Moïse Blanchard

Moïse Blanchard, Patrick Jaillet

We provide algorithms for regression with adversarial responses under large classes of non-i.i.d. instance sequences, on general separable metric spaces, with provably minimal assumptions. We also give characterizations of learnability in this regression context. We consider universal consistency which asks for strong consistency of a learner without restrictions on the value responses. Our analysis shows that such an objective is achievable for a significantly larger class of instance sequences than stationary processes, and unveils a fundamental dichotomy between value spaces: whether finite-horizon mean estimation is achievable or not. We further provide optimistically universal learning rules, i.e., such that if they fail to achieve universal consistency, any other algorithms will fail as well. For unbounded losses, we propose a mild integrability condition under which there exist algorithms for adversarial regression under large classes of non-i.i.d. instance sequences. In addition, our analysis also provides a learning rule for mean estimation in general metric spaces that is consistent under adversarial responses without any moment conditions on the sequence, a result of independent interest.

Moïse Blanchard, Junhui Zhang, Patrick Jaillet

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 $d^{2-δ}$ bits of memory must make $\tildeΩ(d^{1+δ/3})$ queries, for any $δ\in[0,1]$. For the feasibility problem, in which an algorithm only has access to a separation oracle, we show a stronger trade-off: for at most $d^{2-δ}$ memory, the number of queries required is $\tildeΩ(d^{1+δ})$. This resolves a COLT 2019 open problem of Woodworth and Srebro.

Moïse Blanchard, Junhui Zhang, Patrick Jaillet

We propose a family of recursive cutting-plane algorithms to solve feasibility problems with constrained memory, which can also be used for first-order convex optimization. Precisely, in order to find a point within a ball of radius $ε$ with a separation oracle in dimension $d$ -- or to minimize $1$-Lipschitz convex functions to accuracy $ε$ over the unit ball -- our algorithms use $\mathcal O(\frac{d^2}{p}\ln \frac{1}ε)$ bits of memory, and make $\mathcal O((C\frac{d}{p}\ln \frac{1}ε)^p)$ oracle calls, for some universal constant $C \geq 1$. The family is parametrized by $p\in[d]$ and provides an oracle-complexity/memory trade-off in the sub-polynomial regime $\ln\frac{1}ε\gg\ln d$. While several works gave lower-bound trade-offs (impossibility results) -- we explicit here their dependence with $\ln\frac{1}ε$, showing that these also hold in any sub-polynomial regime -- to the best of our knowledge this is the first class of algorithms that provides a positive trade-off between gradient descent and cutting-plane methods in any regime with $ε\leq 1/\sqrt d$. The algorithms divide the $d$ variables into $p$ blocks and optimize over blocks sequentially, with approximate separation vectors constructed using a variant of Vaidya's method. In the regime $ε\leq d^{-Ω(d)}$, our algorithm with $p=d$ achieves the information-theoretic optimal memory usage and improves the oracle-complexity of gradient descent.

Moïse Blanchard, Abhishek Shetty, Alexander Rakhlin

Understanding minimal assumptions that enable learning and generalization is perhaps the central question of learning theory. Several celebrated results in statistical learning theory, such as the VC theorem and Littlestone's characterization of online learnability, establish conditions on the hypothesis class that allow for learning under independent data and adversarial data, respectively. Building upon recent work bridging these extremes, we study sequential decision making under distributional adversaries that can adaptively choose data-generating distributions from a fixed family $U$ and ask when such problems are learnable with sample complexity that behaves like the favorable independent case. We provide a near complete characterization of families $U$ that admit learnability in terms of a notion known as generalized smoothness i.e. a distribution family admits VC-dimension-dependent regret bounds for every finite-VC hypothesis class if and only if it is generalized smooth. Further, we give universal algorithms that achieve low regret under any generalized smooth adversary without explicit knowledge of $U$. Finally, when $U$ is known, we provide refined bounds in terms of a combinatorial parameter, the fragmentation number, that captures how many disjoint regions can carry nontrivial mass under $U$. These results provide a nearly complete understanding of learnability under distributional adversaries. In addition, building upon the surprising connection between online learning and differential privacy, we show that the generalized smoothness also characterizes private learnability under distributional constraints.

