We ask whether distribution-independent SQ learning implies low dimension complexity, and whether anything learnable with (S)GD on a (benign) neural network under any input distribution is also learnable with a linear model.

We tightly characterize the VC dimension of depth-$L$ Transformers with a total of $W$ parameters, mapping an input sequence of length $T$ to a single output, establishing an upper bound of $O(L W \log (T W))$ and a nearly matching lower bound of $\Omega(L W \log (T W / L))$. We further tightly characterize the sample complexity of chain-of-thought learning using such a Transformer, showing teacher forcing (i.e. selecting a predictor consistent with the entire chain-of-thought on training data) learns with sample complexity $O\left(L W \log \left(\left(T+T^{\prime}\right) W\right)\right)$ and that any learning rule that uses chain-of-thought data requires at least $\Omega\left(L W \log \left(\left(T+T^{\prime}\right) W / L\right)\right)$ examples, where $T$ is the input length and $T^{\prime}$ is the number of autoregressive steps.

For a given base class of sequence-to-next-token generators, we consider learning prompt-to-answer mappings obtained by iterating a fixed, time-invariant generator for multiple steps, thus generating a chain-of-thought, and then taking the final token as the answer. We formalize the learning problems both when the chain-of-thought is observed and when training only on prompt-answer pairs, with the chain-of-thought latent. We analyze the sample and computational complexity both in terms of general properties of the base class (e.g. its VC dimension) and for specific base classes such as linear thresholds. We present a simple base class that allows for universal representability and computationally tractable chain-of-thought learning. Central to our development is that time invariance allows for sample complexity that is independent of the length of the chain-of-thought. Attention arises naturally in our construction.

We provide a complete characterization of the entire regularization curve of a modified two-part-code Minimum Description Length (MDL) learning rule for binary classification, based on an arbitrary prior or description language. Gr{ü}nwald and Langford (2004) previously established the lack of asymptotic consistency, from an agnostic PAC (frequentist worst case) perspective, of the MDL rule with a penalty parameter of $\lambda=1$, suggesting that it underegularizes. Driven by interest in understanding how benign or catastrophic under-regularization and overfitting might be, we obtain a precise quantitative description of the worst case limiting error as a function of the regularization parameter $\lambda$ and noise level (or approximation error), significantly tightening the analysis of Gr{ü}nwald and Langford for $\lambda=1$ and extending it to all other choices of $\lambda$.

Weak-to-Strong Generalization (Burns et al.,2024) is the phenomenon whereby a strong student, say GPT-4, learns a task from a weak teacher, say GPT-2, and ends up significantly outperforming the teacher. We show that this phenomenon does not require a complex and pretrained learner like GPT-4, can arise even in simple non-pretrained models, simply due to the size advantage of the student. But, we also show that there are inherint limits to the extent of such weak to strong generalization. We consider students and teachers that are random feature models, described by two-layer networks with a random and fixed bottom layer and trained top layer. A ‘weak’ teacher, with a small number of units (i.e. random features), is trained on the population, and a ‘strong’ student, with a much larger number of units (i.e. random features), is trained only on labels generated by the weak teacher. We demonstrate, prove, and understand how the student can outperform the teacher, even though trained only on data labeled by the teacher. We also explain how such weak-to-strong generalization is enabled by early stopping. We then show the quantitative limits of weak-to-strong generalization in this model, and in fact in a much broader class of models, for arbitrary teacher and student feature spaces and a broad class of learning rules, including when the student features are pre-trained or otherwise more informative. In particular, we show that in such models the student’s error can only approach zero if the teacher’s error approaches zero, and a strong student cannot “boost” a slightly-better-then-chance teacher to obtain a small error.

While state-of-the-art LLMs have demonstrated great promise of using long Chains-of-Thought (CoT) to boost reasoning, scaling it up to more challenging problems is fundamentally limited by suboptimal memory usage — intermediate computations accumulate indefinitely in context even no longer needed for future thoughts. We introduce PENCIL, which incorporates a novel reduction mechanism into the autoregressive generation process that recursively clean up intermediate thoughts based on patterns learned from training. By alternately generating and erasing, PENCIL can think deeper to solve harder problems using shorter context and less computes. Empirically, for example, we demonstrate PENCIL with a small 25M-parameter transformer and 2048 context length solves Einstein's puzzle — a task that challenges much larger models like GPT-4. Theoretically, we prove PENCIL can perform universal efficient computation by simulating any Turing machines with optimal time and space complexity, and thus can solve arbitrary computable tasks that are otherwise intractable for vanilla CoT.

We analyze the generalization gap (gap between the training and test errors) when training a potentially over-parametrized model using a Markovian stochastic training algorithm, initialized from some distribution $\theta_0 \sim p_0$. We focus on Langevin dynamics with a positive temperature $\beta^{-1}$, i.e. gradient descent on a training loss $L$ with infinitesimal step size, perturbed with $\beta^{-1}$-variances Gaussian noise, and lightly regularized or bounded. There, we bound the generalization gap, *at any time during training*, by $\sqrt{(\beta\mathbb{E} L (\theta_0) + \log(1/\delta))/N}$ with probability $1-\delta$ over the dataset, where $N$ is the sample size, and $\mathbb{E} L(\theta_0)=O(1)$ with standard initialization scaling. In contrast to previous guarantees, we have no dependence on either training time or reliance on mixing, nor a dependence on dimensionality, gradient norms, or any other properties of the loss or model. This guarantee follows from a general analysis of any Markov process-based training that has a Gibbs-style stationary distribution. The proof is surprisingly simple, once we observe that the marginal distribution divergence from initialization remains bounded, as implied by a generalized second law of thermodynamics.

We study the problem of learning single-index models, where the label $y \in \mathbb{R}$ depends on the input $\boldsymbol{x} \in \mathbb{R}^d$ only through an unknown one-dimensional projection $\langle \boldsymbol{w_*}, \boldsymbol{x} \rangle$. Prior work has shown that under Gaussian inputs, the statistical and computational complexity of recovering $\boldsymbol{w}_*$ is governed by the Hermite expansion of the link function. In this paper, we propose a new perspective: we argue that *spherical harmonics*---rather than *Hermite polynomials*---provide the natural basis for this problem, as they capture its intrinsic \textit{rotational symmetry}. Building on this insight, we characterize the complexity of learning single-index models under arbitrary spherically symmetric input distributions. We introduce two families of estimators---based on tensor-unfolding and online SGD---that respectively achieve either optimal sample complexity or optimal runtime, and argue that estimators achieving both may not exist in general. When specialized to Gaussian inputs, our theory not only recovers and clarifies existing results but also reveals new phenomena that had previously been overlooked.

Metalearning and multitask learning are two frameworks for solving a group of related learning tasks more efficiently than we could hope to solve each of the individual tasks on their own. In multitask learning, we are given a fixed set of related learning tasks and need to output one accurate model per task, whereas in metalearning we are given tasks that are drawn i.i.d. from a metadistribution and need to output some common information that can be easily specialized to new, previously unseen tasks from the metadistribution. In this work, we consider a binary classification setting where tasks are related by a shared representation, that is, every task $P$ of interest can be solved by a classifier of the form $f_{P} \circ h$ where $h \in \mathcal{H}$ is a map from features to some representation space that is shared across tasks, and $f_{P} \in \mathcal{F}$ is a task-specific classifier from the representation space to labels. The main question we ask in this work is how much data do we need to metalearn a good representation? Here, the amount of data is measured in terms of both the number of tasks $t$ that we need to see and the number of samples $n$ per task. We focus on the regime where the number of samples per task is extremely small. Our main result shows that, in a distribution-free setting where the feature vectors are in $\mathbb{R}^d$, the representation is a linear map from $\mathbb{R}^d \to \mathbb{R}^k$, and the task-specific classifiers are halfspaces in $\mathbb{R}^k$, we can metalearn a representation with error $\varepsilon$ using just $n = k+2$ samples per task, and $d \cdot (1/\varepsilon)^{O(k)}$ tasks. Learning with so few samples per task is remarkable because metalearning would be impossible with $k+1$ samples per task, and because we cannot even hope to learn an accurate task-specific classifier with just $k+2$ samples per task. To obtain this result, we develop a sample-and-task-complexity theory for distribution-free metalearning and multitask learning, which identifies what properties of $\mathcal{F}$ and $\mathcal{H}$ make metalearning possible with few samples per task. Our theory also yields a simple characterization of distribution-free multitask learning. Finally, we give sample-efficient reductions between metalearning and multitask learning, which, when combined with our characterization of multitask learning, give a characterization of metalearning in certain parameter regimes.

We study depth separation in infinite-width neural networks, where complexity is controlled by the overall squared $\ell_2$-norm of the weights (sum of squares of all weights in the network). Whereas previous depth separation results focused on separation in terms of width, such results do not give insight into whether depth determines if it is possible to learn a network that generalizes well even when the network width is unbounded. Here, we study separation in terms of the sample complexity required for learnability. Specifically, we show that there are functions that are learnable with sample complexity polynomial in the input dimension by norm-controlled depth-3 ReLU networks, yet are not learnable with sub-exponential sample complexity by norm-controlled depth-2 ReLU networks (with any value for the norm). We also show that a similar statement in the reverse direction is not possible: any function learnable with polynomial sample complexity by a norm-controlled depth-2 ReLU network with infinite width is also learnable with polynomial sample complexity by a norm-controlled depth-3 ReLU network.

Local SGD is a popular optimization method in distributed learning, often outperforming mini-batch SGD. Despite this practical success, proving the efficiency of local SGD has been difficult, creating a significant gap between theory and practice. We provide new lower bounds for local SGD under existing first-order data heterogeneity assumptions, showing these assumptions can not capture local SGD’s effectiveness. We also demonstrate the min-max optimality of accelerated mini-batch SGD under these assumptions. Our findings emphasize the need for improved modeling of data heterogeneity. Under higher-order assumptions, we provide new upper bounds that verify the dominance of local SGD over mini-batch SGD when data heterogeneity is low.

Understanding how overparameterized neural networks generalize despite perfect interpolation of noisy training data is a fundamental question. Mallinar et. al. (2022) noted that neural networks seem to often exhibit ``tempered overfitting'', wherein the population risk does not converge to the Bayes optimal error, but neither does it approach infinity, yielding non-trivial generalization. However, this has not been studied rigorously. We provide the first rigorous analysis of the overfiting behaviour of regression with minimum norm ($\ell_2$ of weights), focusing on univariate two-layer ReLU networks. We show overfitting is tempered (with high probability) when measured with respect to the $L_1$ loss, but also show that the situation is more complex than suggested by Mallinar et. al., and overfitting is catastrophic with respect to the $L_2$ loss, or when taking an expectation over the training set.

We study the cost of overfitting in noisy kernel ridge regression (KRR), which we define as the ratio between the test error of the interpolating ridgeless model and the test error of the optimally-tuned model. We take an ``agnostic'' view in the following sense: we consider the cost as a function of sample size for any target function, even if the sample size is not large enough for consistency or the target is outside the RKHS. We analyze the cost of overfitting under a Gaussian universality ansatz using recently derived (non-rigorous) risk estimates in terms of the task eigenstructure. Our analysis provides a more refined characterization of benign, tempered and catastrophic overfitting (cf. Mallinar et al. 2022).

A main theoretical puzzle is why over-parameterized Neural Networks (NNs) generalize well when trained to zero loss (i.e., so they interpolate the data). Usually, the NN is trained with Stochastic Gradient Descent (SGD) or one of its variants. However, recent empirical work examined the generalization of a random NN that interpolates the data: the NN was sampled from a seemingly uniform prior over the parameters, conditioned on that the NN perfectly classifying the training set. Interestingly, such a NN sample typically generalized as well as SGD-trained NNs. We prove that such a random NN interpolator typically generalizes well if there exists an underlying narrow “teacher NN" that agrees with the labels. Specifically, we show that such a ‘flat’ prior over the NN parametrization induces a rich prior over the NN functions, due to the redundancy in the NN structure. In particular, this creates a bias towards simpler functions, which require less relevant parameters to represent — enabling learning with a sample complexity approximately proportional to the complexity of the teacher (roughly, the number of non-redundant parameters), rather than the student’s.

