This work explores the non-convex optimization in compressive learning and the performance of heuristics.
problem The challenge of learning from compressed representations in compressive learning.
method Numerical simulations of the non-convex optimization landscape and heuristic performance.
result Properties of the non-convex optimization landscape and heuristic performance are explored.
This work shows neural networks can solve non-convex constraints problems.
problem Training neural networks under non-convex constraints.
method Project stochastic gradient descent with no-regret analysis of online learning.
result Overparameterized neural networks achieve near-optimal and near-feasible solutions.
We consider online forecasting problems for non-convex machine learning models. Forecasting introduces several challenges such as (i) frequent updates are necessary to deal with concept drift issues since the dynamics of the environment change over time, and (ii) the state of the art models are non-convex models. We ad…
Paper estimates differences in multi-attribute Gaussian graphical models using non-convex penalties.
problem Estimating differences in multi-attribute Gaussian graphical models with similar structure.
method Penalized D-trace loss function with non-convex (log-sum and SCAD) penalties, proximal gradient descent methods.
result Theoretical analysis and numerical examples support consistency in support recovery and estimation.
In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying optimization problems are often non-convex. While convex relaxations could lead to polyn…
Improved optimization guarantees for deep learning models with Nesterov acceleration.
problem Optimization in non-convex deep learning landscapes.
method Analysis of Nesterov acceleration in benignly non-convex landscapes.
result Identical guarantees can be obtained in optimization problems with weak geometric assumptions, especially in overparametrized deep learning.
New insights into using momentum for non-convex optimization.
problem Improving training of non-convex models like deep neural networks.
method Developed a Lyapunov analysis of SGD with momentum using stochastic primal averaging.
result Precise conditions under which SGD+M outperforms SGD and optimal hyper-parameter schedules.
Study accelerates optimization methods in non-convex problems, but doesn't improve the algorithm's performance.
problem Understanding the behavior of momentum-based acceleration methods in non-convex, high-dimensional landscapes.
method Used dynamical mean field theory to describe the average dynamics of heavy-ball momentum and Nesterov acceleration in a non-convex model.
result Accelerated dynamics but did not improve the algorithm's performance with respect to gradient descent.
We rigorously prove statistical physics predictions for non-convex GLMs in high dimensions.
problem Analyzing high-dimensional optimization problems in non-convex Generalized Linear Models.
method Developed a systematic framework using the Gaussian Min-Max Theorem and AMP to rigorously prove replica-symmetric formulas.
result Validated statistical physics predictions for non-convex GLMs, aligning with physicist's conjectures.
Compact, non-convex curve flows are created.
problem Creating compact, non-convex ancient solutions for curve shortening flow.
method Constructed an ancient solution asymptotic to Yin-Yang curve.
result Compact, non-convex ancient solutions for curve shortening flow are demonstrated.
AGGLIO optimizes non-convex functions with local convexity guarantees.
problem Optimizing non-convex functions with local convexity.
method Stage-wise, graduated optimization technique for locally convex functions.
result Global convergence to the global optimum for non-convex and locally convex objectives.
Non-convex optimization problems often arise from probabilistic modeling, such as estimation of posterior distributions. Non-convexity makes the problems intractable, and poses various obstacles for us to design efficient algorithms. In this work, we attack non-convexity by first introducing the concept of \emph{probab…
A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-c…
Paper uses integer programming for non-convex boosting in classification.
problem Improving classification performance using non-convex optimization.
method Non-convex boosting via integer programming.
result Results comparable to or better than state-of-the-art.
Develops a new fairness learning approach for multi-task regression models.
problem Fairness in multi-task regression models with biased datasets.
method Uses rank-based non-parametric independence test (Mann Whitney U statistic) and reformulates as non-convex optimization problem.
result Outperforms state-of-the-art methods on fairness metrics.
First order methods can take extremely long to find global minima of non-convex functions.
problem Finding global minimizers of non-convex functions.
method Designing a family of non-convex functions and using statistical lower bounds for parameter estimation.
result First order methods can take exponential time to converge to a global minimizer.
