Proposes a new Armijo's condition for coordinate-wise functions.
problem Finding optimal step sizes in coordinate-wise optimization.
method Introduces a new Armijo's condition for functions defined on product spaces.
result Shows the advantage of the new condition over the standard Armijo's condition.
We develop and analyze efficient "coordinate-wise" methods for finding the leading eigenvector, where each step involves only a vector-vector product. We establish global convergence with overall runtime guarantees that are at least as good as Lanczos's method and dominate it for slowly decaying spectrum. Our methods a…
In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the…
Proposes a new Armijo's condition for general functions and provides an algorithm for its application.
problem Finding suitable step sizes for general functions in optimization.
method Introduces a coordinate-wise Armijo's condition and provides an algorithm to find suitable step sizes.
result Proves convergent results for various functions using the proposed algorithm.
We consider a variant of online convex optimization in which both the instances (input vectors) and the comparator (weight vector) are unconstrained. We exploit a natural scale invariance symmetry in our unconstrained setting: the predictions of the optimal comparator are invariant under any linear transformation of th…
Stochastic methods with coordinate-wise adaptive stepsize (such as RMSprop and Adam) have been widely used in training deep neural networks. Despite their fast convergence, they can generalize worse than stochastic gradient descent. In this paper, by revisiting the design of Adagrad, we propose to split the network par…
Nonconvex penalty methods for sparse modeling in linear regression have been a topic of fervent interest in recent years. Herein, we study a family of nonconvex penalty functions that we call the trimmed Lasso and that offers exact control over the desired level of sparsity of estimators. We analyze its structural prop…
PROBE algorithm efficiently solves sparse high-dimensional linear regression.
problem Sparse high-dimensional linear regression models with complex parameter spaces.
method Partitioned empirical Bayes ECM algorithm for computationally efficient MAP estimation.
result PROBE algorithm provides robust and efficient coordinate-wise optimization.
New DP-CD method outperforms DP-SGD in solving composite DP-ERM problems.
problem Privacy-preserving machine learning with differential privacy.
method Differentially Private proximal Coordinate Descent (DP-CD) for composite Empirical Risk Minimization (ERM).
result DP-CD outperforms DP-SGD due to larger step sizes and better gradient exploitation.
Sparse high dimensional graphical model selection is a popular topic in contemporary machine learning. To this end, various useful approaches have been proposed in the context of ℓ1-penalized estimation in the Gaussian framework. Though many of these inverse covariance estimation approaches are demonstrably scala…
Paper tackles Byzantine attacks in Federated Learning by clustering and robustifying.
problem Adversarial attacks from Byzantine machines in Federated Learning.
method Iterative Federated Clustering Algorithm (IFCA) with trimmed mean and median aggregation.
result Improved convergence rate for strongly convex loss functions in Byzantine-Robust IFCA.
In this work, we consider the resilience of distributed algorithms based on stochastic gradient descent (SGD) in distributed learning with potentially Byzantine attackers, who could send arbitrary information to the parameter server to disrupt the training process. Toward this end, we propose a new Lipschitz-inspired c…
AdaGrad outperforms SGD in non-convex optimization problems by a factor of d.
problem Finding near-stationary points in stochastic non-convex optimization.
method Refined assumptions on smoothness and gradient noise variance, l1-norm stationarity measure. result AdaGrad achieves a convergence rate favorable over SGD in certain non-convex settings.
In this paper we generalize the framework of the feasible descent method (FDM) to a randomized (R-FDM) and a coordinate-wise random feasible descent method (RC-FDM) framework. We show that the famous SDCA algorithm for optimizing the SVM dual problem, or the stochastic coordinate descent method for the LASSO problem, f…
In this short report, we discuss how coordinate-wise descent algorithms can be used to solve minimum variance portfolio (MVP) problems in which the portfolio weights are constrained by lq norms, where 1≤q≤2. A portfolio which weights are regularised by such norms is called a sparse portfolio (Brodie et …
We present an approach for penalized tensor decomposition (PTD) that estimates smoothly varying latent factors in multi-way data. This generalizes existing work on sparse tensor decomposition and penalized matrix decompositions, in a manner parallel to the generalized lasso for regression and smoothing problems. Our ap…
New bound on Rademacher complexity for vector functions.
problem Bounding Rademacher complexity for vector-valued functions.
method Bounding Rademacher complexity by coordinate-wise complexity with a factor of sqrt(K).
result Rademacher complexity is bounded by the maximum coordinate-wise complexity times sqrt(K).
