Amit Daniely

NeurIPS Conference 2025 Conference Paper

Online Learning of Neural Networks

• Amit Daniely
• Idan Mehalel
• Elchanan Mossel

We study online learning of feedforward neural networks with the sign activation function that implement functions from the unit ball in $\mathbb{R}^d$ to a finite label set $\mathcal{Y} = \{1, \ldots, Y \}$. First, we characterize a margin condition that is sufficient and in some cases necessary for online learnability of a neural network: Every neuron in the first hidden layer classifies all instances with some margin $\gamma$ bounded away from zero. Quantitatively, we prove that for any net, the optimal mistake bound is at most approximately $\mathtt{TS}(d, \gamma)$, which is the $(d, \gamma)$-totally-separable-packing number, a more restricted variation of the standard $(d, \gamma)$-packing number. We complement this result by constructing a net on which any learner makes $\mathtt{TS}(d, \gamma)$ many mistakes. We also give a quantitative lower bound of approximately $\mathtt{TS}(d, \gamma) \geq \max\{1/(\gamma \sqrt{d})^d, d\}$ when $\gamma \geq 1/2$, implying that for some nets and input sequences every learner will err for $\exp(d)$ many times, and that a dimension-free mistake bound is almost always impossible. To remedy this inevitable dependence on $d$, it is natural to seek additional natural restrictions to be placed on the network, so that the dependence on $d$ is removed. We study two such restrictions. The first is the multi-index model, in which the function computed by the net depends only on $s \ll d$ orthonormal directions. We prove a mistake bound of approximately $(1. 5/\gamma)^{s + 2}$ in this model. The second is the extended margin assumption. In this setting, we assume that all neurons (in all layers) in the network classify every ingoing input from previous layer with margin $\gamma$ bounded away from zero. In this model, we prove a mistake bound of approximately $(\log Y)/ \gamma^{O(L)}$, where L is the depth of the network.

ICLR Conference 2023 Conference Paper

An Exact Poly-Time Membership-Queries Algorithm for Extracting a Three-Layer ReLU Network

• Amit Daniely
• Elad Granot

We consider the natural problem of learning a ReLU network from queries, which was recently remotivated by model extraction attacks. In this work, we present a polynomial-time algorithm that can learn a depth-two ReLU network from queries under mild general position assumptions. We also present a polynomial-time algorithm that, under mild general position assumptions, can learn a rich class of depth-three ReLU networks from queries. For instance, it can learn most networks where the number of first layer neurons is smaller than the dimension and the number of second layer neurons. These two results substantially improve state-of-the-art: Until our work, polynomial-time algorithms were only shown to learn from queries depth-two networks under the assumption that either the underlying distribution is Gaussian (Chen et al. (2021)) or that the weights matrix rows are linearly independent (Milli et al. (2019)). For depth three or more, there were no known poly-time results.

NeurIPS Conference 2023 Conference Paper

Computational Complexity of Learning Neural Networks: Smoothness and Degeneracy

• Amit Daniely
• Nati Srebro
• Gal Vardi

Understanding when neural networks can be learned efficientlyis 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.

NeurIPS Conference 2023 Conference Paper

Most Neural Networks Are Almost Learnable

• Amit Daniely
• Nati Srebro
• Gal Vardi

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.

NeurIPS Conference 2023 Conference Paper

Multiclass Boosting: Simple and Intuitive Weak Learning Criteria

• Nataly Brukhim
• Amit Daniely
• Yishay Mansour
• Shay Moran

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 simplified algorithms and analysis compared to previous works.

NeurIPS Conference 2021 Conference Paper

Asynchronous Stochastic Optimization Robust to Arbitrary Delays

• Alon Cohen
• Amit Daniely
• Yoel Drori
• Tomer Koren
• Mariano Schain

We consider the problem of stochastic optimization with delayed gradients in which, at each time step $t$, the algorithm makes an update using a stale stochastic gradient from step $t - d_t$ for some arbitrary delay $d_t$. This setting abstracts asynchronous distributed optimization where a central server receives gradient updates computed by worker machines. These machines can experience computation and communication loads that might vary significantly over time. In the general non-convex smooth optimization setting, we give a simple and efficient algorithm that requires $O( \sigma^2/\epsilon^4 + \tau/\epsilon^2 )$ steps for finding an $\epsilon$-stationary point $x$. Here, $\tau$ is the \emph{average} delay $\frac{1}{T}\sum_{t=1}^T d_t$ and $\sigma^2$ is the variance of the stochastic gradients. This improves over previous work, which showed that stochastic gradient decent achieves the same rate but with respect to the \emph{maximal} delay $\max_{t} d_t$, that can be significantly larger than the average delay especially in heterogeneous distributed systems. Our experiments demonstrate the efficacy and robustness of our algorithm in cases where the delay distribution is skewed or heavy-tailed.

