Persistence landscapes map diagrams into function spaces for statistical and machine learning applications.
problem Mapping persistence diagrams into function spaces for statistical and machine learning.
method Introducing persistence landscapes, weighted persistence landscapes, and Poisson-weighted persistence landscape kernels.
result Persistence landscapes allow for the application of statistical and machine learning tools, and are stable and invertible.
Model explains surprising properties of neural network training landscapes.
problem Understanding the geometry of neural network loss landscapes.
method Developed a simple theoretical model of gradients and Hessians.
result Unified model accounts for 4 surprising properties of neural loss landscapes.
Unified model predicts structure of neural network loss landscapes.
problem Understanding the structure of neural network loss landscapes.
method Modeling loss landscape as high-dimensional wedges, analyzing hyperparameters' effects.
result Existence of low-loss subspaces connecting solutions.
Characterizes critical points and landscapes of neural networks.
problem Understanding loss functions and critical points in neural networks.
method Full characterization of analytical forms for critical points and global minimizers of square loss functions.
result Linear networks have no spurious local minima, while ReLU networks have local minima that are not global minima.
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.
The paper characterizes neural network landscapes for gradient dominance and regularity.
problem Understanding the landscape of neural network loss functions.
method Characterization of gradient dominance and regularity conditions for neural networks.
result Explicit characterization of global minimizers and landscape properties for different neural network types.
Machine learning techniques are being increasingly used as flexible non-linear fitting and prediction tools in the physical sciences. Fitting functions that exhibit multiple solutions as local minima can be analysed in terms of the corresponding machine learning landscape. Methods to explore and visualise molecular pot…
Deep nets' optima are surprisingly connected by simple paths.
problem Understanding the loss landscape of deep nets.
method Mathematical explanations based on generic properties of well-trained nets.
result Optima in deep nets are connected by piece-wise linear paths.
Monotonic Linear Interpolation property in neural networks persists despite non-convexity.
problem Understanding the geometric properties of neural network loss landscapes.
method Tools from differential geometry to analyze the monotonicity of neural network weights.
result Sufficient conditions for the Monotonic Linear Interpolation property under mean squared error.
Study energy landscapes in glass models, focusing on Gaussian and spiked-tensor functions.
problem Characterize statistical properties and phase transitions of high-dimensional energy landscapes.
method Developed a Kac-Rice method framework to compute landscape complexity and analyze phase transitions rigorously.
result Characterized the ruggedness and arrangements of local minima in energy landscapes.
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.
Deep neural networks and glassy systems share dynamics but differ in landscape properties.
problem Comparing training dynamics of DNNs and glassy systems.
method Statistical physics methods applied to DNN training.
result DNN dynamics slow down due to many flat directions, diffusing at the loss minimum.
We analyze the optimization landscape of α-loss in logistic models.
problem Optimization landscape of α-loss in logistic models.
method Tools from strictly-locally-quasi-convex functions and geometric techniques.
result Evolution of optimization landscape with respect to α.
This paper studies the landscape of empirical risk of deep neural networks by theoretically analyzing its convergence behavior to the population risk as well as its stationary points and properties. For an l-layer linear neural network, we prove its empirical risk uniformly converges to its population risk at the rat…
Flexible Kernels for Protein Property Prediction
problem Predicting protein properties from sparse experimental data
method Sequence kernels using evolutionary substitution matrices and local linearity
result Data-efficient models of protein property landscapes
Analyzes adversarial training's impact on loss landscape, proposing PAS to improve model performance.
problem Challenges in optimizing models under adversarial training due to loss landscape properties.
method Analytical studies of adversarial loss functions, numerical analyses, PAS strategy.
result Adversarial training impairs optimization, but PAS strategy improves model performance.
Gradient descent variants improve phase retrieval accuracy.
problem Phase retrieval problem in high-dimensional spaces.
method Gradient descent, stochastic gradient descent, Langevin algorithm, dynamical mean-field theory.
result Stochastic variants of gradient descent achieve better generalization in phase retrieval.
