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Let $\mathrm{R}$ be a real closed field, $S \subset \mathrm{R}^n$ a closed and bounded semi-algebraic set and $\mathbf{f} = (f_1,\ldots,f_p):S \rightarrow \mathrm{R}^p$ a continuous semi-algebraic map. We study the poset module structure in homology induced by the simultaneous filtrations of $S$ by the sub-level sets of the functions $f_i$ from an algorithmic and quantitative point of view. For fixed dimensional homology we prove a singly exponential upper bound on the complexity of these modules which are encoded as certain semi-algebraically constructible functions on $\mathrm{R}^p \times \mathrm{R}^p$. We also deduce for semi-algebraic filtrations of bounded complexity, upper bounds on the number of equivalence classes of finite poset modules that such a filtration induces -- establishing a tight analogy with a well-known graph theoretical result on the "speed'' of algebraically defined graphs., Comment: 35 pages, 1 figure. Comments welcome

Generalization and optimization guarantees on the population loss in machine learning often rely on uniform convergence based analysis, typically based on the Rademacher complexity of the predictors. The rich representation power of modern models has led to concerns about this approach. In this paper, we present generalization and optimization guarantees in terms of the complexity of the gradients, as measured by the Loss Gradient Gaussian Width (LGGW). First, we introduce generalization guarantees directly in terms of the LGGW under a flexible gradient domination condition, which we demonstrate to hold empirically for deep models. Second, we show that sample reuse in finite sum (stochastic) optimization does not make the empirical gradient deviate from the population gradient as long as the LGGW is small. Third, focusing on deep networks, we present results showing how to bound their LGGW under mild assumptions. In particular, we show that their LGGW can be bounded (a) by the $L_2$-norm of the loss Hessian eigenvalues, which has been empirically shown to be $\tilde{O}(1)$ for commonly used deep models; and (b) in terms of the Gaussian width of the featurizer, i.e., the output of the last-but-one layer. To our knowledge, our generalization and optimization guarantees in terms of LGGW are the first results of its kind, avoid the pitfalls of predictor Rademacher complexity based analysis, and hold considerable promise towards quantitatively tight bounds for deep models.

Recent works have shown a reduction from contextual bandits to online regression under a realizability assumption [Foster and Rakhlin, 2020, Foster and Krishnamurthy, 2021]. In this work, we investigate the use of neural networks for such online regression and associated Neural Contextual Bandits (NeuCBs). Using existing results for wide networks, one can readily show a ${\mathcal{O}}(\sqrt{T})$ regret for online regression with square loss, which via the reduction implies a ${\mathcal{O}}(\sqrt{K} T^{3/4})$ regret for NeuCBs. Departing from this standard approach, we first show a $\mathcal{O}(\log T)$ regret for online regression with almost convex losses that satisfy QG (Quadratic Growth) condition, a generalization of the PL (Polyak-\L ojasiewicz) condition, and that have a unique minima. Although not directly applicable to wide networks since they do not have unique minima, we show that adding a suitable small random perturbation to the network predictions surprisingly makes the loss satisfy QG with unique minima. Based on such a perturbed prediction, we show a ${\mathcal{O}}(\log T)$ regret for online regression with both squared loss and KL loss, and subsequently convert these respectively to $\tilde{\mathcal{O}}(\sqrt{KT})$ and $\tilde{\mathcal{O}}(\sqrt{KL^*} + K)$ regret for NeuCB, where $L^*$ is the loss of the best policy. Separately, we also show that existing regret bounds for NeuCBs are $\Omega(T)$ or assume i.i.d. contexts, unlike this work. Finally, our experimental results on various datasets demonstrate that our algorithms, especially the one based on KL loss, persistently outperform existing algorithms.

