COLT 2023 Papers

We study repeated bilateral trade where an adaptive $\sigma$-smooth adversary generates the valuations of sellers and buyers. We provide a complete characterization of the regret regimes for fixed-price mechanisms under different feedback models in the two cases where the learner can post either the same or different prices to buyers and sellers.We begin by showing that the minimax regret after $T$ rounds is of order $\sqrt{T}$ in the full-feedback scenario. Under partial feedback, any algorithm that has to post the same price to buyers and sellers suffers worst-case linear regret. However, when the learner can post two different prices at each round, we design an algorithm enjoying regret of order $T^{3/4}$ ignoring log factors.We prove that this rate is optimal by presenting a surprising $T^{3/4}$ lower bound, which is the main technical contribution of the paper.

We study robust community detection in the context of node-corrupted stochastic block model, where an adversary can arbitrarily modify all the edges incident to a fraction of the n vertices. We present the first polynomial-time algorithm that achieves weak recovery at the Kesten-Stigum threshold even in the presence of a small constant fraction of corrupted nodes. Prior to this work, even state-of-the-art robust algorithms were known to break under such node corruption adversaries, when close to the Kesten-Stigum threshold.We further extend our techniques to the $Z_2$ synchronization problem, where our algorithm reaches the optimal recovery threshold in the presence of similar strong adversarial perturbations.The key ingredient of our algorithm is a novel identifiability proof that leverages the push-out effect of the Grothendieck norm of principal submatrices.

The aim of this paper is to design computationally-efficient and optimal algorithms for the online and stochastic exp-concave optimization settings. Typical algorithms for these settings, such as the Online Newton Step (ONS), can guarantee a $O(d\ln T)$ bound on their regret after $T$ rounds, where $d$ is the dimension of the feasible set. However, such algorithms perform so-called \emph{generalized projections} whenever their iterates step outside the feasible set. Such generalized projections require $\Omega(d^3)$ arithmetic operations even for simple sets such a Euclidean ball, making the total runtime of ONS of order $d^3 T$ after $T$ rounds, in the worst-case. In this paper, we side-step generalized projections by using a self-concordant barrier as a regularizer to compute the Newton steps. This ensures that the iterates are always within the feasible set without requiring projections. This approach still requires the computation of the inverse of the Hessian of the barrier at every step. However, using stability properties of the Newton iterates, we show that the inverse of the Hessians can be efficiently approximated via Taylor expansions for most rounds, resulting in a $\widetilde O(d^2 T +d^\omega \sqrt{T})$ total computational complexity, where $\omega\in(2,3]$ is the exponent of matrix multiplication. In the stochastic setting, we show that this translates into a $\widetilde O(d^3/\varepsilon)$ computational complexity for finding an $\varepsilon$-optimal point, answering an open question by Koren 2013. We first prove these new results for the simple case where the feasible set is a Euclidean ball. Then, to move to general convex sets, we use a reduction to Online Convex Optimization over the Euclidean ball. Our final algorithm for general convex sets can be viewed as a more computationally-efficient version of ONS.

In the problem of quantum channel certification, we have black box access to a quantum process and would like to decide if this process matches some predefined specification or is $\eps$-far from this specification. The objective is to achieve this task while minimizing the number of times the black box is used. Note that the state certification problem is a special case where the black box has no input. Here, we focus on two relevant extreme cases. The first one is when the predefined specification is a unitary channel, e.g., a gate in a quantum circuit. In this case, we show that testing whether the black box is described by a fixed unitary or $\eps$-far from it in the trace norm requires $\Theta(d/\eps^2)$ uses of the black box. The second setting we consider is when the predefined specification is a completely depolarizing channels with input dimension $\din$ and output dimension $\dout$. In this case, we prove that, in the non-adaptive setting, $\Tilde{\Theta}(\din^2\dout^{1.5}/\eps^2)$ uses of the channel are necessary and sufficient to verify whether it is equal to the depolarizing channel or $\eps$-far from it in the diamond norm. Finally, we prove a lower bound of $\Omega(\din^2\dout/\eps^2)$ for this problem in the adaptive setting. Note that the special case $\din = 1$ corresponds to the well-studied quantum identity testing problem.

We give query complexity lower bounds for convex optimization and the related feasibility problem. We show that quadratic memory is necessary to achieve the optimal oracle complexity for first-order convex optimization. In particular, this shows that center-of-mass cutting-planes algorithms in dimension $d$ which use $\tilde O(d^2)$ memory and $\tilde O(d)$ queries are Pareto-optimal for both convex optimization and the feasibility problem, up to logarithmic factors. Precisely, we prove that to minimize $1$-Lipschitz convex functions over the unit ball to $1/d^4$ accuracy, any deterministic first-order algorithms using at most $d^{2-\delta}$ bits of memory must make $\tilde\Omega(d^{1+\delta/3})$ queries, for any $\delta\in[0,1]$. For the feasibility problem, in which an algorithm only has access to a separation oracle, we show a stronger trade-off: for at most $d^{2-\delta}$ memory, the number of queries required is $\tilde\Omega(d^{1+\delta})$. This resolves a COLT 2019 open problem of Woodworth and Srebro.