We study the implicit bias of optimization in robust empirical risk minimization (robust ERM) and its connection with robust generalization. In classification settings under adversarial perturbations with linear models, we study what type of regularization should ideally be applied for a given perturbation set to improve (robust) generalization. We then show that the implicit bias of optimization in robust ERM can significantly affect the robustness of the model and identify two ways this can happen; either through the optimization algorithm or the architecture. We verify our predictions in simulations with synthetic data and experimentally study the importance of implicit bias in robust ERM with deep neural networks.

We study the overfitting behavior of fully connected deep Neural Networks (NNs) with binary weights fitted to perfectly classify a noisy training set. We consider interpolation using both the smallest NN (having the minimal number of weights) and a random interpolating NN. For both learning rules, we prove overfitting is tempered. Our analysis rests on a new bound on the size of a threshold circuit consistent with a partial function. To the best of our knowledge, ours are the first theoretical results on benign or tempered overfitting that: (1) apply to deep NNs, and (2) do not require a very high or very low input dimension.

The goal of this paper is to investigate the complexity of gradient algorithms when learning sparse functions (juntas). We introduce a type of Statistical Queries ($\mathsf{SQ}$), which we call Differentiable Learning Queries ($\mathsf{DLQ}$), to model gradient queries on a specified loss with respect to an arbitrary model. We provide a tight characterization of the query complexity of $\mathsf{DLQ}$ for learning the support of a sparse function over generic product distributions. This complexity crucially depends on the loss function. For the squared loss, $\mathsf{DLQ}$ matches the complexity of Correlation Statistical Queries $(\mathsf{CSQ})$—potentially much worse than $\mathsf{SQ}$. But for other simple loss functions, including the $\ell_1$ loss, $\mathsf{DLQ}$ always achieves the same complexity as $\mathsf{SQ}$. We also provide evidence that $\mathsf{DLQ}$ can indeed capture learning with (stochastic) gradient descent by showing it correctly describes the complexity of learning with a two-layer neural network in the mean field regime and linear scaling.

We consider the overfitting behavior of minimum norm interpolating solutions of Gaussian kernel ridge regression (i.e. kernel ridgeless regression), when the bandwidth or input dimension varies with the sample size. For fixed dimensions, we show that even with varying or tuned bandwidth, the ridgeless solution is never consistent and, at least with large enough noise, always worse than the null predictor. For increasing dimension, we give a generic characterization of the overfitting behavior for any scaling of the dimension with sample size. We use this to provide the first example of benign overfitting using the Gaussian kernel with sub-polynomial scaling dimension. All our results are under the Gaussian universality ansatz and the (non-rigorous) risk predictions in terms of the kernel eigenstructure.

Linear classifiers and leaky ReLU networks trained by gradient flow on the logistic loss have an implicit bias towards solutions which satisfy the Karush–Kuhn–Tucker (KKT) conditions for margin maximization. In this work we establish a number of settings where the satisfaction of these KKT conditions implies benign overfitting in linear classifiers and in two-layer leaky ReLU networks: the estimators interpolate noisy training data and simultaneously generalize well to test data. The settings include variants of the noisy class-conditional Gaussians considered in previous work as well as new distributional settings where benign overfitting has not been previously observed. The key ingredient to our proof is the observation that when the training data is nearly-orthogonal, both linear classifiers and leaky ReLU networks satisfying the KKT conditions for their respective margin maximization problems behave like a weighted average of the training examples.

We prove that the Minimum Program Length learning rule exhibits tempered overfitting. We obtain tempered agnostic finite sample learning guarantees and characterize the asymptotic behavior in the presence of random label noise.

We analyze continual learning on a sequence of separable linear classification tasks with binary labels. We show theoretically that learning with weak regularization reduces to solving a sequential max-margin problem, corresponding to a special case of the Projection Onto Convex Sets (POCS) framework. We then develop upper bounds on the forgetting and other quantities of interest under various settings with recurring tasks, including cyclic and random orderings of tasks. We discuss several practical implications to popular training practices like regularization scheduling and weighting. We point out several theoretical differences between our continual classification setting and a recently studied continual regression setting.

We study the problems of *distributed online and bandit convex optimization* against an adaptive adversary. We aim to minimize the average regret on $M$ machines working in parallel over $T$ rounds with $R$ intermittent communications. Assuming the underlying cost functions are convex and can be generated adaptively, our results show that *collaboration is not beneficial when the machines have access to the first-order gradient information at the queried points*. This is in contrast to the case for stochastic functions, where each machine samples the cost functions from a fixed distribution. Furthermore, we delve into the more challenging setting of *federated online optimization with bandit (zeroth-order) feedback*, where the machines can only access values of the cost functions at the queried points. The key finding here is *identifying the high-dimensional regime where collaboration is beneficial and may even lead to a linear speedup in the number of machines*. We further illustrate our findings through federated adversarial linear bandits by developing novel distributed single and two-point feedback algorithms. Our work is the first attempt towards a systematic understanding of federated online optimization with limited feedback, and it attains tight regret bounds in the intermittent communication setting for both first and zeroth-order feedback. Our results thus bridge the gap between stochastic and adaptive settings in federated online optimization.

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.

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.

In this work, we study the implications of the implicit bias of gradient flow on generalization and adversarial robustness in ReLU networks. We focus on a setting where the data consists of clusters and the correlations between cluster means are small, and show that in two-layer ReLU networks gradient flow is biased towards solutions that generalize well, but are vulnerable to adversarial examples. Our results hold even in cases where the network is highly overparameterized. Despite the potential for harmful overfitting in such settings, we prove that the implicit bias of gradient flow prevents it. However, the implicit bias also leads to non-robust solutions (susceptible to small adversarial $\ell_2$-perturbations), even though robust networks that fit the data exist.

We study the problem of agnostic PAC reinforcement learning (RL): given a policy class $\Pi$, how many rounds of interaction with an unknown MDP (with a potentially large state and action space) are required to learn an $\epsilon$-suboptimal policy with respect to \(\Pi\)? Towards that end, we introduce a new complexity measure, called the \emph{spanning capacity}, that depends solely on the set \(\Pi\) and is independent of the MDP dynamics. With a generative model, we show that the spanning capacity characterizes PAC learnability for every policy class $\Pi$. However, for online RL, the situation is more subtle. We show there exists a policy class $\Pi$ with a bounded spanning capacity that requires a superpolynomial number of samples to learn. This reveals a surprising separation for agnostic learnability between generative access and online access models (as well as between deterministic/stochastic MDPs under online access). On the positive side, we identify an additional \emph{sunflower} structure which in conjunction with bounded spanning capacity enables statistically efficient online RL via a new algorithm called POPLER, which takes inspiration from classical importance sampling methods as well as recent developments for reachable-state identification and policy evaluation in reward-free exploration.

We establish generic uniform convergence guarantees for Gaussian data in terms of the Radamacher complexity of the hypothesis class and the Lipschitz constant of the square root of the scalar loss function. We show how these guarantees substantially generalize previous results based on smoothness (Lipschitz constant of the derivative), and allow us to handle the broader class of square-root-Lipschtz losses, which includes also non-smooth loss functions appropriate for studying phase retrieval and ReLU regression, as well as rederive and better understand “optimistic rate” and interpolation learning guarantees.

We study the problem of adversarially robust learning in the transductive setting. For classes H of bounded VC dimension, we propose a simple transductive learner that when presented with a set of labeled training examples and a set of unlabeled test examples (both sets possibly adversarially perturbed), it correctly labels the test examples with a robust error rate that is linear in the VC dimension and is adaptive to the complexity of the perturbation set. This result provides an exponential improvement in dependence on VC dimension over the best known upper bound on the robust error in the inductive setting, at the expense of competing with a more restrictive notion of optimal robust error.

To better understand catastrophic forgetting, we study fitting an overparameterized linear model to a sequence of tasks with different input distributions. We analyze how much the model forgets the true labels of earlier tasks after training on subsequent tasks, obtaining exact expressions and bounds. We establish connections between continual learning in the linear setting and two other research areas – alternating projections and the Kaczmarz method. In specific settings, we highlight differences between forgetting and convergence to the offline solution as studied in those areas. In particular, when $T$ tasks in $d$ dimensions are presented cyclically for $k$ iterations, we prove an upper bound of $T^2\min\{1/\sqrt{k},d/k\}$ on the forgetting. This stands in contrast to the convergence to the offline solution, which can be arbitrarily slow according to existing alternating projection results. We further show that the $T^2$ factor can be lifted when tasks are presented in a random ordering.

Focusing on diagonal linear networks as a model for understanding the implicit bias in underdetermined models, we show how the gradient descent step size can have a large qualitative effect on the implicit bias, and thus on generalization ability. In particular, we show how using large step size for non-centered data can change the implicit bias from a "kernel" type behavior to a "rich" (sparsity-inducing) regime — even when gradient flow, studied in previous works, would not escape the "kernel" regime. We do so by using dynamic stability, proving that convergence to dynamically stable global minima entails a bound on some weighted $\ell_1$-norm of the linear predictor, i.e. a "rich" regime. We prove this leads to good generalization in a sparse regression setting.

We resolve the min-max complexity of distributed stochastic convex optimization (up to a log factor) in the intermittent communication setting, where M machines work in parallel over the course of R rounds of communication to optimize the objective, and during each round of communication, each machine may sequentially compute K stochastic gradient estimates. We present a novel lower bound with a matching upper bound that establishes an optimal algorithm.

We consider linear prediction with a convex Lipschitz loss, or more generally, stochastic convex optimization problems of generalized linear form, i.e.~where each instantaneous loss is a scalar convex function of a linear function. We show that in this setting, early stopped Gradient Descent (GD), without any explicit regularization or projection, ensures excess error at most $\varepsilon$ (compared to the best possible with unit Euclidean norm) with an optimal, up to logarithmic factors, sample complexity of $\tilde{O}(1/\varepsilon^2)$ and only $\tilde{O}(1/\varepsilon^2)$ iterations. This contrasts with general stochastic convex optimization, where $\Omega(1/\varepsilon^4)$ iterations are needed Amir et al. 2021. The lower iteration complexity is ensured by leveraging uniform convergence rather than stability. But instead of uniform convergence in a norm ball, which we show can guarantee suboptimal learning using $\Theta(1/\varepsilon^4)$ samples, we rely on uniform convergence in a distribution-dependent ball.

We provide new insights on eluder dimension, a complexity measure that has been extensively used to bound the regret of algorithms for online bandits and reinforcement learning with function approximation. First, we study the relationship between the eluder dimension for a function class and a generalized notion of \emph{rank}, defined for any monotone ``activation'' $\sigma : \mathbb{R}\to \mathbb{R}$, which corresponds to the minimal dimension required to represent the class as a generalized linear model. It is known that when $\sigma$ has derivatives bounded away from $0$, $\sigma$-rank gives rise to an upper bound on eluder dimension for any function class; we show however that eluder dimension can be exponentially smaller than $\sigma$-rank. We also show that the condition on the derivative is necessary; namely, when $\sigma$ is the $\mathsf{relu}$ activation, the eluder dimension can be exponentially larger than $\sigma$-rank. For Boolean-valued function classes, we obtain a characterization of the eluder dimension in terms of star number and threshold dimension, quantities which are relevant in active learning and online learning respectively.