Generative models have gained popularity for their potential applications in imaging science, such as image reconstruction, posterior sampling and data sharing. Flow-based generative models are particularly attractive due to their ability to tractably provide exact density estimates along with fast, inexpensive and diverse samples. Training such models, however, requires a large, high quality dataset of objects. In applications such as computed imaging, it is often difficult to acquire such data due to requirements such as long acquisition time or high radiation dose, while acquiring noisy or partially observed measurements of these objects is more feasible. In this work, we propose AmbientFlow, a framework for learning flow-based generative models directly from noisy and incomplete data. Using variational Bayesian methods, a novel framework for establishing flow-based generative models from noisy, incomplete data is proposed. Extensive numerical studies demonstrate the effectiveness of AmbientFlow in learning the object distribution. The utility of AmbientFlow in a downstream inference task of image reconstruction is demonstrated., Comment: Accepted to Transactions on Machine Learning Research (TMLR). OpenReview: https://openreview.net/forum?id=txpYITR8oa

The Landsat program is the longest-running Earth observation program in history, with 50+ years of data acquisition by 8 satellites. The multispectral imagery captured by sensors onboard these satellites is critical for a wide range of scientific fields. Despite the increasing popularity of deep learning and remote sensing, the majority of researchers still use decision trees and random forests for Landsat image analysis due to the prevalence of small labeled datasets and lack of foundation models. In this paper, we introduce SSL4EO-L, the first ever dataset designed for Self-Supervised Learning for Earth Observation for the Landsat family of satellites (including 3 sensors and 2 product levels) and the largest Landsat dataset in history (5M image patches). Additionally, we modernize and re-release the L7 Irish and L8 Biome cloud detection datasets, and introduce the first ML benchmark datasets for Landsats 4-5 TM and Landsat 7 ETM+ SR. Finally, we pre-train the first foundation models for Landsat imagery using SSL4EO-L and evaluate their performance on multiple semantic segmentation tasks. All datasets and model weights are available via the TorchGeo (https://github.com/microsoft/torchgeo) library, making reproducibility and experimentation easy, and enabling scientific advancements in the burgeoning field of remote sensing for a multitude of downstream applications.

In this paper, we study utilizing neural networks for the exploitation and exploration of contextual multi-armed bandits. Contextual multi-armed bandits have been studied for decades with various applications. To solve the exploitation-exploration trade-off in bandits, there are three main techniques: epsilon-greedy, Thompson Sampling (TS), and Upper Confidence Bound (UCB). In recent literature, a series of neural bandit algorithms have been proposed to adapt to the non-linear reward function, combined with TS or UCB strategies for exploration. In this paper, instead of calculating a large-deviation based statistical bound for exploration like previous methods, we propose, ``EE-Net,'' a novel neural-based exploitation and exploration strategy. In addition to using a neural network (Exploitation network) to learn the reward function, EE-Net uses another neural network (Exploration network) to adaptively learn the potential gains compared to the currently estimated reward for exploration. We provide an instance-based $\widetilde{\mathcal{O}}(\sqrt{T})$ regret upper bound for EE-Net and show that EE-Net outperforms related linear and neural contextual bandit baselines on real-world datasets., Comment: Journal Version of EE-Net. arXiv admin note: substantial text overlap with arXiv:2110.03177

All machine learning algorithms use a loss, cost, utility or reward function to encode the learning objective and oversee the learning process. This function that supervises learning is a frequently unrecognized hyperparameter that determines how incorrect outputs are penalized and can be tuned to improve performance. This paper shows that training speed and final accuracy of neural networks can significantly depend on the loss function used to train neural networks. In particular derivative values can be significantly different with different loss functions leading to significantly different performance after gradient descent based Backpropagation (BP) training. This paper explores the effect on performance of using new loss functions that are also convex but penalize errors differently compared to the popular Cross-entropy loss. Two new classification loss functions that significantly improve performance on a wide variety of benchmark tasks are proposed. A new loss function call smooth absolute error that outperforms the Squared error, Huber and Log-Cosh losses on datasets with significantly many outliers is proposed. This smooth absolute error loss function is infinitely differentiable and more closely approximates the absolute error loss compared to the Huber and Log-Cosh losses used for robust regression.

