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.
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.
New algorithm reduces optimization complexity in adaptive mirror descent.
problem Optimizing complex, non-smooth, non-convex functions efficiently.
method SVRAMD: Variance Reduced Adaptive Mirror Descent.
result Variance reduction accelerates convergence in adaptive mirror descent.
Develops parameter-free online mirror descent for optimal dynamic regret.
problem Optimal online linear optimization in unbounded domains.
method Modified online mirror descent framework for parameter-free algorithms.
result First unconstrained online linear optimization achieving optimal dynamic regret.
Mirror descent linked to information ratio via Bayesian regret bounds.
problem Understanding stability in mirror descent and its relation to information ratio.
method Developed a connection between mirror descent and information ratio using Bayesian regret bounds.
result Mirror descent with suitable estimators and distributions achieves bounds similar to information-directed sampling.
Continuous-time distributed mirror descent with integral feedback converges to global optimum.
problem Distributed optimization of a global strongly convex function with local convex components.
method Continuous-time distributed mirror descent with integral feedback.
result Asymptotic convergence to global optimum with constant step-size.
This paper explores a new framework for reinforcement learning based on online convex optimization, in particular mirror descent and related algorithms. Mirror descent can be viewed as an enhanced gradient method, particularly suited to minimization of convex functions in highdimensional spaces. Unlike traditional grad…
Mirror descent algorithm recovers low-rank matrices in matrix sensing.
problem Matrix sensing with low-rank matrices under certain conditions.
method Discrete-time mirror descent applied to empirical risk with Bregman divergence analysis.
result Mirror descent converges to a matrix minimizing a specific nuclear norm-related quantity.
The paper connects tempering and entropic mirror descent for sampling.
problem Sampling from a target distribution with known unnormalized density.
method Establishes the connection between tempering SMC and entropic mirror descent, deriving convergence rates and geometric insights.
result Tempering SMC iterates correspond to entropic mirror descent on the reverse KL divergence, providing new optimization perspectives.
Continuous-time mirror descent solves sparse phase retrieval efficiently.
problem Recovering sparse signals from magnitude-only measurements.
method Continuous-time mirror descent applied to unconstrained empirical risk minimization problem.
result Mirror descent recovers k k k -sparse vectors with minimum non-zero entry order of ∥ x ⋆ ∥ 2 / k \| \mathbf{x}^\star \|_2/\sqrt{k} ∥ x ⋆ ∥ 2 / k from k 2 k^2 k 2 Gaussian measurements. Paper studies early-stopped mirror descent for noisy sparse phase retrieval.
problem Recovering a sparse signal from noisy quadratic measurements.
method Early-stopped mirror descent with hyperbolic entropy mirror map.
result Achieves nearly minimax-optimal rate of convergence for k k k -sparse signals. New analysis shows GMD can converge linearly under PL-like conditions.
problem Establishing linear convergence for generalized mirror descent.
method PL-based analysis for time-dependent mirrors, Taylor-series approach for stochastic GMD.
result Linear convergence of stochastic GMD under PL-like conditions.
This paper extends Mirror Descent to Riemannian manifolds for optimization.
problem Optimization on Riemannian manifolds.
method Developed a Riemannian Mirror Descent (RMD) framework and a stochastic variant.
result Established non-asymptotic convergence guarantees for RMD and stochastic RMD.
The paper analyzes mirror descent in measure spaces and its applications.
problem Optimizing convex functionals over measure spaces.
method Derives convergence of mirror descent for relatively smooth and convex pairs of functionals.
result Shows Sinkhorn's primal iterations and EM as mirror descent.
Paper uses Mirror Descent for efficient risk budgeting portfolios.
problem Computing optimal risk budgeting weights for various risk measures.
method Employed Mirror Descent algorithms in deterministic and stochastic settings.
result Established convergence and quantitative rate for averaged Mirror Descent algorithm.
Information geometry applies concepts in differential geometry to probability and statistics and is especially useful for parameter estimation in exponential families where parameters are known to lie on a Riemannian manifold. Connections between the geometric properties of the induced manifold and statistical properti…
The paper improves convergence for linear systems using entropic mirror descent with Polyak stepsizes.
problem Convergence analysis for linear systems with unbounded domain.
method Entropic mirror descent with Polyak stepsizes, sublinear and linear convergence results.
result Generalized convergence result for arbitrary convex functions.
