Raghunathan et al. provide an upper bound on the adversarial loss of two-layer networks and also derive a regularization method to minimize this upper bound. In particular, the authors consider the scoring functions $f^i(x) = V_i^T\sigma(Wx)$ with bounded derivative $\sigma'(z) \in [0,1]$ which holds for Sigmoid and ReLU activation functions. Still, the model is very constrained considering recent, well-performng deep (convolutional) neural networks. The upper bound is then derived by considering $f(A(x))$ where $A(x)$ is the optimal attacker $A(x) = \arg\max_{\tilde{x} \in B_\epsilon(x)} f(\tilde{x})$. For a linear model $f(x) = (W_1 – W_2)^Tx$, an upper bound can be derived as follows:

$f(\tilde{x}) = f(x) + (W_1 – W_2)^T(\tilde{x} – x) \leq f(x) + \epsilon\|W_1 – W_2\|_1$.

For two-layer networks a bound is derived by considering

$f(\tilde{x}) = f(x) + \int_0^1 \nabla f(t\tilde{x} + (1-t)x)^T (\tilde{x} – x) dt \leq f(x) + \max_{\tilde{x}\in B_\epsilon(x)} \epsilon\|\nabla f(\tilde{x})\|_1$.

In this case, Raghunathan rewrite the second term, i.e. $\max_{\tilde{x}\in B_\epsilon(x)} \epsilon\|\nabla f(\tilde{x})\|_1$ to derive an upper bound in the form of a semidefinite program, see the paper for details. For $v = V_1 – V_2$, this semidefinite program is based on the matrix

$M(v,W) = \left[\begin{array}0 & 0 & 1^T W^R \text{diag}(v)\\0 & 0 & W^T\text{diag}(v)\\ \text{diag}(v)^T W 1 & \text{diag}(v)^T W & 0\end{array}\right]$.

By deriving the dual objective, the upper bound can then be minimized by constraining the eigenvalues of $M(v, W)$ (specifically, the largest eigenvalue; note that the dual also involves dual variables – see the paper for details). Overall, the proposed regularize involves minimizing the largest eigenvalue of $M(v, W) – D$ where $D$ is a diagonal matrix based on the dual variables. In practice, this is implemented using SciPy's implementation of the Lanczos algorithm.

Also view this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

**Summary**: This paper presents three tricks that make model-based reinforcement more reliable when tested in tasks that require walking and balancing. The tricks are 1) are planning based on features, 2) using a recursive network that mixes probabilistic and deterministic information, and 3) looking forward multiple steps.

**Longer summary**

Imagine playing pool, armed with a tablet that can predict exactly where the ball will bounce, and the next bounce, and so on. That would be a huge advantage to someone learning pool, however small inaccuracies in the model could mislead you especially when thinking ahead to the 2nd and third bounce. 

The tablet is analogous to the dynamics model in model-based reinforcement learning (RL). Model based RL promises to solve a lot of the open problems with RL, letting the agent learn with less experience, transfer well, dream, and many others advantages. Despite the promise, dynamics models are hard to get working: they often suffer from even small inaccuracies, and need to be redesigned for specific tasks.

Enter PlaNet, a clever name, and a net that plans well in range of environments. To increase the challenge the model must predict directly from pixels in fairly difficult tasks such as teaching a cheetah to run or balancing a ball in a cup.


How do they do this? Three main tricks.

- Planning in latest space: this means that the policy network doesn't need to look at the raw image, but looks at a summary of it as represented by a feature vector.
- Recurrent state space models: They found that probabilistic information helps describe the space of possibilities but makes it harder for their RNN based model to look back multiple steps. However mixing probabilistic information and deterministic information gives it the best of both worlds, and they have results that show a starting performance increase when both compared to just one.
- Latent overshooting: They train the model to look more than one step ahead, this helps prevent errors that build up over time

Overall this paper shows great results that tackle the shortfalls of model based RL. I hope the results remain when tested on different and more complex environments.

