convergence rates on the residual norm. Motivated by applications in Game Theory, Optimization, and Generative Adversarial Networks, recent work of Daskalakis et al \cite{DISZ17} and follow-up work of Liang and Stokes \cite{LiangS18} have established that a variant of the widely used Gradient Descent/Ascent procedure, called "Optimistic Gradient Descent/Asce…
New algorithm for solving minimax problems over distributions converges to Nash equilibrium.
problem Solving minimax problems over probability distributions.
method Symmetric Mean-field Langevin Dynamics (MFL-AG and MFL-ABR) with weighted averaging and best response dynamics.
result Converges to mixed Nash equilibrium with average-iterate and last-iterate convergence.
In statistical learning theory, generalization error is used to quantify the degree to which a supervised machine learning algorithm may overfit to training data. Recent work [Xu and Raginsky (2017)] has established a bound on the generalization error of empirical risk minimization based on the mutual information $I(S;…
This paper advances extragradient methods for solving inclusions under co-hypomonotonicity.
problem Solving inclusions with non-Lipschitz mappings.
method Unified and generalized extragradient methods under co-hypomonotonicity.
result Achieved O ( 1 / k ) \mathcal{O}(1/k) O ( 1/ k ) convergence rates for residual norms. Paper achieves ε − 2 ε^{-2} ε − 2 sample complexity for actor-critic methods with minimal assumptions.
problem Achieving ε − 2 ε^{-2} ε − 2 sample complexity for actor-critic methods under minimal assumptions. method Single-loop, single-timescale implementation; coupled Lyapunov drift framework.
result First i l d e O ( ε − 2 ) ilde{\mathcal{O}}(ε^{-2}) i l d e O ( ε − 2 ) sample complexity guarantee for finding an ε ε ε -optimal policy. New study shows faster convergence of SGD and Kaczmarz methods.
problem Improving convergence rates of iterative linear system solvers.
method Last-iterate convergence analysis of SGD with greedy step size over smooth quadratics.
result The t t t -th iterate attains an O ( 1 / t 3 / 4 ) O(1/t^{3/4}) O ( 1/ t 3/4 ) convergence rate. Paper analyzes normal approximation for two-timescale stochastic algorithms, revealing interaction between fast and slow timescales.
problem Non-asymptotic bounds for accuracy of normal approximation in linear two-timescale stochastic approximation algorithms.
method Established bounds for normal approximation in terms of convex distance, focusing on last iterate and Polyak-Ruppert averaging.
result Normal approximation rate for the last iterate improves with increased timescale separation, while it decreases in the averaged setting.
The paper analyzes convergence rates for SGD and SHB methods.
problem Analyzing convergence rates for stochastic gradient descent and heavy ball methods.
method Stochastic gradient descent and stochastic heavy ball method for general stochastic approximation problems.
result The last iterate of SHB converges almost surely to a minimizer and has faster convergence rates than SGD.
Gradient methods converge exponentially in concave network games.
problem Finding Nash equilibria in concave network zero-sum games.
method Gradient Ascent and Optimistic Gradient Ascent analyses.
result Exponential convergence rates in various game settings.
Paper provides tail bounds for stochastic mirror descent in heavy-tailed noise.
problem Optimizing convex and Lipschitz functions with heavy-tailed noise.
method Develops tail bounds for optimization error of Stochastic Mirror Descent.
result Tail bounds extend to heavier-tailed noise regimes without diameter constraints.
In this paper, we study the online learning algorithm without explicit regularization terms. This algorithm is essentially a stochastic gradient descent scheme in a reproducing kernel Hilbert space (RKHS). The polynomially decaying step size in each iteration can play a role of regularization to ensure the generalizati…
Algorithm converges to Nash equilibria in competitive games.
problem Finding Nash equilibria in decentralized, competitive Markov games.
method Decentralized Optimistic Gradient Descent/Ascent with a critic.
result Converges to the set of Nash equilibria under self-play.
Non-affine aggregation rules cannot preserve monotonicity in convex learning.
problem Designing non-affine aggregation rules that maintain monotonicity in convex learning.
method Proving that monotonicity of aggregated gradients is preserved only if the aggregation rule is positively affine.
result Non-affine aggregation prevents steady convergence and substantially degrades algorithmic stability.