Paper accelerates Bayesian few-shot classification using mirror descent.
problem Non-conjugate inference in Bayesian few-shot classification.
method Integrates mirror descent-based variational inference into Gaussian process-based few-shot classification.
result Accelerated convergence and improved uncertainty quantification.
We consider the problem of multi-class classification and a stochastic opti- mization approach to it. We derive risk bounds for stochastic mirror descent algorithm and provide examples of set geometries that make the use of the algorithm efficient in terms of error in k.
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.
Unified framework for solving MDPs with stochastic mirror descent.
problem Approximately solving infinite-horizon Markov decision processes (MDPs).
method Primal-dual stochastic mirror descent for MDPs with a unified framework.
result Computes ε-optimal policies with expected samples for both average-reward and discounted MDPs.
Mirror descent method improved RL algorithms.
problem Improving RL algorithms for better performance.
method Mirror descent method applied to RL, solving trust-region problems.
result MDPO outperforms or matches other RL algorithms in continuous control tasks.
Recently there has been a surge of interest in understanding implicit regularization properties of iterative gradient-based optimization algorithms. In this paper, we study the statistical guarantees on the excess risk achieved by early-stopped unconstrained mirror descent algorithms applied to the unregularized empiri…
Study shows Stochastic Mirror Descent optimizes convex problems with infinite noise variance.
problem Optimizing convex problems with infinite noise variance.
method Stochastic Mirror Descent algorithm with uniformly convex mirror maps.
result Demonstrates convergence rate quantified in terms of iterations, dimensionality, and geometric parameters.
This paper balances bias and variance in adaptive importance sampling using mirror descent.
problem Large variance in adaptive importance sampling weights.
method Regularization strategy with power raised importance weights connected to mirror descent.
result The regularization parameter balances bias and variance.
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…
New method improves convergence and reduces variance in noisy optimization problems.
problem Computing exact minimizers with noisy gradient information.
method Stochastic mirror descent with interacting particles.
result Interaction helps improve convergence and reduce variance.
New algorithm for reinforcement learning reduces complexity and guarantees convergence.
problem Reinforcement learning problems with convex occupancy measures.
method MD-CURL, inspired by mirror descent, uses non-standard regularization.
result Achieves convergence guarantees and simple closed-form solution.
Stochastic mirror descent improves performance on ensemble models.
problem Improving performance of ensemble models using stochastic mirror descent.
method Utilizes mirror potential to influence training algorithm's implicit bias, mapping evolution to continuous time process.
result Converges to a nonlinear PDE in asymptotic regime of large networks, with mirror potential affecting gradient flow.
Matrix SMD converges to unique solution minimizing Bregman divergence.
problem High-dimensional multi-output classification and matrix completion problems.
method Stochastic Mirror Descent with matrix parameters and matrix mirror functions.
result Matrix SMD converges exponentially to the unique solution minimizing Bregman divergence.
Paper uses Stochastic Mirror Descent for large-scale sparse recovery problems.
problem Statistical estimation of high-dimensional sparse parameters.
method Non-Euclidean Composite Stochastic Mirror Descent (CSMD) algorithm for solving penalized stochastic optimization problems.
result The proposed algorithm achieves optimal convergence in sparse Generalized Linear Regression problems.
DDO-RM improves reward-based policies by converting reward scores into a target distribution.
problem Improving reward-based policies when the reward function is simpler than the policy.
method Converts reward scores into a target distribution and uses KL-regularized mirror-descent updates.
result DDO-RM outperforms DPO in pair accuracy and mean margin.
EM algorithm converges in KL divergence for exponential families via mirror descent.
problem Lack of understanding of EM's non-asymptotic convergence properties.
method Viewing EM as a mirror descent algorithm, showing convergence rates in KL divergence.
result KL divergence rates for EM in exponential families, invariant to parametrization.