NeurIPS Conference 2020 Conference Paper

Hardness of Learning Neural Networks with Natural Weights

• Amit Daniely
• Gal Vardi

Neural networks are nowadays highly successful despite strong hardness results. The existing hardness results focus on the network architecture, and assume that the network's weights are arbitrary. A natural approach to settle the discrepancy is to assume that the network's weights are ``well-behaved" and posses some generic properties that may allow efficient learning. This approach is supported by the intuition that the weights in real-world networks are not arbitrary, but exhibit some ''random-like" properties with respect to some ''natural" distributions. We prove negative results in this regard, and show that for depth-$2$ networks, and many ``natural" weights distributions such as the normal and the uniform distribution, most networks are hard to learn. Namely, there is no efficient learning algorithm that is provably successful for most weights, and every input distribution. It implies that there is no generic property that holds with high probability in such random networks and allows efficient learning.

NeurIPS Conference 2020 Conference Paper

Learning Parities with Neural Networks

• Amit Daniely
• Eran Malach

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.

NeurIPS Conference 2020 Conference Paper

Most ReLU Networks Suffer from $\ell^2$ Adversarial Perturbations

• Amit Daniely
• Hadas Shacham

We consider ReLU networks with random weights, in which the dimension decreases at each layer. We show that for most such networks, most examples $x$ admit an adversarial perturbation at an Euclidean distance of $O\left(\frac{\|x\|}{\sqrt{d}}\right)$, where $d$ is the input dimension. Moreover, this perturbation can be found via gradient flow, as well as gradient descent with sufficiently small steps. This result can be seen as an explanation to the abundance of adversarial examples, and to the fact that they are found via gradient descent.

NeurIPS Conference 2020 Conference Paper

Neural Networks Learning and Memorization with (almost) no Over-Parameterization

• Amit Daniely

Many results in recent years established polynomial time learnability of various models via neural networks algorithms (e. g. \cite{andoni2014learning, daniely2016toward, daniely2017sgd, cao2019generalization, ziwei2019polylogarithmic, zou2019improved, ma2019comparative, du2018gradient, arora2019fine, song2019quadratic, oymak2019towards, ge2019mildly, brutzkus2018sgd}). However, unless the model is linear separable~\cite{brutzkus2018sgd}, or the activation is a polynomial~\cite{ge2019mildly}, these results require very large networks -- much more than what is needed for the mere existence of a good predictor. In this paper we prove that SGD on depth two neural networks can memorize samples, learn polynomials with bounded weights, and learn certain kernel spaces, with {\em near optimal} network size, sample complexity, and runtime. In particular, we show that SGD on depth two network with $\tilde{O}\left(\frac{m}{d}\right)$ hidden neurons (and hence $\tilde{O}(m)$ parameters) can memorize $m$ random labeled points in $\sphere^{d-1}$.

ICLR Conference 2020 Conference Paper

The Implicit Bias of Depth: How Incremental Learning Drives Generalization

• Daniel Gissin
• Shai Shalev-Shwartz
• Amit Daniely

A leading hypothesis for the surprising generalization of neural networks is that the dynamics of gradient descent bias the model towards simple solutions, by searching through the solution space in an incremental order of complexity. We formally define the notion of incremental learning dynamics and derive the conditions on depth and initialization for which this phenomenon arises in deep linear models. Our main theoretical contribution is a dynamical depth separation result, proving that while shallow models can exhibit incremental learning dynamics, they require the initialization to be exponentially small for these dynamics to present themselves. However, once the model becomes deeper, the dependence becomes polynomial and incremental learning can arise in more natural settings. We complement our theoretical findings by experimenting with deep matrix sensing, quadratic neural networks and with binary classification using diagonal and convolutional linear networks, showing all of these models exhibit incremental learning.

