Deep networks can interpolate noisy data without losing generalization.
problem Characterizing the relationship between interpolation and generalization in overparameterized deep networks.
method Analyzing the loss landscape of neural network functions over volumes around training data points, varying model parameters and training epochs.
result Loss sharpness in the input space follows a double descent, with large models predicting noisy targets over larger volumes around training data points.
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.
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.
Paper analyzes noisy low-rank matrix optimization, improving RIP bounds and convergence rates.
problem Noisy low-rank matrix optimization with general objective functions.
method Develops new mathematical framework and proves convergence rate under RIP condition.
result Any spurious local solution is close to ground truth when RIP constant is less than 1/3.
New method tackles rugged optimization landscapes in contact-rich scenarios.
problem Optimization challenges in dynamic environments with deformable objects.
method Combines Bayesian optimization with semi-local 'leaps' for global search.
result Outperforms gradient-based and gradient-free baselines in simulation and real robot experiments.
Deep learning with noisy gradient descent outperforms linear estimators in high dimensions.
problem Theoretical explanation of deep learning's superiority over linear methods.
method Theoretical analysis of excess risk of a deep learning estimator trained by noisy gradient descent.
result Deep learning achieves a faster learning rate than linear estimators, especially in high dimensions.
New method finds minimum in noisy data, useful for model selection.
problem Finding the index of the minimum value in noisy observations.
method Developed an asymptotically normal test statistic integrating cross-validation and differential privacy.
result Achieves a favorable bias-variance trade-off in practical scenarios.
This work classifies SOC loss functions based on their gradient properties.
problem Optimizing noisy systems in stochastic optimal control.
method Grouping loss functions into classes with the same gradient expectation.
result Different loss functions have the same optimization landscape but differ in gradient variance.
New method improves training of PINNs for PDEs by adding noisy supervision terms.
problem Slow or failed convergence of PINNs on challenging PDEs.
method Operator preconditioning using Feynman-Kac supervision and non-asymptotic error bounds.
result Non-asymptotic error bounds for FK-PINNs, showing improved performance over standard PINNs.
SGD transitions between maxima and minima with varying time scales.
problem Understanding SGD's behavior near critical points in noisy landscapes.
method Analyzing SGD convergence and escape dynamics in 1D landscapes with infinite- and finite-variance noise.
result SGD reliably moves to the basin's minimum unless close to a local maximum, where it can linger.
Discovering quasipotential equations from data using machine learning.
problem Understanding escape mechanisms from metastable states in nonlinear systems.
method Combining neural networks and sparse regression to symbolically reconstruct quasipotential equations.
result Model-unbiased analytical forms of quasipotential discovered directly from data.
Gradient-descent-based algorithms and their stochastic versions have widespread applications in machine learning and statistical inference. In this work we perform an analytic study of the performances of one of them, the Langevin algorithm, in the context of noisy high-dimensional inference. We employ the Langevin alg…
Quantum control is valuable for various quantum technologies such as high-fidelity gates for universal quantum computing, adaptive quantum-enhanced metrology, and ultra-cold atom manipulation. Although supervised machine learning and reinforcement learning are widely used for optimizing control parameters in classical …
Multi-layer neural networks are among the most powerful models in machine learning, yet the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a non-convex high-dimensional objective (risk function), a problem which is usually attacked using stochastic g…
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…
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.
New method reduces memorization in diffusion models without sacrificing image quality.
problem Diffusion models often memorize training data, especially with small datasets.
method Train models using noisy data at large noise scales to reduce memorization.
result Significant reduction in memorization without compromising image quality.
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.
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.
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.
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.
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.
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.
The paper analyzes conditions for solving low-rank matrix recovery problems with noisy measurements.
problem Low-rank matrix recovery with corrupted measurements.
method Analysis of the restricted isometry property (RIP) and local search methods.
result Sharp bounds on the maximum distance between local minimizers and the ground truth.
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.
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…
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.
