Pub Date : 2022-07-15DOI: 10.48550/arXiv.2207.07721
T. McElroy, A. Roy, Gaurab Hore
Guaranteeing privacy in released data is an important goal for data-producing agencies. There has been extensive research on developing suitable privacy mechanisms in recent years. Particularly notable is the idea of noise addition with the guarantee of differential privacy. There are, however, concerns about compromising data utility when very stringent privacy mechanisms are applied. Such compromises can be quite stark in correlated data, such as time series data. Adding white noise to a stochastic process may significantly change the correlation structure, a facet of the process that is essential to optimal prediction. We propose the use of all-pass filtering as a privacy mechanism for regularly sampled time series data, showing that this procedure preserves utility while also providing sufficient privacy guarantees to entity-level time series.
{"title":"FLIP: A Utility Preserving Privacy Mechanism for Time Series","authors":"T. McElroy, A. Roy, Gaurab Hore","doi":"10.48550/arXiv.2207.07721","DOIUrl":"https://doi.org/10.48550/arXiv.2207.07721","url":null,"abstract":"Guaranteeing privacy in released data is an important goal for data-producing agencies. There has been extensive research on developing suitable privacy mechanisms in recent years. Particularly notable is the idea of noise addition with the guarantee of differential privacy. There are, however, concerns about compromising data utility when very stringent privacy mechanisms are applied. Such compromises can be quite stark in correlated data, such as time series data. Adding white noise to a stochastic process may significantly change the correlation structure, a facet of the process that is essential to optimal prediction. We propose the use of all-pass filtering as a privacy mechanism for regularly sampled time series data, showing that this procedure preserves utility while also providing sufficient privacy guarantees to entity-level time series.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"35 1","pages":"111:1-111:29"},"PeriodicalIF":0.0,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87110381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-12DOI: 10.48550/arXiv.2207.05740
T. Fritz, Andreas Klingler
The d-separation criterion detects the compatibility of a joint probability distribution with a directed acyclic graph through certain conditional independences. In this work, we study this problem in the context of categorical probability theory by introducing a categorical definition of causal models, a categorical notion of d-separation, and proving an abstract version of the d-separation criterion. This approach has two main benefits. First, categorical d-separation is a very intuitive criterion based on topological connectedness. Second, our results apply both to measure-theoretic probability (with standard Borel spaces) and beyond probability theory, including to deterministic and possibilistic networks. It therefore provides a clean proof of the equivalence of local and global Markov properties with causal compatibility for continuous and mixed random variables as well as deterministic and possibilistic variables.
{"title":"The d-separation criterion in Categorical Probability","authors":"T. Fritz, Andreas Klingler","doi":"10.48550/arXiv.2207.05740","DOIUrl":"https://doi.org/10.48550/arXiv.2207.05740","url":null,"abstract":"The d-separation criterion detects the compatibility of a joint probability distribution with a directed acyclic graph through certain conditional independences. In this work, we study this problem in the context of categorical probability theory by introducing a categorical definition of causal models, a categorical notion of d-separation, and proving an abstract version of the d-separation criterion. This approach has two main benefits. First, categorical d-separation is a very intuitive criterion based on topological connectedness. Second, our results apply both to measure-theoretic probability (with standard Borel spaces) and beyond probability theory, including to deterministic and possibilistic networks. It therefore provides a clean proof of the equivalence of local and global Markov properties with causal compatibility for continuous and mixed random variables as well as deterministic and possibilistic variables.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"7 1","pages":"46:1-46:49"},"PeriodicalIF":0.0,"publicationDate":"2022-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91083100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-02DOI: 10.48550/arXiv.2207.00713
Yanwei Jia, X. Zhou
We study the continuous-time counterpart of Q-learning for reinforcement learning (RL) under the entropy-regularized, exploratory diffusion process formulation introduced by Wang et al. (2020). As the conventional (big) Q-function collapses in continuous time, we consider its first-order approximation and coin the term ``(little) q-function". This function is related to the instantaneous advantage rate function as well as the Hamiltonian. We develop a ``q-learning"theory around the q-function that is independent of time discretization. Given a stochastic policy, we jointly characterize the associated q-function and value function by martingale conditions of certain stochastic processes, in both on-policy and off-policy settings. We then apply the theory to devise different actor-critic algorithms for solving underlying RL problems, depending on whether or not the density function of the Gibbs measure generated from the q-function can be computed explicitly. One of our algorithms interprets the well-known Q-learning algorithm SARSA, and another recovers a policy gradient (PG) based continuous-time algorithm proposed in Jia and Zhou (2022b). Finally, we conduct simulation experiments to compare the performance of our algorithms with those of PG-based algorithms in Jia and Zhou (2022b) and time-discretized conventional Q-learning algorithms.
