We consider minimizing functions for which it is expensive to compute the (possibly stochastic) gradient. Such functions are prevalent in reinforcement learning, imitation learning and adversarial training. Our target optimization framework uses the (expensive) gradient computation to construct surrogate functions in a \emph{target space} (e.g. the logits output by a linear model for classification) that can be minimized efficiently. This allows for multiple parameter updates to the model, amortizing the cost of gradient computation. In the full-batch setting, we prove that our surrogate is a global upper-bound on the loss, and can be (locally) minimized using a black-box optimization algorithm. We prove that the resulting majorization-minimization algorithm ensures convergence to a stationary point of the loss. Next, we instantiate our framework in the stochastic setting and propose the $SSO$ algorithm, which can be viewed as projected stochastic gradient descent in the target space. This connection enables us to prove theoretical guarantees for $SSO$ when minimizing convex functions. Our framework allows the use of standard stochastic optimization algorithms to construct surrogates which can be minimized by any deterministic optimization method. To evaluate our framework, we consider a suite of supervised learning and imitation learning problems. Our experiments indicate the benefits of target optimization and the effectiveness of $SSO$.
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Generative Adversarial Networks (GANs) have shown immense potential in fields such as text and image generation. Only very recently attempts to exploit GANs to statistical-mechanics models have been reported. Here we quantitatively test this approach by applying it to a prototypical stochastic process on a lattice. By suitably adding noise to the original data we succeed in bringing both the Generator and the Discriminator loss functions close to their ideal value. Importantly, the discreteness of the model is retained despite the noise. As typical for adversarial approaches, oscillations around the convergence limit persist also at large epochs. This undermines model selection and the quality of the generated trajectories. We demonstrate that a simple multi-model procedure where stochastic trajectories are advanced at each step upon randomly selecting a Generator leads to a remarkable increase in accuracy. This is illustrated by quantitative analysis of both the predicted equilibrium probability distribution and of the escape-time distribution. Based on the reported findings, we believe that GANs are a promising tool to tackle complex statistical dynamics by machine learning techniques
The stochastic compositional minimax problem has attracted a surge of attention in recent years since it covers many emerging machine learning models. Meanwhile, due to the emergence of distributed data, optimizing this kind of problem under the decentralized setting becomes badly needed. However, the compositional structure in the loss function brings unique challenges to designing efficient decentralized optimization algorithms. In particular, our study shows that the standard gossip communication strategy cannot achieve linear speedup for decentralized compositional minimax problems due to the large consensus error about the inner-level function. To address this issue, we developed a novel decentralized stochastic compositional gradient descent ascent with momentum algorithm to reduce the consensus error in the inner-level function. As such, our theoretical results demonstrate that it is able to achieve linear speedup with respect to the number of workers. We believe this novel algorithmic design could benefit the development of decentralized compositional optimization. Finally, we applied our methods to the imbalanced classification problem. The extensive experimental results provide evidence for the effectiveness of our algorithm.
Assigning importance weights to adversarial data has achieved great success in training adversarially robust networks under limited model capacity. However, existing instance-reweighted adversarial training (AT) methods heavily depend on heuristics and/or geometric interpretations to determine those importance weights, making these algorithms lack rigorous theoretical justification/guarantee. Moreover, recent research has shown that adversarial training suffers from a severe non-uniform robust performance across the training distribution, e.g., data points belonging to some classes can be much more vulnerable to adversarial attacks than others. To address both issues, in this paper, we propose a novel doubly-robust instance reweighted AT framework, which allows to obtain the importance weights via exploring distributionally robust optimization (DRO) techniques, and at the same time boosts the robustness on the most vulnerable examples. In particular, our importance weights are obtained by optimizing the KL-divergence regularized loss function, which allows us to devise new algorithms with a theoretical convergence guarantee. Experiments on standard classification datasets demonstrate that our proposed approach outperforms related state-of-the-art baseline methods in terms of average robust performance, and at the same time improves the robustness against attacks on the weakest data points. Codes will be available soon.
Finding a solution to the linear system $Ax = b$ with various minimization properties arises from many engineering and computer science applications, including compressed sensing, image processing, and machine learning. In the age of big data, the scalability of stochastic optimization algorithms has made it increasingly important to solve problems of unprecedented sizes. This paper focuses on the problem of minimizing a strongly convex objective function subject to linearly constraints. We consider the dual formulation of this problem and adopt the stochastic coordinate descent to solve it. The proposed algorithmic framework, called fast stochastic dual coordinate descent, utilizes an adaptive variation of Polyak's heavy ball momentum and user-defined distributions for sampling. Our adaptive heavy ball momentum technique can efficiently update the parameters by using iterative information, overcoming the limitation of the heavy ball momentum method where prior knowledge of certain parameters, such as singular values of a matrix, is required. We prove that, under strongly admissible of the objective function, the propose method converges linearly in expectation. By varying the sampling matrix, we recover a comprehensive array of well-known algorithms as special cases, including the randomized sparse Kaczmarz method, the randomized regularized Kaczmarz method, the linearized Bregman iteration, and a variant of the conjugate gradient (CG) method. Numerical experiments are provided to confirm our results.