We propose an optimistic model-based algorithm, dubbed SMRL, for finite-horizon episodic reinforcement learning (RL) when the transition model is specified by exponential family distributions with $d$ parameters and the reward is bounded and known. SMRL uses score matching, an unnormalized density estimation technique that enables efficient estimation of the model parameter by ridge regression. Under standard regularity assumptions, SMRL achieves $\tilde O(d\sqrt{H^3T})$ online regret, where $H$ is the length of each episode and $T$ is the total number of interactions (ignoring polynomial dependence on structural scale parameters).

We present a family $\{\widehat{\pi}_p\}_{p\ge 1}$ of pessimistic learning rules for offline learning of linear contextual bandits, relying on confidence sets with respect to different $\ell_p$ norms, where $\widehat{\pi}_2$ corresponds to Bellman-consistent pessimism (BCP), while $\widehat{\pi}_\infty$ is a novel generalization of lower confidence bound (LCB) to the linear setting. We show that the novel $\widehat{\pi}_\infty$ learning rule is, in a sense, adaptively optimal, as it achieves the minimax performance (up to log factors) against all $\ell_q$-constrained problems, and as such it strictly dominates all other predictors in the family, including $\widehat{\pi}_2$.

We present a minimax optimal learner for the problem of learning predictors robust to adversarial examples at test-time. Interestingly, we find that this requires new algorithmic ideas and approaches to adversarially robust learning. In particular, we show, in a strong negative sense, the suboptimality of the robust learner proposed by Montasser, Hanneke, and Srebro [2019] and a broader family of learners we identify as local learners. Our results are enabled by adopting a global perspective, specifically, through a key technical contribution: the the global one-inclusion graph, which may be of independent interest, that generalizes the classical one-inclusion graph due to Haussler, Littlestone, and Warmuth [1994]. Finally, as a byproduct, we identify a dimension characterizing qualitatively and quantitatively what classes of predictors $\mathcal{H}$ are robustly learnable. This resolves an open problem due to Montasser et al. [2019], and closes a (potentially) infinite gap between the established upper and lower bounds on the sample complexity of adversarially robust learning.

We study the problem of distributed stochastic non-convex optimization with intermittent communication. We consider the full participation setting where $M$ machines work in parallel over $R$ communication rounds and the partial participation setting where $M$ machines are sampled independently every round from some meta-distribution over machines. We propose and analyze a new algorithm that improves existing methods by requiring fewer and lighter variance reduction operations. We also present lower bounds, showing our algorithm is either $\textit{optimal}$ or $\textit{almost optimal}$ in most settings. Numerical experiments demonstrate the superior performance of our algorithm.

We study norm-based uniform convergence bounds for neural networks, aiming at a tight understanding of how these are affected by the architecture and type of norm constraint, for the simple class of scalar-valued one-hidden-layer networks, and inputs bounded in Euclidean norm. We begin by proving that in general, controlling the spectral norm of the hidden layer weight matrix is insufficient to get uniform convergence guarantees (independent of the network width), while a stronger Frobenius norm control is sufficient, extending and improving on previous work. Motivated by the proof constructions, we identify and analyze two important settings where (perhaps surprisingly) a mere spectral norm control turns out to be sufficient: First, when the network's activation functions are sufficiently smooth (with the result extending to deeper networks); and second, for certain types of convolutional networks. In the latter setting, we study how the sample complexity is additionally affected by parameters such as the amount of overlap between patches and the overall number of patches.

The implicit bias of neural networks has been extensively studied in recent years. Lyu and Li (2019) showed that in homogeneous networks trained with the exponential or the logistic loss, gradient flow converges to a KKT point of the max margin problem in parameter space. However, that leaves open the question of whether this point will generally be an actual optimum of the max margin problem. In this paper, we study this question in detail, for several neural network architectures involving linear and ReLU activations. Perhaps surprisingly, we show that in many cases, the KKT point is not even a local optimum of the max margin problem. On the flip side, we identify multiple settings where a local or global optimum can be guaranteed.

We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss $\ell$ can control the test error under all Moreau envelopes of the loss $\ell$ . We use our finite-sample bound to directly recover the “optimistic rate” of Zhou et al. (2021) for linear regression with the square loss, which is known to be tight for minimal $\ell_2$-norm interpolation, but we also handle more general settings where the label is generated by a potentially misspecified multi-index model. The same argument can analyze noisy interpolation of max-margin classifiers through the squared hinge loss, and establishes consistency results in spiked-covariance settings. More generally, when the loss is only assumed to be Lipschitz, our bound effectively improves Talagrand’s well-known contraction lemma by a factor of two, and we prove uniform convergence of interpolators (Koehler et al. 2021) for all smooth, non-negative losses. Finally, we show that application of our generalization bound using localized Gaussian width will generally be sharp for empirical risk minimizers, establishing a non-asymptotic Moreau envelope theory for generalization that applies outside of proportional scaling regimes, handles model misspecification, and complements existing asymptotic Moreau envelope theories for M-estimation.

We present a direct (primal only) derivation of Mirror Descent as a “partial” discretization of gradient flow on a Riemannian manifold where the metric tensor is the Hessian of the Mirror Descent potential function. We contrast this discretization to Natural Gradient Descent, which is obtained by a “full” forward Euler discretization. This view helps shed light on the relationship between the methods and allows generalizing Mirror Descent to any Riemannian geometry in $\mathbb{R}^d$, even when the metric tensor is not a Hessian, and thus there is no “dual.”

We show that the Invariant Risk Minimization (IRM) formulation of Arjovsky et al. (2019) can fail to capture "natural" invariances, at least when used in its practical "linear" form, and even on very simple problems which directly follow the motivating examples for IRM. This can lead to worse generalization on new environments, even when compared to unconstrained ERM. The issue stems from a significant gap between the linear variant (as in their concrete method IRMv1) and the full non-linear IRM formulation. Additionally, even when capturing the "right" invariances, we show that it is possible for IRM to learn a sub-optimal predictor, due to the loss function not being invariant across environments. The issues arise even when measuring invariance on the population distributions, but are exacerbated by the fact that IRM is extremely fragile to sampling.

We study the problem of learning predictors that are robust to adversarial examples with respect to an unknown perturbation set, relying instead on interaction with an adversarial attacker or access to attack oracles, examining different models for such interactions. We obtain upper bounds on the sample complexity and upper and lower bounds on the number of required interactions, or number of successful attacks, in different interaction models, in terms of the VC and Littlestone dimensions of the hypothesis class of predictors, and without any assumptions on the perturbation set.

We resolve the min-max complexity of distributed stochastic convex optimization (up to a log factor) in the intermittent communication setting, where $M$ machines work in parallel over the course of $R$ rounds of communication to optimize the objective, and during each round of communication, each machine may sequentially compute $K$ stochastic gradient estimates. We present a novel lower bound with a matching upper bound that establishes an optimal algorithm.

We investigate the capacity control provided by dropout in various machine learning problems. First, we study dropout for matrix completion, where it induces a distribution-dependent regularizer that equals the weighted trace-norm of the product of the factors. In deep learning, we show that the distribution-dependent regularizer due to dropout directly controls the Rademacher complexity of the underlying class of deep neural networks. These developments enable us to give concrete generalization error bounds for the dropout algorithm in both matrix completion as well as training deep neural networks.

Recent work has highlighted the role of initialization scale in determining the structure of the solutions that gradient methods converge to. In particular, it was shown that large initialization leads to the neural tangent kernel regime solution, whereas small initialization leads to so called “rich regimes”. However, the initialization structure is richer than the overall scale alone and involves relative magnitudes of different weights and layers in the network. Here we show that these relative scales, which we refer to as initialization shape, play an important role in determining the learned model. We develop a novel technique for deriving the inductive bias of gradient-flow and use it to obtain closed-form implicit regularizers for multiple cases of interest.

We present and analyze a momentum-based gradient method for training linear classifiers with an exponentially-tailed loss (e.g., the exponential or logistic loss), which maximizes the classification margin on separable data at a rate of O(1/t^2). This contrasts with a rate of O(1/log(t)) for standard gradient descent, and O(1/t) for normalized gradient descent. The momentum-based method is derived via the convex dual of the maximum-margin problem, and specifically by applying Nesterov acceleration to this dual, which manages to result in a simple and intuitive method in the primal. This dual view can also be used to derive a stochastic variant, which performs adaptive non-uniform sampling via the dual variables.

We study the relative power of learning with gradient descent on differentiable models, such as neural networks, versus using the corresponding tangent kernels. We show that under certain conditions, gradient descent achieves small error only if a related tangent kernel method achieves a non-trivial advantage over random guessing (a.k.a. weak learning), though this advantage might be very small even when gradient descent can achieve arbitrarily high accuracy. Complementing this, we show that without these conditions, gradient descent can in fact learn with small error even when no kernel method, in particular using the tangent kernel, can achieve a non-trivial advantage over random guessing.

We study the power of learning via mini-batch stochastic gradient descent (SGD) on the loss of a differentiable model or neural network, and ask what learning problems can be learnt using this paradigm. We show that SGD can always simulate learning with statistical queries (SQ), but its ability to go beyond that depends on the precision $\rho$ of the gradients and the minibatch size $b$. With fine enough precision relative to minibatch size, namely when $b \rho$ is small enough, SGD can go beyond SQ learning and simulate any sample-based learning algorithm and thus its learning power is equivalent to that of PAC learning; this extends prior work that achieved this result for $b=1$. Moreover, with polynomially many bits of precision (i.e. when $\rho$ is exponentially small), SGD can simulate PAC learning regardless of the batch size. On the other hand, when $b \rho^2$ is large enough, the power of SGD is equivalent to that of SQ learning.

We propose and analyze a stochastic Newton algorithm for homogeneous distributed stochastic convex optimization, where each machine can calculate stochastic gradients of the same population objective, as well as stochastic Hessian-vector products (products of an independent unbiased estimator of the Hessian of the population objective with arbitrary vectors), with many such stochastic computations performed between rounds of communication. We show that our method can reduce the number, and frequency, of required communication rounds, compared to existing methods without hurting performance, by proving convergence guarantees for quasi-self-concordant objectives (e.g., logistic regression), alongside empirical evidence.

For different parameterizations (mappings from parameters to predictors), we study the regularization cost in predictor space induced by $l_2$ regularization on the parameters (weights). We focus on linear neural networks as parameterizations of linear predictors. We identify the representation cost of certain sparse linear ConvNets and residual networks. In order to get a better understanding of how the architecture and parameterization affect the representation cost, we also study the reverse problem, identifying which regularizers on linear predictors (e.g., $l_p$ norms, group norms, the $k$-support-norm, elastic net) can be the representation cost induced by simple $l_2$ regularization, and designing the parameterizations that do so.

We consider interpolation learning in high-dimensional linear regression with Gaussian data, and prove a generic uniform convergence guarantee on the generalization error of interpolators in an arbitrary hypothesis class in terms of the class’s Gaussian width. Applying the generic bound to Euclidean norm balls recovers the consistency result of Bartlett et al. (2020) for minimum-norm interpolators, and confirms a prediction of Zhou et al. (2020) for near-minimal-norm interpolators in the special case of Gaussian data. We demonstrate the generality of the bound by applying it to the simplex, obtaining a novel consistency result for minimum $\ell_1$-norm interpolators (basis pursuit). Our results show how norm-based generalization bounds can explain and be used to analyze benign overfitting, at least in some settings.

We present and analyze an algorithm for optimizing smooth and convex or strongly convex objectives using minibatch stochastic gradient estimates. The algorithm is optimal with respect to its dependence on both the minibatch size and minimum expected loss simultaneously. This improves over the optimal method of Lan, which is insensitive to the minimum expected loss; over the optimistic acceleration of Cotter et al., which has suboptimal dependence on the minibatch size; and over the algorithm of Liu and Belkin, which is limited to least squares problems and is also similarly suboptimal. Applied to interpolation learning, the improvement over Cotter et al.~and Liu and Belkin translates to a linear, rather than square-root, parallelization speedup.