{"title":"q-Learning in Continuous Time","authors":"Yanwei Jia, X. Zhou","doi":"10.48550/arXiv.2207.00713","DOIUrl":"https://doi.org/10.48550/arXiv.2207.00713","url":null,"abstract":"We study the continuous-time counterpart of Q-learning for reinforcement learning (RL) under the entropy-regularized, exploratory diffusion process formulation introduced by Wang et al. (2020). As the conventional (big) Q-function collapses in continuous time, we consider its first-order approximation and coin the term ``(little) q-function\". This function is related to the instantaneous advantage rate function as well as the Hamiltonian. We develop a ``q-learning\"theory around the q-function that is independent of time discretization. Given a stochastic policy, we jointly characterize the associated q-function and value function by martingale conditions of certain stochastic processes, in both on-policy and off-policy settings. We then apply the theory to devise different actor-critic algorithms for solving underlying RL problems, depending on whether or not the density function of the Gibbs measure generated from the q-function can be computed explicitly. One of our algorithms interprets the well-known Q-learning algorithm SARSA, and another recovers a policy gradient (PG) based continuous-time algorithm proposed in Jia and Zhou (2022b). Finally, we conduct simulation experiments to compare the performance of our algorithms with those of PG-based algorithms in Jia and Zhou (2022b) and time-discretized conventional Q-learning algorithms.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"17 1","pages":"161:1-161:61"},"PeriodicalIF":0.0,"publicationDate":"2022-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81941524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-26DOI: 10.48550/arXiv.2206.12963
Zhuotong Chen, Qianxiao Li, Zheng Zhang
Despite the wide applications of neural networks, there have been increasing concerns about their vulnerability issue. While numerous attack and defense techniques have been developed, this work investigates the robustness issue from a new angle: can we design a self-healing neural network that can automatically detect and fix the vulnerability issue by itself? A typical self-healing mechanism is the immune system of a human body. This biology-inspired idea has been used in many engineering designs but is rarely investigated in deep learning. This paper considers the post-training self-healing of a neural network, and proposes a closed-loop control formulation to automatically detect and fix the errors caused by various attacks or perturbations. We provide a margin-based analysis to explain how this formulation can improve the robustness of a classifier. To speed up the inference of the proposed self-healing network, we solve the control problem via improving the Pontryagin Maximum Principle-based solver. Lastly, we present an error estimation of the proposed framework for neural networks with nonlinear activation functions. We validate the performance on several network architectures against various perturbations. Since the self-healing method does not need a-priori information about data perturbations/attacks, it can handle a broad class of unforeseen perturbations.
{"title":"Self-Healing Robust Neural Networks via Closed-Loop Control","authors":"Zhuotong Chen, Qianxiao Li, Zheng Zhang","doi":"10.48550/arXiv.2206.12963","DOIUrl":"https://doi.org/10.48550/arXiv.2206.12963","url":null,"abstract":"Despite the wide applications of neural networks, there have been increasing concerns about their vulnerability issue. While numerous attack and defense techniques have been developed, this work investigates the robustness issue from a new angle: can we design a self-healing neural network that can automatically detect and fix the vulnerability issue by itself? A typical self-healing mechanism is the immune system of a human body. This biology-inspired idea has been used in many engineering designs but is rarely investigated in deep learning. This paper considers the post-training self-healing of a neural network, and proposes a closed-loop control formulation to automatically detect and fix the errors caused by various attacks or perturbations. We provide a margin-based analysis to explain how this formulation can improve the robustness of a classifier. To speed up the inference of the proposed self-healing network, we solve the control problem via improving the Pontryagin Maximum Principle-based solver. Lastly, we present an error estimation of the proposed framework for neural networks with nonlinear activation functions. We validate the performance on several network architectures against various perturbations. Since the self-healing method does not need a-priori information about data perturbations/attacks, it can handle a broad class of unforeseen perturbations.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"15 1","pages":"319:1-319:54"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84672069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-22DOI: 10.48550/arXiv.2206.11128
Mikhail (Misha) Usvyatsov, R. Ballester-Ripoll, K. Schindler
We present tntorch, a tensor learning framework that supports multiple decompositions (including Candecomp/Parafac, Tucker, and Tensor Train) under a unified interface. With our library, the user can learn and handle low-rank tensors with automatic differentiation, seamless GPU support, and the convenience of PyTorch's API. Besides decomposition algorithms, tntorch implements differentiable tensor algebra, rank truncation, cross-approximation, batch processing, comprehensive tensor arithmetics, and more.
