Second-order Stochastic Optimization methods for Machine Learning

Analysis of the Hessian #

1. Empirical Analysis of the Hessian of Over-Parametrized Neural Networks #

Year: 2017
Authors: Levent Sagun, Utku Evci, V. Ugur Guney, Yann Dauphin, Leon Bottou
ArXiv ID: arXiv:1706.04454
URL: https://arxiv.org/abs/1706.04454

Abstract: We study the properties of common loss surfaces through their Hessian matrix. In particular, in the context of deep learning, we empirically show that the spectrum of the Hessian is composed of two parts: (1) the bulk centered near zero, (2) and outliers away from the bulk. We present numerical evidence and mathematical justifications to the following conjectures laid out by Sagun et al. (2016): Fixing data, increasing the number of parameters merely scales the bulk of the spectrum; fixing the dimension and changing the data (for instance adding more clusters or making the data less separable) only affects the outliers. We believe that our observations have striking implications for non-convex optimization in high dimensions. First, the flatness of such landscapes (which can be measured by the singularity of the Hessian) implies that classical notions of basins of attraction may be quite misleading. And that the discussion of wide/narrow basins may be in need of a new perspective around over-parametrization and redundancy that are able to create large connected components at the bottom of the landscape. Second, the dependence of small number of large eigenvalues to the data distribution can be linked to the spectrum of the covariance matrix of gradients of model outputs. With this in mind, we may reevaluate the connections within the data-architecture-algorithm framework of a model, hoping that it would shed light into the geometry of high-dimensional and non-convex spaces in modern applications. In particular, we present a case that links the two observations: small and large batch gradient descent appear to converge to different basins of attraction but we show that they are in fact connected through their flat region and so belong to the same basin.

Source Code: No explicit source code information found

2. The Full Spectrum of Deepnet Hessians at Scale: Dynamics with SGD Training and Sample Size #

Year: 2018
Authors: Vardan Papyan
ArXiv ID: arXiv:1811.07062
URL: https://arxiv.org/abs/1811.07062

Abstract: We apply state-of-the-art tools in modern high-dimensional numerical linear algebra to approximate efficiently the spectrum of the Hessian of modern deepnets, with tens of millions of parameters, trained on real data. Our results corroborate previous findings, based on small-scale networks, that the Hessian exhibits “spiked” behavior, with several outliers isolated from a continuous bulk. We decompose the Hessian into different components and study the dynamics with training and sample size of each term individually.

Source Code: No explicit source code information found

3. PyHessian: Neural Networks Through the Lens of the Hessian #

Year: 2019
Authors: Zhewei Yao, Amir Gholami, Kurt Keutzer, Michael W. Mahoney
ArXiv ID: arXiv:1912.07145
URL: https://arxiv.org/abs/1912.07145

Abstract: We present PYHESSIAN, a new scalable framework that enables fast computation of Hessian (i.e., second-order derivative) information for deep neural networks. PYHESSIAN enables fast computations of the top Hessian eigenvalues, the Hessian trace, and the full Hessian eigenvalue/spectral density, and it supports distributed-memory execution on cloud/supercomputer systems and is available as open source. This general framework can be used to analyze neural network models, including the topology of the loss landscape (i.e., curvature information) to gain insight into the behavior of different models/optimizers. To illustrate this, we analyze the effect of residual connections and Batch Normalization layers on the trainability of neural networks. One recent claim, based on simpler first-order analysis, is that residual connections and Batch Normalization make the loss landscape smoother, thus making it easier for Stochastic Gradient Descent to converge to a good solution. Our extensive analysis shows new finer-scale insights, demonstrating that, while conventional wisdom is sometimes validated, in other cases it is simply incorrect. In particular, we find that Batch Normalization does not necessarily make the loss landscape smoother, especially for shallower networks.

