We prove constant degree polynomial algorithms cannot optimize pure spherical p-spin Hamiltonians beyond the algorithmic threshold $\operatorname{ALG} (p) = 2\sqrt {\frac{{p - 1}}{p}} $. The proof goes by transforming any hypothetical such algorithm into a Lipschitz one, for which hardness was shown previously by the author and B. Huang.

In the incentivized exploration model, a principal aims to explore and learn over time by interacting with a sequence of self-interested agents. It has been recently understood that the main challenge in designing incentive-compatible algorithms for this problem is to gather a moderate amount of initial data, after which one can obtain near-optimal regret via posterior sampling. With high-dimensional contexts, however, this \emph{initial exploration} phase requires exponential sample complexity in some cases, which prevents efficient learning unless initial data can be acquired exogenously. We show that these barriers to exploration disappear under mild geometric conditions on the set of available actions, in which case incentive-compatibility does not preclude regret-optimality. Namely, we consider the linear bandit model with actions in the Euclidean unit ball, and give an incentive-compatible exploration algorithm with sample complexity that scales polynomially with the dimension and other parameters.

The existence of “lottery tickets” (Frankle & Carbin, 2018) at or near initialization raises the tantalizing question of whether large models are necessary in deep learning, or whether sparse networks can be quickly identified and trained without ever training the dense models that contain them. However, efforts to find these sparse subnetworks without training the dense model (“pruning at initialization”) have been broadly unsuccessful (Frankle et al., 2020b). We put forward a theoretical explanation for this, based on the model’s effective parameter count, $p_\text{eff}$, given by the sum of the number of non-zero weights in the final network and the mutual information between the sparsity mask and the data. We show the Law of Robustness of (Bubeck & Sellke, 2023) extends to sparse networks with the usual parameter count replaced by $p_\text{eff}$, meaning a sparse neural network which robustly interpolates noisy data requires a heavily data-dependent mask. We posit that pruning during and after training outputs masks with higher mutual information than those produced by pruning at initialization. Thus two networks may have the same sparsities, but differ in effective parameter count based on how they were trained. This suggests that pruning near initialization may be infeasible and explains why lottery tickets exist, but cannot be found fast (i.e. without training the full network). Experiments on neural networks confirm that information gained during training may indeed affect model capacity.

We introduce a new linear-algebraic tool based on group representation theory, and use it to address three key problems in machine learning. 1. Past researchers have proposed fast attention algorithms for LLMs by approximating or replace softmax attention with other functions, such as low-degree polynomials. The key property of these functions is that, when applied entry-wise to the matrix $QK^{\top}$, the result is a low rank matrix when $Q$ and $K$ are $n \times d$ matrices and $n \gg d$. This suggests a natural question: what are all functions $f$ with this property? If other $f$ exist and are quickly computable, they can be used in place of softmax for fast subquadratic attention algorithms. It was previously known that low-degree polynomials have this property. We prove that low-degree polynomials are the only piecewise continuous functions with this property. This suggests that the low-rank fast attention only works for functions approximable by polynomials. Our work gives a converse to the polynomial method in algorithm design. 2. We prove the first full classification of all positive definite kernels that are functions of Manhattan or $\ell_1$ distance. Our work generalizes an existing theorem at the heart of all kernel methods in machine learning: the classification of all positive definite kernels that are functions of Euclidean distance. 3. The key problem in metric transforms, a mathematical theory used in geometry and machine learning, asks what functions transform pairwise distances in semi-metric space $M$ to semi-metric space $N$ for specified $M$ and $N$. We provide the first full classification of functions that transform Manhattan distances to Manhattan distances. Our work generalizes the foundational work of Schoenberg, which fully classifies functions that transform Euclidean to Euclidean distances. We additionally prove results about stable-rank preserving functions that are potentially useful in algorithmic design, and more. Our core new tool is called the representation theory of the hyperrectangle.

We consider the classic question of state tomography: given copies of an unknown quantum state $\rho \in \mathbb{C}^{d \times d}$, output $\widehat{\rho}$ which is close to $\rho$ in some sense, e.g. trace distance or fidelity. When one is allowed to make coherent measurements entangled across all copies, $\Theta\left(d^{2} / \varepsilon^{2}\right)$ copies are necessary and sufficient to get trace distance $\varepsilon$ [18], [29]. Unfortunately, the protocols achieving this rate incur large quantum memory overheads that preclude implementation on near-term devices. On the other hand, the best known protocol using incoherent (single-copy) measurements uses $O\left(d^{3} / \varepsilon^{2}\right)$ copies [24], and multiple papers have posed it as an open question to understand whether or not this rate is tight [6], [18]. In this work, we fully resolve this question, by showing that any protocol using incoherent measurements, even if they are chosen adaptively, requires $\Omega\left(d^{3} / \varepsilon^{2}\right)$ copies, matching the upper bound of [24]. We do so by a new proof technique which directly bounds the “tilt” of the posterior distribution after measurements, which yields a surprisingly short proof of our lower bound, and which we believe may be of independent interest. While this implies that adaptivity does not help for tomography with respect to trace distance, we show that it actually does help for tomography with respect to infidelity. We give an adaptive algorithm that outputs a state which is $\gamma$-close in infidelity to $\rho$ using only $\widetilde{O}\left(d^{3} / \gamma\right)$ copies, which is optimal for incoherent measurements. In contrast, it is known [18] that any nonadaptive algorithm requires $\Omega\left(d^{3} / \gamma^{2}\right)$ copies. While it is folklore that in 2 dimensions, one can achieve a scaling of $O(1 / \gamma)$, to the best of our knowledge, our algorithm is the first to achieve the optimal rate in all dimensions.