We study the problem of representational transfer in RL, where an agent first pretrains in a number of \emph{source tasks} to discover a shared representation, which is subsequently used to learn a good policy in a \emph{target task}. We propose a new notion of task relatedness between source and target tasks, and develop a novel approach for representational transfer under this assumption. Concretely, we show that given a generative access to source tasks, we can discover a representation, using which subsequent linear RL techniques quickly converge to a near-optimal policy in the target task. The sample complexity is close to knowing the ground truth features in the target task, and comparable to prior representation learning results in the source tasks. We complement our positive results with lower bounds without generative access, and validate our findings with empirical evaluation on rich observation MDPs that require deep exploration. In our experiments, we observe speed up in learning in the target by pre-training, and also validate the need for generative access in source tasks.

Proper losses (or proper scoring rules) have been used for over half a century to elicit users’ subjective probability from the observations. In the recent machine learning community, we often tackle downstream tasks such as classification and bipartite ranking with the elicited probabilities. Here, we engage in assessing the quality of the elicited probabilities with different proper losses, which can be characterized by surrogate regret bounds to describe the convergence speed of an estimated probability to the optimal one when optimizing a proper loss. This work contributes to a sharp analysis of surrogate regret bounds in two ways. First, we provide general surrogate regret bounds for proper losses measured by the $L^1$ distance. This abstraction eschews a tailor-made analysis of each downstream task and delineates how universally a loss function operates. Our analysis relies on a classical mathematical tool known as the moduli of convexity, which is of independent interest per se. Second, we evaluate the surrogate regret bounds with polynomials to identify the quantitative convergence rate. These devices enable us to compare different losses, with which we can confirm that the lower bound of the surrogate regret bounds is $\Omega(\epsilon^{1/2})$ for popular loss functions.

We consider the setting of online convex optimization (OCO) with \textit{exp-concave} losses. The best regret bound known for this setting is $O(n\log{}T)$, where $n$ is the dimension and $T$ is the number of prediction rounds (treating all other quantities as constants and assuming $T$ is sufficiently large), and is attainable via the well-known Online Newton Step algorithm (ONS). However, ONS requires on each iteration to compute a projection (according to some matrix-induced norm) onto the feasible convex set, which is often computationally prohibitive in high-dimensional settings and when the feasible set admits a non-trivial structure. In this work we consider projection-free online algorithms for exp-concave and smooth losses, where by projection-free we refer to algorithms that rely only on the availability of a linear optimization oracle (LOO) for the feasible set, which in many applications of interest admits much more efficient implementations than a projection oracle. We present an LOO-based ONS-style algorithm, which using overall $O(T)$ calls to a LOO, guarantees in worst case regret bounded by $\widetilde{O}(n^{2/3}T^{2/3})$ (ignoring all quantities except for $n,T$). However, our algorithm is most interesting in an important and plausible low-dimensional data scenario: if the gradients (approximately) span a subspace of dimension at most $\rho$, $\rho << n$, the regret bound improves to $\widetilde{O}(\rho^{2/3}T^{2/3})$, and by applying standard deterministic sketching techniques, both the space and average additional per-iteration runtime requirements are only $O(\rho{}n)$ (instead of $O(n^2)$). This improves upon recently proposed LOO-based algorithms for OCO which, while having the same state-of-the-art dependence on the horizon $T$, suffer from regret/oracle complexity that scales with $\sqrt{n}$ or worse.

Online prediction from experts is a fundamental problem in machine learning and several works have studied this problem under privacy constraints. We propose and analyze new algorithms for this problem that improve over the regret bounds of the best existing algorithms for non-adaptive adversaries. For approximate differential privacy, our algorithms achieve regret bounds of $\wt O(\sqrt{T \log d} + \log d/\eps)$ for the stochastic setting and $\wt O(\sqrt{T \log d} + T^{1/3} \log d/\eps)$ for oblivious adversaries (where $d$ is the number of experts). For pure DP, our algorithms are the first to obtain sub-linear regret for oblivious adversaries in the high-dimensional regime $d \ge T$. Moreover, we prove new lower bounds for adaptive adversaries. Our results imply that unlike the non-private setting, there is a strong separation between the optimal regret for adaptive and non-adaptive adversaries for this problem. Our lower bounds also show a separation between pure and approximate differential privacy for adaptive adversaries where the latter is necessary to achieve the non-private $O(\sqrt{T})$ regret.