We improve the theoretical and empirical performance of neural-network(NN)-based active learning algorithms for the non-parametric streaming setting. In particular, we introduce two regret metrics by minimizing the population loss that are more suitable in active learning than the one used in state-of-the-art (SOTA) related work. Then, the proposed algorithm leverages the powerful representation of NNs for both exploitation and exploration, has the query decision-maker tailored for $k$-class classification problems with the performance guarantee, utilizes the full feedback, and updates parameters in a more practical and efficient manner. These careful designs lead to an instance-dependent regret upper bound, roughly improving by a multiplicative factor $O(\log T)$ and removing the curse of input dimensionality. Furthermore, we show that the algorithm can achieve the same performance as the Bayes-optimal classifier in the long run under the hard-margin setting in classification problems. In the end, we use extensive experiments to evaluate the proposed algorithm and SOTA baselines, to show the improved empirical performance., Comment: Published on NeurIPS 2022

We consider the problem of optimization of deep learning models with smooth activation functions. While there exist influential results on the problem from the ``near initialization'' perspective, we shed considerable new light on the problem. In particular, we make two key technical contributions for such models with $L$ layers, $m$ width, and $\sigma_0^2$ initialization variance. First, for suitable $\sigma_0^2$, we establish a $O(\frac{\text{poly}(L)}{\sqrt{m}})$ upper bound on the spectral norm of the Hessian of such models, considerably sharpening prior results. Second, we introduce a new analysis of optimization based on Restricted Strong Convexity (RSC) which holds as long as the squared norm of the average gradient of predictors is $\Omega(\frac{\text{poly}(L)}{\sqrt{m}})$ for the square loss. We also present results for more general losses. The RSC based analysis does not need the ``near initialization" perspective and guarantees geometric convergence for gradient descent (GD). To the best of our knowledge, ours is the first result on establishing geometric convergence of GD based on RSC for deep learning models, thus becoming an alternative sufficient condition for convergence that does not depend on the widely-used Neural Tangent Kernel (NTK). We share preliminary experimental results supporting our theoretical advances.

Let $k$ be a field, let $A$ and $B$ be polynomial rings over $k$, and let $S= A \otimes_k B$. Let $I \subseteq A$ and $J \subseteq B$ be monomial ideals. We establish a binomial expansion for rational powers of $I+J \subseteq S$ in terms of those of $I$ and $J$. Particularly, for a positive rational number $u$, we prove that $(I+J)_u = \sum_{0 \le \omega \le u, \ \omega \in \mathbb{Q}} I_\omega J_{u-\omega},$ and that the sum on the right hand side is a finite sum. This finite sum can be made more precise using jumping numbers of rational powers of $I$ and $J$. We further give sufficient conditions for this formula to hold for the integral closures of powers of $I+J$ in terms of those of $I$ and $J$. Under these conditions, we provide explicit formulas for the depth and regularity of $\overline{(I+J)^k}$ in terms of those of powers of $I$ and $J$., Comment: revision added connections to jumping numbers; 14 pages

We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an $\mathfrak{m}$-primary ideal exists in a Noetherian local ring $(R,\mathfrak{m})$ with prime characteristic $p>0.$ This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring $R$ of a simplicial complex and an ideal $J$ generated by pure powers of the variables, the generalized Hilbert-Kunz function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of $J$ in terms of Hilbert-Samuel multiplicity of $J.$ We conclude by giving a counter-example to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal., Comment: 16 pages

