The Connection Between Approximation, Depth Separation and Learnability in Neural Networks

Several recent works have shown separation results between deep neural networks, and hypothesis classes with inferior approximation capacity such as shallow networks or kernel classes. On the other hand, the fact that deep networks can efficiently express a target function does not mean this target function can be learned efficiently by deep neural networks. In this work we study the intricate connection between learnability and approximation capacity. We show that learnability with deep networks of a target function depends on the ability of simpler classes to approximate the target. Specifically, we show that a necessary condition for a function to be learnable by gradient descent on deep neural networks is to be able to approximate the function, at least in a weak sense, with shallow neural networks. We also show that a class of functions can be learned by an efficient statistical query algorithm if and only if it can be approximated in a weak sense by some kernel class. We give several examples of functions which demonstrate depth separation, and conclude that they cannot be efficiently learned, even by a hypothesis class that can efficiently approximate them.

Computational Separation Between Convolutional and Fully-Connected Networks

Convolutional neural networks (CNN) exhibit unmatched performance in a multitude of computer vision tasks. However, the advantage of using convolutional networks over fully-connected networks is not understood from a theoretical perspective. In this work, we show how convolutional networks can leverage locality in the data, and thus achieve a computational advantage over fully-connected networks. Specifically, we show a class of problems that can be efficiently solved using convolutional networks trained with gradient-descent, but at the same time is hard to learn using a polynomial-size fully-connected network.

When Hardness of Approximation Meets Hardness of Learning

A supervised learning algorithm has access to a distribution of labeled examples, and needs to return a function (hypothesis) that correctly labels the examples. The hypothesis of the learner is taken from some fixed class of functions (e.g., linear classifiers, neural networks etc.). A failure of the learning algorithm can occur due to two possible reasons: wrong choice of hypothesis class (hardness of approximation), or failure to find the best function within the hypothesis class (hardness of learning). Although both approximation and learnability are important for the success of the algorithm, they are typically studied separately. In this work, we show a single hardness property that implies both hardness of approximation using linear classes and shallow networks, and hardness of learning using correlation queries and gradient-descent. This allows us to obtain new results on hardness of approximation and learnability of parity functions, DNF formulas and $AC^0$ circuits.

Learning Parities with Neural Networks

In recent years we see a rapidly growing line of research which shows learnability of various models via common neural network algorithms. Yet, besides a very few outliers, these results show learnability of models that can be learned using linear methods. Namely, such results show that learning neural-networks with gradient-descent is competitive with learning a linear classifier on top of a data-independent representation of the examples. This leaves much to be desired, as neural networks are far more successful than linear methods. Furthermore, on the more conceptual level, linear models don't seem to capture the ``deepness" of deep networks. In this paper we make a step towards showing leanability of models that are inherently non-linear. We show that under certain distributions, sparse parities are learnable via gradient decent on depth-two network. On the other hand, under the same distributions, these parities cannot be learned efficiently by linear methods.

Proving the Lottery Ticket Hypothesis: Pruning is All You Need

The lottery ticket hypothesis (Frankle and Carbin, 2018), states that a randomly-initialized network contains a small subnetwork such that, when trained in isolation, can compete with the performance of the original network. We prove an even stronger hypothesis (as was also conjectured in Ramanujan et al., 2019), showing that for every bounded distribution and every target network with bounded weights, a sufficiently over-parameterized neural network with random weights contains a subnetwork with roughly the same accuracy as the target network, without any further training.

Learning Boolean Circuits with Neural Networks

Training neural-networks is computationally hard. However, in practice they are trained efficiently using gradient-based algorithms, achieving remarkable performance on natural data. To bridge this gap, we observe the property of local correlation: correlation between small patterns of the input and the target label. We focus on learning deep neural-networks with a variant of gradient-descent, when the target function is a tree-structured Boolean circuit. We show that in this case, the existence of correlation between the gates of the circuit and the target label determines whether the optimization succeeds or fails. Using this result, we show that neural-networks can learn the (log n)-parity problem for most product distributions. These results hint that local correlation may play an important role in differentiating between distributions that are hard or easy to learn.

On the Optimality of Trees Generated by ID3

Since its inception in the 1980s, ID3 has become one of the most successful and widely used algorithms for learning decision trees. However, its theoretical properties remain poorly understood. In this work, we analyze the heuristic of growing a decision tree with ID3 for a limited number of iterations $t$ and given that nodes are split as in the case of exact information gain and probability computations. In several settings, we provide theoretical and empirical evidence that the TopDown variant of ID3, introduced by Kearns and Mansour (1996), produces trees with optimal or near-optimal test error among all trees with $t$ internal nodes. We prove optimality in the case of learning conjunctions under product distributions and learning read-once DNFs with 2 terms under the uniform distribition. Using efficient dynamic programming algorithms, we empirically show that TopDown generates trees that are near-optimal ($\sim \%1$ difference from optimal test error) in a large number of settings for learning read-once DNFs under product distributions.

ID3 Learns Juntas for Smoothed Product Distributions

In recent years, there are many attempts to understand popular heuristics. An example of such a heuristic algorithm is the ID3 algorithm for learning decision trees. This algorithm is commonly used in practice, but there are very few theoretical works studying its behavior. In this paper, we analyze the ID3 algorithm, when the target function is a $k$-Junta, a function that depends on $k$ out of $n$ variables of the input. We prove that when $k = \log n$, the ID3 algorithm learns in polynomial time $k$-Juntas, in the smoothed analysis model of Kalai & Teng. That is, we show a learnability result when the observed distribution is a "noisy" variant of the original distribution.

Decoupling Gating from Linearity

ReLU neural-networks have been in the focus of many recent theoretical works, trying to explain their empirical success. Nonetheless, there is still a gap between current theoretical results and empirical observations, even in the case of shallow (one hidden-layer) networks. For example, in the task of memorizing a random sample of size $m$ and dimension $d$, the best theoretical result requires the size of the network to be $\tilde{\Omega}(\frac{m^2}{d})$, while empirically a network of size slightly larger than $\frac{m}{d}$ is sufficient. To bridge this gap, we turn to study a simplified model for ReLU networks. We observe that a ReLU neuron is a product of a linear function with a gate (the latter determines whether the neuron is active or not), where both share a jointly trained weight vector. In this spirit, we introduce the Gated Linear Unit (GaLU), which simply decouples the linearity from the gating by assigning different vectors for each role. We show that GaLU networks allow us to get optimization and generalization results that are much stronger than those available for ReLU networks. Specifically, we show a memorization result for networks of size $\tilde{\Omega}(\frac{m}{d})$, and improved generalization bounds. Finally, we show that in some scenarios, GaLU networks behave similarly to ReLU networks, hence proving to be a good choice of a simplified model.