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Amit Daniely

Publications and source records attributed to Amit Daniely.

At least 37 records · Page 2Linked to original sources
arXiv2020

Hardness of Learning Neural Networks with Natural Weights

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.

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arXiv2020

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.

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arXiv2020

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 introduce a novel metric of a decision tree algorithm's performance, called mean iteration statistical consistency (MIC), which measures optimality of trees generated by ID3. As opposed to previous metrics, MIC can differentiate between different decision tree algorithms and compare their performance. We provide theoretical and empirical evidence that the TopDown variant of ID3, introduced by Kearns and Mansour (1996), has near-optimal MIC in various settings for learning read-once DNFs under product distributions. In contrast, another widely used variant of ID3 has MIC which is not near-optimal. We show that the MIC analysis predicts well the performance of these algorithms in practice. Our results present a novel view of decision tree algorithms which may lead to better and more practical guarantees for these algorithms.

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arXiv2020

On the Complexity of Minimizing Convex Finite Sums Without Using the Indices of the Individual Functions

Recent advances in randomized incremental methods for minimizing $L$-smooth $μ$-strongly convex finite sums have culminated in tight complexity of $\tilde{O}((n+\sqrt{n L/μ})\log(1/ε))$ and $O(n+\sqrt{nL/ε})$, where $μ>0$ and $μ=0$, respectively, and $n$ denotes the number of individual functions. Unlike incremental methods, stochastic methods for finite sums do not rely on an explicit knowledge of which individual function is being addressed at each iteration, and as such, must perform at least $Ω(n^2)$ iterations to obtain $O(1/n^2)$-optimal solutions. In this work, we exploit the finite noise structure of finite sums to derive a matching $O(n^2)$-upper bound under the global oracle model, showing that this lower bound is indeed tight. Following a similar approach, we propose a novel adaptation of SVRG which is both \emph{compatible with stochastic oracles}, and achieves complexity bounds of $\tilde{O}((n^2+n\sqrt{L/μ})\log(1/ε))$ and $O(n\sqrt{L/ε})$, for $μ>0$ and $μ=0$, respectively. Our bounds hold w.h.p. and match in part existing lower bounds of $\tildeΩ(n^2+\sqrt{nL/μ}\log(1/ε))$ and $\tildeΩ(n^2+\sqrt{nL/ε})$, for $μ>0$ and $μ=0$, respectively.

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arXiv2019

The Implicit Bias of Depth: How Incremental Learning Drives Generalization

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.

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arXiv2019

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

Many results in recent years established polynomial time learnability of various models via neural networks algorithms. However, unless the model is linear separable, or the activation is a polynomial, 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 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 $\mathbb{S}^{d-1}$.

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arXiv2019

Locally Private Learning without Interaction Requires Separation

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. 2011). 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. (2011). 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.

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arXiv2019

Generalization Bounds for Neural Networks via Approximate Description Length

We investigate the sample complexity of networks with bounds on the magnitude of its weights. In particular, we consider the class \[ H=\left\{W_t\circρ\circ \ldots\circρ\circ W_{1} :W_1,\ldots,W_{t-1}\in M_{d, d}, W_t\in M_{1,d}\right\} \] where the spectral norm of each $W_i$ is bounded by $O(1)$, the Frobenius norm is bounded by $R$, and $ρ$ is the sigmoid function $\frac{e^x}{1+e^x}$ or the smoothened ReLU function $ \ln (1+e^x)$. We show that for any depth $t$, if the inputs are in $[-1,1]^d$, the sample complexity of $H$ is $\tilde O\left(\frac{dR^2}{ε^2}\right)$. This bound is optimal up to log-factors, and substantially improves over the previous state of the art of $\tilde O\left(\frac{d^2R^2}{ε^2}\right)$. 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 at the onset of the training process, we get sample complexity bounds that are sub-linear in the number of parameters, in many typical regimes of parameters. To establish our results we develop a new technique to analyze the sample complexity of families $H$ of predictors. We start by defining a new notion of a randomized approximate description of functions $f:X\to\mathbb{R}^d$. We then show that if there is a way to approximately describe functions in a class $H$ using $d$ bits, then $d/ε^2$ examples suffices to guarantee uniform convergence. Namely, that the empirical loss of all the functions in the class is $ε$-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.

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arXiv2019

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.

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arXiv2019

Learning Rules-First Classifiers

Complex classifiers may exhibit "embarassing" failures in cases where humans can easily provide a justified classification. Avoiding such failures is obviously of key importance. In this work, we focus on one such setting, where a label is perfectly predictable if the input contains certain features, or rules, and otherwise it is predictable by a linear classifier. We define a hypothesis class that captures this notion and determine its sample complexity. We also give evidence that efficient algorithms cannot achieve this sample complexity. We then derive a simple and efficient algorithm and show that its sample complexity is close to optimal, among efficient algorithms. Experiments on synthetic and sentiment analysis data demonstrate the efficacy of the method, both in terms of accuracy and interpretability.