{"title":"tntorch: Tensor Network Learning with PyTorch","authors":"Mikhail (Misha) Usvyatsov, R. Ballester-Ripoll, K. Schindler","doi":"10.48550/arXiv.2206.11128","DOIUrl":"https://doi.org/10.48550/arXiv.2206.11128","url":null,"abstract":"We present tntorch, a tensor learning framework that supports multiple decompositions (including Candecomp/Parafac, Tucker, and Tensor Train) under a unified interface. With our library, the user can learn and handle low-rank tensors with automatic differentiation, seamless GPU support, and the convenience of PyTorch's API. Besides decomposition algorithms, tntorch implements differentiable tensor algebra, rank truncation, cross-approximation, batch processing, comprehensive tensor arithmetics, and more.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"62 1","pages":"208:1-208:6"},"PeriodicalIF":0.0,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83210335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-21DOI: 10.48550/arXiv.2206.10189
Yann Fraboni, Richard Vidal, Laetitia Kameni, Marco Lorenzi
We propose a novel framework to study asynchronous federated learning optimization with delays in gradient updates. Our theoretical framework extends the standard FedAvg aggregation scheme by introducing stochastic aggregation weights to represent the variability of the clients update time, due for example to heterogeneous hardware capabilities. Our formalism applies to the general federated setting where clients have heterogeneous datasets and perform at least one step of stochastic gradient descent (SGD). We demonstrate convergence for such a scheme and provide sufficient conditions for the related minimum to be the optimum of the federated problem. We show that our general framework applies to existing optimization schemes including centralized learning, FedAvg, asynchronous FedAvg, and FedBuff. The theory here provided allows drawing meaningful guidelines for designing a federated learning experiment in heterogeneous conditions. In particular, we develop in this work FedFix, a novel extension of FedAvg enabling efficient asynchronous federated training while preserving the convergence stability of synchronous aggregation. We empirically demonstrate our theory on a series of experiments showing that asynchronous FedAvg leads to fast convergence at the expense of stability, and we finally demonstrate the improvements of FedFix over synchronous and asynchronous FedAvg.
{"title":"A General Theory for Federated Optimization with Asynchronous and Heterogeneous Clients Updates","authors":"Yann Fraboni, Richard Vidal, Laetitia Kameni, Marco Lorenzi","doi":"10.48550/arXiv.2206.10189","DOIUrl":"https://doi.org/10.48550/arXiv.2206.10189","url":null,"abstract":"We propose a novel framework to study asynchronous federated learning optimization with delays in gradient updates. Our theoretical framework extends the standard FedAvg aggregation scheme by introducing stochastic aggregation weights to represent the variability of the clients update time, due for example to heterogeneous hardware capabilities. Our formalism applies to the general federated setting where clients have heterogeneous datasets and perform at least one step of stochastic gradient descent (SGD). We demonstrate convergence for such a scheme and provide sufficient conditions for the related minimum to be the optimum of the federated problem. We show that our general framework applies to existing optimization schemes including centralized learning, FedAvg, asynchronous FedAvg, and FedBuff. The theory here provided allows drawing meaningful guidelines for designing a federated learning experiment in heterogeneous conditions. In particular, we develop in this work FedFix, a novel extension of FedAvg enabling efficient asynchronous federated training while preserving the convergence stability of synchronous aggregation. We empirically demonstrate our theory on a series of experiments showing that asynchronous FedAvg leads to fast convergence at the expense of stability, and we finally demonstrate the improvements of FedFix over synchronous and asynchronous FedAvg.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"1 1","pages":"110:1-110:43"},"PeriodicalIF":0.0,"publicationDate":"2022-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83351939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-07DOI: 10.48550/arXiv.2206.03345
G. Zhang, S. Fattahi, Richard Y. Zhang
We consider using gradient descent to minimize the nonconvex function $f(X)=phi(XX^{T})$ over an $ntimes r$ factor matrix $X$, in which $phi$ is an underlying smooth convex cost function defined over $ntimes n$ matrices. While only a second-order stationary point $X$ can be provably found in reasonable time, if $X$ is additionally rank deficient, then its rank deficiency certifies it as being globally optimal. This way of certifying global optimality necessarily requires the search rank $r$ of the current iterate $X$ to be overparameterized with respect to the rank $r^{star}$ of the global minimizer $X^{star}$. Unfortunately, overparameterization significantly slows down the convergence of gradient descent, from a linear rate with $r=r^{star}$ to a sublinear rate when $r>r^{star}$, even when $phi$ is strongly convex. In this paper, we propose an inexpensive preconditioner that restores the convergence rate of gradient descent back to linear in the overparameterized case, while also making it agnostic to possible ill-conditioning in the global minimizer $X^{star}$.