Source Code: Mentions ‘available’ in abstract; Mentions ‘open source’ in abstract; Known repository: https://github.com/amirgholami/PyHessian

4. A Deeper Look at the Hessian Eigenspectrum of Deep Neural Networks and its Applications to Regularization #

Year: 2020
Authors: Adepu Ravi Sankar, Yash Khasbage, Rahul Vigneswaran, Vineeth N Balasubramanian
ArXiv ID: arXiv:2012.03801
URL: https://arxiv.org/abs/2012.03801

Abstract: Loss landscape analysis is extremely useful for a deeper understanding of the generalization ability of deep neural network models. In this work, we propose a layerwise loss landscape analysis where the loss surface at every layer is studied independently and also on how each correlates to the overall loss surface. We study the layerwise loss landscape by studying the eigenspectra of the Hessian at each layer. In particular, our results show that the layerwise Hessian geometry is largely similar to the entire Hessian. We also report an interesting phenomenon where the Hessian eigenspectrum of middle layers of the deep neural network are observed to most similar to the overall Hessian eigenspectrum. We also show that the maximum eigenvalue and the trace of the Hessian (both full network and layerwise) reduce as training of the network progresses. We leverage on these observations to propose a new regularizer based on the trace of the layerwise Hessian. Penalizing the trace of the Hessian at every layer indirectly forces Stochastic Gradient Descent to converge to flatter minima, which are shown to have better generalization performance. In particular, we show that such a layerwise regularizer can be leveraged to penalize the middlemost layers alone, which yields promising results. Our empirical studies on well-known deep nets across datasets support the claims of this work

Source Code: No explicit source code information found

Diagonal Scaling #

1. AdaHessian: An Adaptive Second Order Optimizer for Machine Learning #

Year: 2020
Authors: Zhewei Yao, Amir Gholami, Sheng Shen, Mustafa Mustafa, Kurt Keutzer, Michael W. Mahoney
ArXiv ID: arXiv:2006.00719
Algorithm: AdaHessian
URL: https://arxiv.org/abs/2006.00719

Abstract: We introduce ADAHESSIAN, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the HESSIAN. Second order algorithms are among the most powerful optimization algorithms with superior convergence properties as compared to first order methods such as SGD and Adam. The main disadvantage of traditional second order methods is their heavier per iteration computation and poor accuracy as compared to first order methods. To address these, we incorporate several novel approaches in ADAHESSIAN, including: (i) a fast Hutchinson based method to approximate the curvature matrix with low computational overhead; (ii) a root-mean-square exponential moving average to smooth out variations of the Hessian diagonal across different iterations; and (iii) a block diagonal averaging to reduce the variance of Hessian diagonal elements. We show that ADAHESSIAN achieves new state-of-the-art results by a large margin as compared to other adaptive optimization methods, including variants of Adam. In particular, we perform extensive tests on CV, NLP, and recommendation system tasks and find that ADAHESSIAN: (i) achieves 1.80%/1.45% higher accuracy on ResNets20/32 on Cifar10, and 5.55% higher accuracy on ImageNet as compared to Adam; (ii) outperforms AdamW for transformers by 0.13/0.33 BLEU score on IWSLT14/WMT14 and 2.7/1.0 PPL on PTB/Wikitext-103; (iii) outperforms AdamW for SqueezeBert by 0.41 points on GLUE; and (iv) achieves 0.032% better score than Adagrad for DLRM on the Criteo Ad Kaggle dataset. Importantly, we show that the cost per iteration of ADAHESSIAN is comparable to first order methods, and that it exhibits robustness towards its hyperparameters.

Source Code: Known repository: https://github.com/amirgholami/adahessian

2. Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training #

Year: 2023
Authors: Hong Liu, Zhiyuan Li, David Hall, Percy Liang, Tengyu Ma
ArXiv ID: arXiv:2305.14342
Algorithm: Sophia
URL: https://arxiv.org/abs/2305.14342