We design a general framework for answering adaptive statistical queries that focuses on providing explicit confidence intervals along with point estimates. Prior work in this area has either focused on providing tight confidence intervals for specific analyses, or providing general worst-case bounds for point estimates. Unfortunately, as we observe, these worst-case bounds are loose in many settings — often not even beating simple baselines like sample splitting. Our main contribution is to design a framework for providing valid, instance-specific confidence intervals for point estimates that can be generated by heuristics. When paired with good heuristics, this method gives guarantees that are orders of magnitude better than the best worst-case bounds. We provide a Python library implementing our method.

We establish matching upper and lower complexity bounds for gradient descent and stochastic gradient descent on quadratic functions, when the gradients are delayed and reflect iterates from $\tau$ rounds ago. First, we show that without stochastic noise, delays strongly affect the attainable optimization error: In fact, the error can be as bad as non-delayed gradient descent ran on only $1/\tau$ of the gradients. In sharp contrast, we quantify how stochastic noise makes the effect of delays negligible, improving on previous work which only showed this phenomenon asymptotically or for much smaller delays. Also, in the context of distributed optimization, the results indicate that the performance of gradient descent with delays is competitive with synchronous approaches such as mini-batching. Our results are based on a novel technique for analyzing convergence of optimization algorithms using generating functions.

We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to {\em approximate}, rather then exactly represent, a given hypothesis class. We show that such notions are not only sufficient for learning using linear predictors or a kernel, but unlike the exact variants, are also necessary. Thus they are better suited for discussing limitations of linear or kernel methods.

A recent line of work studies overparametrized neural networks in the “kernel regime,” i.e. when during training the network behaves as a kernelized linear predictor, and thus, training with gradient descent has the effect of finding the corresponding minimum RKHS norm solution. This stands in contrast to other studies which demonstrate how gradient descent on overparametrized networks can induce rich implicit biases that are not RKHS norms. Building on an observation by \citet{chizat2018note}, we show how the \textbf{\textit{scale of the initialization}} controls the transition between the “kernel” (aka lazy) and “rich” (aka active) regimes and affects generalization properties in multilayer homogeneous models. We provide a complete and detailed analysis for a family of simple depth-$D$ linear networks that exhibit an interesting and meaningful transition between the kernel and rich regimes, and highlight an interesting role for the \emph{width} of the models. We further demonstrate this transition empirically for matrix factorization and multilayer non-linear networks.

We study the problem of learning adversarially robust halfspaces in the distribution-independent setting. In the realizable setting, we provide necessary and sufficient conditions on the adversarial perturbation sets under which halfspaces are efficiently robustly learnable. In the presence of random label noise, we give a simple computationally efficient algorithm for this problem with respect to any $\ell_p$-perturbation.

Sensitive attributes such as race are rarely available to learners in real world settings as their collection is often restricted by laws and regulations. We give a scheme that allows individuals to release their sensitive information privately while still allowing any downstream entity to learn non-discriminatory predictors. We show how to adapt non-discriminatory learners to work with privatized protected attributes giving theoretical guarantees on performance. Finally, we highlight how the methodology could apply to learning fair predictors in settings where protected attributes are only available for a subset of the data.

We study local SGD (also known as parallel SGD and federated SGD), a natural and frequently used distributed optimization method. Its theoretical foundations are currently lacking and we highlight how all existing error guarantees in the convex setting are dominated by a simple baseline, minibatch SGD. (1) For quadratic objectives we prove that local SGD strictly dominates minibatch SGD and that accelerated local SGD is minmax optimal for quadratics; (2) For general convex objectives we provide the first guarantee that at least \emph{sometimes} improves over minibatch SGD, but our guarantee does not always improve over, nor even match, minibatch SGD; (3) We show that indeed local SGD does \emph{not} dominate minibatch SGD by presenting a lower bound on the performance of local SGD that is worse than the minibatch SGD guarantee.

We study the problem of reducing adversarially robust learning to standard PAC learning, i.e. the complexity of learning adversarially robust predictors using access to only a black-box non-robust learner. We give a reduction that can robustly learn any hypothesis class C using any non-robust learner A for C. The number of calls to A depends logarithmically on the number of allowed adversarial perturbations per example, and we give a lower bound showing this is unavoidable.

We provide a detailed asymptotic study of gradient flow trajectories and their implicit optimization bias when minimizing the exponential loss over "diagonal linear networks". This is the simplest model displaying a transition between "kernel" and non-kernel ("rich" or "active") regimes. We show how the transition is controlled by the relationship between the initialization scale and how accurately we minimize the training loss. Our results indicate that some limit behavior of gradient descent only kick in at ridiculous training accuracies (well beyond 10^-100). Moreover, the implicit bias at reasonable initialization scales and training accuracies is more complex and not captured by these limits.

We analyze Local SGD (aka parallel or federated SGD) and Minibatch SGD in the heterogeneous distributed setting, where each machine has access to stochastic gradient estimates for a different, machine-specific, convex objective; the goal is to optimize w.r.t.~the average objective; and machines can only communicate intermittently. We argue that, (i) Minibatch SGD (even without acceleration) dominates all existing analysis of Local SGD in this setting, (ii) accelerated Minibatch SGD is optimal when the heterogeneity is high, and (iii) present the first upper bound for Local SGD that improves over Minibatch SGD in a non-homogeneous regime.

We consider an underdetermined noisy linear regression model where the minimum-norm interpolating predictor is known to be consistent, and ask: can uniform convergence in a norm ball, or at least (following Nagarajan and Kolter) the subset of a norm ball that the algorithm selects on a typical input set, explain this success? We show that uniformly bounding the difference between empirical and population errors cannot show any learning in the norm ball, and cannot show consistency for any set, even one depending on the exact algorithm and distribution. But we argue we can explain the consistency of the minimal-norm interpolator with a slightly weaker, yet standard, notion: uniform convergence of zero-error predictors in a norm ball. We use this to bound the generalization error of low- (but not minimal-)norm interpolating predictors.

We provide a detailed study on the implicit bias of gradient descent when optimizing loss functions with strictly monotone tails, such as the logistic loss, over separable datasets. We look at two basic questions: (a) what are the conditions on the tail of the loss function under which gradient descent converges in the direction of the $L_2$ maximum-margin separator? (b) how does the rate of margin convergence depend on the tail of the loss function and the choice of the step size? We show that for a large family of super-polynomial tailed losses, gradient descent iterates on linear networks of any depth converge in the direction of $L_2$ maximum-margin solution, while this does not hold for losses with heavier tails. Within this family, for simple linear models we show that the optimal rates with fixed step size is indeed obtained for the commonly used exponentially tailed losses such as logistic loss. However, with a fixed step size the optimal convergence rate is extremely slow as $1/\log(t)$, as also proved in Soudry et al (2018). For linear models with exponential loss, we further prove that the convergence rate could be improved to $\log (t) /\sqrt{t}$ by using aggressive step sizes that compensates for the rapidly vanishing gradients. Numerical results suggest this method might be useful for deep networks.

Stochastic Gradient Descent (SGD) is a central tool in machine learning. We prove that SGD converges to zero loss, even with a fixed (non-vanishing) learning rate — in the special case of homogeneous linear classifiers with smooth monotone loss functions, optimized on linearly separable data. Previous works assumed either a vanishing learning rate, iterate averaging, or loss assumptions that do not hold for monotone loss functions used for classification, such as the logistic loss. We prove our result on a fixed dataset, both for sampling with or without replacement. Furthermore, for logistic loss (and similar exponentially-tailed losses), we prove that with SGD the weight vector converges in direction to the $L_2$ max margin vector as $O(1/\log(t))$ for almost all separable datasets, and the loss converges as $O(1/t)$ — similarly to gradient descent. Lastly, we examine the case of a fixed learning rate proportional to the minibatch size. We prove that in this case, the asymptotic convergence rate of SGD (with replacement) does not depend on the minibatch size in terms of epochs, if the support vectors span the data. These results may suggest an explanation to similar behaviors observed in deep networks, when trained with SGD.

We study stochastic optimization of nonconvex loss functions, which are typical objectives for training neural networks. We propose stochastic approximation algorithms which optimize a series of regularized, nonlinearized losses on large minibatches of samples, using only first-order gradient information. Our algorithms provably converge to an approximate critical point of the expected objective with faster rates than minibatch stochastic gradient descent, and facilitate better parallelization by allowing larger minibatches.

We give nearly matching upper and lower bounds on the oracle complexity of finding $\epsilon$-stationary points $(\|\nabla F(x)\|\leq\epsilon$ in stochastic convex optimization. We jointly analyze the oracle complexity in both the local stochastic oracle model and the global oracle (or, statistical learning) model. This allows us to decompose the complexity of finding near-stationary points into optimization complexity and sample complexity, and reveals some surprising differences between the complexity of stochastic optimization versus learning. Notably, we show that in the global oracle/statistical learning model, only logarithmic dependence on smoothness is required to find a near-stationary point, whereas polynomial dependence on smoothness is necessary in the local stochastic oracle model. In other words, the separation in complexity between the two models can be exponential, and the folklore understanding that smoothness is required to find stationary points is only weakly true for statistical learning. Our upper bounds are based on extensions of a recent “recursive regularization” technique proposed by Allen-Zhu (2018). We show how to extend the technique to achieve near-optimal rates, and in particular show how to leverage the extra information available in the global oracle model. Our algorithm for the global model can be implemented efficiently through finite sum methods, and suggests an interesting new computational-statistical tradeoff.

We study the question of learning an adversarially robust predictor. We show that any hypothesis class $\mathcal{H}$ with finite VC dimension is robustly PAC learnable with an \emph{improper} learning rule. The requirement of being improper is necessary as we exhibit examples of hypothesis classes $\mathcal{H}$ with finite VC dimension that are \emph{not} robustly PAC learnable with any \emph{proper} learning rule.

We consider the question of what functions can be captured by ReLU networks with an unbounded number of units (infinite width), but where the overall network Euclidean norm (sum of squares of all weights in the system, except for an unregularized bias term for each unit) is bounded; or equivalently what is the minimal norm required to approximate a given function. For functions $f:\mathbb R \rightarrow\mathbb R$ and a single hidden layer, we show that the minimal network norm for representing $f$ is $\max(\int \lvert f”(x) \rvert \mathrm{d} x, \lvert f’(-\infty) + f’(+\infty) \rvert)$, and hence the minimal norm fit for a sample is given by a linear spline interpolation.

We note that known methods achieving the optimal oracle complexity for first order convex optimization require quadratic memory, and ask whether this is necessary, and more broadly seek to characterize the minimax number of first order queries required to optimize a convex Lipschitz function subject to a memory constraint.

Classifiers can be trained with data-dependent constraints to satisfy fairness goals, reduce churn, achieve a targeted false positive rate, or other policy goals. We study the generalization performance for such constrained optimization problems, in terms of how well the constraints are satisfied at evaluation time, given that they are satisfied at training time. To improve generalization, we frame the problem as a two-player game where one player optimizes the model parameters on a training dataset, and the other player enforces the constraints on an independent validation dataset. We build on recent work in two-player constrained optimization to show that if one uses this two-dataset approach, then constraint generalization can be significantly improved. As we illustrate experimentally, this approach works not only in theory, but also in practice.

We consider convex SGD updates with a block-cyclic structure, i.e., where each cycle consists of a small number of blocks, each with many samples from a possibly different, block-specific, distribution. This situation arises, e.g., in Federated Learning where the mobile devices available for updates at different times during the day have different characteristics. We show that such block-cyclic structure can significantly deteriorate the performance of SGD, but propose a simple approach that allows prediction with the same guarantees as for i.i.d., non-cyclic, sampling.