Recent years have seen advances in generalization bounds for noisy stochastic algorithms, especially stochastic gradient Langevin dynamics (SGLD) based on stability (Mou et al., 2018; Li et al., 2020) and information theoretic approaches (Xu and Raginsky, 2017; Negrea et al., 2019; Steinke and Zakynthinou, 2020). In this paper, we unify and substantially generalize stability based generalization bounds and make three technical contributions. First, we bound the generalization error in terms of expected (not uniform) stability which arguably leads to quantitatively sharper bounds. Second, as our main contribution, we introduce Exponential Family Langevin Dynamics (EFLD), a substantial generalization of SGLD, which includes noisy versions of Sign-SGD and quantized SGD as special cases. We establish data-dependent expected stability based generalization bounds for any EFLD algorithm with a O(1/n) sample dependence and dependence on gradient discrepancy rather than the norm of gradients, yielding significantly sharper bounds. Third, we establish optimization guarantees for special cases of EFLD. Further, empirical results on benchmarks illustrate that our bounds are non-vacuous, quantitatively sharper than existing bounds, and behave correctly under noisy labels.

Remotely sensed geospatial data are critical for applications including precision agriculture, urban planning, disaster monitoring and response, and climate change research, among others. Deep learning methods are particularly promising for modeling many remote sensing tasks given the success of deep neural networks in similar computer vision tasks and the sheer volume of remotely sensed imagery available. However, the variance in data collection methods and handling of geospatial metadata make the application of deep learning methodology to remotely sensed data nontrivial. For example, satellite imagery often includes additional spectral bands beyond red, green, and blue and must be joined to other geospatial data sources that can have differing coordinate systems, bounds, and resolutions. To help realize the potential of deep learning for remote sensing applications, we introduce TorchGeo, a Python library for integrating geospatial data into the PyTorch deep learning ecosystem. TorchGeo provides data loaders for a variety of benchmark datasets, composable datasets for generic geospatial data sources, samplers for geospatial data, and transforms that work with multispectral imagery. TorchGeo is also the first library to provide pre-trained models for multispectral satellite imagery (e.g., models that use all bands from the Sentinel-2 satellites), allowing for advances in transfer learning on downstream remote sensing tasks with limited labeled data. We use TorchGeo to create reproducible benchmark results on existing datasets and benchmark our proposed method for preprocessing geospatial imagery on the fly. TorchGeo is open source and available on GitHub: https://github.com/microsoft/torchgeo.

In this paper, we propose a novel neural exploration strategy in contextual bandits, EE-Net, distinct from the standard UCB-based and TS-based approaches. Contextual multi-armed bandits have been studied for decades with various applications. To solve the exploitation-exploration tradeoff in bandits, there are three main techniques: epsilon-greedy, Thompson Sampling (TS), and Upper Confidence Bound (UCB). In recent literature, linear contextual bandits have adopted ridge regression to estimate the reward function and combine it with TS or UCB strategies for exploration. However, this line of works explicitly assumes the reward is based on a linear function of arm vectors, which may not be true in real-world datasets. To overcome this challenge, a series of neural bandit algorithms have been proposed, where a neural network is used to learn the underlying reward function and TS or UCB are adapted for exploration. Instead of calculating a large-deviation based statistical bound for exploration like previous methods, we propose "EE-Net", a novel neural-based exploration strategy. In addition to using a neural network (Exploitation network) to learn the reward function, EE-Net uses another neural network (Exploration network) to adaptively learn potential gains compared to the currently estimated reward for exploration. Then, a decision-maker is constructed to combine the outputs from the Exploitation and Exploration networks. We prove that EE-Net can achieve $\mathcal{O}(\sqrt{T\log T})$ regret and show that EE-Net outperforms existing linear and neural contextual bandit baselines on real-world datasets., Comment: Published on ICLR 2022