Benchmarking Deep Learning Hardware and Frameworks: Qualitative Metrics 

Previous papers on benchmarking deep neural networks offer knowledge of deep learning hardware devices and software frameworks. This paper introduces benchmarking principles, surveys machine learning devices including GPUs, FPGAs, and ASICs, and reviews deep learning software frameworks. It also qualitatively compares these technologies with respect to benchmarking from the angles of our 7-metric approach to deep learning frameworks and 12-metric approach to machine learning hardware platforms. 

After reading the paper, the audience will understand seven benchmarking principles, generally know that differential characteristics of mainstream artificial intelligence devices,  qualitatively compare deep learning hardware through the 12-metric approach for benchmarking neural network hardware, and read benchmarking results of 16 deep learning frameworks via our 7-metric set for benchmarking  frameworks.

Lee et al. propose a regularizer to increase the size of linear regions of rectified deep networks around training and test points. Specifically, they assume piece-wise linear networks, in its most simplistic form consisting of linear layers (fully connected layers, convolutional layers) and ReLU activation functions. In these networks, linear regions are determined by activation patterns, i.e., a pattern indicating which neurons have value greater than zero. Then, the goal is to compute, and later to increase, the size $\epsilon$ such that the $L_p$-ball of radius $\epsilon$ around a sample $x$, denoted $B_{\epsilon,p}(x)$ is contained within one linear region (corresponding to one activation pattern). Formally, letting $S(x)$ denote the set of feasible inputs $x$ for a given activation pattern, the task is to determine

$\hat{\epsilon}_{x,p} = \max_{\epsilon \geq 0, B_{\epsilon,p}(x) \subset S(x)} \epsilon$.

For $p = 1, 2, \infty$, the authors show how $\hat{\epsilon}_{x,p}$ can be computed efficiently. For $p = 2$, for example, it results in

$\hat{\epsilon}_{x,p} = \min_{(i,j) \in I} \frac{|z_j^i|}{\|\nabla_x z_j^i\|_2}$.

Here, $z_j^i$ corresponds to the $j$th neuron in the $i$th layer of a multi-layer perceptron with ReLU activations; and $I$ contains all the indices of hidden neurons. This analytical form can then used to add a regularizer to encourage the network to learn larger linear regions:

$\min_\theta \sum_{(x,y) \in D} \left[\mathcal{L}(f_\theta(x), y) - \lambda \min_{(i,j) \in I} \frac{|z_j^i|}{\|\nabla_x z_j^i\|_2}\right]$

where $f_\theta$ is the neural network with paramters $\theta$. In the remainder of the paper, the authors propose a relaxed version of this training procedure that resembles a max-margin formulation and discuss efficient computation of the involved derivatives $\nabla_x z_j^i$ without too many additional forward/backward passes.

https://i.imgur.com/jSc9zbw.jpg
Figure 1: Visualization of locally linear regions for three different models on toy 2D data.

On toy data and datasets such as MNIST and CalTech-256, it is shown that the training procedure is effective in the sense that larger linear regions around training and test points are learned. For example, on a 2D toy dataset, Figure 1 visualizes the linear regions for the optimal regularizer as well as the proposed relaxed version.

Also find this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

Nagarajan and Kolter show that neural networks are implicitly regularized by stochastic gradient descent to have small distance from their initialization. This implicit regularization may explain the good generalization performance of over-parameterized neural networks; specifically, more complex models usually generalize better, which contradicts the general trade-off between expressivity and generalization in machine learning. On MNIST, the authors show that the distance of the network’s parameters to the original initialization (as measured using the $L_2$ norm on the flattened parameters) reduces with increasing width, and increases with increasing sample size. Additionally, the distance increases significantly when fitting corrupted labels, which may indicate that memorization requires to travel a larger distance in parameter space.

Also find this summary at [davidstutz.de](https://davidstutz.de/category/reading/).