In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Careful tuning of the learning rate, or even schedules thereof, can be crucial to effective neural net training. There has been much recent interest in gradient-based meta-optimization, where one tunes hyperparameters, or even learns an optimizer, in order to minimize the expected loss when the training procedure is unrolled. But because the training procedure must be unrolled thousands of times, the meta-objective must be defined with an orders-of-magnitude shorter time horizon than is typical for neural net training. We show that such short-horizon meta-objectives cause a serious bias towards small step sizes, an effect we term short-horizon bias. We introduce a toy problem, a noisy quadratic cost function, on which we analyze short-horizon bias by deriving and comparing the optimal schedules for short and long time horizons. We then run meta-optimization experiments (both offline and online) on standard benchmark datasets, showing that meta-optimization chooses too small a learning rate by multiple orders of magnitude, even when run with a moderately long time horizon (100 steps) typical of work in the area. We believe short-horizon bias is a fundamental problem that needs to be addressed if meta-optimization is to scale to practical neural net training regimes.
We investigate the bias in the short-horizon meta-optimization objective.

Understanding the behavior of stochastic gradient descent (SGD) in the context of deep neural networks has raised lots of concerns recently. Along this line, we theoretically study a general form of gradient based optimization dynamics with unbiased noise, which unifies SGD and standard Langevin dynamics. Through investigating this general optimization dynamics, we analyze the behavior of SGD on escaping from minima and its regularization effects. A novel indicator is derived to characterize the efficiency of escaping from minima through measuring the alignment of noise covariance and the curvature of loss function. Based on this indicator, two conditions are established to show which type of noise structure is superior to isotropic noise in term of escaping efficiency. We further show that the anisotropic noise in SGD satisfies the two conditions, and thus helps to escape from sharp and poor minima effectively, towards more stable and flat minima that typically generalize well. We verify our understanding through comparing
this anisotropic diffusion with full gradient descent plus isotropic diffusion (i.e. Langevin dynamics) and other types of position-dependent noise.
We provide theoretical and empirical analysis on the role of anisotropic noise introduced by stochastic gradient on escaping from minima.

We present the first verification that a neural network for perception tasks produces
a correct output within a specified tolerance for every input of interest. We define
correctness relative to a specification which identifies 1) a state space consisting of
all relevant states of the world and 2) an observation process that produces neural
network inputs from the states of the world. Tiling the state and input spaces with
a finite number of tiles, obtaining ground truth bounds from the state tiles and
network output bounds from the input tiles, then comparing the ground truth and
network output bounds delivers an upper bound on the network output error for
any input of interest. Results from two case studies highlight the ability of our
technique to deliver tight error bounds for all inputs of interest and show how the
error bounds vary over the state and input spaces.
We present the first verification that a neural network for perception tasks produces a correct output within a specified tolerance for every input of interest.