Sub-seasonal climate forecasting (SSF) is the prediction of key climate variables such as temperature and precipitation on the 2-week to 2-month time horizon. Skillful SSF would have substantial societal value in areas such as agricultural productivity, hydrology and water resource management, and emergency planning for extreme events such as droughts and wildfires. Despite its societal importance, SSF has stayed a challenging problem compared to both short-term weather forecasting and long-term seasonal forecasting. Recent studies have shown the potential of machine learning (ML) models to advance SSF. In this paper, for the first time, we perform a fine-grained comparison of a suite of modern ML models with start-of-the-art physics-based dynamical models from the Subseasonal Experiment (SubX) project for SSF in the western contiguous United States. Additionally, we explore mechanisms to enhance the ML models by using forecasts from dynamical models. Empirical results illustrate that, on average, ML models outperform dynamical models while the ML models tend to be conservatives in their forecasts compared to the SubX models. Further, we illustrate that ML models make forecasting errors under extreme weather conditions, e.g., cold waves due to the polar vortex, highlighting the need for separate models for extreme events. Finally, we show that suitably incorporating dynamical model forecasts as inputs to ML models can substantially improve the forecasting performance of the ML models. The SSF dataset constructed for the work, dynamical model predictions, and code for the ML models are released along with the paper for the benefit of the broader machine learning community.

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of $I(\mathcal{D})$ coincide when $\mathcal{D}$ is a weighted oriented certain class of tree. Finally, we give necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented lines., Comment: 7 pages

Recent empirical work on stochastic gradient descent (SGD) applied to over-parameterized deep learning has shown that most gradient components over epochs are quite small. Inspired by such observations, we rigorously study properties of Truncated SGD (T-SGD), that truncates the majority of small gradient components to zeros. Considering non-convex optimization problems, we show that the convergence rate of T-SGD matches the order of vanilla SGD. We also establish the generalization error bound for T-SGD. Further, we propose Noisy Truncated SGD (NT-SGD), which adds Gaussian noise to the truncated gradients. We prove that NT-SGD has the same convergence rate as T-SGD for non-convex optimization problems. We demonstrate that with the help of noise, NT-SGD can provably escape from saddle points and requires less noise compared to previous related work. We also prove that NT-SGD achieves better generalization error bound compared to T-SGD because of the noise. Our generalization analysis is based on uniform stability and we show that additional noise in the gradient update can boost the stability. Our experiments on a variety of benchmark datasets (MNIST, Fashion-MNIST, CIFAR-10, and CIFAR-100) with various networks (VGG and ResNet) validate the theoretical properties of NT-SGD, i.e., NT-SGD matches the speed and accuracy of vanilla SGD while effectively working with sparse gradients, and can successfully escape poor local minima.

In spite of advances in understanding lazy training, recent work attributes the practical success of deep learning to the rich regime with complex inductive bias. In this paper, we study rich regime training empirically with benchmark datasets, and find that while most parameters are lazy, there is always a small number of active parameters which change quite a bit during training. We show that re-initializing (resetting to their initial random values) the active parameters leads to worse generalization. Further, we show that most of the active parameters are in the bottom layers, close to the input, especially as the networks become wider. Based on such observations, we study static Layer-Wise Sparse (LWS) SGD, which only updates some subsets of layers. We find that only updating the top and bottom layers have good generalization and, as expected, only updating the top layers yields a fast algorithm. Inspired by this, we investigate probabilistic LWS-SGD, which mostly updates the top layers and occasionally updates the full network. We show that probabilistic LWS-SGD matches the generalization performance of vanilla SGD and the back-propagation time can be 2-5 times more efficient.

A generic formulation for the optimal control of a single wave-energy converter (WEC) is proposed. The formulation involves hard and soft constraints on the motion of the WEC to promote reduced damage and fatigue to the device during operation. Most of the WEC control literature ignores the cost of the control and could therefore result in generating less power than expected, or even negative power. Therefore, to ensure actual power gains in practice, we incorporate a penalty term in the objective function to approximate the cost of applying the control force. A discretization of the resulting optimal control problem is a quadratic optimization problem that can be solved efficiently using state-of-the-art solvers. Using hydrodynamic coefficients estimated by simulations made in WEC-Sim, numerical illustrations are provided of the trade-off between careful operation of the device and power generated. Finally, a demonstration of the real-time use of the approach is provided., Comment: Submitted for journal publication (under review)