With an eye toward understanding complexity control in deep learning, we study how infinitesimal regularization or gradient descent optimization lead to margin maximizing solutions in both homogeneous and non homogeneous models, extending previous work that focused on infinitesimal regularization only in homogeneous models. To this end we study the limit of loss minimization with a diverging norm constraint (the “constrained path”), relate it to the limit of a “margin path” and characterize the resulting solution. For non-homogeneous ensemble models, which output is a sum of homogeneous sub-models, we show that this solution discards the shallowest sub-models if they are unnecessary. For homogeneous models, we show convergence to a “lexicographic max-margin solution”, and provide conditions under which max-margin solutions are also attained as the limit of unconstrained gradient descent.

We study the sample complexity of canonical correlation analysis (CCA), i.e., the number of samples needed to estimate the population canonical correlation and directions up to arbitrarily small error. With mild assumptions on the data distribution, we show that in order to achieve $\epsilon$-suboptimality in a properly defined measure of alignment between the estimated canonical directions and the population solution, we can solve the empirical objective exactly with $N(\epsilon, \Delta, \gamma)$ samples, where $\Delta$ is the singular value gap of the whitened cross-covariance matrix and $1/\gamma$ is an upper bound of the condition number of auto-covariance matrices. Moreover, we can achieve the same learning accuracy by drawing the same level of samples and solving the empirical objective approximately with a stochastic optimization algorithm; this algorithm is based on the shift-and-invert power iterations and only needs to process the dataset for $\mathcal{O} \left(\log \frac{1}{\epsilon} \right)$ passes. Finally, we show that, given an estimate of the canonical correlation, the streaming version of the shift-and-invert power iterations achieves the same learning accuracy with the same level of sample complexity, by processing the data only once.

We develop and analyze efficient "coordinate-wise" methods for finding the leading eigenvector, where each step involves only a vector-vector product. We establish global convergence with overall runtime guarantees that are at least as good as Lanczos’s method and dominate it for slowly decaying spectrum. Our methods are based on combining a shift-and-invert approach with coordinate-wise algorithms for linear regression.

We study the bias of generic optimization methods, including Mirror Descent, Natural Gradient Descent and Steepest Descent with respect to different potentials and norms, when optimizing underdetermined linear models or separable linear classification problems. We ask the question of whether the global minimum (among the many possible global minima) reached by optimization can be characterized in terms of the potential or norm, and indecently of hyper-parameter choices, such as stepsize and momentum.

We study the interplay between sequential decision making and avoiding discrimination against protected groups, when examples arrive online and do not follow distributional assumptions. We consider the most basic extension of classical online learning: Given a class of predictors that are individually non-discriminatory with respect to a particular metric, how can we combine them to perform as well as the best predictor, while preserving non-discrimination? Surprisingly we show that this task is unachievable for the prevalent notion of "equalized odds" that requires equal false negative rates and equal false positive rates across groups. On the positive side, for another notion of non-discrimination, "equalized error rates", we show that running separate instances of the classical multiplicative weights algorithm for each group achieves this guarantee. Interestingly, even for this notion, we show that algorithms with stronger performance guarantees than multiplicative weights cannot preserve non-discrimination.

We show that gradient descent on full-width linear convolutional networks of depth $L$ converges to a linear predictor related to the $\ell_{2/L}$ bridge penalty in the frequency domain. This is in contrast to linearly fully connected networks, where gradient descent converges to the hard margin linear SVM solution, regardless of depth.

The problem of handling adaptivity in data analysis, intentional or not, permeates a variety of fields, including test-set overfitting in ML challenges and the accumulation of invalid scientific discoveries. We propose a mechanism for answering an arbitrarily long sequence of potentially adaptive statistical queries, by charging a price for each query and using the proceeds to collect additional samples. Crucially, we guarantee statistical validity without any assumptions on how the queries are generated. We also ensure with high probability that the cost for $M$ non-adaptive queries is $O(\log M)$, while the cost to a potentially adaptive user who makes $M$ queries that do not depend on any others is $O(\sqrt{M})$.

We suggest a general oracle-based framework that captures parallel stochastic optimization in different parallelization settings described by a dependency graph, and derive generic lower bounds in terms of this graph. We then use the framework and derive lower bounds to study several specific parallel optimization settings, including delayed updates and parallel processing with intermittent communication. We highlight gaps between lower and upper bounds on the oracle complexity, and cases where the ``natural'' algorithms are not known to be optimal.

We examine gradient descent on unregularized logistic regression problems, with homogeneous linear predictors on linearly separable datasets. We show the predictor converges to the direction of the max-margin (hard margin SVM) solution. The result also generalizes to other monotone decreasing loss functions with an infimum at infinity, to multi-class problems, and to training a weight layer in a deep network in a certain restricted setting. Furthermore, we show this convergence is very slow, and only logarithmic in the convergence of the loss itself. This can help explain the benefit of continuing to optimize the logistic or cross-entropy loss even after the training error is zero and the training loss is extremely small, and, as we show, even if the validation loss increases. Our methodology can also aid in understanding implicit regularization in more complex models and with other optimization methods.

Sketching techniques scale up machine learning algorithms by reducing the sample size or dimensionality of massive data sets, without sacrificing their statistical properties. In this paper, we study sketching from an optimization point of view. We first show that the iterative Hessian sketch is an optimization process with \emphpreconditioning and develop an \emphaccelerated version using this insight together with conjugate gradient descent. Next, we establish a primal-dual connection between the Hessian sketch and dual random projection, which allows us to develop an \emphaccelerated iterative dual random projection method by applying the preconditioned conjugate gradient descent on the dual problem. Finally, we tackle the problems of large sample size and high-dimensionality in massive data sets by developing the \emphprimal-dual sketch. The primal-dual sketch iteratively sketches the primal and dual formulations and requires only a logarithmic number of calls to solvers of small sub-problems to recover the optimum of the original problem up to \empharbitrary precision. Extensive experiments on synthetic and real data sets complement our theoretical results.

We present and analyze statistically optimal, communication and memory efficient distributed stochastic optimization algorithms with near-linear speedups (up to $\log$-factors). This improves over prior work which includes methods with near-linear speedups but polynomial communication requirements (accelerated minibatch SGD) and communication efficient methods which do not exhibit any runtime speedups over a naive single-machine approach. We first analyze a distributed SVRG variant as a distributed stochastic optimization method and show that it can achieve near-linear speedups with logarithmic rounds of communication, at the cost of high memory requirements. We then present a novel method, MB-DSVRG, which trades off memory for communication and still allows for optimization with communication which scales only logarithmically with the desired accuracy while also being memory efficient. MB-DSVRG is based on a minibatch prox procedure, solving a non-linearized subproblem on a minibatch at each iteration. We provide a novel analysis for this procedure which achieves the statistical optimal rate regardless of minibatch size and smoothness, and thus significantly improving on prior work.

We consider learning a predictor which is non-discriminatory with respect to a “protected attribute” according to the notion of “equalized odds” proposed by Hardt et al. (2016). We study the problem of learning such a non-discriminatory predictor from a finite training set, both statistically and computationally. We show that a post-hoc correction approach, as suggested by Hardt et al, can be highly suboptimal, present a nearly-optimal statistical procedure, argue that the associated computational problem is intractable, and suggest a second moment relaxation of the non-discrimination definition for which learning is tractable.

We study the fundamental problem of Principal Component Analysis in a statistical distributed setting in which each machine out of m stores a sample of n points sampled i.i.d. from a single unknown distribution. We study algorithms for estimating the leading principal component of the population covariance matrix that are both communication-efficient and achieve estimation error of the order of the centralized ERM solution that uses all mn samples. On the negative side, we show that in contrast to results obtained for distributed estimation under convexity assumptions, for the PCA objective, simply averaging the local ERM solutions cannot guarantee error that is consistent with the centralized ERM. We show that this unfortunate phenomena can be remedied by performing a simple correction step which correlates between the individual solutions, and provides an estimator that is consistent with the centralized ERM for sufficiently-large n. We also introduce an iterative distributed algorithm that is applicable in any regime of n, which is based on distributed matrix-vector products. The algorithm gives significant acceleration in terms of communication rounds over previous distributed algorithms, in a wide regime of parameters.

We propose a novel, efficient approach for distributed sparse learning with observations randomly partitioned across machines. In each round of the proposed method, worker machines compute the gradient of the loss on local data and the master machine solves a shifted $\ell_1$ regularized loss minimization problem. After a number of communication rounds that scales only logarithmically with the number of machines, and independent of other parameters of the problem, the proposed approach provably matches the estimation error bound of centralized methods.

We propose novel first-order stochastic approximation algorithms for canonical correlation analysis (CCA). Algorithms presented are instances of inexact matrix stochastic gradient (MSG) and inexact matrix exponentiated gradient (MEG), and achieve $\epsilon$-suboptimality in the population objective in $\operatorname{poly}(\frac{1}{\epsilon})$ iterations. We also consider practical variants of the proposed algorithms and compare them with other methods for CCA both theoretically and empirically.

We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of X. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.

With a goal of understanding what drives generalization in deep networks, we consider several recently suggested explanations, including norm-based control, sharpness and robustness. We study how these measures can ensure generalization, highlighting the importance of scale normalization, and making a connection between sharpness and PAC-Bayes theory. We then investigate how well the measures explain different observed phenomena.

Adaptive optimization methods, which perform local optimization with a metric constructed from the history of iterates, are becoming increasingly popular for training deep neural networks. Examples include AdaGrad, RMSProp, and Adam. We show that for simple overparameterized problems, adaptive methods often find drastically different solutions than gradient descent (GD) or stochastic gradient descent (SGD). We construct an illustrative binary classification problem where the data is linearly separable, GD and SGD achieve zero test error, and AdaGrad, Adam, and RMSProp attain test errors arbitrarily close to half. We additionally study the empirical generalization capability of adaptive methods on several state-of-the-art deep learning models. We observe that the solutions found by adaptive methods generalize worse (often significantly worse) than SGD, even when these solutions have better training performance. These results suggest that practitioners should reconsider the use of adaptive methods to train neural networks.

We present an efficient method for training slack-rescaled structural SVM. Although finding the most violating label in a margin-rescaled formulation is often easy since the target function decomposes with respect to the structure, this is not the case for a slack-rescaled formulation, and finding the most violated label might be very difficult. Our core contribution is an efficient method for finding the most-violating-label in a slack-rescaled formulation, given an oracle that returns the most-violating-label in a (slightly modified) margin-rescaled formulation. We show that our method enables accurate and scalable training for slack-rescaled SVMs, reducing runtime by an order of magnitude compared to previous approaches to slack-rescaled SVMs.

We consider the problem of distributed multi-task learning, where each machine learns a separate, but related, task. Specifically, each machine learns a linear predictor in high-dimensional space, where all tasks share the same small support. We present a communication-efficient estimator based on the debiased lasso and show that it is comparable with the optimal centralized method.

We show that there are no spurious local minima in the non-convex factorized parametrization of low-rank matrix recovery from incoherent linear measurements. With noisy measurements we show all local minima are very close to a global optimum. Together with a curvature bound at saddle points, this yields a polynomial time global convergence guarantee for stochastic gradient descent {\em from random initialization}.

We investigate the parameter-space geometry of recurrent neural networks (RNNs), and develop an adaptation of path-SGD optimization method, attuned to this geometry, that can learn plain RNNs with ReLU activations. On several datasets that require capturing long-term dependency structure, we show that path-SGD can significantly improve trainability of ReLU RNNs compared to RNNs trained with SGD, even with various recently suggested initialization schemes.