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 investigate the structure of the profit landscape obtained from the most basic, fluctuation based, trading strategy applied for the daily stock price data. The strategy is parameterized by only two variables, p and q. Stocks are sold and bought if the log return is bigger than p and less than -q, respectively. Repet…
In this paper we investigate the properties of series of vacua in the string theory landscape. In particular, we study minima to the flux potential in type IIB compactifications on the mirror quintic. Using geometric transitions, we embed its one dimensional complex structure moduli space in that of another Calabi-Yau …
Neural networks' optimization dynamics are confined to a single basin despite connected basins in the loss landscape.
problem Neural networks' optimization dynamics are confined to a single basin despite connected basins in the loss landscape.
method Identifying entropic barriers arising from the interplay between curvature variations along low-loss paths and noise in optimization dynamics.
result Curvature-induced entropic forces bias noisy dynamics back toward the endpoints, explaining the confinement and connectivity of solutions.
Novel approach embeds loss tunnels in neural networks, revealing insights into their structure.
problem Understanding the structure of neural network loss surfaces, especially low-loss tunnels.
method Directly embedding loss tunnels into the loss landscape of neural networks.
result Improved insights into the length and structure of loss tunnels, and better subspace inference in Bayesian neural networks.
The paper proves skip connections help neural networks avoid shallow local minima.
problem Understanding how skip connections affect the loss landscape of deep neural networks.
method Theoretical analysis of the topology of loss landscapes of deep ReLU neural networks with skip connections.
result Skip connections help control the connectedness of sub-level sets, avoiding shallow local minima.
Deep neural networks' loss surfaces contain every low-dimensional pattern.
problem Finding arbitrary low-dimensional patterns in neural network loss surfaces.
method Empirical and theoretical analysis of loss landscapes of deep neural networks.
result Deep universal approximators exhibit a property where arbitrary smooth patterns exist in their loss surfaces.
Study geometric properties of loss functions to understand neural network performance.
problem Understanding the geometric properties of high-dimensional loss functions to improve neural network performance.
method Combine concepts from high-dimensional probability and differential geometry to study curvature properties in lower-dimensional loss representations.
result Mean curvature in the original loss space determines if saddle points appear as minima, maxima, or flat regions.
Three training regimes found for scale-invariant neural networks on the sphere.
problem Training scale-invariant neural networks on the sphere with varying effective learning rate.
method Investigated three regimes of training: convergence, chaotic equilibrium, and divergence.
result Discovered three distinct training regimes with unique characteristics.
Deep learning dynamics exhibit anomalous superdiffusion initially, aiding escape from local minima.
problem Understanding the dynamics of learning in deep neural networks.
method Novel analysis of SGD dynamics and loss landscape structure.
result SGD exhibits anomalous superdiffusion initially, transitioning to subdiffusion as learning progresses.
We study the landscape of Lagrangian functions for nonconvex optimization problems.
problem Understanding the stable equilibria of nonconvex optimization problems.
method We define a special class of Lagrangian functions and propose a stochastic primal-dual algorithm.
result We establish an asymptotic convergence rate and sample complexity for solving the online GEV problem.
Study optimization landscapes for overcomplete representations, showing benign geometric structures.
problem Optimizing overcomplete representations in high-dimensional data analysis.
method Formulate as ℓ4-norm optimization problems with spherical constraint, analyze geometric properties. result Nonconvex objectives have benign geometric structures, ensuring local search algorithms find target solutions.
Most high-dimensional estimation and prediction methods propose to minimize a cost function (empirical risk) that is written as a sum of losses associated to each data point. In this paper we focus on the case of non-convex losses, which is practically important but still poorly understood. Classical empirical process …
Unified framework explains geometric properties of CNNs.
problem Understanding why encoder-decoder CNNs perform well.
method Unified mathematical framework based on recent neural network theories.
result Encoder-decoder CNNs are related to nonlinear basis representation using combinatorial convolution frames.