NeurIPS Conference 2019 Conference Paper

Generalization Bounds for Neural Networks via Approximate Description Length

• Amit Daniely
• Elad Granot

We investigate the sample complexity of networks with bounds on the magnitude of its weights. In particular, we consider the class \[ \cn = \left\{W_t\circ\rho\circ W_{t-1}\circ\rho\ldots\circ \rho\circ W_{1}: W_1, \ldots, W_{t-1}\in M_{d\times d}, W_t\in M_{1, d} \right\} \] where the spectral norm of each $W_i$ is bounded by $O(1)$, the Frobenius norm is bounded by $R$, and $\rho$ is the sigmoid function $\frac{e^x}{1 + e^x}$ or the smoothened ReLU function $ \ln\left(1 + e^x\right)$. We show that for any depth $t$, if the inputs are in $[-1, 1]^d$, the sample complexity of $\cn$ is $\tilde O\left(\frac{dR^2}{\epsilon^2}\right)$. This bound is optimal up to log-factors, and substantially improves over the previous state of the art of $\tilde O\left(\frac{d^2R^2}{\epsilon^2}\right)$, that was established in a recent line of work. We furthermore show that this bound remains valid if instead of considering the magnitude of the $W_i$'s, we consider the magnitude of $W_i - W_i^0$, where $W_i^0$ are some reference matrices, with spectral norm of $O(1)$. By taking the $W_i^0$ to be the matrices in the onset of the training process, we get sample complexity bounds that are sub-linear in the number of parameters, in many {\em typical} regimes of parameters. To establish our results we develop a new technique to analyze the sample complexity of families $\ch$ of predictors. We start by defining a new notion of a randomized approximate description of functions $f: \cx\to\reals^d$. We then show that if there is a way to approximately describe functions in a class $\ch$ using $d$ bits, then $\frac{d}{\epsilon^2}$ examples suffices to guarantee uniform convergence. Namely, that the empirical loss of all the functions in the class is $\epsilon$-close to the true loss. Finally, we develop a set of tools for calculating the approximate description length of classes of functions that can be presented as a composition of linear function classes and non-linear functions.

NeurIPS Conference 2019 Conference Paper

Locally Private Learning without Interaction Requires Separation

• Amit Daniely
• Vitaly Feldman

We consider learning under the constraint of local differential privacy (LDP). For many learning problems known efficient algorithms in this model require many rounds of communication between the server and the clients holding the data points. Yet multi-round protocols are prohibitively slow in practice due to network latency and, as a result, currently deployed large-scale systems are limited to a single round. Despite significant research interest, very little is known about which learning problems can be solved by such non-interactive systems. The only lower bound we are aware of is for PAC learning an artificial class of functions with respect to a uniform distribution (Kasiviswanathan et al. , 2008). We show that the margin complexity of a class of Boolean functions is a lower bound on the complexity of any non-interactive LDP algorithm for distribution-independent PAC learning of the class. In particular, the classes of linear separators and decision lists require exponential number of samples to learn non-interactively even though they can be learned in polynomial time by an interactive LDP algorithm. This gives the first example of a natural problem that is significantly harder to solve without interaction and also resolves an open problem of Kasiviswanathan et al. ~(2008). We complement this lower bound with a new efficient learning algorithm whose complexity is polynomial in the margin complexity of the class. Our algorithm is non-interactive on labeled samples but still needs interactive access to unlabeled samples. All of our results also apply to the statistical query model and any model in which the number of bits communicated about each data point is constrained.

NeurIPS Conference 2017 Conference Paper

SGD Learns the Conjugate Kernel Class of the Network

• Amit Daniely

We show that the standard stochastic gradient decent (SGD) algorithm is guaranteed to learn, in polynomial time, a function that is competitive with the best function in the conjugate kernel space of the network, as defined in Daniely, Frostig and Singer. The result holds for log-depth networks from a rich family of architectures. To the best of our knowledge, it is the first polynomial-time guarantee for the standard neural network learning algorithm for networks of depth more that two. As corollaries, it follows that for neural networks of any depth between 2 and log(n), SGD is guaranteed to learn, in polynomial time, constant degree polynomials with polynomially bounded coefficients. Likewise, it follows that SGD on large enough networks can learn any continuous function (not in polynomial time), complementing classical expressivity results.

STOC Conference 2016 Conference Paper

Complexity theoretic limitations on learning halfspaces

• Amit Daniely

We study the problem of agnostically learning halfspaces which is defined by a fixed but unknown distribution D on Q^n X {-1,1}. We define Err_H(D) as the least error of a halfspace classifier for D. A learner who can access D has to return a hypothesis whose error is small compared to Err_H(D). Using the recently developed method of Daniely, Linial and Shalev-Shwartz we prove hardness of learning results assuming that random K-XOR formulas are hard to (strongly) refute. We show that no efficient learning algorithm has non-trivial worst-case performance even under the guarantees that Err_H(D) 0, and that D is supported in the Boolean cube. Namely, even under these favorable conditions, and for every c>0, it is hard to return a hypothesis with error 0. These results substantially improve on previously known results, that only show hardness of exact learning.

