Most theoretical studies explaining the regularization effect in deep learning have only focused on gradient descent with a sufficient small learning rate or even gradient flow (infinitesimal learning rate). Such researches, however, have neglected a reasonably large learning rate applied in most practical applications. In this work, we characterize the implicit bias effect of deep linear networks for binary classification using the logistic loss in the large learning rate regime, inspired by the seminal work by Lewkowycz et al. [26] in a regression setting with squared loss. They found a learning rate regime with a large stepsize named the catapult phase, where the loss grows at the early stage of training and eventually converges to a minimum that is flatter than those found in the small learning rate regime. We claim that depending on the separation conditions of data, the gradient descent iterates will converge to a flatter minimum in the catapult phase. We rigorously prove this claim under the assumption of degenerate data by overcoming the difficulty of the non-constant Hessian of logistic loss and further characterize the behavior of loss and Hessian for non-separable data. Finally, we demonstrate that flatter minima in the space spanned by non-separable data along with the learning rate in the catapult phase can lead to better generalization empirically.

In this paper we are concerned with the approximation of functions by single hidden layer neural networks with ReLU activation functions on the unit circle. In particular, we are interested in the case when the number of data-points exceeds the number of nodes. We first study the convergence to equilibrium of the stochastic gradient flow associated with the cost function with a quadratic penalization. Specifically, we prove a Poincar\'e inequality for a penalized version of the cost function with explicit constants that are independent of the data and of the number of nodes. As our penalization biases the weights to be bounded, this leads us to study how well a network with bounded weights can approximate a given function of bounded variation (BV). Our main contribution concerning approximation of BV functions, is a result which we call the localization theorem. Specifically, it states that the expected error of the constrained problem, where the length of the weights are less than $R$, is of order $R^{-1/9}$ with respect to the unconstrained problem (the global optimum). The proof is novel in this topic and is inspired by techniques from regularity theory of elliptic partial differential equations. Finally we quantify the expected value of the global optimum by proving a quantitative version of the universal approximation theorem.

Conditional Value-at-Risk ($\mathrm{CV@R}$) is one of the most popular measures of risk, which has been recently considered as a performance criterion in supervised statistical learning, as it is related to desirable operational features in modern applications, such as safety, fairness, distributional robustness, and prediction error stability. However, due to its variational definition, $\mathrm{CV@R}$ is commonly believed to result in difficult optimization problems, even for smooth and strongly convex loss functions. We disprove this statement by establishing noisy (i.e., fixed-accuracy) linear convergence of stochastic gradient descent for sequential $\mathrm{CV@R}$ learning, for a large class of not necessarily strongly-convex (or even convex) loss functions satisfying a set-restricted Polyak-Lojasiewicz inequality. This class contains all smooth and strongly convex losses, confirming that classical problems, such as linear least squares regression, can be solved efficiently under the $\mathrm{CV@R}$ criterion, just as their risk-neutral versions. Our results are illustrated numerically on such a risk-aware ridge regression task, also verifying their validity in practice.

We study scalable alternatives to robust gradient descent (RGD) techniques that can be used when the losses and/or gradients can be heavy-tailed, though this will be unknown to the learner. The core technique is simple: instead of trying to robustly aggregate gradients at each step, which is costly and leads to sub-optimal dimension dependence in risk bounds, we instead focus computational effort on robustly choosing (or newly constructing) a strong candidate based on a collection of cheap stochastic sub-processes which can be run in parallel. The exact selection process depends on the convexity of the underlying objective, but in all cases, our selection technique amounts to a robust form of boosting the confidence of weak learners. In addition to formal guarantees, we also provide empirical analysis of robustness to perturbations to experimental conditions, under both sub-Gaussian and heavy-tailed data, along with applications to a variety of benchmark datasets. The overall take-away is an extensible procedure that is simple to implement, trivial to parallelize, which keeps the formal merits of RGD methods but scales much better to large learning problems.

A key challenge with machine learning approaches for ranking is the gap between the performance metrics of interest and the surrogate loss functions that can be optimized with gradient-based methods. This gap arises because ranking metrics typically involve a sorting operation which is not differentiable w.r.t. the model parameters. Prior works have proposed surrogates that are loosely related to ranking metrics or simple smoothed versions thereof. We propose PiRank, a new class of differentiable surrogates for ranking, which employ a continuous, temperature-controlled relaxation to the sorting operator. We show that PiRank exactly recovers the desired metrics in the limit of zero temperature and scales favorably with the problem size, both in theory and practice. Empirically, we demonstrate that PiRank significantly improves over existing approaches on publicly available internet-scale learning-to-rank benchmarks.