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.
flacco simplifies feature-based landscape analysis for optimization problems.
problem Choosing the best optimizer from a portfolio of algorithms.
method Developed an R-package for feature-based landscape analysis.
result Makes landscape analysis accessible and comprehensible.
Spatially constrained clustering divides landscapes into homogeneous regions with spatial contiguity.
problem Dividing landscapes into homogeneous patches with spatial contiguity and hierarchy.
method Developed a spatially constrained spectral clustering framework using a flexible kernel and recursive bisection.
result The proposed framework outperforms baseline methods in balancing region contiguity and homogeneity.
This paper addresses the issue of feature importance landscapes in complex images and proposes a regularisation technique to improve network performance.
problem Feature importance landscapes in complex images are not as uniform as assumed, affecting network performance.
method Developed the PILCRO objective to regularize weight configurations, making importance landscapes smoother and more data-driven.
result P-regularised networks have a flat importance landscape, train faster, and perform better in accuracy and robustness.
Adaptor 'E' extends gradient-based optimizers to explore loss landscapes, improving generalization.
problem Finding lower and better-generalizing minima in deep learning.
method Proposes an adaptor 'E' to extend gradient-based optimizers, encouraging exploration along landscape valleys.
result Adapted optimizers increase test accuracy by an average of 2.5% in large-batch training tasks.
The paper studies the loss landscape of regularized deep matrix factorization, revealing unique and sharp minimizers.
problem Understanding the loss landscape and minimizers of regularized deep matrix factorization problems.
method Theoretical analysis of ℓ2-regularized deep matrix factorization/deep linear network training problems with squared-error loss. result The unique end-to-end minimizer exists for all target matrices except for a set of Lebesgue measure zero.
Study reveals properties of local minima in ReLU networks.
problem Understanding the loss landscape of neural networks.
method Theoretical analysis of one-hidden-layer ReLU networks.
result All differentiable local minima are global within certain regions.
Adaptive methods produce less robust models to adversarial examples.
problem Robustness of machine learning models to adversarial examples.
method Comparison of adaptive and non-adaptive optimization methods in linear regression.
result Non-adaptive methods consistently produce more robust models.
Deep RL policies share adversarial features across different MDPs.
problem Understanding decision boundaries and loss landscapes in neural policies.
method Investigating similarities in high sensitivity directions across MDPs using Arcade Learning Environment.
result High sensitivity directions for neural policies are correlated across MDPs, suggesting shared non-robust features.
Designs a non-convex objective function to learn one-hidden-layer neural networks.
problem Learning one-hidden-layer neural networks with Gaussian input and nonnegative label.
method Analytic formula for population risk, landscape design of non-convex objective function G(⋅), stochastic gradient descent. result Stochastic gradient descent on G converges to the global minimum and learns the ground-truth parameters. 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.
The paper analyzes adaptive algorithms in non-convex optimization landscapes.
problem Analyzing adaptive algorithms in non-convex optimization landscapes.
method Stochastic algorithms with decreasing step-size, considering mini-batches and noise.
result Established almost sure convergence to critical points and minimizers.
Automated model selects best algorithm for continuous problems efficiently.
problem Optimizing continuous black-box problems with limited resources.
method Combining ELA features with machine learning for algorithm selection.
result Average resource requirement is less than half compared to best single solver.
The paper studies quadratic neural networks, proving existence of spurious minima and saddle points.
problem Understanding the loss landscape of neural networks with quadratic activations.
method Theoretical analysis of mean squared error loss for neural networks with quadratic activations.
result Proves existence of spurious local minima and saddle points in the training landscape of deep overparameterized quadratic neural networks.
Study on loss landscapes of deep linear networks using algebraic geometry.
problem Understanding the loss landscapes of deep linear networks.
method Algebraic geometry methods to analyze the properties of the optimization landscapes.
result Deep linear networks have local minima that are not global minima.
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.
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.