Adversarial training makes logistic regression weight loss landscapes sharper.
problem Understanding why adversarial training sharpens the weight loss landscape in logistic regression.
method Theoretical analysis of linear logistic regression model with L2 norm constraints, and experiments on ResNet18.
result Adversarial training sharpens the weight loss landscape in linear logistic regression models.
Paper characterizes optimization landscape of Tucker decomposition.
problem Finding exact Tucker decomposition is a nonconvex optimization problem.
method Characterized the optimization landscape and provided a local search algorithm.
result All local minima are globally optimal if tensor has an exact Tucker decomposition.
Experimental fractal landscape dynamics observed in emulsions.
problem Understanding anomalous motions in soft glassy materials.
method Quantitative analysis of oil droplet trajectories in dense emulsions.
result Experimental fractal geometry matches computational model of soft glassy dynamics.
FPCA optimizes fairness in target vectors' span.
problem Fairness in principal component analysis for multiple target vectors.
method Non-concave maximization of worst projected target norm using sub-gradient descent.
result Optimization landscape is benign with globally optimal local minima.
The paper reveals surprising star-shaped connectivity in neural networks.
problem Understanding mode connectivity in neural network landscapes.
method Fine-grained analysis of connectivity in overparameterized and finite minima cases.
result Star-shaped connectivity exists in neural network landscapes, suggesting near convexity.
We explore the evolution of daily returns of four major US stock market indices during the technology crash of 2000, and the financial crisis of 2007-2009. Our methodology is based on topological data analysis (TDA). We use persistence homology to detect and quantify topological patterns that appear in multidimensional…
Optimizes MMD learning for generative models with theoretical guarantees.
problem Theoretical guarantees for optimizing non-convex MMD objectives.
method Analyzes MMD optimization landscape for specific distributions.
result Gradient-based methods globally minimize MMD objective for certain distributions.
Choosing the best-performing optimizer(s) out of a portfolio of optimization algorithms is usually a difficult and complex task. It gets even worse, if the underlying functions are unknown, i.e., so-called Black-Box problems, and function evaluations are considered to be expensive. In the case of continuous single-obje…
Paper analyzes Transformer learning dynamics, proving benign landscape for in-context learning.
problem Understanding how Transformers learn in context with nonlinear features.
method Mean-field and two-timescale analysis of Transformer dynamics, proving nonconvex but benign landscape.
result Proves mean-field dynamics avoid saddle points, leading to improved optimization.
The study tests inferences about neural network optimization from linear interpolation of loss landscapes.
problem Understanding the difficulty of neural network optimization problems.
method Linear interpolation of neural network loss landscapes, systematic evaluation of various factors.
result Linear interpolation does not correlate with model performance, challenging prior intuition.
Study identifies new stable climate states in climate model.
problem Understanding multistability and transitions in climate models.
method Combination of quasipotential theory and manifold learning.
result Discovery of a third stable climate state not previously known.
SGD vs quasi-Newton optimization in neural networks: different landscapes, different generalizability.
problem Understanding neural network optimization and generalizability.
method Comparison of stochastic gradient descent (SGD) and quasi-Newton optimization methods using computational tools.
result SGD solutions are separated by lower barriers than quasi-Newton solutions, but quasi-Newton solutions are deeper and more isolated.
New function class characterizes loss landscape of deep neural networks without over-parametrization.
problem Complex loss landscape of deep neural networks without over-parametrization.
method Proposed a novel class of functions to characterize loss landscape without over-parametrization.
result Gradient-based optimizers possess theoretical guarantees of convergence under the new function class assumption.
This work justifies neural collapse under MSE loss and analyzes the optimization landscape.
problem Understanding neural collapse in deep neural networks under MSE loss.
method Global landscape analysis of vanilla nonconvex MSE loss.
result The only global minimizers are neural collapse solutions.
Riemannian geometry improves protein dynamics analysis.
problem Efficient analysis of protein dynamics data in non-linear spaces.
method Developed a local approximation technique for geodesics and a smooth manifold of protein conformations.
result Geodesics approximate molecular dynamics trajectories and provide realistic summary statistics.