Note on subgaussian bounds for sign-quantized linear maps.
problem Understanding subgaussian behavior of sign-quantized linear maps.
method Developed a dimension-independent subgaussian concentration bound for Gaussian vectors under nonlinear mappings.
result Answered a question about sign-quantized linear maps using a new subgaussian bound.
FedDuA adapts global learning rate for federated learning.
problem Slow convergence in federated learning due to dataset and parameter space heterogeneity.
method FedDuA uses mirror descent to adaptively select global learning rate based on inter-client and coordinate-wise heterogeneity.
result FedDuA achieves minimax optimal convergence for convex objectives and outperforms baselines in various settings.
New algorithm optimizes Bayesian network learning from Gaussian data.
problem Learning Bayesian networks from Gaussian observational data.
method Proposes a coordinate descent algorithm for ℓ0-penalized maximum likelihood estimation. result The algorithm converges to a coordinate-wise minimum and achieves optimal objective value as sample size increases.
Recently, new defense techniques have been developed to tolerate Byzantine failures for distributed machine learning. The Byzantine model captures workers that behave arbitrarily, including malicious and compromised workers. In this paper, we break two prevailing Byzantine-tolerant techniques. Specifically we show robu…
A new algorithm improves posterior sampling for linear inverse problems.
problem Efficiently sampling from posterior distributions in noisy linear inverse problems.
method Proposes \pddim, a DDIM-type sampler that separately samples along singular directions of the measurement operator.
result The method converges to the Bayesian posterior conditioned on the measurements.
Paper tackles distributed linear regression with compositional covariates.
problem Solving distributed statistical methodology and computing for massive compositional data.
method Proposes two distributed optimization techniques based on ADMM and CDMM for solving constrained convex optimization problems.
result Established convergence theories for the proposed algorithms under regularity conditions.
Decentralized learning for GLMs with feature distribution and network connectivity.
problem Optimizing generalized linear models in a decentralized network with feature partitioning.
method Chambolle--Pock primal--dual algorithm applied to an equivalent saddle-point formulation.
result Convergence rates for empirical risk minimization under Lipschitz and square root Lipschitz assumptions.
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 …
New algorithms for collaborative learning in uncertain, decentralized environments.
problem Byzantine collaborative learning in uncertain, decentralized environments.
method Two asynchronous solutions to averaging agreement, each optimal according to some dimension.
result New algorithms achieve optimal Byzantine resilience and collaborative learning.
New method speeds up optimization over probability measures.
problem High computational overhead in optimizing probability measures.
method Randomized coordinate descent on Wasserstein space.
result Significant speedups over full-gradient methods.
The paper analyzes high-dimensional linear regression using parametric empirical Bayes methods.
problem Estimation of i.i.d. priors in high-dimensional Bayesian linear regression with random design.
method Parametric empirical Bayes estimation, variational lower bound maximization, phase transition analysis.
result The vEB estimator is information theoretically optimal up to p=o(n2/3) but sub-optimal in higher dimensions. A new method for optimizing functions without gradients, improving efficiency and convergence.
problem Optimizing functions without gradient information in machine learning.
method Hybrid Gradient Descent (HGE) using random and coordinate-wise gradient estimates.
result The proposed method achieves optimal convergence rates in convex cases and generalizes to non-convex cases.
Generative AutoEncoders require a chosen probability distribution in latent space, usually multivariate Gaussian. The original Variational AutoEncoder (VAE) uses randomness in encoder - causing problematic distortion, and overlaps in latent space for distinct inputs. It turned out unnecessary: we can instead use determ…
We improve MoE models for classification with rigorous guarantees and practical methods.
problem Limited guarantees for stable maximum-likelihood training and model selection in softmax-gated MoE models.
method Derived a batch MM algorithm with closed-form updates, proved finite-sample rates, and developed a dendrogram selector.
result Achieved near-parametric optimal rates for parameter recovery and improved accuracy over baselines.
Privacy preserving machine learning algorithms are crucial for learning models over user data to protect sensitive information. Motivated by this, differentially private stochastic gradient descent (SGD) algorithms for training machine learning models have been proposed. At each step, these algorithms modify the gradie…
In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, including ℓ0, bridge, smoothly clipped absolute deviation, capped ℓ1 and mini…
The paper introduces a new method for feature selection without explicit sparsification.
problem Feature selection with implicit sparsity-inducing mechanisms.
method Optimization over a family of kernels with gradient descent.
result The method achieves exact sparsity without explicit penalization techniques.