Abstract: Given the massive cost of language model pre-training, a non-trivial improvement of the optimization algorithm would lead to a material reduction on the time and cost of training. Adam and its variants have been state-of-the-art for years, and more sophisticated second-order (Hessian-based) optimizers often incur too much per-step overhead. In this paper, we propose Sophia, Second-order Clipped Stochastic Optimization, a simple scalable second-order optimizer that uses a light-weight estimate of the diagonal Hessian as the pre-conditioner. The update is the moving average of the gradients divided by the moving average of the estimated Hessian, followed by element-wise clipping. The clipping controls the worst-case update size and tames the negative impact of non-convexity and rapid change of Hessian along the trajectory. Sophia only estimates the diagonal Hessian every handful of iterations, which has negligible average per-step time and memory overhead. On language modeling with GPT models of sizes ranging from 125M to 1.5B, Sophia achieves a 2x speed-up compared to Adam in the number of steps, total compute, and wall-clock time, achieving the same perplexity with 50% fewer steps, less total compute, and reduced wall-clock time. Theoretically, we show that Sophia, in a much simplified setting, adapts to the heterogeneous curvatures in different parameter dimensions, and thus has a run-time bound that does not depend on the condition number of the loss.

Source Code: Known repository: https://github.com/Liuhong99/Sophia

Hessian-free Optimization #

1. Learning Recurrent Neural Networks with Hessian-Free Optimization #

Year: 2011
Authors: James Martens, Ilya Sutskever
ArXiv ID:
URL: https://www.cs.toronto.edu/~jmartens/docs/RNN_HF.pdf

Abstract: In this work we resolve the long-outstanding problem of how to effectively train recurrent neural networks (RNNs) on complex and difficult sequence modeling problems which may contain long-term data dependencies. Utilizing recent advances in the Hessian-free optimization approach (Martens, 2010), together with a novel damping scheme, we successfully train RNNs on two sets of challenging problems. First, a collection of pathological synthetic datasets which are known to be impossible for standard optimization approaches (due to their extremely long-term dependencies), and second, on three natural and highly complex real-world sequence datasets where we find that our method significantly outperforms the previous state-of-the-art method for training neural sequence models: the Long Short-term Memory approach of Hochreiter and Schmidhuber (1997). Additionally, we offer a new interpretation of the generalized Gauss-Newton matrix of Schraudolph (2002) which is used within the HF approach of Martens.

Source Code: No explicit source code information found

2. Training Neural Networks with Stochastic Hessian-Free Optimization #

Year: 2013
Authors: Ryan Kiros
ArXiv ID: arXiv:1301.3641
Algorithm: SHF
URL: https://arxiv.org/abs/1301.3641

Abstract: Hessian-free (HF) optimization has been successfully used for training deep autoencoders and recurrent networks. HF uses the conjugate gradient algorithm to construct update directions through curvature-vector products that can be computed on the same order of time as gradients. In this paper we exploit this property and study stochastic HF with gradient and curvature mini-batches independent of the dataset size. We modify Martens’ HF for these settings and integrate dropout, a method for preventing co-adaptation of feature detectors, to guard against overfitting. Stochastic Hessian-free optimization gives an intermediary between SGD and HF that achieves competitive performance on both classification and deep autoencoder experiments.

Source Code: Mentions ‘code’ in abstract

Quasi-Newton #

1. A Stochastic Quasi-Newton Method for Large-Scale Optimization #

Year: 2014
Authors: R.H. Byrd, S.L. Hansen, J. Nocedal, Y. Singer
ArXiv ID: arXiv:1401.7020
URL: https://arxiv.org/abs/1401.7020

Abstract: The question of how to incorporate curvature information in stochastic approximation methods is challenging. The direct application of classical quasi- Newton updating techniques for deterministic optimization leads to noisy curvature estimates that have harmful effects on the robustness of the iteration. In this paper, we propose a stochastic quasi-Newton method that is efficient, robust and scalable. It employs the classical BFGS update formula in its limited memory form, and is based on the observation that it is beneficial to collect curvature information pointwise, and at regular intervals, through (sub-sampled) Hessian-vector products. This technique differs from the classical approach that would compute differences of gradients, and where controlling the quality of the curvature estimates can be difficult. We present numerical results on problems arising in machine learning that suggest that the proposed method shows much promise.