We consider the problem of approximating a $d \times d$ covariance matrix $M$ with a rank-$k$ matrix under $(\varepsilon,\delta)$-differential privacy. We present and analyze a complex variant of the Gaussian mechanism and show that the Frobenius norm of the difference between the matrix output by this mechanism and the best rank-$k$ approximation to $M$ is bounded by roughly $\tilde{O}(\sqrt{kd})$, whenever there is an appropriately large gap between the $k$’th and the $k+1$’th eigenvalues of $M$. This improves on previous work that requires that the gap between every pair of top-$k$ eigenvalues of $M$ is at least $\sqrt{d}$ for a similar bound. Our analysis leverages the fact that the eigenvalues of complex matrix Brownian motion repel more than in the real case, and uses Dyson’s stochastic differential equations governing the evolution of its eigenvalues to show that the eigenvalues of the matrix $M$ perturbed by complex Gaussian noise have large gaps with high probability. Our results contribute to the analysis of low-rank approximations under average-case perturbations and to an understanding of eigenvalue gaps for random matrices, which may be of independent interest.

We provide exact asymptotic expressions for the performance of regression by an $L-$layer deep random feature (RF) model, where the input is mapped through multiple random embedding and non-linear activation functions. For this purpose, we establish two key steps: First, we prove a novel universality result for RF models and deterministic data, by which we demonstrate that a deep random feature model is equivalent to a deep linear Gaussian model that matches it in the first and second moments, at each layer. Second, we make use of the convex Gaussian Min-Max theorem multiple times to obtain the exact behavior of deep RF models. We further characterize the variation of the eigendistribution in different layers of the equivalent Gaussian model, demonstrating that depth has a tangible effect on model performance despite the fact that only the last layer of the model is being trained.

Goldwasser et al. (2021) recently proposed the setting of PAC verification, where a hypothesis (machine learning model) that purportedly satisfies the agnostic PAC learning objective is verified using an interactive proof. In this paper we develop this notion further in a number of ways. First, we prove a lower bound of $\Omega(\sqrt{d}/\varepsilon^2)$ i.i.d. samples for PAC verification of hypothesis classes of VC dimension $d$. Second, we present a protocol for PAC verification of unions of intervals over $\mathbb{R}$ that improves upon their proposed protocol for that task, and matches our lower bound’s dependence on $d$. Third, we introduce a natural generalization of their definition to verification of general statistical algorithms, which is applicable to a wider variety of settings beyond agnostic PAC learning. Showcasing our proposed definition, our final result is a protocol for the verification of statistical query algorithms that satisfy a combinatorial constraint on their queries.

We revisit the canonical problem of learning a single neuron with ReLU activation under Gaussian input with square loss. We particularly focus on the over-parameterization setting where the student network has $n\ge 2$ neurons. We prove the global convergence of randomly initialized gradient descent with a $O\left(T^{-3}\right)$ rate. This is the first global convergence result for this problem beyond the exact-parameterization setting ($n=1$) in which the gradient descent enjoys an $\exp(-\Omega(T))$ rate. Perhaps surprisingly, we further present an $\Omega\left(T^{-3}\right)$ lower bound for randomly initialized gradient flow in the over-parameterization setting. These two bounds jointly give an exact characterization of the convergence rate and imply, for the first time, that \emph{over-parameterization can exponentially slow down the convergence rate}. To prove the global convergence, we need to tackle the interactions among student neurons in the gradient descent dynamics, which are not present in the exact-parameterization case. We use a three-phase structure to analyze GD’s dynamics. Along the way, we prove gradient descent automatically balances student neurons and uses this property to deal with the non-smoothness of the objective function. To prove the convergence rate lower bound, we construct a novel potential function that characterizes the pairwise distances between the student neurons (which cannot be done in the exact-parameterization case). We show this potential function converges slowly, which implies the slow convergence rate of the loss function.

We consider the problem of minimizing a non-convex function over a smooth manifold M. We propose a novel algorithm, the Orthogonal Directions Constrained Gradient Method (ODCGM), which only requires computing a projection onto a vector space. ODCGM is infeasible but the iterates are constantly pulled towards the manifold, ensuring the convergence of ODCGM towards M. ODCGM is much simpler to implement than the classical methods which require the computation of a retraction. Moreover, we show that ODCGM exhibits the near-optimal oracle complexities O(1/ε^{-2}) and O(1/ε^{-4}) in the deterministic and stochastic cases, respectively. Furthermore, we establish that, under an appropriate choice of the projection metric, our method recovers the landing algorithm of Ablin and Peyré (2022), a recently introduced algorithm for optimization over the Stiefel manifold. As a result, we significantly extend the analysis of Ablin and Peyré (2022), establishingnear-optimal rates both in deterministic and stochastic frameworks. Finally, we perform numerical experiments, which shows the efficiency of ODCGM in a high-dimensional setting.