In this work, we study an optimizer, Grad-Avg to optimize error functions. We establish the convergence of the sequence of iterates of Grad-Avg mathematically to a minimizer (under boundedness assumption). We apply Grad-Avg along with some of the popular optimizers on regression as well as classification tasks. In regression tasks, it is observed that the behaviour of Grad-Avg is almost identical with Stochastic Gradient Descent (SGD). We present a mathematical justification of this fact. In case of classification tasks, it is observed that the performance of Grad-Avg can be enhanced by suitably scaling the parameters. Experimental results demonstrate that Grad-Avg converges faster than the other state-of-the-art optimizers for the classification task on two benchmark datasets.

Analyzing data owned by several parties while achieving a good trade-off between utility and privacy is a key challenge in federated learning and analytics. In this work, we introduce a novel relaxation of local differential privacy (LDP) that naturally arises in fully decentralized protocols, i.e. participants exchange information by communicating along the edges of a network graph. This relaxation, that we call network DP, captures the fact that users have only a local view of the decentralized system. To show the relevance of network DP, we study a decentralized model of computation where a token performs a walk on the network graph and is updated sequentially by the party who receives it. For tasks such as real summation, histogram computation and gradient descent, we propose simple algorithms and prove privacy amplification results on ring and complete topologies. The resulting privacy-utility trade-off significantly improves upon LDP, and in some cases even matches what can be achieved with approaches based on secure aggregation and secure shuffling. Our experiments confirm the practical significance of the gains compared to LDP.

The winners of the NeurIPS 2020 awards have been announced. This year, three papers have received Best Paper Awards. There was also one Test of Time Award; this recognises a paper that has had significant and lasting impact on the community. No-Regret Learning Dynamics for Extensive-Form Correlated Equilibrium Andrea Celli, Alberto Marchesi, Gabriele Farina, Nicola Gatti Abstract: The existence of simple, uncoupled no-regret dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium.

Predicting the dynamics of neural network parameters during training is one of the key challenges in building a theoretical foundation for deep learning. A central obstacle is that the motion of a network in high-dimensional parameter space undergoes discrete finite steps along complex stochastic gradients derived from real-world datasets. We circumvent this obstacle through a unifying theoretical framework based on intrinsic symmetries embedded in a network's architecture that are present for any dataset. We show that any such symmetry imposes stringent geometric constraints on gradients and Hessians, leading to an associated conservation law in the continuous-time limit of stochastic gradient descent (SGD), akin to Noether's theorem in physics. We further show that finite learning rates used in practice can actually break these symmetry induced conservation laws. We apply tools from finite difference methods to derive modified gradient flow, a differential equation that better approximates the numerical trajectory taken by SGD at finite learning rates. We combine modified gradient flow with our framework of symmetries to derive exact integral expressions for the dynamics of certain parameter combinations. We empirically validate our analytic predictions for learning dynamics on VGG-16 trained on Tiny ImageNet. Overall, by exploiting symmetry, our work demonstrates that we can analytically describe the learning dynamics of various parameter combinations at finite learning rates and batch sizes for state of the art architectures trained on any dataset. Just like the fundamental laws of classical and quantum mechanics taught us how to control and optimize the physical world for engineering purposes, a better understanding of the laws governing neural network learning dynamics can have a profound impact on the optimization of artificial neural networks.

Neural Tangent Kernel (NTK) theory is widely used to study the dynamics of infinitely-wide deep neural networks (DNNs) under gradient descent. But do the results for infinitely-wide networks give us hints about the behaviour of real finite-width ones? In this paper we study empirically when NTK theory is valid in practice for fully-connected ReLu and sigmoid networks. We find out that whether a network is in the NTK regime depends on the hyperparameters of random initialization and network's depth. In particular, NTK theory does not explain behaviour of sufficiently deep networks initialized so that their gradients explode: the kernel is random at initialization and changes significantly during training, contrary to NTK theory. On the other hand, in case of vanishing gradients DNNs are in the NTK regime but become untrainable rapidly with depth. We also describe a framework to study generalization properties of DNNs by means of NTK theory and discuss its limits.