{"title":"Preconditioned Gradient Descent for Overparameterized Nonconvex Burer-Monteiro Factorization with Global Optimality Certification","authors":"G. Zhang, S. Fattahi, Richard Y. Zhang","doi":"10.48550/arXiv.2206.03345","DOIUrl":"https://doi.org/10.48550/arXiv.2206.03345","url":null,"abstract":"We consider using gradient descent to minimize the nonconvex function $f(X)=phi(XX^{T})$ over an $ntimes r$ factor matrix $X$, in which $phi$ is an underlying smooth convex cost function defined over $ntimes n$ matrices. While only a second-order stationary point $X$ can be provably found in reasonable time, if $X$ is additionally rank deficient, then its rank deficiency certifies it as being globally optimal. This way of certifying global optimality necessarily requires the search rank $r$ of the current iterate $X$ to be overparameterized with respect to the rank $r^{star}$ of the global minimizer $X^{star}$. Unfortunately, overparameterization significantly slows down the convergence of gradient descent, from a linear rate with $r=r^{star}$ to a sublinear rate when $r>r^{star}$, even when $phi$ is strongly convex. In this paper, we propose an inexpensive preconditioner that restores the convergence rate of gradient descent back to linear in the overparameterized case, while also making it agnostic to possible ill-conditioning in the global minimizer $X^{star}$.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"23 1","pages":"163:1-163:55"},"PeriodicalIF":0.0,"publicationDate":"2022-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87649367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-30DOI: 10.48550/arXiv.2205.15368
A. Ganguly, Riten Mitra, Jin Zhou
The paper has two major themes. The first part of the paper establishes certain general results for infinite-dimensional optimization problems on Hilbert spaces. These results cover the classical representer theorem and many of its variants as special cases and offer a wider scope of applications. The second part of the paper then develops a systematic approach for learning the drift function of a stochastic differential equation by integrating the results of the first part with Bayesian hierarchical framework. Importantly, our Baysian approach incorporates low-cost sparse learning through proper use of shrinkage priors while allowing proper quantification of uncertainty through posterior distributions. Several examples at the end illustrate the accuracy of our learning scheme.
{"title":"Infinite-dimensional optimization and Bayesian nonparametric learning of stochastic differential equations","authors":"A. Ganguly, Riten Mitra, Jin Zhou","doi":"10.48550/arXiv.2205.15368","DOIUrl":"https://doi.org/10.48550/arXiv.2205.15368","url":null,"abstract":"The paper has two major themes. The first part of the paper establishes certain general results for infinite-dimensional optimization problems on Hilbert spaces. These results cover the classical representer theorem and many of its variants as special cases and offer a wider scope of applications. The second part of the paper then develops a systematic approach for learning the drift function of a stochastic differential equation by integrating the results of the first part with Bayesian hierarchical framework. Importantly, our Baysian approach incorporates low-cost sparse learning through proper use of shrinkage priors while allowing proper quantification of uncertainty through posterior distributions. Several examples at the end illustrate the accuracy of our learning scheme.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"41 1","pages":"159:1-159:39"},"PeriodicalIF":0.0,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84050824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-19DOI: 10.48550/arXiv.2205.09680
Imanol Arrieta Ibarra, Paman Gujral, Jonathan Tannen, M. Tygert, Cherie Xu
Predictions are often probabilities; e.g., a prediction could be for precipitation tomorrow, but with only a 30% chance. Given such probabilistic predictions together with the actual outcomes,"reliability diagrams"help detect and diagnose statistically significant discrepancies -- so-called"miscalibration"-- between the predictions and the outcomes. The canonical reliability diagrams histogram the observed and expected values of the predictions; replacing the hard histogram binning with soft kernel density estimation is another common practice. But, which widths of bins or kernels are best? Plots of the cumulative differences between the observed and expected values largely avoid this question, by displaying miscalibration directly as the slopes of secant lines for the graphs. Slope is easy to perceive with quantitative precision, even when the constant offsets of the secant lines are irrelevant; there is no need to bin or perform kernel density estimation. The existing standard metrics of miscalibration each summarize a reliability diagram as a single scalar statistic. The cumulative plots naturally lead to scalar metrics for the deviation of the graph of cumulative differences away from zero; good calibration corresponds to a horizontal, flat graph which deviates little from zero. The cumulative approach is currently unconventional, yet offers many favorable statistical properties, guaranteed via mathematical theory backed by rigorous proofs and illustrative numerical examples. In particular, metrics based on binning or kernel density estimation unavoidably must trade-off statistical confidence for the ability to resolve variations as a function of the predicted probability or vice versa. Widening the bins or kernels averages away random noise while giving up some resolving power. Narrowing the bins or kernels enhances resolving power while not averaging away as much noise.