Extends PD-NJ-ODE to noisy observations and dependent observation times.
problem Predicting continuous-time stochastic processes with irregular and noisy observations.
method Extends PD-NJ-ODE to handle conditional independence and noisy observations.
result Theoretical guarantees and empirical examples for handling noisy observations and dependent observation times.
Study linear regression with missing or corrupted data, showing error bounds.
problem Linear regression under missing or corrupted data.
method Information-theoretic lower bounds and efficient algorithms.
result Error bounds match in missing and corruption settings.
This paper analyzes convergence of RMSProp and Adam in non-convex optimization with tight complexity bounds.
problem Analyzing convergence of RMSProp and Adam in non-convex optimization with relaxed assumptions.
method Developed new convergence analyses for RMSProp and Adam, considering adaptive learning rates and affine noise variance.
result RMSProp and Adam converge to ε-stationary points with iteration complexities of O(ε^(-4)) under proper hyperparameters.
Adam converges with high probability under unconstrained non-convex smooth stochastic optimizations.
problem Theoretical limitations of Adam's convergence under unconstrained non-convex smooth stochastic optimizations.
method Deep analysis of Adam's convergence rate under affine variance noise, without bounded gradient assumptions.
result Adam converges to the stationary point with a high probability rate of $\mathcal{O}\left({
m poly}(\log T)/\sqrt{T}
ight)$.
Differentially private random block coordinate descent improves utility in machine learning.
problem Lack of privacy in classical CD methods when handling sensitive information.
method Proposes a differentially private random block coordinate descent method using sketch matrices and importance sampling.
result Demonstrates improved convergence rates and utility guarantees compared to non-private methods.
New framework improves EM algorithm convergence under log-Sobolev inequality.
problem Improving convergence of the EM algorithm.
method Extending gradient flow techniques to EM algorithm, using free energy representation.
result Exponential convergence of EM algorithm under log-Sobolev inequality.
AdAdaGrad optimizes batch sizes for deep learning models, reducing the generalization gap.
problem The generalization gap between large-batch and small-batch training in deep learning.
method AdAdaGrad introduces adaptive batch size strategies derived from adaptive sampling methods.
result AdAdaGradNorm converges to a first-order stationary point with a rate of O(1/K) in K iterations.
A new method for streaming PCA provides confidence intervals for eigenvector entries.
problem Uncertainty quantification for individual entries in streaming PCA.
method Oja's algorithm, Bernstein-type concentration bound, Central Limit Theorem, subsampling algorithm.
result Sharp concentration bound and Central Limit Theorem for streaming PCA entries.
New method predicts state evolution for non-first-order algorithms on nonconvex problems.
problem Analyzing nonconvex optimization problems with random data.
method Developed a state evolution for a broader class of algorithms including first-order and saddle point updates.
result Established rigorous state evolution predictions and finite-sample guarantees for non-first-order methods.
New optimizer designs respect symmetry, improving deep learning models.
problem Optimizers lack respect for symmetry in neural networks.
method Introduce symmetry-compatible principle for optimizer design.
result Symmetry-compatible optimizers improve model performance.
For the problem of high-dimensional sparse linear regression, it is known that an ℓ0-based estimator can achieve a 1/n "fast" rate on the prediction error without any conditions on the design matrix, whereas in absence of restrictive conditions on the design matrix, popular polynomial-time methods only guarante…
Estimates isotonic functions under unknown permutations, achieving optimal statistical and computational efficiency.
problem Estimating isotonic functions with unknown permutations in multiway comparison data.
method Mirsky partition estimator for minimax optimal and adaptive estimation.
result Achieves optimal worst-case statistical performance and computational efficiency.
Muon outperforms GD in associative memory learning by balancing frequency components.
problem Training dynamics and scaling behavior of Muon in associative memory learning.
method Study of Muon in a linear associative memory model with softmax retrieval and hierarchical frequency spectrum over query-answer pairs.
result Muon achieves exponential speedup over GD in noiseless case and superior scaling efficiency in noisy case.
A method for constructing tight prediction intervals for multiple numerical outputs.
problem Constructing tight prediction intervals for multiple related numerical outputs.
method A novel coordinate-wise standardization procedure that makes residuals comparable across output dimensions, estimating suitable scaling parameters using calibration data.
result The method produces tighter prediction intervals than existing baselines while maintaining valid simultaneous coverage.