Source Code: No explicit source code information found

2. A Multi-Batch L-BFGS Method for Machine Learning #

Year: 2016
Authors: Albert S. Berahas, Jorge Nocedal, Martin Takáč
ArXiv ID: arXiv:1605.06049
URL: https://arxiv.org/abs/1605.06049

Abstract: The question of how to parallelize the stochastic gradient descent (SGD) method has received much attention in the literature. In this paper, we focus instead on batch methods that use a sizeable fraction of the training set at each iteration to facilitate parallelism, and that employ second-order information. In order to improve the learning process, we follow a multi-batch approach in which the batch changes at each iteration. This can cause difficulties because L-BFGS employs gradient differences to update the Hessian approximations, and when these gradients are computed using different data points the process can be unstable. This paper shows how to perform stable quasi-Newton updating in the multi-batch setting, illustrates the behavior of the algorithm in a distributed computing platform, and studies its convergence properties for both the convex and nonconvex cases.

Source Code: No explicit source code information found

3. Stochastic Quasi-Newton with Line-Search Regularization #

Year: 2019
Authors: Adrian Wills, Thomas Schön
ArXiv ID: arXiv:1909.01238
Algorithm: SQN
URL: https://arxiv.org/abs/1909.01238

Abstract: In this paper we present a novel quasi-Newton algorithm for use in stochastic optimisation. Quasi-Newton methods have had an enormous impact on deterministic optimisation problems because they afford rapid convergence and computationally attractive algorithms. In essence, this is achieved by learning the second-order (Hessian) information based on observing first-order gradients. We extend these ideas to the stochastic setting by employing a highly flexible model for the Hessian and infer its value based on observing noisy gradients. In addition, we propose a stochastic counterpart to standard line-search procedures and demonstrate the utility of this combination on maximum likelihood identification for general nonlinear state space models.

Source Code: No explicit source code information found

4. Practical Quasi-Newton Methods for Training Deep Neural Networks #

Year: 2020
Authors: Donald Goldfarb, Yi Ren, Achraf Bahamou
ArXiv ID: arXiv:2006.08877
URL: https://arxiv.org/abs/2006.08877

Abstract: We consider the development of practical stochastic quasi-Newton, and in particular Kronecker-factored block-diagonal BFGS and L-BFGS methods, for training deep neural networks (DNNs). In DNN training, the number of variables and components of the gradient $n$ is often of the order of tens of millions and the Hessian has $n^2$ elements. Consequently, computing and storing a full $n \times n$ BFGS approximation or storing a modest number of (step, change in gradient) vector pairs for use in an L-BFGS implementation is out of the question. In our proposed methods, we approximate the Hessian by a block-diagonal matrix and use the structure of the gradient and Hessian to further approximate these blocks, each of which corresponds to a layer, as the Kronecker product of two much smaller matrices. This is analogous to the approach in KFAC, which computes a Kronecker-factored block-diagonal approximation to the Fisher matrix in a stochastic natural gradient method. Because the indefinite and highly variable nature of the Hessian in a DNN, we also propose a new damping approach to keep the upper as well as the lower bounds of the BFGS and L-BFGS approximations bounded. In tests on autoencoder feed-forward neural network models with either nine or thirteen layers applied to three datasets, our methods outperformed or performed comparably to KFAC and state-of-the-art first-order stochastic methods.

Source Code: Mentions ‘code’ in abstract; Mentions ‘implementation’ in abstract

Gauss-Newton #

1. Efficient Subsampled Gauss-Newton and Natural Gradient Methods for Training Neural Networks #

Year: 2019
Authors: Yi Ren, Donald Goldfarb
ArXiv ID: arXiv:1906.02353
Algorithm: SWM-GN, SWM-NG
URL: https://arxiv.org/abs/1906.02353

Abstract: We present practical Levenberg-Marquardt variants of Gauss-Newton and natural gradient methods for solving non-convex optimization problems that arise in training deep neural networks involving enormous numbers of variables and huge data sets. Our methods use subsampled Gauss-Newton or Fisher information matrices and either subsampled gradient estimates (fully stochastic) or full gradients (semi-stochastic), which, in the latter case, we prove convergent to a stationary point. By using the Sherman-Morrison-Woodbury formula with automatic differentiation (backpropagation) we show how our methods can be implemented to perform efficiently. Finally, numerical results are presented to demonstrate the effectiveness of our proposed methods.

