RedEx: Beyond Fixed Representation Methods via Convex Optimization

Optimizing Neural networks is a difficult task which is still not well understood. On the other hand, fixed representation methods such as kernels and random features have provable optimization guarantees but inferior performance due to their inherent inability to learn the representations. In this paper, we aim at bridging this gap by presenting a novel architecture called RedEx (Reduced Expander Extractor) that is as expressive as neural networks and can also be trained in a layer-wise fashion via a convex program with semi-definite constraints and optimization guarantees. We also show that RedEx provably surpasses fixed representation methods, in the sense that it can efficiently learn a family of target functions which fixed representation methods cannot.

Locally Optimal Descent for Dynamic Stepsize Scheduling

We introduce a novel dynamic learning-rate scheduling scheme grounded in theory with the goal of simplifying the manual and time-consuming tuning of schedules in practice. Our approach is based on estimating the locally-optimal stepsize, guaranteeing maximal descent in the direction of the stochastic gradient of the current step. We first establish theoretical convergence bounds for our method within the context of smooth non-convex stochastic optimization, matching state-of-the-art bounds while only assuming knowledge of the smoothness parameter. We then present a practical implementation of our algorithm and conduct systematic experiments across diverse datasets and optimization algorithms, comparing our scheme with existing state-of-the-art learning-rate schedulers. Our findings indicate that our method needs minimal tuning when compared to existing approaches, removing the need for auxiliary manual schedules and warm-up phases and achieving comparable performance with drastically reduced parameter tuning.

Multiclass Boosting: Simple and Intuitive Weak Learning Criteria

We study a generalization of boosting to the multiclass setting. We introduce a weak learning condition for multiclass classification that captures the original notion of weak learnability as being "slightly better than random guessing". We give a simple and efficient boosting algorithm, that does not require realizability assumptions and its sample and oracle complexity bounds are independent of the number of classes. In addition, we utilize our new boosting technique in several theoretical applications within the context of List PAC Learning. First, we establish an equivalence to weak PAC learning. Furthermore, we present a new result on boosting for list learners, as well as provide a novel proof for the characterization of multiclass PAC learning and List PAC learning. Notably, our technique gives rise to a simplified analysis, and also implies an improved error bound for large list sizes, compared to previous results.

Most Neural Networks Are Almost Learnable

We present a PTAS for learning random constant-depth networks. We show that for any fixed $\epsilon>0$ and depth $i$, there is a poly-time algorithm that for any distribution on $\sqrt{d} \cdot \mathbb{S}^{d-1}$ learns random Xavier networks of depth $i$, up to an additive error of $\epsilon$. The algorithm runs in time and sample complexity of $(\bar{d})^{\mathrm{poly}(\epsilon^{-1})}$, where $\bar d$ is the size of the network. For some cases of sigmoid and ReLU-like activations the bound can be improved to $(\bar{d})^{\mathrm{polylog}(\epsilon^{-1})}$, resulting in a quasi-poly-time algorithm for learning constant depth random networks.

Efficiently Learning Neural Networks: What Assumptions May Suffice?

Understanding when neural networks can be learned efficiently is a fundamental question in learning theory. Existing hardness results suggest that assumptions on both the input distribution and the network's weights are necessary for obtaining efficient algorithms. Moreover, it was previously shown that depth-$2$ networks can be efficiently learned under the assumptions that the input distribution is Gaussian, and the weight matrix is non-degenerate. In this work, we study whether such assumptions may suffice for learning deeper networks and prove negative results. We show that learning depth-$3$ ReLU networks under the Gaussian input distribution is hard even in the smoothed-analysis framework, where a random noise is added to the network's parameters. It implies that learning depth-$3$ ReLU networks under the Gaussian distribution is hard even if the weight matrices are non-degenerate. Moreover, we consider depth-$2$ networks, and show hardness of learning in the smoothed-analysis framework, where both the network parameters and the input distribution are smoothed. Our hardness results are under a well-studied assumption on the existence of local pseudorandom generators.

On the Sample Complexity of Two-Layer Networks: Lipschitz vs. Element-Wise Lipschitz Activation

We investigate the sample complexity of bounded two-layer neural networks using different activation functions. In particular, we consider the class \[ \mathcal{H} = \left\{\textbf{x}\mapsto \langle \textbf{v}, \sigma \circ W\textbf{x} + \textbf{b} \rangle : \textbf{b}\in\mathbb{R}^d, W \in \mathbb{R}^{T\times d}, \textbf{v} \in \mathbb{R}^{T}\right\} \] where the spectral norm of $W$ and $\textbf{v}$ is bounded by $O(1)$, the Frobenius norm of $W$ is bounded from its initialization by $R > 0$, and $\sigma$ is a Lipschitz activation function. We prove that if $\sigma$ is element-wise, then the sample complexity of $\mathcal{H}$ is width independent and that this complexity is tight. Moreover, we show that the element-wise property of $\sigma$ is essential for width-independent bound, in the sense that there exist non-element-wise activation functions whose sample complexity is provably width-dependent. For the upper bound, we use the recent approach for norm-based bounds named Approximate Description Length (ADL) by arXiv:1910.05697. We further develop new techniques and tools for this approach, that will hopefully inspire future works.

Approximate Description Length, Covering Numbers, and VC Dimension

Recently, Daniely and Granot [arXiv:1910.05697] introduced a new notion of complexity called Approximate Description Length (ADL). They used it to derive novel generalization bounds for neural networks, that despite substantial work, were out of reach for more classical techniques such as discretization, Covering Numbers and Rademacher Complexity. In this paper we explore how ADL relates to classical notions of function complexity such as Covering Numbers and VC Dimension. We find that for functions whose range is the reals, ADL is essentially equivalent to these classical complexity measures. However, this equivalence breaks for functions with high dimensional range.

Monotone Learning

The amount of training-data is one of the key factors which determines the generalization capacity of learning algorithms. Intuitively, one expects the error rate to decrease as the amount of training-data increases. Perhaps surprisingly, natural attempts to formalize this intuition give rise to interesting and challenging mathematical questions. For example, in their classical book on pattern recognition, Devroye, Gyorfi, and Lugosi (1996) ask whether there exists a {monotone} Bayes-consistent algorithm. This question remained open for over 25 years, until recently Pestov (2021) resolved it for binary classification, using an intricate construction of a monotone Bayes-consistent algorithm. We derive a general result in multiclass classification, showing that every learning algorithm A can be transformed to a monotone one with similar performance. Further, the transformation is efficient and only uses a black-box oracle access to A. This demonstrates that one can provably avoid non-monotonic behaviour without compromising performance, thus answering questions asked by Devroye et al (1996), Viering, Mey, and Loog (2019), Viering and Loog (2021), and by Mhammedi (2021). Our transformation readily implies monotone learners in a variety of contexts: for example it extends Pestov's result to classification tasks with an arbitrary number of labels. This is in contrast with Pestov's work which is tailored to binary classification. In addition, we provide uniform bounds on the error of the monotone algorithm. This makes our transformation applicable in distribution-free settings. For example, in PAC learning it implies that every learnable class admits a monotone PAC learner. This resolves questions by Viering, Mey, and Loog (2019); Viering and Loog (2021); Mhammedi (2021).