The algorithmic advancement of synchronizing maps is important in order to solve a wide range of practice problems with possible large-scale dataset. In this paper, we provide theoretical justifications for spectral techniques for the map synchronization problem, i.e., it takes as input a collection of objects and noisy maps estimated between pairs of objects, and outputs clean maps between all pairs of objects. We show that a simple normalized spectral method that projects the blocks of the top eigenvectors of a data matrix to the map space leads to surprisingly good results. As the noise is modelled naturally as random permutation matrix, this algorithm NormSpecSync leads to competing theoretical guarantees as state-of-the-art convex optimization techniques, yet it is much more efficient. We demonstrate the usefulness of our algorithm in a couple of applications, where it is optimal in both complexity and exactness among existing methods.

We study the stochastic optimization of canonical correlation analysis (CCA), whose objective is nonconvex and does not decouple over training samples. Although several stochastic gradient based optimization algorithms have been recently proposed to solve this problem, no global convergence guarantee was provided by any of them. Inspired by the alternating least squares/power iterations formulation of CCA, and the shift-and-invert preconditioning method for PCA, we propose two globally convergent meta-algorithms for CCA, both of which transform the original problem into sequences of least squares problems that need only be solved approximately. We instantiate the meta-algorithms with state-of-the-art SGD methods and obtain time complexities that significantly improve upon that of previous work. Experimental results demonstrate their superior performance.

We provide tight upper and lower bounds on the complexity of minimizing the average of m convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs randomized optimization. For smooth functions, we show that accelerated gradient descent (AGD) and an accelerated variant of SVRG are optimal in the deterministic and randomized settings respectively, and that a gradient oracle is sufficient for the optimal rate. For non-smooth functions, having access to prox oracles reduces the complexity and we present optimal methods based on smoothing that improve over methods using just gradient accesses.

Structured output prediction is a powerful framework for jointly predicting interdependent output labels. Learning the parameters of structured predictors is a central task in machine learning applications. However, training the model from data often becomes computationally expensive. Several methods have been proposed to exploit the model structure, or decomposition, in order to obtain efficient training algorithms. In particular, methods based on linear programming relaxation, or dual decomposition, decompose the prediction task into multiple simpler prediction tasks and enforce agreement between overlapping predictions. In this work we observe that relaxing these agreement constraints and replacing them with soft constraints yields a much easier optimization problem. Based on this insight we propose an alternative training objective, analyze its theoretical properties, and derive an algorithm for its optimization. Our method, based on the Frank-Wolfe algorithm, achieves significant speedups over existing state-of-the-art methods without hurting prediction accuracy.

We investigate the capacity, convexity and characterization of a general family of norm-constrained feed-forward networks.

We consider the problem of designing locality sensitive hashes (LSH) for inner product similarity, and of the power of asymmetric hashes in this context. Shrivastava and Li (2014a) argue that there is no symmetric LSH for the problem and propose an asymmetric LSH based on different mappings for query and database points. However, we show there does exist a simple symmetric LSH that enjoys stronger guarantees and better empirical performance than the asymmetric LSH they suggest. We also show a variant of the settings where asymmetry is in-fact needed, but there a different asymmetric LSH is required.

We revisit the choice of SGD for training deep neural networks by reconsidering the appropriate geometry in which to optimize the weights. We argue for a geometry invariant to rescaling of weights that does not affect the output of the network, and suggest Path-SGD, which is an approximate steepest descent method with respect to a path-wise regularizer related to max-norm regularization. Path-SGD is easy and efficient to implement and leads to empirical gains over SGD and AdaGrad.

We present a novel Newton-type method for distributed optimization, which is particularly well suited for stochastic optimization and learning problems. For quadratic objectives, the method enjoys a linear rate of convergence which provably \emphimproves with the data size, requiring an essentially constant number of iterations under reasonable assumptions. We provide theoretical and empirical evidence of the advantages of our method compared to other approaches, such as one-shot parameter averaging and ADMM.

We consider the problem of Collaborative Permutation Recovery, i.e. recovering multiple permutations over objects (e.g. preference rankings over different options) from limited pairwise comparisons. We tackle both the problem of how to recover multiple related permutations from limited observations, and the active learning problem of which pairwise comparison queries to ask so as to allow better recovery. There has been much work on recovering single permutations from pairwise comparisons, but we show that considering several related permutations jointly we can leverage their relatedness so as to reduce the number of comparisons needed compared to reconstructing each permutation separately. To do so, we take a collaborative filtering / matrix completion approach and use a trace-norm or max-norm regularized matrix learning model. Our approach can also be seen as a collaborative learning version of Jamieson and Nowak's recent work on constrained permutation recovery, where instead of basing the recovery on known features, we learn the best features de novo.

We show how to train SVMs with an optimal guarantee on the number of support vectors (up to constants), and with sample complexity and training runtime bounds matching the best known for kernel SVM optimization (i.e. without any additional asymptotic cost beyond standard SVM training). Our method is simple to implement and works well in practice.

We address the issue of using mini-batches in stochastic optimization of SVMs. We show that the same quantity, the spectral norm of the data, controls the parallelization speedup obtained for both primal stochastic subgradient descent(SGD) and stochastic dual coordinate ascent (SCDA) methods and use it to derive novel variants of mini-batched SDCA. Our guarantees for both methods are expressed in terms of the original nonsmooth primal problem based on the hinge-loss.

We study PCA as a stochastic optimization problem and propose a novel stochastic approximation algorithm which we refer to as Matrix Stochastic Gradient'' (MSG), as well as a practical variant, Capped MSG. We study the method both theoretically and empirically. "

When approximating binary similarity using the hamming distance between short binary hashes, we shown that even if the similarity is symmetric, we can have shorter and more accurate hashes by using two distinct code maps. I.e.~by approximating the similarity between $x$ and $x'$ as the hamming distance between $f(x)$ and $g(x')$, for two distinct binary codes $f,g$, rather than as the hamming distance between $f(x)$ and $f(x')$.

We propose a learning setting in which unlabeled data is free, and the cost of a label depends on its value, which is not known in advance. We study binary classification in an extreme case, where the algorithm only pays for negative labels. Our motivation are applications such as fraud detection, in which investigating an honest transaction should be avoided if possible. We term the setting auditing, and consider the auditing complexity of an algorithm: The number of negative points it labels to learn a hypothesis with low relative error. We design auditing algorithms for thresholds on the line and axis-aligned rectangles, and show that with these algorithms, the auditing complexity can be significantly lower than the active label complexity. We discuss a general approach for auditing for a general hypothesis class, and describe several interesting directions for future work.

We derive a novel norm that corresponds to the tightest convex relaxation of sparsity combined with an $\ell_2$ penalty. We show that this new norm provides a tighter relaxation than the elastic net, and is thus a good replacement for the Lasso or the elastic net in sparse prediction problems. But through studying our new norm, we also bound the looseness of the elastic net, thus shedding new light on it and providing justification for its use.

We introduce a new family of matrix norms, the ''local max'' norms, generalizing existing methods such as the max norm, the trace norm (nuclear norm), and the weighted or smoothed weighted trace norms, which have been extensively used in the literature as regularizers for matrix reconstruction problems. We show that this new family can be used to interpolate between the (weighted or unweighted) trace norm and the more conservative max norm. We test this interpolation on simulated data and on the large-scale Netflix and MovieLens ratings data, and find improved accuracy relative to the existing matrix norms. We also provide theoretical results showing learning guarantees for some of the new norms.

We present a method for efficiently training binary and multiclass kernelized SVMs on a Graphics Processing Unit (GPU). Our methods apply to a broad range of kernels, including the popular Gaus- sian kernel, on datasets as large as the amount of available memory on the graphics card. Our approach is distinguished from earlier work in that it cleanly and efficiently handles sparse datasets through the use of a novel clustering technique. Our optimization algorithm is also specifically designed to take advantage of the graphics hardware. This leads to different algorithmic choices then those preferred in serial implementations. Our easy-to-use library is orders of magnitude faster then existing CPU libraries, and several times faster than prior GPU approaches.

Ranking is one of the key problems in information retrieval. Recently, there has been significant interest in a class of ranking algorithms based on the assumption that data is sampled from a low dimensional manifold embedded in a higher dimensional Euclidean space.

Mini-batch algorithms have recently received significant attention as a way to speed-up stochastic convex optimization problems. In this paper, we study how such algorithms can be improved using accelerated gradient methods. We provide a novel analysis, which shows how standard gradient methods may sometimes be insufficient to obtain a significant speed-up. We propose a novel accelerated gradient algorithm, which deals with this deficiency, and enjoys a uniformly superior guarantee. We conclude our paper with experiments on real-world datasets, which validates our algorithm and substantiates our theoretical insights.

We provide rigorous guarantees on learning with the weighted trace-norm under arbitrary sampling distributions. We show that the standard weighted-trace norm might fail when the sampling distribution is not a product distribution (i.e. when row and column indexes are not selected independently), present a corrected variant for which we establish strong learning guarantees, and demonstrate that it works better in practice. We provide guarantees when weighting by either the true or empirical sampling distribution, and suggest that even if the true distribution is known (or is uniform), weighting by the empirical distribution may be beneficial.

We present an optimization approach for linear SVMs based on a stochastic primal-dual approach, where the primal step is akin to an importance-weighted SGD, and the dual step is a stochastic update on the importance weights. This yields an optimization method with a sublinear dependence on the training set size, and the first method for learning linear SVMs with runtime less then the size of the training set required for learning!

We show that for a general class of convex online learning problems, Mirror Descent can always achieve a (nearly) optimal regret guarantee.

The max-norm was proposed as a convex matrix regularizer by Srebro et al (2004) and was shown to be empirically superior to the trace-norm for collaborative filtering problems. Although the max-norm can be computed in polynomial time, there are currently no practical algorithms for solving large-scale optimization problems that incorporate the max-norm. The present work uses a factorization technique of Burer and Monteiro (2003) to devise scalable first-order algorithms for convex programs involving the max-norm. These algorithms are applied to solve huge collaborative filtering, graph cut, and clustering problems. Empirically, the new methods outperform mature techniques from all three areas.

We obtain a tight distribution-specific characterization of the sample complexity of large-margin classification with L2 regularization: We introduce the gamma-adapted-dimension, which is a simple function of the spectrum of a distribution's covariance matrix, and show distribution-specific upper and lower bounds on the sample complexity, both governed by the gamma-adapted-dimension of the source distribution. We conclude that this new quantity tightly characterizes the true sample complexity of large-margin classification. The bounds hold for a rich family of sub-Gaussian distributions.

We show that matrix completion with trace-norm regularization can be significantly hurt when entries of the matrix are sampled non-uniformly, but that a properly weighted version of the trace-norm regularizer works well with non-uniform sampling. We show that the weighted trace-norm regularization indeed yields significant gains on the highly non-uniformly sampled Netflix dataset.

We establish an excess risk bound of O(H Rn^2 + sqrt{H L*} Rn) for ERM with an H-smooth loss function and a hypothesis class with Rademacher complexity Rn, where L* is the best risk achievable by the hypothesis class. For typical hypothesis classes where Rn = sqrt{R/n}, this translates to a learning rate of ̃ O(RH/n) in the separable (L* = 0) case and O(RH/n + sqrt{L* RH/n}) more generally. We also provide similar guarantees for online and stochastic convex optimization of a smooth non-negative objective.

We study the behavior of the popular Laplacian Regularization method for Semi-Supervised Learning at the regime of a fixed number of labeled points but a large number of unlabeled points. We show that in $\R^d$, $d \geq 2$, the method is actually not well-posed, and as the number of unlabeled points increases the solution degenerates to a noninformative function. We also contrast the method with the Laplacian Eigenvector method, and discuss the ``smoothness assumptions associated with this alternate method.

We discuss how the runtime of SVM optimization should decrease as the size of the training data increases. We present theoretical and empirical results demonstrating how a simple subgradient descent approach indeed displays such behavior, at least for linear kernels.