We study the homological algebra of edge ideals of Erd\"{o}s-R\'enyi random graphs. These random graphs are generated by deleting edges of a complete graph on $n$ vertices independently of each other with probability $1-p$. We focus on some aspects of these random edge ideals - linear resolution, unmixedness and algebraic invariants like the Castelnuovo-Mumford regularity, projective dimension and depth. We first show a double phase transition for existence of linear presentation and resolution and determine the critical windows as well. As a consequence, we obtain that except for a very specific choice of parameters (i.e., $n,p := p(n)$), with high probability, a random edge ideal has linear presentation if and only if it has linear resolution. This shows certain conjectures hold true for large random graphs with high probability even though the conjectures were shown to fail for determinstic graphs. Next, we study asymptotic behaviour of some algebraic invariants - the Castelnuovo-Mumford regularity, projective dimension and depth - of such random edge ideals in the sparse regime (i.e., $p = \frac{\lambda}{n}, \lambda \in (0,\infty)$). These invariants are studied using local weak convergence (or Benjamini-Schramm convergence) and relating them to invariants on Galton-Watson trees. We also show that when $p \to 0$ or $p \to 1$ fast enough, then with high probability the edge ideals are unmixed and for most other choices of $p$, these ideals are not unmixed with high probability. This is further progress towards the conjecture that random monomial ideals are unlikely to have Cohen-Macaulay property (see De Loera et al. 2019a,2019b) in the setting when the number of variables goes to infinity but the degree is fixed., Comment: 24 pages. An error in the theorem on regularity is now corrected and the theorem is also extended to projective dimension and depth. Parts of the introduction are re-written and some sections are re-arranged

Differentially private SGD (DP-SGD) is one of the most popular methods for solving differentially private empirical risk minimization (ERM). Due to its noisy perturbation on each gradient update, the error rate of DP-SGD scales with the ambient dimension $p$, the number of parameters in the model. Such dependence can be problematic for over-parameterized models where $p \gg n$, the number of training samples. Existing lower bounds on private ERM show that such dependence on $p$ is inevitable in the worst case. In this paper, we circumvent the dependence on the ambient dimension by leveraging a low-dimensional structure of gradient space in deep networks -- that is, the stochastic gradients for deep nets usually stay in a low dimensional subspace in the training process. We propose Projected DP-SGD that performs noise reduction by projecting the noisy gradients to a low-dimensional subspace, which is given by the top gradient eigenspace on a small public dataset. We provide a general sample complexity analysis on the public dataset for the gradient subspace identification problem and demonstrate that under certain low-dimensional assumptions the public sample complexity only grows logarithmically in $p$. Finally, we provide a theoretical analysis and empirical evaluations to show that our method can substantially improve the accuracy of DP-SGD in the high privacy regime (corresponding to low privacy loss $\epsilon$).

We study differentially private (DP) algorithms for stochastic non-convex optimization. In this problem, the goal is to minimize the population loss over a $p$-dimensional space given $n$ i.i.d. samples drawn from a distribution. We improve upon the population gradient bound of ${\sqrt{p}}/{\sqrt{n}}$ from prior work and obtain a sharper rate of $\sqrt[4]{p}/\sqrt{n}$. We obtain this rate by providing the first analyses on a collection of private gradient-based methods, including adaptive algorithms DP RMSProp and DP Adam. Our proof technique leverages the connection between differential privacy and adaptive data analysis to bound gradient estimation error at every iterate, which circumvents the worse generalization bound from the standard uniform convergence argument. Finally, we evaluate the proposed algorithms on two popular deep learning tasks and demonstrate the empirical advantages of DP adaptive gradient methods over standard DP SGD., Comment: In the current version, we drop the experimental results on CIFAR-10 dataset due to an implementation error