Noise is a part of data whether the data is from measurement, experiment or ... A few techniques are suggested for noise reduction to improve the data quality in recent years some of which are based on wavelet, orthogonalization and neural networks. The computational cost of existing methods are more than expected and that's why their application in some cases is not beneficial. In this paper, we suggest a low cost techniques based on special linear algebra structures (tridiagonal systems) to improve the signal quality. In this method, we suggest a tridiagonal model for the noise around the most noisy elements. To update the predicted noise, the algorithm is equipped with a learning/feedback approach. The details are described below and based on presented numerical results this algorithm is successful in computing the noise with lower MSE (mean squared error) in computation time specially when the data size is lower than 5000. Our algorithm is used for low-range noise while for high-range noise it is sufficient to use the presented algorithm in hybrid with moving average. The algorithm is implemented in MATLAB 2019b on a computer with Windows 11 having 8GB RAM. It is then tested over many randomly generated experiments. The numerical results confirm the efficiency of presented algorithm in most cases in comparison with existing methods.
In Causal Bayesian Optimization (CBO), an agent intervenes on an unknown structural causal model to maximize a downstream reward variable. In this paper, we consider the generalization where other agents or external events also intervene on the system, which is key for enabling adaptiveness to non-stationarities such as weather changes, market forces, or adversaries. We formalize this generalization of CBO as Adversarial Causal Bayesian Optimization (ACBO) and introduce the first algorithm for ACBO with bounded regret: Causal Bayesian Optimization with Multiplicative Weights (CBO-MW). Our approach combines a classical online learning strategy with causal modeling of the rewards. To achieve this, it computes optimistic counterfactual reward estimates by propagating uncertainty through the causal graph. We derive regret bounds for CBO-MW that naturally depend on graph-related quantities. We further propose a scalable implementation for the case of combinatorial interventions and submodular rewards. Empirically, CBO-MW outperforms non-causal and non-adversarial Bayesian optimization methods on synthetic environments and environments based on real-word data. Our experiments include a realistic demonstration of how CBO-MW can be used to learn users' demand patterns in a shared mobility system and reposition vehicles in strategic areas.
This paper proposes a non-centered parameterization based infinite-dimensional mean-field variational inference (NCP-iMFVI) approach for solving the hierarchical Bayesian inverse problems. This method can generate available estimates from the approximated posterior distribution efficiently. To avoid the mutually singular obstacle that occurred in the infinite-dimensional hierarchical approach, we propose a rigorous theory of the non-centered variational Bayesian approach. Since the non-centered parameterization weakens the connection between the parameter and the hyper-parameter, we can introduce the hyper-parameter to all terms of the eigendecomposition of the prior covariance operator. We also show the relationships between the NCP-iMFVI and infinite-dimensional hierarchical approaches with centered parameterization. The proposed algorithm is applied to three inverse problems governed by the simple smooth equation, the Helmholtz equation, and the steady-state Darcy flow equation. Numerical results confirm our theoretical findings, illustrate the efficiency of solving the iMFVI problem formulated by large-scale linear and nonlinear statistical inverse problems, and verify the mesh-independent property.
Adversarial attacks are usually expressed in terms of a gradient-based operation on the input data and model, this results in heavy computations every time an attack is generated. In this work, we solidify the idea of representing adversarial attacks as a trainable function, without further gradient computation. We first motivate that the theoretical best attacks, under proper conditions, can be represented as smooth piece-wise functions (piece-wise H\"older functions). Then we obtain an approximation result of such functions by a neural network. Subsequently, we emulate the ideal attack process by a neural network and reduce the adversarial training to a mathematical game between an attack network and a training model (a defense network). We also obtain convergence rates of adversarial loss in terms of the sample size $n$ for adversarial training in such a setting.
Theorems from universal algebra such as that of Murski\u{i} from the 1970s have a striking similarity to universal approximation results for neural nets along the lines of Cybenko's from the 1980s. We consider here a discrete analogue of the classical notion of a neural net which places these results in a unified setting. We introduce a learning algorithm based on polymorphisms of relational structures and show how to use it for a classical learning task.
Momentum is known to accelerate the convergence of gradient descent in strongly convex settings without stochastic gradient noise. In stochastic optimization, such as training neural networks, folklore suggests that momentum may help deep learning optimization by reducing the variance of the stochastic gradient update, but previous theoretical analyses do not find momentum to offer any provable acceleration. Theoretical results in this paper clarify the role of momentum in stochastic settings where the learning rate is small and gradient noise is the dominant source of instability, suggesting that SGD with and without momentum behave similarly in the short and long time horizons. Experiments show that momentum indeed has limited benefits for both optimization and generalization in practical training regimes where the optimal learning rate is not very large, including small- to medium-batch training from scratch on ImageNet and fine-tuning language models on downstream tasks.
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