Apoorva Nandini Saridena

Devansh Bisla, Apoorva Nandini Saridena, Anna Choromanska

This paper focuses on understanding how the generalization error scales with the amount of the training data for deep neural networks (DNNs). Existing techniques in statistical learning require computation of capacity measures, such as VC dimension, to provably bound this error. It is however unclear how to extend these measures to DNNs and therefore the existing analyses are applicable to simple neural networks, which are not used in practice, e.g., linear or shallow ones or otherwise multi-layer perceptrons. Moreover, many theoretical error bounds are not empirically verifiable. We derive estimates of the generalization error that hold for deep networks and do not rely on unattainable capacity measures. The enabling technique in our approach hinges on two major assumptions: i) the network achieves zero training error, ii) the probability of making an error on a test point is proportional to the distance between this point and its nearest training point in the feature space and at a certain maximal distance (that we call radius) it saturates. Based on these assumptions we estimate the generalization error of DNNs. The obtained estimate scales as O(1/(δN^{1/d})), where N is the size of the training data and is parameterized by two quantities, the effective dimensionality of the data as perceived by the network (d) and the aforementioned radius (δ), both of which we find empirically. We show that our estimates match with the experimentally obtained behavior of the error on multiple learning tasks using benchmark data-sets and realistic models. Estimating training data requirements is essential for deployment of safety critical applications such as autonomous driving etc. Furthermore, collecting and annotating training data requires a huge amount of financial, computational and human resources. Our empirical estimates will help to efficiently allocate resources.

Naman Patel, Apoorva Nandini Saridena, Anna Choromanska et al.

The paper proposes an on-line monitoring framework for continuous real-time safety/security in learning-based control systems (specifically application to a unmanned ground vehicle). We monitor validity of mappings from sensor inputs to actuator commands, controller-focused anomaly detection (CFAM), and from actuator commands to sensor inputs, system-focused anomaly detection (SFAM). CFAM is an image conditioned energy based generative adversarial network (EBGAN) in which the energy based discriminator distinguishes between proper and anomalous actuator commands. SFAM is based on an action condition video prediction framework to detect anomalies between predicted and observed temporal evolution of sensor data. We demonstrate the effectiveness of the approach on our autonomous ground vehicle for indoor environments and on Udacity dataset for outdoor environments.

Benjamin Cowen, Apoorva Nandini Saridena, Anna Choromanska

We propose an efficient algorithm for the generalized sparse coding (SC) inference problem. The proposed framework applies to both the single dictionary setting, where each data point is represented as a sparse combination of the columns of one dictionary matrix, as well as the multiple dictionary setting as given in morphological component analysis (MCA), where the goal is to separate a signal into additive parts such that each part has distinct sparse representation within a corresponding dictionary. Both the SC task and its generalization via MCA have been cast as $\ell_1$-regularized least-squares optimization problems. To accelerate traditional acquisition of sparse codes, we propose a deep learning architecture that constitutes a trainable time-unfolded version of the Split Augmented Lagrangian Shrinkage Algorithm (SALSA), a special case of the Alternating Direction Method of Multipliers (ADMM). We empirically validate both variants of the algorithm, that we refer to as LSALSA (learned-SALSA), on image vision tasks and demonstrate that at inference our networks achieve vast improvements in terms of the running time, the quality of estimated sparse codes, and visual clarity on both classic SC and MCA problems. Finally, we present a theoretical framework for analyzing LSALSA network: we show that the proposed approach exactly implements a truncated ADMM applied to a new, learned cost function with curvature modified by one of the learned parameterized matrices. We extend a very recent Stochastic Alternating Optimization analysis framework to show that a gradient descent step along this learned loss landscape is equivalent to a modified gradient descent step along the original loss landscape. In this framework, the acceleration achieved by LSALSA could potentially be explained by the network's ability to learn a correction to the gradient direction of steeper descent.