{"title":"Metrics of calibration for probabilistic predictions","authors":"Imanol Arrieta Ibarra, Paman Gujral, Jonathan Tannen, M. Tygert, Cherie Xu","doi":"10.48550/arXiv.2205.09680","DOIUrl":"https://doi.org/10.48550/arXiv.2205.09680","url":null,"abstract":"Predictions are often probabilities; e.g., a prediction could be for precipitation tomorrow, but with only a 30% chance. Given such probabilistic predictions together with the actual outcomes,\"reliability diagrams\"help detect and diagnose statistically significant discrepancies -- so-called\"miscalibration\"-- between the predictions and the outcomes. The canonical reliability diagrams histogram the observed and expected values of the predictions; replacing the hard histogram binning with soft kernel density estimation is another common practice. But, which widths of bins or kernels are best? Plots of the cumulative differences between the observed and expected values largely avoid this question, by displaying miscalibration directly as the slopes of secant lines for the graphs. Slope is easy to perceive with quantitative precision, even when the constant offsets of the secant lines are irrelevant; there is no need to bin or perform kernel density estimation. The existing standard metrics of miscalibration each summarize a reliability diagram as a single scalar statistic. The cumulative plots naturally lead to scalar metrics for the deviation of the graph of cumulative differences away from zero; good calibration corresponds to a horizontal, flat graph which deviates little from zero. The cumulative approach is currently unconventional, yet offers many favorable statistical properties, guaranteed via mathematical theory backed by rigorous proofs and illustrative numerical examples. In particular, metrics based on binning or kernel density estimation unavoidably must trade-off statistical confidence for the ability to resolve variations as a function of the predicted probability or vice versa. Widening the bins or kernels averages away random noise while giving up some resolving power. Narrowing the bins or kernels enhances resolving power while not averaging away as much noise.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"54 1","pages":"351:1-351:54"},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80185762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-11DOI: 10.48550/arXiv.2205.05428
Wei Liu, Xin Liu, Xiaojun Chen
The leaky ReLU network with a group sparse regularization term has been widely used in the recent years. However, training such a network yields a nonsmooth nonconvex optimization problem and there exists a lack of approaches to compute a stationary point deterministically. In this paper, we first resolve the multi-layer composite term in the original optimization problem by introducing auxiliary variables and additional constraints. We show the new model has a nonempty and bounded solution set and its feasible set satisfies the Mangasarian-Fromovitz constraint qualification. Moreover, we show the relationship between the new model and the original problem. Remarkably, we propose an inexact augmented Lagrangian algorithm for solving the new model and show the convergence of the algorithm to a KKT point. Numerical experiments demonstrate that our algorithm is more efficient for training sparse leaky ReLU neural networks than some well-known algorithms.
{"title":"An Inexact Augmented Lagrangian Algorithm for Training Leaky ReLU Neural Network with Group Sparsity","authors":"Wei Liu, Xin Liu, Xiaojun Chen","doi":"10.48550/arXiv.2205.05428","DOIUrl":"https://doi.org/10.48550/arXiv.2205.05428","url":null,"abstract":"The leaky ReLU network with a group sparse regularization term has been widely used in the recent years. However, training such a network yields a nonsmooth nonconvex optimization problem and there exists a lack of approaches to compute a stationary point deterministically. In this paper, we first resolve the multi-layer composite term in the original optimization problem by introducing auxiliary variables and additional constraints. We show the new model has a nonempty and bounded solution set and its feasible set satisfies the Mangasarian-Fromovitz constraint qualification. Moreover, we show the relationship between the new model and the original problem. Remarkably, we propose an inexact augmented Lagrangian algorithm for solving the new model and show the convergence of the algorithm to a KKT point. Numerical experiments demonstrate that our algorithm is more efficient for training sparse leaky ReLU neural networks than some well-known algorithms.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"21 1","pages":"212:1-212:43"},"PeriodicalIF":0.0,"publicationDate":"2022-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82242391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：