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arXiv2019

Competitive ratio versus regret minimization: achieving the best of both worlds

We consider online algorithms under both the competitive ratio criteria and the regret minimization one. Our main goal is to build a unified methodology that would be able to guarantee both criteria simultaneously. For a general class of online algorithms, namely any Metrical Task System (MTS), we show that one can simultaneously guarantee the best known competitive ratio and a natural regret bound. For the paging problem we further show an efficient online algorithm (polynomial in the number of pages) with this guarantee. To this end, we extend an existing regret minimization algorithm (specifically, Kapralov and Panigrahy) to handle movement cost (the cost of switching between states of the online system). We then show how to use the extended regret minimization algorithm to combine multiple online algorithms. Our end result is an online algorithm that can combine a "base" online algorithm, having a guaranteed competitive ratio, with a range of online algorithms that guarantee a small regret over any interval of time. The combined algorithm guarantees both that the competitive ratio matches that of the base algorithm and a low regret over any time interval. As a by product, we obtain an expert algorithm with close to optimal regret bound on every time interval, even in the presence of switching costs. This result is of independent interest.

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arXiv2017

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

We develop a general duality between neural networks and compositional kernels, striving towards a better understanding of deep learning. We show that initial representations generated by common random initializations are sufficiently rich to express all functions in the dual kernel space. Hence, though the training objective is hard to optimize in the worst case, the initial weights form a good starting point for optimization. Our dual view also reveals a pragmatic and aesthetic perspective of neural networks and underscores their expressive power.

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arXiv2017

SGD Learns the Conjugate Kernel Class of the Network

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.

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arXiv2017

Random Features for Compositional Kernels

We describe and analyze a simple random feature scheme (RFS) from prescribed compositional kernels. The compositional kernels we use are inspired by the structure of convolutional neural networks and kernels. The resulting scheme yields sparse and efficiently computable features. Each random feature can be represented as an algebraic expression over a small number of (random) paths in a composition tree. Thus, compositional random features can be stored compactly. The discrete nature of the generation process enables de-duplication of repeated features, further compacting the representation and increasing the diversity of the embeddings. Our approach complements and can be combined with previous random feature schemes.

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arXiv2017

Depth Separation for Neural Networks

Let $f:\mathbb{S}^{d-1}\times \mathbb{S}^{d-1}\to\mathbb{S}$ be a function of the form $f(\mathbf{x},\mathbf{x}') = g(\langle\mathbf{x},\mathbf{x}'\rangle)$ for $g:[-1,1]\to \mathbb{R}$. We give a simple proof that shows that poly-size depth two neural networks with (exponentially) bounded weights cannot approximate $f$ whenever $g$ cannot be approximated by a low degree polynomial. Moreover, for many $g$'s, such as $g(x)=\sin(πd^3x)$, the number of neurons must be $2^{Ω\left(d\log(d)\right)}$. Furthermore, the result holds w.r.t.\ the uniform distribution on $\mathbb{S}^{d-1}\times \mathbb{S}^{d-1}$. As many functions of the above form can be well approximated by poly-size depth three networks with poly-bounded weights, this establishes a separation between depth two and depth three networks w.r.t.\ the uniform distribution on $\mathbb{S}^{d-1}\times \mathbb{S}^{d-1}$.

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arXiv2016

Behavior-Based Machine-Learning: A Hybrid Approach for Predicting Human Decision Making

A large body of work in behavioral fields attempts to develop models that describe the way people, as opposed to rational agents, make decisions. A recent Choice Prediction Competition (2015) challenged researchers to suggest a model that captures 14 classic choice biases and can predict human decisions under risk and ambiguity. The competition focused on simple decision problems, in which human subjects were asked to repeatedly choose between two gamble options. In this paper we present our approach for predicting human decision behavior: we suggest to use machine learning algorithms with features that are based on well-established behavioral theories. The basic idea is that these psychological features are essential for the representation of the data and are important for the success of the learning process. We implement a vanilla model in which we train SVM models using behavioral features that rely on the psychological properties underlying the competition baseline model. We show that this basic model captures the 14 choice biases and outperforms all the other learning-based models in the competition. The preliminary results suggest that such hybrid models can significantly improve the prediction of human decision making, and are a promising direction for future research.

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arXiv2016

Sketching and Neural Networks

High-dimensional sparse data present computational and statistical challenges for supervised learning. We propose compact linear sketches for reducing the dimensionality of the input, followed by a single layer neural network. We show that any sparse polynomial function can be computed, on nearly all sparse binary vectors, by a single layer neural network that takes a compact sketch of the vector as input. Consequently, when a set of sparse binary vectors is approximately separable using a sparse polynomial, there exists a single-layer neural network that takes a short sketch as input and correctly classifies nearly all the points. Previous work has proposed using sketches to reduce dimensionality while preserving the hypothesis class. However, the sketch size has an exponential dependence on the degree in the case of polynomial classifiers. In stark contrast, our approach of using improper learning, using a larger hypothesis class allows the sketch size to have a logarithmic dependence on the degree. Even in the linear case, our approach allows us to improve on the pesky $O({1}/{γ^2})$ dependence of random projections, on the margin $γ$. We empirically show that our approach leads to more compact neural networks than related methods such as feature hashing at equal or better performance.

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