The increasing availability of large but noisy data sets with a large number of heterogeneous variables leads to the increasing interest in the automation of common tasks for data analysis. The most time-consuming part of this process is the Exploratory Data Analysis, crucial for better domain understanding, data clean…
Develops a generalized version of Chung's Lemma for stochastic optimization methods.
problem Establishing asymptotic convergence rates for stochastic optimization methods under various step size rules.
method Generalized version of Chung's Lemma for a broader family of step size rules.
result Demonstrates tight non-asymptotic convergence rates for various stochastic methods.
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 improves HNNs by learning optimal curvature for better generalization.
problem Inappropriate curvatures in HNNs lead to suboptimal performance.
method Sharpness-aware curvature learning method to smooth loss landscape.
result Proposed method improves HNNs' generalization across various settings.
Study reveals sharp characterisation of local minima in neural network loss landscapes.
problem Characterizing local minima in high-dimensional two-layer ReLU neural networks.
method Exact low-dimensional representation of local minima using summary statistics and link with one-pass SGD dynamics.
result Local minima in overparameterized neural networks form discrete families with varying stability and reachability.
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.
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.
Study visualizes actor-critic loss landscapes for inventory optimization.
problem Difficulties in solving multi-store dynamic inventory control problems.
method Low-dimensional visualizations of actor loss function.
result Loss landscapes favor optimal policies in reinforcement learning.
We study rough high-dimensional landscapes in which an increasingly stronger preference for a given configuration emerges. Such energy landscapes arise in glass physics and inference. In particular we focus on random Gaussian functions, and on the spiked-tensor model and generalizations. We thoroughly analyze the stati…
Considering the creation of persistence landscape on a parametrized curve and structure of sampling, there exists a random process for which a finite mixture model of persistence landscape (FMMPL) can provide a better description for a given dataset. In this paper, a nonparametric approach for computing integrated mean…
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.
LSAM optimizes deep learning training with improved efficiency.
problem Inefficiency in distributed large-batch training with Sharpness-Aware Minimization (SAM).
method Integrates SAM's adversarial steps with an asynchronous distributed sampling strategy.
result Higher final accuracy compared to data-parallel SAM.
We characterize stationary solutions to McKean-Vlasov equations on the circle.
problem Stationary solutions of McKean-Vlasov equations on the circle.
method Exact equivalence to an infinite-dimensional quadratic system of equations over Fourier coefficients, leading to explicit characterization of stationary states.
result Analytic expressions for the emergence, form, and shape of bifurcations involving multiple Fourier modes, and connections with discontinuous phase transitions.
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.
Researchers improve NCE by addressing its flat loss landscape issues.
problem NCE's poor performance due to an ill-behaved loss landscape.
method Introduced eNCE with an exponential loss and normalized gradient descent.
result Proven that landscape issues arise from inappropriate noise distribution.
Embedding principle explains loss landscape of deep neural networks.
problem Understanding the structure of loss landscapes in deep neural networks.
method Proposed an embedding principle that critical points of narrower DNNs can be embedded to critical points of wider DNNs.
result Wide DNNs are often attracted by highly-degenerate critical points embedded from narrower DNNs.
There are many surprising and perhaps counter-intuitive properties of optimization of deep neural networks. We propose and experimentally verify a unified phenomenological model of the loss landscape that incorporates many of them. High dimensionality plays a key role in our model. Our core idea is to model the loss la…
The landscape of empirical risk has been widely studied in a series of machine learning problems, including low-rank matrix factorization, matrix sensing, matrix completion, and phase retrieval. In this work, we focus on the situation where the corresponding population risk is a degenerate non-convex loss function, nam…
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.
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. Deep ReLU networks with extra parameters have mostly good loss landscapes.
problem Finding good local minima in the loss landscape of deep neural networks.
method Analyzing shallow and deep ReLU networks with extra parameters on a generic dataset.
result Most activation patterns correspond to regions with no bad local minima.