Eigendecomposition of symmetric matrices is at the heart of many computer vision algorithms. However, the derivatives of the eigenvectors tend to be numerically unstable, whether using the SVD to compute them analytically or using the Power Iteration (PI) method to approximate them. This instability arises in the presence of eigenvalues that are close to each other. This makes integrating eigendecomposition into deep networks difficult and often results in poor convergence, particularly when dealing with large matrices. While this can be mitigated by partitioning the data into small arbitrary groups, doing so has no theoretical basis and makes it impossible to exploit the full power of eigendecomposition. In previous work, we mitigated this using SVD during the forward pass and PI to compute the gradients during the backward pass. However, the iterative deflation procedure required to compute multiple eigenvectors using PI tends to accumulate errors and yield inaccurate gradients. Here, we show that the Taylor expansion of the SVD gradient is theoretically equivalent to the gradient obtained using PI without relying in practice on an iterative process and thus yields more accurate gradients. We demonstrate the benefits of this increased accuracy for image classification and style transfer.
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In this paper we derive a new capability for robots to measure relative direction, or Angle-of-Arrival (AOA), to other robots operating in non-line-of-sight and unmapped environments with occlusions, without requiring external infrastructure. We do so by capturing all of the paths that a WiFi signal traverses as it travels from a transmitting to a receiving robot, which we term an AOA profile. The key intuition is to "emulate antenna arrays in the air" as the robots move in 3D space, a method akin to Synthetic Aperture Radar (SAR). The main contributions include development of i) a framework to accommodate arbitrary 3D trajectories, as well as continuous mobility all robots, while computing AOA profiles and ii) an accompanying analysis that provides a lower bound on variance of AOA estimation as a function of robot trajectory geometry based on the Cramer Rao Bound. This is a critical distinction with previous work on SAR that restricts robot mobility to prescribed motion patterns, does not generalize to 3D space, and/or requires transmitting robots to be static during data acquisition periods. Our method results in more accurate AOA profiles and thus better AOA estimation, and formally characterizes this observation as the informativeness of the trajectory; a computable quantity for which we derive a closed form. All theoretical developments are substantiated by extensive simulation and hardware experiments. We also show that our formulation can be used with an off-the-shelf trajectory estimation sensor. Finally, we demonstrate the performance of our system on a multi-robot dynamic rendezvous task.
This paper investigates the role of dimensionality reduction in efficient communication and differential privacy (DP) of the local datasets at the remote users for over-the-air computation (AirComp)-based federated learning (FL) model. More precisely, we consider the FL setting in which clients are prompted to train a machine learning model by simultaneous channel-aware and limited communications with a parameter server (PS) over a Gaussian multiple-access channel (GMAC), so that transmissions sum coherently at the PS globally aware of the channel coefficients. For this setting, an algorithm is proposed based on applying federated stochastic gradient descent (FedSGD) for training the minimum of a given loss function based on the local gradients, Johnson-Lindenstrauss (JL) random projection for reducing the dimension of the local updates, and artificial noise to further aid user's privacy. For this scheme, our results show that the local DP performance is mainly improved due to injecting noise of greater variance on each dimension while keeping the sensitivity of the projected vectors unchanged. This is while the convergence rate is slowed down compared to the case without dimensionality reduction. As the performance outweighs for the slower convergence, the trade-off between privacy and convergence is higher but is shown to lessen in high-dimensional regime yielding almost the same trade-off with much less communication cost.
Recovering a signal (function) from finitely many binary or Fourier samples is one of the core problems in modern medical imaging, and by now there exist a plethora of methods for recovering a signal from such samples. Examples of methods, which can utilise wavelet reconstruction, include generalised sampling, infinite-dimensional compressive sensing, the parameterised-background data-weak (PBDW) method etc. However, for any of these methods to be applied in practice, accurate and fast modelling of an $N \times M$ section of the infinite-dimensional change-of-basis matrix between the sampling basis (Fourier or Walsh-Hadamard samples) and the wavelet reconstruction basis is paramount. In this work, we derive an algorithm, which bypasses the $NM$ storage requirement and the $\mathcal{O}(NM)$ computational cost of matrix-vector multiplication with this matrix when using Walsh-Hadamard samples and wavelet reconstruction. The proposed algorithm computes the matrix-vector multiplication in $\mathcal{O}(N\log N)$ operations and has a storage requirement of $\mathcal{O}(2^q)$, where $N=2^{dq} M$, (usually $q \in \{1,2\}$) and $d=1,2$ is the dimension. As matrix-vector multiplications is the computational bottleneck for iterative algorithms used by the mentioned reconstruction methods, the proposed algorithm speeds up the reconstruction of wavelet coefficients from Walsh-Hadamard samples considerably.
We explore contemporary robust classification algorithms for overcoming class-dependant labelling noise: Forward, Importance Re-weighting and T-revision. The classifiers are trained and evaluated on class-conditional random label noise data while the final test data is clean. We demonstrate methods for estimating the transition matrix in order to obtain better classifier performance when working with noisy data. We apply deep learning to three data-sets and derive an end-to-end analysis with unknown noise on the CIFAR data-set from scratch. The effectiveness and robustness of the classifiers are analysed, and we compare and contrast the results of each experiment are using top-1 accuracy as our criterion.