Source Code: No explicit source code information found

2. On the Promise of the Stochastic Generalized Gauss-Newton Method for Training DNNs #

Year: 2020
Authors: Matilde Gargiani, et al.
ArXiv ID: arXiv:2006.02409
Algorithm: SGN
URL: https://arxiv.org/abs/2006.02409

Abstract: Following early work on Hessian-free methods for deep learning, we study a stochastic generalized Gauss-Newton method (SGN) for training DNNs. SGN is a second-order optimization method, with efficient iterations, that we demonstrate to often require substantially fewer iterations than standard SGD to converge. As the name suggests, SGN uses a Gauss-Newton approximation for the Hessian matrix, and, in order to compute an approximate search direction, relies on the conjugate gradient method combined with forward and reverse automatic differentiation. Despite the success of SGD and its first-order variants, and despite Hessian-free methods based on the Gauss-Newton Hessian approximation having been already theoretically proposed as practical methods for training DNNs, we believe that SGN has a lot of undiscovered and yet not fully displayed potential in big mini-batch scenarios. For this setting, we demonstrate that SGN does not only substantially improve over SGD in terms of the number of iterations, but also in terms of runtime. This is made possible by an efficient, easy-to-use and flexible implementation of SGN we propose in the Theano deep learning platform, which, unlike Tensorflow and Pytorch, supports forward automatic differentiation. This enables researchers to further study and improve this promising optimization technique and hopefully reconsider stochastic second-order methods as competitive optimization techniques for training DNNs; we also hope that the promise of SGN may lead to forward automatic differentiation being added to Tensorflow or Pytorch. Our results also show that in big mini-batch scenarios SGN is more robust than SGD with respect to its hyperparameters (we never had to tune its step-size for our benchmarks!), which eases the expensive process of hyperparameter tuning that is instead crucial for the performance of first-order methods.

Source Code: Mentions ‘implementation’ in abstract

3. Stochastic Gauss-Newton Algorithms for Nonconvex Compositional Optimization #

Year: 2020
Authors: Quoc Tran-Dinh, et al.
ArXiv ID: arXiv:2002.07290
Algorithm: SGN with SARAH estimators
URL: https://arxiv.org/abs/2002.07290

Abstract: We develop two new stochastic Gauss-Newton algorithms for solving a class of non-convex stochastic compositional optimization problems frequently arising in practice. We consider both the expectation and finite-sum settings under standard assumptions, and use both classical stochastic and SARAH estimators for approximating function values and Jacobians. In the expectation case, we establish $\mathcal{O}(\varepsilon^{-2})$ iteration-complexity to achieve a stationary point in expectation and estimate the total number of stochastic oracle calls for both function value and its Jacobian, where $\varepsilon$ is a desired accuracy. In the finite sum case, we also estimate $\mathcal{O}(\varepsilon^{-2})$ iteration-complexity and the total oracle calls with high probability. To our best knowledge, this is the first time such global stochastic oracle complexity is established for stochastic Gauss-Newton methods. Finally, we illustrate our theoretical results via two numerical examples on both synthetic and real datasets.

Source Code: No explicit source code information found

4. Nonlinear Least Squares for Large-Scale Machine Learning using Stochastic Jacobian Estimates #

Year: 2021
Authors: Johannes J. Brust
ArXiv ID: arXiv:2107.05598
Algorithm: NLLS1, NLLSL
URL: https://arxiv.org/abs/2107.05598

Abstract: For large nonlinear least squares loss functions in machine learning we exploit the property that the number of model parameters typically exceeds the data in one batch. This implies a low-rank structure in the Hessian of the loss, which enables effective means to compute search directions. Using this property, we develop two algorithms that estimate Jacobian matrices and perform well when compared to state-of-the-art methods.

Source Code: No explicit source code information found

5. Improving Levenberg-Marquardt Algorithm for Neural Networks #

Year: 2022
Authors: Omead Pooladzandi, Yiming Zhou
ArXiv ID: arXiv:2212.08769
Algorithm: LM
URL: https://arxiv.org/abs/2212.08769

Abstract: We explore the usage of the Levenberg-Marquardt (LM) algorithm for regression (non-linear least squares) and classification (generalized Gauss-Newton methods) tasks in neural networks. We compare the performance of the LM method with other popular first-order algorithms such as SGD and Adam, as well as other second-order algorithms such as L-BFGS , Hessian-Free and KFAC. We further speed up the LM method by using adaptive momentum, learning rate line search, and uphill step acceptance.