Unified framework for non-Euclidean CPD under scalable stochastic mirror descent.
problem Handling non-Euclidean losses in tensor decomposition.
method Tensor fiber sampling strategy-based stochastic mirror descent.
result Global convergence to a stationary point under reasonable conditions.
Paper tackles private optimization for non-smooth objectives efficiently.
problem Private stochastic convex optimization for non-smooth objectives.
method Noisy mirror descent algorithm.
result Achieves optimal rates in statistical complexity and number of queries.
We accelerate PMD algorithms for reinforcement learning using functional methods.
problem Improving reinforcement learning algorithms for large-scale optimization.
method Proposed a momentum-based update for PMD algorithms leveraging duality.
result The proposed method accelerates reinforcement learning algorithms.
This paper extends mirror descent to Banach spaces with reproducing kernels.
problem Optimizing in Banach spaces with reproducing kernels.
method Mirror descent algorithm adapted for Banach spaces with reproducing kernels.
result Mirror descent achieves linear convergence in certain conditions and standard convergence in a constrained setting.
GPMD solves regularized RL with linear convergence, promoting structural policies.
problem Regularized reinforcement learning to encourage exploration and structural policies.
method Policy mirror descent with generalized convex regularizers and Bregman divergence.
result GPMD converges linearly to the global solution over a wide range of learning rates.
Paper proposes ZO-SMD for MERO, achieving optimal convergence rates.
problem Minimizing excess risk across all test distributions.
method Zeroth-order stochastic mirror descent algorithm for both smooth and non-smooth MERO.
result Converges at optimal rates of O ( 1 / t ) \mathcal{O}(1/\sqrt{t}) O ( 1/ t ) for estimates and optimization errors. Paper proposes a new method for training nonconvex models.
problem Training nonconvex models like neural networks.
method Successive functional gradient optimization using mirror descent in a function space.
result The method leads to better performance than standard training techniques.
A new parallel algorithm for learning optimal policies in MDPs with low communication costs.
problem Learning optimal policies for infinite-horizon MDPs.
method Primal-Dual Stochastic Mirror Descent for convex programming problems with inexact constraints.
result First parallel algorithm for average-reward MDPs with generative model and low communication costs.
The paper analyzes EM for Mixtures of Experts and shows its equivalence to projected Mirror Descent.
problem Training Mixtures of Experts (MoE) models.
method Rigorously analyzes Expectation Maximization (EM) for MoE models using a Mirror Descent perspective.
result Derives new convergence results and identifies conditions for local linear convergence.
Unified analysis of online optimization with self-concordant barriers, improving regret bounds.
problem Online convex optimization with specific loss functions.
method Online mirror descent with self-concordant barriers and logarithmic loss.
result Improved regret bounds for online portfolio selection and quantum state learning.
Optimizes distributions robustly with Sinkhorn distance.
problem Distributionally robust optimization with Wasserstein distance.
method Convex programming dual reformulation, stochastic mirror descent algorithm.
result Demonstrates superior performance in synthetic and real data.
The MAP estimate's log-likelihood sub-optimality is hard to bound in general.
problem Bounding the expected log-likelihood sub-optimality of MAP for exponential families.
method Interpreting MAP as stochastic mirror descent and analyzing convergence rates.
result Current convergence results do not apply to standard examples of exponential families.
Bayesian methods are appealing in their flexibility in modeling complex data and ability in capturing uncertainty in parameters. However, when Bayes' rule does not result in tractable closed-form, most approximate inference algorithms lack either scalability or rigorous guarantees. To tackle this challenge, we propose …
New method estimates mixture distributions using stochastic mirror descent.
problem Estimating unknown distributions from samples.
method Stochastic Mirror Descent (SMD) algorithm for mixture models.
result Near-optimal convergence rates in KL divergence and ℓ 2 \ell_2 ℓ 2 -norm. This work closes the theory-practice gap for distributed optimization methods by introducing a new regularity condition.
problem Existing convergence conditions for distributed optimization methods are violated by nearly all kernels used in practice.
method Introduces Hessian relative uniform continuity (HRUC) to guarantee convergence under mild conditions.
result Derives convergence guarantees for mirror descent-based gradient tracking without restrictive assumptions.