The paper analyzes phase retrieval under limited samples, ensuring a benign local landscape for convergence.
problem Ensuring a benign local landscape for phase retrieval under limited samples.
method Fine-grained analysis of local landscape properties under the regime of limited samples.
result Gradient descent can converge to an od(1)-loss solution exponentially fast under certain conditions. New method simplifies optimization landscapes by transforming saddle points.
problem Saddle points hinder non-convex optimization in machine learning.
method Variable elimination algorithms, like VarPro, are compared to reveal geometric insights.
result Variable elimination reshapes critical point structure, creating local maxima from saddle points.
Study uses topological signatures to quantify financial market complexity.
problem Capturing temporal organization beyond volatility measures.
method Null validated topological approach using L1 norm of persistence landscapes. result Persistence landscape norms reveal dynamical structure during market stress.
Paper shows no spurious local minima in a specific matrix factorization problem.
problem Optimization of ℓ1-norm rank-one symmetric matrix factorization. method Second-order variational analysis to study the landscape of the problem.
result Any second-order stationary point is globally optimal.
We analyze the landscape of empirical risk minimization for high-dimensional models, predicting phase transitions and critical point properties.
problem Understanding the complexity and structure of high-dimensional empirical risk landscapes.
method Using the Kac-Rice formula, we analyze the expected number of critical points and their spectral properties, providing detailed predictions.
result We derive complete topological phase diagrams for the phase retrieval problem, predicting BBP-type transitions and critical point stability.
We solve the optimization of two-layer ReLU networks using convex math.
problem Optimizing two-layer ReLU neural networks.
method Exact characterization of optimal solutions via convex optimization.
result We prove that all globally optimal solutions can be found via convex optimization.
Equivalent formulations for low-rank matrix optimization are proven.
problem Low-rank matrix optimization with rank constraints.
method Established geometric landscape connections between manifold and factorization formulations.
result Equivalence between manifold and factorization formulations at FOSPs, SOSPs, and strict saddles.
We present multi-point optimization: an optimization technique that allows to train several models simultaneously without the need to keep the parameters of each one individually. The proposed method is used for a thorough empirical analysis of the loss landscape of neural networks. By extensive experiments on FashionM…
Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine lea…
We study nonconvex optimization landscapes for learning overcomplete representations, including learning (i) sparsely used overcomplete dictionaries and (ii) convolutional dictionaries, where these unsupervised learning problems find many applications in high-dimensional data analysis. Despite the empirical success of …
In this paper, we build upon previous work on designing informative and efficient Exploratory Landscape Analysis features for characterizing problems' landscapes and show their effectiveness in automatically constructing algorithm selection models in continuous black-box optimization problems. Focussing on algorithm pe…
Reviews recent findings on neural network landscapes.
problem Non-convexity of loss functions causing bad landscapes.
method Rigorous geometric analysis and empirical exploration.
result Wide neural nets may have sub-optimal local minima.
TDA-based portfolios show better risk-adjusted returns than classical methods.
problem Traditional portfolio selection methods fail to capture complex asset dynamics.
method Topological Data Analysis (TDA) using persistence landscapes to quantify portfolio risk.
result TDA-based portfolios outperform classical models in excess mean return and financial ratios.
This paper presents approximate confidence intervals for each function of parameters in a Banach space based on a bootstrap algorithm. We apply kernel density approach to estimate the persistence landscape. In addition, we evaluate the quality distribution function estimator of random variables using integrated mean sq…
We analyze the loss landscape and expressiveness of practical deep convolutional neural networks (CNNs) with shared weights and max pooling layers. We show that such CNNs produce linearly independent features at a "wide" layer which has more neurons than the number of training samples. This condition holds e.g. for the…
GGA improves untrustworthy prediction detection in neural networks without retraining.
problem Susceptibility of neural networks to untrustworthy predictions, especially adversarial attacks and out-of-distribution data.
method Geometric Gradient Analysis (GGA) analyzes the geometry of neural network loss landscapes based on saliency maps.
result GGA outperforms existing methods in detecting untrustworthy predictions, including adversarial and out-of-distribution data.
SGD with machine learning noise converges to global minimum exponentially fast.
problem Optimizing machine learning models with stochastic gradient descent.
method Analysis of SGD with machine learning noise, focusing on energy landscapes and gradient noise.
result SGD converges to the global minimum exponentially fast under certain conditions.
Sampling strategies significantly affect feature approximations in ELA, impacting classifier accuracy.
problem The impact of sampling strategies on feature approximations in ELA.
method Analysis of feature approximations from different sampling strategies and sample sizes.
result Feature approximations from different sampling strategies do not converge, affecting classifier accuracy.