New approach for distributed online optimization of non-convex losses with sublinear regret.
problem Regret evaluation and consensus in distributed, multi-agent systems with non-convex losses.
method Composite regret metric and consensus-based online normalized gradient (CONGD) approach for pseudo-convex losses; offline optimization oracle for general non-convex losses.
result First sublinear regret bound for general distributed online non-convex learning.
Paper proposes a working set algorithm for non-convex sparse regression with provable convergence.
problem Estimating sparse linear models from high-dimensional data using non-convex regularizers.
method FireWorks algorithm based on non-convex reformulation and leveraging residual geometry.
result Convergence to a stationary point of the full problem with provable guarantees.
New algorithm improves convergence for non-convex problems with boundaries.
problem Optimizing non-convex problems with constraints.
method Reflected Gradient Langevin Dynamics with probabilistic representation.
result Promising convergence rates, faster than existing methods.
Non-convex SGD learns halfspaces with adversarial label noise efficiently.
problem Agnostically learning halfspaces in adversarial label noise settings.
method Non-convex SGD optimization for halfspace learning.
result Non-convex SGD achieves misclassification error close to optimal with adversarial noise.
Unified analysis of multi-attribute graph learning with non-convex penalties.
problem Graph inference from multi-attribute data.
method Penalized log-likelihood objective function with ADMM and local linear approximation.
result Local consistency in support recovery and precision matrix estimation for non-convex penalties.
We investigate 1) the rate at which refined properties of the empirical risk---in particular, gradients---converge to their population counterparts in standard non-convex learning tasks, and 2) the consequences of this convergence for optimization. Our analysis follows the tradition of norm-based capacity control. We p…
Matrix completion is a well-studied problem with many machine learning applications. In practice, the problem is often solved by non-convex optimization algorithms. However, the current theoretical analysis for non-convex algorithms relies heavily on the assumption that every entry is observed with exactly the same pro…
The popular Lasso approach for sparse estimation can be derived via marginalization of a joint density associated with a particular stochastic model. A different marginalization of the same probabilistic model leads to a different non-convex estimator where hyperparameters are optimized. Extending these arguments to pr…
This paper reviews techniques for distributed learning with non-convex models.
problem Optimizing non-convex models in distributed learning.
method Selective review of recent techniques for batch and streaming data.
result Trade-offs between computation and communication costs in distributed learning.
New method estimates GGLM parameters, overcoming non-convexity.
problem Estimating parameters in GGLM with dependencies.
method Monotone operator-based variational inequality method.
result Guarantees for parameter recovery in GLM and GGLM.
We consider online learning in an adversarial, non-convex setting under the assumption that the learner has an access to an offline optimization oracle. In the general setting of prediction with expert advice, Hazan et al. (2016) established that in the optimization-oracle model, online learning requires exponentially …
New sampler tackles complex discrete energy landscapes efficiently.
problem Stagnation in gradient-based discrete samplers for non-convex settings.
method DREXEL sampler with Replica Exchange and Adjusted Metropolis.
result Proves samplers satisfy detailed balance and converge to target distribution.
Study uses weak transport for non-convex costs in fixed-income markets.
problem Characterizing optimal caplet pricing in fixed-income markets.
method Introduced weak optimal transport for non-convex costs, reduced general costs to convex problems.
result Established robust super-replication results for fixed-income markets.
Paper develops robust SGLD for solving non-convex DRO problems.
problem Solving non-convex distributionally robust optimisation problems with adversarially corrupted samples.
method Developed a Stochastic Gradient Langevin Dynamics (SGLD) algorithm with non-asymptotic convergence bounds.
result The robust SGLD estimator outperforms vanilla SGLD in terms of test accuracy.
SGD converges to global minimum for structured non-convex functions.
problem Optimizing non-convex functions using SGD with slow convergence rates.
method Convergence theorems for SGD on structured non-convex functions, including Quasar and PL conditions.
result SGD converges to global minimum for specific non-convex functions under certain conditions.