We show that the empirical minimizer of a stochastic strongly convex objective, where the stochastic component is linear, converges to the population minimizer with rate $O(1/n)$. The result applies, in particular, to the SVM objective. Thus, we get a rate of $O(1/n)$ on the convergence of the SVM objective to its infinite data limit. We demonstrate how this is essential for obtaining tight oracle inequalities for SVMs. The results extend also to strong convexity with respect to other $\ellnorm_p$ norms, and so also to objectives regularized using other norms.

This paper suggests a method for multiclass learning with many classes by simultaneously learning shared characteristics common to the classes, and predictors for the classes in terms of these characteristics. We cast this as a convex optimization problem, using trace-norm regularization and study gradient-based optimization both for the linear case and the kernelized setting.

We describe and analyze a simple and effective iterative algorithm for solving the optimization problem cast by Support Vector Machines (SVM). Our method alternates between stochastic gradient descent steps and projection steps. We prove that the number of iterations required to obtain a solution of accuracy ε is Õ(1/ε). In contrast, previous analyses of stochastic gradient descent methods require Ω (1/ε2) iterations. As in previously devised SVM solvers, the number of iterations also scales linearly with 1/λ, where λ is the regularization parameter of SVM. For a linear kernel, the total run-time of our method is Õ (d/(λε)), where d is a bound on the number of non-zero features in each example. Since the run-time does not depend directly on the size of the training set, the resulting algorithm is especially suited for learning from large datasets. Our approach can seamlessly be adapted to employ non-linear kernels while working solely on the primal objective function. We demonstrate the efficiency and applicability of our approach by conducting experiments on large text classification problems, comparing our solver to existing state-of-the-art SVM solvers. For example, it takes less than 5 seconds for our solver to converge when solving a text classification problem from Reuters Corpus Volume 1 (RCV1) with 800,000 training examples.

We investigate under what conditions clus-tering by learning a mixture of spherical Gaussians is (a) computationally tractable; and (b) statistically possible. We show that using principal component projection greatly aids in recovering the clustering using EM; present empirical evidence that even using such a projection, there is still a large gap between the number of samples needed to re-cover the clustering using EM, and the num-ber of samples needed without computational restrictions; and characterize the regime in which such a gap exists. 1.

Maximum Margin Matrix Factorization (MMMF) was recently suggested (Srebro et al., 2005) as a convex, infinite dimensional alternative to low-rank approximations and standard factor models. MMMF can be formulated as a semi-definite programming (SDP) and learned using standard SDP solvers. However, current SDP solvers can only handle MMMF problems on matrices of dimensionality up to a few hundred. Here, we investigate a direct gradient-based optimization method for MMMF and demonstrate it on large collaborative prediction problems. We compare against results obtained by Marlin (2004) and find that MMMF substantially outperforms all nine methods he tested.