Moïse Blanchard, Samory Kpotufe

There has been much recent interest in understanding the continuum from adversarial to stochastic settings in online learning, with various frameworks including smoothed settings proposed to bridge this gap. We consider the more general and flexible framework of distributionally constrained adversaries in which instances are drawn from distributions chosen by an adversary within some constrained distribution class [RST11]. Compared to smoothed analysis, we consider general distributional classes which allows for a fine-grained understanding of learning settings between fully stochastic and fully adversarial for which a learner can achieve non-trivial regret. We give a characterization for which distribution classes are learnable in this context against both oblivious and adaptive adversaries, providing insights into the types of interplay between the function class and distributional constraints on adversaries that enable learnability. In particular, our results recover and generalize learnability for known smoothed settings. Further, we show that for several natural function classes including linear classifiers, learning can be achieved without any prior knowledge of the distribution class -- in other words, a learner can simultaneously compete against any constrained adversary within learnable distribution classes.

Jialin Yu, Moïse Blanchard

We study a sequential prediction problem in which an adversary is allowed to inject arbitrarily many adversarial instances in a stream of i.i.d.\ instances, but at each round, the learner may also \emph{abstain} from making a prediction without incurring any penalty if the instance was indeed corrupted. This semi-adversarial setting naturally sits between the classical stochastic case with i.i.d.\ instances for which function classes with finite VC dimension are learnable; and the adversarial case with arbitrary instances, known to be significantly more restrictive. For this problem, Goel et al. (2023) showed that, if the learner knows the distribution $μ$ of clean samples in advance, learning can be achieved for all VC classes without restrictions on adversary corruptions. This is, however, a strong assumption in both theory and practice: a natural question is whether similar learning guarantees can be achieved without prior distributional knowledge, as is standard in classical learning frameworks (e.g., PAC learning or asymptotic consistency) and other non-i.i.d.\ models (e.g., smoothed online learning). We therefore focus on the distribution-free setting where $μ$ is \emph{unknown} and propose an algorithm \textsc{AbstainBoost} based on a boosting procedure of weak learners, which guarantees sublinear error for general VC classes in \emph{distribution-free} abstention learning for oblivious adversaries. These algorithms also enjoy similar guarantees for adaptive adversaries, for structured function classes including linear classifiers. These results are complemented with corresponding lower bounds, which reveal an interesting polynomial trade-off between misclassification error and number of erroneous abstentions.

Moïse Blanchard, Vineet Goyal

We study the impact of sharing exploration in multi-armed bandits in a grouped setting where a set of groups have overlapping feasible action sets [Baek and Farias '24]. In this grouped bandit setting, groups share reward observations, and the objective is to minimize the collaborative regret, defined as the maximum regret across groups. This naturally captures applications in which one aims to balance the exploration burden between groups or populations -- it is known that standard algorithms can lead to significantly imbalanced exploration cost between groups. We address this problem by introducing an algorithm Col-UCB that dynamically coordinates exploration across groups. We show that Col-UCB achieves both optimal minimax and instance-dependent collaborative regret up to logarithmic factors. These bounds are adaptive to the structure of shared action sets between groups, providing insights into when collaboration yields significant benefits over each group learning their best action independently.

Moïse Blanchard

We study the subject of universal online learning with non-i.i.d. processes for bounded losses. The notion of an universally consistent learning was defined by Hanneke in an effort to study learning theory under minimal assumptions, where the objective is to obtain low long-run average loss for any target function. We are interested in characterizing processes for which learning is possible and whether there exist learning rules guaranteed to be universally consistent given the only assumption that such learning is possible. The case of unbounded losses is very restrictive, since the learnable processes almost surely visit a finite number of points and as a result, simple memorization is optimistically universal. We focus on the bounded setting and give a complete characterization of the processes admitting strong and weak universal learning. We further show that k-nearest neighbor algorithm (kNN) is not optimistically universal and present a novel variant of 1NN which is optimistically universal for general input and value spaces in both strong and weak setting. This closes all COLT 2021 open problems posed by Hanneke on universal online learning.

Moïse Blanchard, Romain Cosson

We study universal consistency of non-i.i.d. processes in the context of online learning. A stochastic process is said to admit universal consistency if there exists a learner that achieves vanishing average loss for any measurable response function on this process. When the loss function is unbounded, Blanchard et al. showed that the only processes admitting strong universal consistency are those taking a finite number of values almost surely. However, when the loss function is bounded, the class of processes admitting strong universal consistency is much richer and its characterization could be dependent on the response setting (Hanneke). In this paper, we show that this class of processes is independent from the response setting thereby closing an open question (Hanneke, Open Problem 3). Specifically, we show that the class of processes that admit universal online learning is the same for binary classification as for multiclass classification with countable number of classes. Consequently, any output setting with bounded loss can be reduced to binary classification. Our reduction is constructive and practical. Indeed, we show that the nearest neighbor algorithm is transported by our construction. For binary classification on a process admitting strong universal learning, we prove that nearest neighbor successfully learns at least all finite unions of intervals.