New algorithms converge faster to Nash equilibrium in zero-sum games with bandit feedback.
problem Learning in zero-sum games with bandit feedback without communication.
method Developed two uncoupled algorithms achieving optimal rate of Ω ( T − 1 / 4 ) Ω(T^{-1/4}) Ω ( T − 1/4 ) . result Achieved optimal rate of Ω ( T − 1 / 4 ) Ω(T^{-1/4}) Ω ( T − 1/4 ) for convergence of policy profiles to Nash equilibrium. 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.
The paper shows how data and algorithm interactions affect overparameterized linear regression generalization.
problem Understanding generalization in overparameterized linear regression.
method Introducing data-algorithm compatibility and performing data-dependent trajectory analysis with gradient descent.
result Early stopping iterates lead to better generalization than last-iterate analysis, with weaker restrictions.
ROOT-SGD solves convex optimization problems with optimal nonasymptotic and near-optimal asymptotic performance.
problem Solving strongly convex and smooth unconstrained optimization problems using stochastic first-order algorithms.
method ROOT-SGD: Recursive One-Over-T SGD, averaging past stochastic gradients.
result Achieves state-of-the-art performance in both nonasymptotic and asymptotic senses.
Unified algorithm solves convex optimization problems with optimal rates.
problem Solving nonsmooth constrained convex optimization problems.
method Unified randomized block-coordinate primal-dual algorithm.
result Achieves optimal convergence rates of O ( n / k ) \mathcal{O}(n/k) O ( n / k ) and O ( n 2 / k 2 ) \mathcal{O}(n^2/k^2) O ( n 2 / k 2 ) . A new method for RLHF using proximal point Nash learning.
problem Capturing real human preferences in RLHF.
method Proximal point Nash learning, embedding self-play updates into a proximal point framework.
result High-probability last-iterate convergence for the combined method.
SGD converges to optimal solution in perfect data fitting problem.
problem Noiseless least-squares with perfect data fitting.
method Last iterate of SGD with constant step-size.
result Convergence of SGD final iterate for non-strongly convex problem.
WSqD extends learning rate schedules for large model training without fixed horizons.
problem Fixed learning rate schedules limit training horizon extension.
method WSqD replaces constant stable phase with a shifted inverse-square-root base, retaining linear cooldown.
result WSqD achieves minimax-optimal convergence rate and horizon-independence.
Improved rates for continual learning using SGD and last-iterate analysis.
problem Forgetting in overparameterized models after fitting multiple tasks.
method Developed novel SGD upper bounds for continual linear models and analyzed their performance.
result Established universal forgetting rates for continual learning.
Despite remarkable empirical success, the training dynamics of generative adversarial networks (GAN), which involves solving a minimax game using stochastic gradients, is still poorly understood. In this work, we analyze last-iterate convergence of simultaneous gradient descent (simGD) and its variants under the assump…
Improves training GANs by escaping limit cycles.
problem Limit cycling behavior in training GANs.
method Predictive Centripetal Acceleration Algorithm (PCAA) combined with Adam.
result PCAA improves convergence rates and effectively trains GANs.
Counterfactual Regret Minimization (CFR) has found success in settings like poker which have both terminal states and perfect recall. We seek to understand how to relax these requirements. As a first step, we introduce a simple algorithm, local no-regret learning (LONR), which uses a Q-learning-like update rule to allo…
Stochastic gradient descent's long-term fluctuations are described by a diffusion limit.
problem Long-term behavior of stochastic gradient descent in non-smooth settings.
method Functional central limit theorem applied to rescaled trajectory of SGD.
result Characterization of long-term fluctuations around the minimizer.
We revisit Spakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology. We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with…
Improved SGD for non-strongly-convex regression with faster convergence.
problem Non-strongly-convex least squares regression problems.
method Modified accelerated gradient descent.
result Achieves optimal prediction error rates of O ( d / t ) O(d/t) O ( d / t ) and forgets initial conditions faster to O ( d / t 2 ) O(d/t^2) O ( d / t 2 ) .