The study examines when MAML's objective has a benign landscape.
problem Understanding when MAML's objective landscape is benign.
method Analyzing the landscape of MAML objective on LQR tasks.
result The benign landscape of the MAML objective depends on task similarities.
This paper introduces SRPR for robust phase retrieval with smoothed loss functions.
problem Robust phase retrieval from noisy quadratic measurements with corruptions.
method Smoothed robust phase retrieval (SRPR) using convolution-type smoothed loss functions.
result SRPR has no spurious local solutions and benign landscape under corruptions.
Black holes offer insights into machine learning's loss landscapes.
problem Understanding the loss landscape in machine learning.
method Comparing machine learning loss landscapes to black hole entropy.
result Black holes provide an infinite family of potential landscapes with known minima.
In this paper we develop a new framework that captures the common landscape underlying the common non-convex low-rank matrix problems including matrix sensing, matrix completion and robust PCA. In particular, we show for all above problems (including asymmetric cases): 1) all local minima are also globally optimal; 2) …
New framework to understand and exploit curvature in deep learning loss landscapes.
problem Understanding and optimizing the loss landscape in deep learning models.
method New conceptual framework and techniques to estimate and exploit curvature of expected loss changes.
result Alice algorithm optimizes training by incorporating curvature terms and step bounds.
Deep learning dynamics and NTK evolution studied through diverse measures.
problem Understanding the training dynamics of deep neural networks and their loss landscapes.
method Phenomenological analysis of training dynamics in multiple architectures and datasets.
result Training dynamics exhibit a chaotic initial transient followed by a stable phase, with the NTK evolving to match full network performance.
AWP improves robustness by flattening weight loss landscape.
problem Improving robustness of deep neural networks against adversarial examples.
method Explicitly regularizes the flatness of weight loss landscape through adversarial weight perturbation.
result AWP forms a double-perturbation mechanism in adversarial training, leading to flatter weight loss landscape.
Efficiently infers graph edges from genetic similarity data in landscape genetics.
problem Inferring unknown graph edges from genetic similarity data in a heterogeneous landscape.
method Developed an efficient first-order optimization method to solve the inverse landscape genetics problem.
result Our method provides fast and reliable convergence, significantly outperforming existing heuristics.
We study how the behavior of deep policy gradient algorithms reflects the conceptual framework motivating their development. To this end, we propose a fine-grained analysis of state-of-the-art methods based on key elements of this framework: gradient estimation, value prediction, and optimization landscapes. Our result…
Smoothed fitness landscape improves protein optimization.
problem Infeasibility of combinatorially large protein sequence space.
method Formulate protein fitness as a graph signal, smooth using Tikunov regularization, and optimize with Gibbs sampling.
result 2.5 fold fitness improvement over training set.
We analyze neural collapse in neural networks, showing that features collapse to vertices of a Simplex ETF.
problem Understanding and optimizing the features learned in the last layer of neural networks during training.
method Simplified unconstrained feature model, studying the global optimization landscape of cross-entropy loss with weight decay.
result The global minimizers of the loss are Simplex ETFs, and other critical points are strict saddles with negative curvature.
Dimensionality reduction is ubiquitous in analysis of complex dynamics. The conventional dimensionality reduction techniques, however, focus on reproducing the underlying configuration space, rather than the dynamics itself. The constructed low-dimensional space does not provide complete and accurate description of the…
Researchers improve visualization of neural network loss landscapes.
problem Understanding neural network generalization performance.
method Novel 'jump and retrain' procedure, non-linear dimensionality reduction (PHATE), computational homology.
result Improved visualization and quantification of neural network generalization performance.
Research examines motivations and factors influencing retailers' payment method choices.
problem Understanding motivations and factors affecting retailers' payment method choices.
method Qualitative and quantitative analysis of various factors including regulatory constraints, merchant service providers, and demographic variables.
result Lower interchange fees and regulatory constraints make card payment adoption financially feasible for merchants.
Deeper models have a more favorable optimization landscape, making them more robust to noise.
problem Characterizing the effect of depth on the optimization landscape of linear regression models.
method Robust and over-parameterized setting, simple sub-gradient method.
result A simple sub-gradient method converges to a balanced solution that is close to the ground truth and enjoys a flat local landscape.