The TREX is a recently introduced method for performing sparse high-dimensional regression. Despite its statistical promise as an alternative to the lasso, square-root lasso, and scaled lasso, the TREX is computationally challenging in that it requires solving a non-convex optimization problem. This paper shows a remar…
We show that many machine learning goals, such as improved fairness metrics, can be expressed as constraints on the model's predictions, which we call rate constraints. We study the problem of training non-convex models subject to these rate constraints (or any non-convex and non-differentiable constraints). In the non…
Meta-learning can perform well on non-convex models even with few samples, contrary to convex models.
problem Understanding the sample complexity of meta-learning for non-convex models.
method Constructing a simple meta-learning instance and analyzing the training dynamics of Reptile and multi-task representation learning.
result Meta-learning can achieve new task sample complexity of O(1) for non-convex models, unlike convex models which require Ω(d) samples. New method certifies generative models' robustness.
problem Certifying generative models' robustness is challenging due to non-convex sets.
method ApproxLine, a scalable certification method capturing infinite sets or distributions over them.
result ApproxLine provides sound deterministic and probabilistic guarantees.
New algorithms optimize non-smooth, non-convex objectives with improved complexity.
problem Optimizing non-smooth, non-convex stochastic objectives.
method Reduction to online learning, applying optimistic online learning techniques.
result Improved complexity for finding (δ,ε)-stationary points. NSGLD improves SGLD for non-convex optimization problems.
problem Optimizing non-convex objectives efficiently.
method Introducing non-reversible SGLD by adding an anti-symmetric matrix to the drift term of the Langevin diffusion.
result NSGLD converges faster to the same stationary distribution with non-asymptotic guarantees.
New method improves efficiency of non-convex matrix reconstruction.
problem Reconstructing a low-rank matrix from few linear measurements.
method Factorized gradient descent with spectral initialization.
result Non-convex approaches can match the sample complexity of convex methods.
SGD's uncertainty quantified in non-convex learning problems.
problem Uncertainty quantification in non-convex learning problems.
method Asymptotic normality of SGD iterates and bias characterization.
result SGD iterates are asymptotically normally distributed around the expected value of the invariant distribution.
Optimizers find approximate global minima in non-convex problems.
problem Understanding why local methods solve non-convex optimization problems.
method Formalizing the hypothesis that many local minima are approximately global minima.
result Most local minima of practical non-convex objectives are approximately global minima.
Here we study non-convex composite optimization: first, a finite-sum of smooth but non-convex functions, and second, a general function that admits a simple proximal mapping. Most research on stochastic methods for composite optimization assumes convexity or strong convexity of each function. In this paper, we extend t…
Generalizes smoothness conditions for optimization methods.
problem Optimization under non-uniform smoothness conditions.
method Develops a new analysis technique for bounding gradients.
result Obtains convergence rates for gradient descent and Nesterov's method.
Proposes a non-convex optimization method for a parsimonious weighted naive Bayes classifier.
problem Improving naïve Bayes classifier performance with a large number of input variables.
method Sparse regularization of model log-likelihood for direct estimation of variable weights.
result Optimization-based weighted naïve Bayes classifiers achieve equivalent performance to averaging-based classifiers.
This thesis explores how submodularity aids in optimizing non-convex functions and validating algorithms.
problem Understanding which functions can be optimized efficiently in non-convex settings.
method Introducing continuous submodularity and developing algorithms for maximizing these functions.
result Characterization and optimization of continuous submodular functions with strong guarantees.
New method finds near-optimal solutions for non-convex optimization problems.
problem Finding near-optimal solutions for non-convex optimization problems.
method Riemannian stochastic recursive momentum method
result Achieves a near-optimal complexity of ildeO(ε−3). Paper tackles non-convex constrained DRO with a stochastic algorithm for large-scale applications.
problem Training robust models against data distribution shifts with non-convex loss functions.
method Developed a stochastic algorithm for non-convex constrained DRO with a complexity independent of dataset size.
result Algorithm finds ε-stationary points with computational complexity of O(ε^(-3k_*-5)) for general Cressie-Read divergence.
This paper improves convergence guarantees for SGD algorithms in non-convex smooth functions.
problem Theoretical convergence properties of SGD algorithms for non-convex smooth functions.
method Analysis of SGD algorithms with arbitrary data ordering for non-convex smooth functions.
result Enhanced convergence guarantees for incremental gradient and single shuffle SGD, improving the optimization term of convergence guarantee.
We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict saddle property for non-convex problem that allo…