Smoothed online learning has emerged as a popular framework to mitigate the substantial loss in statistical and computational complexity that arises when one moves from classical to adversarial learning. Unfortunately, for some spaces, it has been shown that efficient algorithms suffer an exponentially worse regret than that which is minimax optimal, even when the learner has access to an optimization oracle over the space. To mitigate that exponential dependence, this work introduces a new notion of complexity, the generalized bracketing numbers, which marries constraints on the adversary to the size of the space, and shows that an instantiation of Follow-the-Perturbed-Leader can attain low regret with the number of calls to the optimization oracle scaling optimally with respect to average regret. We then instantiate our bounds in several problems of interest, including online prediction and planning of piecewise continuous functions, which has many applications in fields as diverse as econometrics and robotics.

This paper develops a framework for the design of scoring rules to optimally incentivize an agent to exert a multi-dimensional effort. This framework is a generalization to strategic agents of the classical knapsack problem (cf. Briest, Krysta, and Vocking, 2005; Singer, 2010) and it is foundational to applying algorithmic mechanism design to the classroom. The paper identifies two simple families of scoring rules that guarantee constant approximations to the optimal scoring rule. The truncated separate scoring rule is the sum of single dimensional scoring rules that is truncated to the bounded range of feasible scores. The threshold scoring rule gives the maximum score if reports exceed a threshold and zero otherwise. Approximate optimality of one or the other of these rules is similar to the bundling or selling separately result of Babaioff, Immorlica, Lucier, and Weinberg (2014). Finally, we show that the approximate optimality of the best of those two simple scoring rules is robust when the agent’s choice of effort is made sequentially.

A classical result in online learning characterizes the optimal mistake bound achievable by deterministic learners using the Littlestone dimension (Littlestone ’88).We prove an analogous result for randomized learners: we show that the optimal expected mistake bound in learning a class $\mathcal{H}$ equals its randomized Littlestone dimension, which we define as follows: it is the largest $d$ for which there exists a tree shattered by $\mathcal{H}$ whose average depth is $2d$.We further study optimal mistake bounds in the agnostic case, as a function of the number of mistakes made by the best function in $\mathcal{H}$, denoted by $k$. Towards this end we introduce the $k$-Littlestone dimension and its randomized variant, and use them to characterize the optimal deterministic and randomized mistake bounds.Quantitatively, we show that the optimal randomized mistake bound for learning a class with Littlestone dimension $d$ is $k + \Theta (\sqrt{k d} + d )$ (equivalently, the optimal regret is $\Theta(\sqrt{kd} + d$). This also implies an optimal deterministic mistake bound of $2k + O (\sqrt{k d} + d )$, thus resolving an open question which was studied by Auer and Long [’99]. As an application of our theory, we revisit the classical problem of prediction using expert advice: about 30 years ago Cesa-Bianchi, Freund, Haussler, Helmbold, Schapire and Warmuth studied prediction using expert advice, provided that the best among the $n$ experts makes at most $k$ mistakes, and asked what are the optimal mistake bounds (as a function of $n$ and $k$). Cesa-Bianchi, Freund, Helmbold, and Warmuth [’93, ’96] provided a nearly optimal bound for deterministic learners, and left the randomized case as an open problem. We resolve this question by providing an optimal learning rule in the randomized case, and showing that its expected mistake bound equals half of the deterministic bound, up to negligible additive terms. This improves upon previous works by Cesa-Bianchi, Freund, Haussler, Helmbold, Schapire and Warmuth [’93, ’97], by Abernethy, Langford, and Warmuth [’06], and by Brânzei and Peres [’19], which handled the regimes $k \ll \log n$ or $k\gg \log n$. In contrast, our result applies to all pairs $n,k$, and does so via a unified analysis using the randomized Littlestone dimension.In our proofs we develop and use optimal learning rules, which can be seen as natural variants of the Standard Optimal Algorithm ($\mathsf{SOA}$) of Littlestone: a weighted variant in the agnostic case, and a probabilistic variant in the randomized case. We conclude the paper with suggested directions for future research and open questions.

Can the log(n) factor in the upper bound of Cohen and Kontorovich (COLT, 2023)be removed?

Multi-distribution learning is a natural generalization of PAC learning to settings with multiple data distributions. There remains a significant gap between the known upper and lower bounds for PAC-learnable classes. In particular, though we understand the sample complexity of learning a VC dimension $d$ class on $k$ distributions to be $O(\epsilon^{-2} \ln(k) (d + k) + \min \{\epsilon^{-1} d k, \epsilon^{-4} \ln(k) d\})$, the best lower bound is $\Omega(\epsilon^{-2}(d + k \ln(k)))$. We discuss recent progress on this problem and some hurdles that are fundamental to the use of game dynamics in statistical learning.