We prove generalization error bounds for predicting entries in a partially observed matrix by fitting the observed entries with a low-rank matrix. In justifying the analysis approach we take to obtain the bounds, we present an example of a class of functions of finite pseudodimension such that the sums of functions from this class have unbounded pseudodimension. 1 Introduction "Collaborative filtering" refers to the general task of providing users with information on what items they might like, or dislike, based on their preferences so far and how they relate to the preferences of other users. This approach contrasts with a more traditional feature- based approach where predictions are made based on features of the items. For feature-based approaches, we are accustomed to studying prediction methods in terms of probabilistic post-hoc generalization error bounds. Such results provide us a (proba- bilistic) bound on the performance of our predictor on future examples, in terms of its performance on the training data. These bounds hold without any assumptions on the true "model", that is the true dependence of the labels on the features, other than the central assumptions that the training examples are drawn i.i.d. from the distribution of interest. In this paper we suggest studying the generalization ability of collaborative prediction methods. By "collaborative prediction" we indicate that the objective is to be able to pre- dict user preferences for items, that is, entries in some unknown target matrix Y of user- item "ratings", based on observing a subset YS of the entries in this matrix1. We present 1In other collaborative filtering tasks, the objective is to be able to provide each user with a few items that overlap his top-rated items, while it is not important to be able to correctly predict the users ratings for other items. Note that it is possible to derive generalization error bounds for this objective based on bounds for the "prediction" objective. arbitrary source distribution target matrix Y random training set random set S of observed entries hypothesis predicted matrix X training error observed discrepancy DS(X; Y ) generalization error true discrepancy D(X; Y ) Figure 1: Correspondence with post-hoc bounds on the generalization error for standard feature-based prediction tasks bounds on the true average overall error D(X; Y ) = 1 n m loss(X nm i=1 a=1 ia; Yia) of the predictions X in terms of the average error over the observed entries DS(X; Y ) = 1 loss(X |S| iaS ia; Yia), without making any assumptions on the true nature of the pref- erences Y . What we do assume is that the subset S of entries that we observe is chosen uniformly at random. This strong assumption parallels the i.i.d. source assumption for feature-based prediction. In particular, we present generalization error bounds on prediction using low-rank models. Collaborative prediction using low-rank models is fairly straight forward. A low-rank ma- trix X is sought that minimizes the average observed error DS(X; Y ). Unobserved entries in Y are then predicted according to X. The premise behind such a model is that there are only a small number of factors influencing the preferences, and that a user's preference vector is determined by how each factor applies to that user. Different methods differ in how they relate real-valued entries in X to preferences in Y , and in the associated measure of discrepancy. For example, entries in X can be seen as parameters for a probabilistic models of the entries in Y , either mean parameters [1] or natural parameters [2], and a maximum likelihood criterion used. Or, other loss functions, such as squared error [3, 2], or zero-one loss versus the signs of entries in X, can be minimized. Prior Work Previous results bounding the error of collaborative prediction using a low- rank matrix all assume the true target matrix Y is well-approximated by a low-rank matrix. This corresponds to a large eigengap between the top few singular values of Y and the remaining singular values. Azar et al [3] give asymptotic results on the convergence of the predictions to the true preferences, assuming they have an eigengap. Drineas et al [4] analyze the sample complexity needed to be able to predict a matrix with an eigengap, and suggests strategies for actively querying entries in the target matrix. To our knowledge, this is the first analysis of the generalization error of low-rank methods that do not make any assumptions on the true target matrix. Generalization error bounds (and related online learning bounds) were previously discussed for collaborative prediction applications, but only when prediction was done for each user separately, using a feature-based method, with the other user's preferences as features [5, 6]. Although these address a collaborative prediction application, the learning setting is a standard feature-based setting. These methods are also limited, in that learning must be performed separately for each user. Shaw-Taylor et al [7] discuss assumption-free post-hoc bounds on the residual errors of low-rank approximation. These results apply to a different setting, where a subset of the rows are fully observed, and bound a different quantity--the distance between rows and the learned subspace, rather then the distance to predicted entries. Organization In Section 2 we present a generalization error bound for zero-one loss, based on a combinatorial result which we prove in Section 3. In Section 4 we generalize the bound to arbitrary loss functions. Finally, in Section 5 we justify the combinatorial approach taken, by considering an alternate approach (viewing rank-k matrices as combi- nation of k rank-1 matrices) and showing why it does not work. 2 Generalization Error Bound for Zero-One Error We begin by considering binary labels Yia and a zero-one sign agreement loss: loss(Xia; Yia) = 1YiaXia0 (1) Theorem 1. For any matrix Y {1}nm, n, m > 2, > 0 and integer k, with proba- bility at least 1 - over choosing a subset S of entries in Y uniformly among all subsets of |S| entries, the discrepancy with respect to the zero-one sign agreement loss satisfies2: k(n + m) log 16em - log k X,rank X<k D(X ; Y ) < D (X ; Y ) + S 2|S| To prove the theorem we employ standard arguments about the generalization error for finite hypothesis classes with bounded cardinality. First fix Y as well as X nm R . When an index pair (i, a) is chosen uniformly at random, loss(Xia; Yia) is a Bernoulli random variable with probability D(X; Y ) of being one. If the entries of S are chosen independently and uniformly, |S|D(X; Y ) is Binomially S distributed with mean |S|D(X; Y ) and using Chernoff's inequality: Pr D(X; Y ) D(X; Y ) + e-2|S| 2 (2) S S The distribution of S in Theorem 1 is slightly different, as S is chosen without repetitions. The mean of D(X; Y ) is the same, but it is more concentrated, and (2) still holds. S Now consider all rank-k matrices. Noting that loss(Xia; Yia) depends only on the sign of Xia, it is enough to consider the equivalence classes of matrices with the same sign patterns. Let f (n, m, k) be the number of such equivalence classes, i.e. the number of possible sign configurations of n m matrices of rank at most k: F (n, m, k) = {sign X {-, 0, +}nm|X nm R , rank X k} f (n, m, k) = F (n, m, k) 1 If Xia > 0 where sign X denotes the element-wise sign matrix (sign X)ia = 0 If Xia = 0 . -1 If Xia < 1 For all matrices in an equivalence class, the random variable D(X; Y ) is the same, and S taking a union bound of the events D(X; Y ) D(X; Y )+ for each of these f (n, m, k) S random variables we have: log f (n, m, k) - log Pr X,rank XkD(X; Y ) D(X; Y ) + (3) S S 2|S| by using (2) and setting = log f (n,m,k)-log . The proof of Theorem 1 rests on bounding 2|S| f (n, m, k), which we will do in the next section. Note that since the equivalence classes we defined do not depend on the sample set, no symmetrization argument is necessary. 2All logarithms are base two 3 Sign Configurations of a Low-Rank Matrix In this section, we bound the number f (n, m, k) of sign configurations of n m rank- k matrices over the reals. Such a bound was previously considered in the context of unbounded error communication complexity. Alon, Frankl and Rodl [8] showed that f (n, m, k) minh (8 nm/h )(n+m)k+h+m, and used counting arguments to establish that some (in fact, most) binary matrices can only be realized by high-rank matrices, and therefore correspond to functions with high unbounded error communication complexity. Here, we follow a general course outlined by Alon [9] to obtain a simpler, and slightly tighter, bound based on the following result due to Warren: Let P1, . . . , Pr be real polynomials in q variables, and let C be the complement of the variety defined by iPi, i.e. the set of points in which all the m polynomials are non-zero: C = {x q R |iPi(x) = 0} Theorem 2 (Warren [10]). If all r polynomials are of degree at most d, then the number of connected components of C is at most: q q r 4edr c(C) 2(2d)q 2i i q i=0 where the second inequality holds when r > q > 2. The signs of the polynomials P1, . . . , Pr are fixed inside each connected component of C. And so, c(C) bounds the number of sign configurations of P1, . . . , Pr that do not contain zeros. To bound the overall number of sign configurations the polynomials are modified slightly (see Appendix), yielding: Corollary 3 ([9, Proposition 5.5]). The number of -/0/+ sign configurations of r polyno- mials, each of degree at most d, over q variables, is at most (8edr/q)q (for r > q > 2). In order to apply these bounds to low-rank matrices, recall that any matrix X of rank at most k can be written as a product X = U V where U nk km R and V R . Consider the k(n+m) entries of U, V as variables, and the nm entries of X as polynomials of degree two over these variables: k Xia = UiVa =1 Applying Corollary 3 we obtain: k(n+m) Lemma 4. f (n, m, k) 8e2nm (16em/k)k(n+m) k(n+m) Substituting this bound in (3) establishes Theorem 1. The upper bound on f (n, m, k) is tight up to a multiplicative factor in the exponent: 1 Lemma 5. For m > k2, f (n, m, k) m (k-1)n 2 Proof. Fix any matrix V mk R with rows in general position, and consider the number f (n, V, k) of sign configurations of matrices U V , where U varies over all n k matrices. Focusing only on +/- sign configurations (no zeros in U V ), each row of sign U V is a homogeneous linear classification of the rows of V , i.e. of m vectors in general position in k m R . There are exactly 2 k-1 possible homogeneous linear classifications of m i=0 i vectors in general position in k R , and so these many options for each row of sign U V . We can therefore bound: n k-1 n n(k-1) 1 f (n, m, k) f (n, V, k) 2 m m m = m (k-1)n 2 i k-1 k-1 i=0 4 Generalization Error Bounds for Other Loss Functions In Section 2 we considered generalization error bounds for a zero-one loss function. More commonly, though, other loss functions are used, and it is desirable to obtain generalization error bounds for general loss functions. When dealing with other loss functions, the magnitude of the entries in the matrix are important, and not only their signs. It is therefore no longer enough to bound the number of sign configurations. Instead, we will bound not only the number of ways low rank matrices behave with regards to a threshold of zero, but the number of possible ways low- rank matrices can behave relative to any set of thresholds. That is, for any threshold matrix T nm R , we will show that the number of possible sign configurations of (X - T ), where X is low-rank, is small. Intuitively, this captures the complexity of the class of low-rank matrices not only around zero, but throughout all possible values. We then use standard results from statistical machine learning to obtain generalization error bounds from the bound on the number of relative sign configurations. The number of rela- tive sign configurations serves as a bound on the pseudodimension--the maximum number of entries for which there exists a set of thresholds such that all relative sign configurations (limited to these entries) is possible. The pseudodimension can in turn be used to show the existence of a small -net, which is used to obtain generalization error bounds. Recall the definition of the pseudodimension of a class of real-valued functions: Definition 1. A class F of real-valued functions pseudo-shatters the points x1, . . . , xn with thresholds t1, . . . , tn if for every binary labeling of the points (s1, . . . , sn) {+, -}n there exists f F s.t. f (xi) ti iff si = -. The pseudodimension of a class F is the supremum over n for which there exist n points and thresholds that can be shattered. In order to apply known results linking the pseudodimension to covering numbers, we consider matrices X nm R as real-valued functions X : [n] [m] R over index pairs to entries in the matrix. The class Xk of rank-k matrices can now be seen as a class of real-valued functions over the domain [n] [m]. We bound the pseudodimension of this class by bounding, for any threshold matrix T nm R the number of relative sign matrices: F nm T (n, m, k) = {sign (X - T ) {-, 0, +}nm|X R , rank X k} fT (n, m, k) = FT (n, m, k) k(n+m) Lemma 6. For any T nm R , we have fT (n, m, k) 16em . k Proof. We take a similar approach to that of Lemma 4, writing rank-k matrices as a product X = U V where U nk km R and V R . Consider the k(n + m) entries of U, V as variables, and the nm entries of X - T as polynomials of degree two over these variables: k (X - T )ia = UiVa - Tia =1 Applying Corollary 10 yields the desired bound. Corollary 7. The pseudodimension of the class Xk of n m matrices over the reals of rank at most k, is at most k(n + m) log 16em . k We can now invoke standard generalization error bounds in terms of the pseudodimension (Theorem 11 in the Appendix) to obtain: Theorem 8. For any monotone loss function with |loss| M , any matrix Y {1}nm, n, m > 2, > 0 and integer k, with probability at least 1 - over choosing a subset S of entries in Y uniformly among all subsets of |S| entries: k(n + m) log 16em log M|S| - log k k(n+m) X,rank X<k D(X ; Y ) < DS (X ; Y ) + 6 |S| 5 Low-Rank Matrices as Combined Classifiers Rank-k matrices are those matrices which are a sum of k rank-1 matrices. If we view matrices as functions from pairs of indices to the reals, we can think of rank-k matrices as "combined" classifiers, and attempt to bound their complexity as such, based on the low complexity of the "basis" functions, i.e. rank-1 matrices. A similar approach is taken in related work on learning with low-norm (maximum margin) matrix factorization [11, 12], where the hypothesis class can be viewed as a convex combi- nation of rank-1 unit-norm matrices. Scale-sensitive (i.e. dependent on the margin, or the slope of the loss function) generalization error bounds for this class are developed based on the graceful behavior of scale-sensitive complexity measures (e.g. log covering numbers and the Rademacher complexity) with respect to convex combinations. Taking a similar view, it is possible to obtain scale-sensitive generalization error bounds for low-rank ma- trices. In this Section we question whether it is possible to obtain scale-insensitive bounds, similar to Theorems 1 and 8, by viewing low-rank matrices as combined classifiers. It cannot be expected that scale-insensitive complexity would be preserved when taking convex combinations of an unbounded number of base functions. However, the VC- dimension, a scale-insensitive measure of complexity, does scale gracefully when taking linear combinations of a bounded number of functions from a low VC-dimension class of indicator function. Using this, we can obtain generalization error bounds for linear com- binations of signs of rank-one matrices, but not signs of linear combinations of rank-one matrices. An alternate candidate scale-insensitive complexity measure is the pseudodi- mension of a class of real-valued functions. If we could bound the pseudodimension of the class of sums of k functions from a bounded-pseudodimension base class of real valued functions, we could avoid the sign-configuration counting and obtain generalization error bounds for rank-k matrices. Unfortunately, the following counterexample shows that this is not possible. Theorem 9. There exists a family F closed under scalar multiplication whose pseudodi- mension is at most five, and such that {f1 + f2|f1, f2 F } does not have a finite pseu- dodimension. Proof. We describe a class F of real-valued functions over the positive integers N. To do so, consider a one-to-one mapping of finite sets of positive integers to the positive integers. For each A N define two functions3, fA(x) = 2xA + 1xA and gA(x) = 2xA. Let F be the set of all scalar multiplications of these functions. For every A N , fA - gA is the indicator function of A, implying that every finite subset can be shattered, and the pseudodimension of {f1 + f2 : f1, f2 F } is unbounded. It remains to show that the pseudodimension of F is less than six. To do so, we note that there are no positive integers A < B and x < y and positive reals , > 0 such that (2xB + 1) > 2xA and 2yB < (2yA + 1). It follows that for any A < B and any , > 0, on an initial segment (possibly empty) of N we have gB fB gA fA while on the rest of N we have gA fA < gB fB. In particular, any pair of 3We use A to refer both to a positive integer and the finite set it maps to. functions (fA, fB) or (fA, gB) or (gA, gB) in F that are not associated with the same subset (i.e. A = B), cross each other at most once. This holds also when or are negative, as the functions never change signs. For any six naturals x1 < x2 < < x6 and six thresholds, consider the three labellings (+, -, +, -, +, -), (-, +, -, +, -, +), (+, +, -, -, +, +). The three functions realizing these labellings must cross each other at least twice, but by the above arguments, there are no three functions in F such that every pair crosses each other at least twice.4 6 Discussion Alon, Frankl and Rodl [8] use a result of Milnor similar to Warren's Theorem 2. Milnor's and Warren's theorems were previously used for bounding the VC-dimension of certain geometric classes [13], and of general concept classes parametrized by real numbers, in terms of the complexity of the boolean formulas over polynomials used to represent them [14]. This last general result can be used to bound the VC-dimension of signs of nm rank- k matrices by 2k(n + m) log(48enm), yielding a bound similar to Theorem 1 with an extra log |S| term. In this paper, we take a simpler path, applying Warren's theorem directly, and thus avoiding the log |S| term and reducing the other logarithmic term. Applying Warren's theorem directly also enables us to bound the pseudodimension and obtain the bound of Theorem 8 for general loss functions. Another notable application of Milnor's result, which likely inspired these later uses, is for bounding the number of configurations of n points in d R with different possible linear clas- sifications [15, 16]. Viewing signs of rank-k n m matrices as n linear classification of m points in k R , this bound can be used to bound f (n, m, k) < 2km log 2n+k(k+1)n log n with- out using Warren's Theorem directly [8, 12]. The bound of Lemma 4 avoids the quadratic dependence on k in the exponent. Acknowledgments We would like to thank Peter Bartlett for pointing out [13, 14]. N.S. and T.J. would like to thank Erik Demaine for introducing them to oriented matroids. A Proof of Corollary 3 Consider a set R q R containing one variable configuration for each possible sign pattern. Set . = 1 min (x) = P 2 1iq,xRPi(x)=0 |Pi(x)| > 0. Now consider the 2q polynomials P + i i(x) + and P -(x) = P q | (x) = 0, P -(x) = 0 . Different points in R i i(x) - and C = x R iP + i i (representing all sign configurations) lie in different connected components of C . Invoking Theorem 2 on C establishes Corollary 3. The count in Corollary 3 differentiates between positive, negative and zero signs. However, we are only concerned with the positivity of YiaXia (in the proof of Theorem 1) or of Xia - Tia (in the proof of Theorem 8), and do not need to differentiate between zero and negative values. Invoking Theorem 2 on C+ = x q R |iP +(x) = 0 , yields: i Corollary 10. The number of -/+ sign configurations (where zero is considered negative) of r poly- nomials, each of degree at most d, over q variables, is at most (4edr/q)q (for r > q > 2). Applying Corollary 10 on the nm degree-two polynomials Y k ia U =1 iVa establishes that for any Y , the number of configurations of sign agreements of rank-k matrices with Y is bounded by (8em/k)k(n+m) and yields a constant of 8 instead of 16 inside the logarithm in Theorem 1. Applying Corollary 10 instead of Corollary 3 allows us to similarly tighten in the bounds in Corollary 7 and in Theorem 8. 4A more careful analysis shows that F has pseudodimension three. B Generalization Error Bound in terms of the Pseudodimension Theorem 11. Let F be a class of real-valued functions f : X R with pseudodimension d, and loss : R Y R be a bounded monotone loss function (i.e. for all y, loss(x, y) is mono- tone in x), with loss < M . For any joint distribution over (X, Y ), consider an i.i.d. sample S = (X1, Y1), . . . , (Xn, Yn). Then for any > 0: n d 1 32eM 2 n Pr fF EX,Y [loss(f (X), Y )] > loss(f (Xi), Yi) + < 4e(d + 1) e- 32 S n i=1 The bound is a composition of a generalization error bound in terms of the L1 covering number [17, Theorem 17.1], a bound on the L1 covering number in terms of the pseudodimension [18] and the observation that composition with a monotone function does not increase the pseudodimension [17, Theorem 12.3].

We present a novel approach to collaborative prediction, using low-norm instead of low-rank factorizations. The approach is inspired by, and has strong connections to, large-margin linear discrimination. We show how to learn low-norm factorizations by solving a semi-definite program, and discuss generalization error bounds for them.

We formulate linear dimensionality reduction as a semi-parametric esti- mation problem, enabling us to study its asymptotic behavior. We gen- eralize the problem beyond additive Gaussian noise to (unknown) non- Gaussian additive noise, and to unbiased non-additive models.

MOTIVATION: A major challenge in gene expression analysis is effective data organization and visualization. One of the most popular tools for this task is hierarchical clustering. Hierarchical clustering allows a user to view relationships in scales ranging from single genes to large sets of genes, while at the same time providing a global view of the expression data. However, hierarchical clustering is very sensitive to noise, it usually lacks of a method to actually identify distinct clusters, and produces a large number of possible leaf orderings of the hierarchical clustering tree. In this paper we propose a new hierarchical clustering algorithm which reduces susceptibility to noise, permits up to k siblings to be directly related, and provides a single optimal order for the resulting tree. RESULTS: We present an algorithm that efficiently constructs a k-ary tree, where each node can have up to k children, and then optimally orders the leaves of that tree. By combining k clusters at each step our algorithm becomes more robust against noise and missing values. By optimally ordering the leaves of the resulting tree we maintain the pairwise relationships that appear in the original method, without sacrificing the robustness. Our k-ary construction algorithm runs in O(n(3)) regardless of k and our ordering algorithm runs in O(4(k)n(3)). We present several examples that show that our k-ary clustering algorithm achieves results that are superior to the binary tree results in both global presentation and cluster identification. AVAILABILITY: We have implemented the above algorithms in C++ on the Linux operating system.