We investigate online nonlinear regression with continually running recurrent neural network networks (RNNs), i.e., RNN-based online learning. For RNN-based online learning, we introduce an efficient first-order training algorithm that theoretically guarantees to converge to the optimum network parameters. Our algorithm is truly online such that it does not make any assumption on the learning environment to guarantee convergence. Through numerical simulations, we verify our theoretical results and illustrate significant performance improvements achieved by our algorithm with respect to the state-of-the-art RNN training methods.
We propose a Safe Pontryagin Differentiable Programming (Safe PDP) methodology, which establishes a theoretical and algorithmic safe differentiable framework to solve a broad class of safety-critical learning and control tasks -- problems that require the guarantee of both immediate and long-term constraint satisfaction at any stage of the learning and control progress. In the spirit of interior-point methods, Safe PDP handles different types of state and input constraints by incorporating them into the cost and loss through barrier functions. We prove the following fundamental features of Safe PDP: first, both the constrained solution and its gradient in backward pass can be approximated by solving a more efficient unconstrained counterpart; second, the approximation for both the solution and its gradient can be controlled for arbitrary accuracy using a barrier parameter; and third, importantly, any intermediate results throughout the approximation and optimization are strictly respecting all constraints, thus guaranteeing safety throughout the entire learning and control process. We demonstrate the capabilities of Safe PDP in solving various safe learning and control tasks, including safe policy optimization, safe motion planning, and learning MPCs from demonstrations, on different challenging control systems such as 6-DoF maneuvering quadrotor and 6-DoF rocket powered landing.
System design tools are often only available as blackboxes with complex nonlinear relationships between inputs and outputs. Blackboxes typically run in the forward direction: for a given design as input they compute an output representing system behavior. Most cannot be run in reverse to produce an input from requirements on output. Thus, finding a design satisfying a requirement is often a trial-and-error process without assurance of optimality. Finding designs concurrently satisfying multiple requirements is harder because designs satisfying individual requirements may conflict with each other. Compounding the hardness are the facts that blackbox evaluations can be expensive and sometimes fail to produce an output due to non-convergence of underlying numerical algorithms. This paper presents CNMA (Constrained optimization with Neural networks, MILP solvers and Active Learning), a new optimization method for blackboxes. It is conservative in the number of blackbox evaluations. Any designs it finds are guaranteed to satisfy all requirements. It is resilient to the failure of blackboxes to compute outputs. It tries to sample only the part of the design space relevant to solving the design problem, leveraging the power of neural networks, MILPs, and a new learning-from-failure feedback loop. The paper also presents parallel CNMA that improves the efficiency and quality of solutions over the sequential version, and tries to steer it away from local optima. CNMA's performance is evaluated for seven nonlinear design problems of 8 (2 problems), 10, 15, 36 and 60 real-valued dimensions and one with 186 binary dimensions. It is shown that CNMA improves the performance of stable, off-the-shelf implementations of Bayesian Optimization and Nelder Mead and Random Search by 1%-87% for a given fixed time and function evaluation budget. Note, that these implementations did not always return solutions.
Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.
Asynchronous momentum stochastic gradient descent algorithms (Async-MSGD) is one of the most popular algorithms in distributed machine learning. However, its convergence properties for these complicated nonconvex problems is still largely unknown, because of the current technical limit. Therefore, in this paper, we propose to analyze the algorithm through a simpler but nontrivial nonconvex problem - streaming PCA, which helps us to understand Aync-MSGD better even for more general problems. Specifically, we establish the asymptotic rate of convergence of Async-MSGD for streaming PCA by diffusion approximation. Our results indicate a fundamental tradeoff between asynchrony and momentum: To ensure convergence and acceleration through asynchrony, we have to reduce the momentum (compared with Sync-MSGD). To the best of our knowledge, this is the first theoretical attempt on understanding Async-MSGD for distributed nonconvex stochastic optimization. Numerical experiments on both streaming PCA and training deep neural networks are provided to support our findings for Async-MSGD.
Machine Learning is a widely-used method for prediction generation. These predictions are more accurate when the model is trained on a larger dataset. On the other hand, the data is usually divided amongst different entities. For privacy reasons, the training can be done locally and then the model can be safely aggregated amongst the participants. However, if there are only two participants in \textit{Collaborative Learning}, the safe aggregation loses its power since the output of the training already contains much information about the participants. To resolve this issue, they must employ privacy-preserving mechanisms, which inevitably affect the accuracy of the model. In this paper, we model the training process as a two-player game where each player aims to achieve a higher accuracy while preserving its privacy. We introduce the notion of \textit{Price of Privacy}, a novel approach to measure the effect of privacy protection on the accuracy of the model. We develop a theoretical model for different player types, and we either find or prove the existence of a Nash Equilibrium with some assumptions. Moreover, we confirm these assumptions via a Recommendation Systems use case: for a specific learning algorithm, we apply three privacy-preserving mechanisms on two real-world datasets. Finally, as a complementary work for the designed game, we interpolate the relationship between privacy and accuracy for this use case and present three other methods to approximate it in a real-world scenario.
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