NeurIPS Conference 2016 Conference Paper

Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity

• Amit Daniely
• Roy Frostig
• Yoram Singer

We develop a general duality between neural networks and compositional kernel Hilbert spaces. We introduce the notion of a computation skeleton, an acyclic graph that succinctly describes both a family of neural networks and a kernel space. Random neural networks are generated from a skeleton through node replication followed by sampling from a normal distribution to assign weights. The kernel space consists of functions that arise by compositions, averaging, and non-linear transformations governed by the skeleton's graph topology and activation functions. We prove that random networks induce representations which approximate the kernel space. In particular, it follows that random weight initialization often yields a favorable starting point for optimization despite the worst-case intractability of training neural networks.

STOC Conference 2015 Conference Paper

Inapproximability of Truthful Mechanisms via Generalizations of the VC Dimension

• Amit Daniely
• Michael Schapira
• Gal Shahaf

JMLR Journal 2015 Journal Article

Multiclass Learnability and the ERM Principle

• Amit Daniely
• Sivan Sabato
• Shai Ben-David
• Shai Shalev-Shwartz

We study the sample complexity of multiclass prediction in several learning settings. For the PAC setting our analysis reveals a surprising phenomenon: In sharp contrast to binary classification, we show that there exist multiclass hypothesis classes for which some Empirical Risk Minimizers (ERM learners) have lower sample complexity than others. Furthermore, there are classes that are learnable by some ERM learners, while other ERM learners will fail to learn them. We propose a principle for designing good ERM learners, and use this principle to prove tight bounds on the sample complexity of learning symmetric multiclass hypothesis classes---classes that are invariant under permutations of label names. We further provide a characterization of mistake and regret bounds for multiclass learning in the online setting and the bandit setting, using new generalizations of Littlestone's dimension. [abs] [ pdf ][ bib ] &copy JMLR 2015. ( edit, beta )

ICML Conference 2015 Conference Paper

Strongly Adaptive Online Learning

• Amit Daniely
• Alon Gonen
• Shai Shalev-Shwartz

Strongly adaptive algorithms are algorithms whose performance on every time interval is close to optimal. We present a reduction that can transform standard low-regret algorithms to strongly adaptive. As a consequence, we derive simple, yet efficient, strongly adaptive algorithms for a handful of problems.

STOC Conference 2014 Conference Paper

From average case complexity to improper learning complexity

• Amit Daniely
• Nati Linial
• Shai Shalev-Shwartz

The basic problem in the PAC model of computational learning theory is to determine which hypothesis classes are effficiently learnable. There is presently a dearth of results showing hardness of learning problems. Moreover, the existing lower bounds fall short of the best known algorithms.

NeurIPS Conference 2013 Conference Paper

More data speeds up training time in learning halfspaces over sparse vectors

• Amit Daniely
• Nati Linial
• Shai Shalev-Shwartz

The increased availability of data in recent years led several authors to ask whether it is possible to use data as a {\em computational} resource. That is, if more data is available, beyond the sample complexity limit, is it possible to use the extra examples to speed up the computation time required to perform the learning task? We give the first positive answer to this question for a {\em natural supervised learning problem} --- we consider agnostic PAC learning of halfspaces over $3$-sparse vectors in $\{-1, 1, 0\}^n$. This class is inefficiently learnable using $O\left(n/\epsilon^2\right)$ examples. Our main contribution is a novel, non-cryptographic, methodology for establishing computational-statistical gaps, which allows us to show that, under a widely believed assumption that refuting random $\mathrm{3CNF}$ formulas is hard, efficiently learning this class using $O\left(n/\epsilon^2\right)$ examples is impossible. We further show that under stronger hardness assumptions, even $O\left(n^{1. 499}/\epsilon^2\right)$ examples do not suffice. On the other hand, we show a new algorithm that learns this class efficiently using $\tilde{\Omega}\left(n^2/\epsilon^2\right)$ examples. This formally establishes the tradeoff between sample and computational complexity for a natural supervised learning problem.

NeurIPS Conference 2012 Conference Paper

Multiclass Learning Approaches: A Theoretical Comparison with Implications

• Amit Daniely
• Sivan Sabato
• Shai Shwartz

We theoretically analyze and compare the following five popular multiclass classification methods: One vs. All, All Pairs, Tree-based classifiers, Error Correcting Output Codes (ECOC) with randomly generated code matrices, and Multiclass SVM. In the first four methods, the classification is based on a reduction to binary classification. We consider the case where the binary classifier comes from a class of VC dimension $d$, and in particular from the class of halfspaces over $\reals^d$. We analyze both the estimation error and the approximation error of these methods. Our analysis reveals interesting conclusions of practical relevance, regarding the success of the different approaches under various conditions. Our proof technique employs tools from VC theory to analyze the \emph{approximation error} of hypothesis classes. This is in sharp contrast to most, if not all, previous uses of VC theory, which only deal with estimation error.