Source Code: No explicit source code information found

6. Rethinking Gauss-Newton for learning over-parameterized models #

Year: 2023
Authors: Michael Arbel, et al.
ArXiv ID: arXiv:2302.02904
URL: https://arxiv.org/abs/2302.02904

Abstract: This work studies the global convergence and implicit bias of Gauss Newton’s (GN) when optimizing over-parameterized one-hidden layer networks in the mean-field regime. We first establish a global convergence result for GN in the continuous-time limit exhibiting a faster convergence rate compared to GD due to improved conditioning. We then perform an empirical study on a synthetic regression task to investigate the implicit bias of GN’s method. While GN is consistently faster than GD in finding a global optimum, the learned model generalizes well on test data when starting from random initial weights with a small variance and using a small step size to slow down convergence. Specifically, our study shows that such a setting results in a hidden learning phenomenon, where the dynamics are able to recover features with good generalization properties despite the model having sub-optimal training and test performances due to an under-optimized linear layer. This study exhibits a trade-off between the convergence speed of GN and the generalization ability of the learned solution.

Source Code: No explicit source code information found

7. Exact Gauss-Newton Optimization for Training Deep Neural Networks #

Year: 2024
Authors: Mikalai Korbit, Adeyemi D. Adeoye, Alberto Bemporad, Mario Zanon
ArXiv ID: arXiv:2405.14402
Algorithm: EGN
URL: https://arxiv.org/abs/2405.14402

Abstract: We present EGN, a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges to an $\epsilon$-stationary point at a linear rate. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, and SGN optimizers across various supervised and reinforcement learning tasks.

Source Code: No explicit source code information found

Fisher Information #

1. Optimizing Neural Networks with Kronecker-factored Approximate Curvature #

Year: 2015
Authors: James Martens, Roger Grosse
ArXiv ID: arXiv:1503.05671
Algorithm: K-FAC
URL: https://arxiv.org/abs/1503.05671

Abstract: We propose an efficient method for approximating natural gradient descent in neural networks which we call Kronecker-Factored Approximate Curvature (K-FAC). K-FAC is based on an efficiently invertible approximation of a neural network’s Fisher information matrix which is neither diagonal nor low-rank, and in some cases is completely non-sparse. It is derived by approximating various large blocks of the Fisher (corresponding to entire layers) as being the Kronecker product of two much smaller matrices. While only several times more expensive to compute than the plain stochastic gradient, the updates produced by K-FAC make much more progress optimizing the objective, which results in an algorithm that can be much faster than stochastic gradient descent with momentum in practice. And unlike some previously proposed approximate natural-gradient/Newton methods which use high-quality non-diagonal curvature matrices (such as Hessian-free optimization), K-FAC works very well in highly stochastic optimization regimes. This is because the cost of storing and inverting K-FAC’s approximation to the curvature matrix does not depend on the amount of data used to estimate it, which is a feature typically associated only with diagonal or low-rank approximations to the curvature matrix.

Source Code: Known repository: Various implementations available

Other #

1. Second-order optimization with lazy Hessians #

Year: 2022
Authors: Nikita Doikov, El Mahdi Chayti, Martin Jaggi
ArXiv ID: arXiv:2212.00781
URL: https://arxiv.org/abs/2212.00781

Abstract: We analyze Newton’s method with lazy Hessian updates for solving general possibly non-convex optimization problems. We propose to reuse a previously seen Hessian for several iterations while computing new gradients at each step of the method. This significantly reduces the overall arithmetical complexity of second-order optimization schemes. By using the cubic regularization technique, we establish fast global convergence of our method to a second-order stationary point, while the Hessian does not need to be updated each iteration. For convex problems, we justify global and local superlinear rates for lazy Newton steps with quadratic regularization, which is easier to compute. The optimal frequency for updating the Hessian is once every $d$ iterations, where $d$ is the dimension of the problem. This provably improves the total arithmetical complexity of second-order algorithms by a factor $\sqrt{d}$.

Source Code: No explicit source code information found