We investigate one of the most basic problems in streaming algorithms: approximating the number of elements in the stream. Famously, [Mor78] gave a randomized algorithm achieving a constant-factor approximation error for streams of length at most N in space $O(\log\log N)$. We investigate the pseudo-deterministic complexity of the problem and prove a tight $\Omega(\log N)$ lower bound, thus resolving a problem of [GGMW20].

We advance the study of incentivized bandit exploration, in which arm choices are viewed as recommendations and are required to be Bayesian incentive compatible. Recent work of Sellke-Slivkins (Operations Research 2022) has shown that for the special case of independent arms, after collecting enough initial samples, the popular Thompson sampling algorithm becomes incentive compatible. This was generalized to the combinatorial semibandit in Hu-Ngo-Slivkins-Wu (NeurIPS 2022). We give an analog of this result for linear bandits, where the independence of the prior is replaced by a natural convexity condition. This opens up the possibility of efficient and regret-optimal incentivized exploration in high-dimensional action spaces. In the semibandit model, we also improve the sample complexity for the pre-Thompson sampling phase of initial data collection.

We study pure exploration with infinitely many bandit arms generated \iid from an unknown distribution. Our goal is to efficiently select a single high quality arm whose average reward is, with probability $1-\delta$, within $\varepsilon$ of being with the top $\eta$-fraction of arms; this is a natural adaptation of the classical PAC guarantee for infinite action sets. We consider both the fixed confidence and fixed budget settings, aiming respectively for optimal \emph{expected} and \emph{fixed} sample complexity. For fixed confidence, we give an algorithm with expected sample complexity $O\left(\frac{\log (1/\eta)\log (1/\delta)}{\eta\varepsilon^2}\right)$. This is optimal except for the $\log (1/\eta)$ factor, and the $\delta$-dependence closes a quadratic gap in the literature. For fixed budget, we show the asymptotically optimal sample complexity as $\delta\to 0$ is $c^{-1}\log(1/\delta)\big(\log\log(1/\delta)\big)^2$ to leading order; equivalently, the optimal failure probability with exactly $N$ samples decays as $\exp\big(-(1\pm o(1))\frac{cN}{\log^2 N}\big)$. The value of $c$ depends explicitly on the problem parameters (including the unknown arm distribution) through a certain Fisher information distance. Even the strictly super-linear dependence on $\log(1/\delta)$ was not known and resolves a question of Grossman-Moshkovitz (FOCS 2015).

Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in deep learning is that models are trained with many more parameters than what this classical theory would suggest. We propose a partial theoretical explanation for this phenomenon. We prove that for a broad class of data distributions and model classes, overparametrization is necessary if one wants to interpolate the data smoothly . Namely we show that smooth interpolation requires d times more parameters than mere interpolation, where d is the ambient data dimension. We prove this universal law of robustness for any smoothly parametrized function class with polynomial size weights, and any covariate distribution verifying isoperimetry (or a mixture thereof). In the case of two-layer neural networks and Gaussian covariates, this law was conjectured in prior work by Bubeck, Li, and Nagaraj. We also give an interpretation of our result as an improved generalization bound for model classes consisting of smooth functions.

We address online combinatorial optimization when the player has a prior over the adversary’s sequence of losses. In this setting, Russo and Van Roy proposed an information theoretic analysis of Thompson Sampling based on the information ratio, allowing for elegant proofs of Bayesian regret bounds. In this paper we introduce three novel ideas to this line of work. First we propose a new quantity, the scale-sensitive information ratio, which allows us to obtain more refined first-order regret bounds (i.e., bounds of the form$O(\sqrt {L^{*}})$where$L^{*}$is the loss of the best combinatorial action). Second we replace the entropy over combinatorial actions by a coordinate entropy, which allows us to obtain the first optimal worst-case bound for Thompson Sampling in the combinatorial setting. We additionally introduce a novel link between Bayesian agents and frequentist confidence intervals. Combining these ideas we show that the classical multi-armed bandit first-order regret bound$ \widetilde {O}(\sqrt {d L^{*}})$still holds true in the more challenging and more general semi-bandit scenario. This latter result improves the previous state of the art bound$ \widetilde {O}(\sqrt {(d+m^{3})L^{*}})$by Lykouris, Sridharan and Tardos. Moreover we sharpen these results with two technical ingredients. The first leverages a recent insight of Zimmert and Lattimore to replace Shannon entropy with more refined potential functions in the analysis. The second is a Thresholded Thompson Sampling algorithm, which slightly modifies the original algorithm by never playing low-probability actions. This thresholding results in fully$T$-independent regret bounds when$L^{*}\leq \overline {L} ^{*}$is almost surely upper-bounded, which we show does not hold for ordinary Thompson Sampling.

We study the stochastic multi-player multi-armed bandit problem. In this problem, there are $m$ players and $K > m$ arms and the players cooperate to maximize their total reward. However the players cannot communicate and are penalized (e.g. receive no reward) if they pull the same arm at the same time. We ask whether it is possible to obtain optimal instance-dependent regret $\wt{O}(1/\Delta)$ where $\Delta$ is the gap between the $m$-th and $m+1$-st best arms. Such guarantees were recently achieved by \cite{pacchiano2021instance, huang2021towards} in a model in which the players are able to implicitly communicate through intentional collisions. We show that with no communication at all, such guarantees are, surprisingly, not achievable. In fact, obtaining the optimal $\wt{O}(1/\Delta)$ regret for some regimes of $\Delta$ necessarily implies strictly sub-optimal regret in other regimes. Our main result is a complete characterization of the Pareto optimal instance-dependent trade-offs that are possible with no communication. Our algorithm generalizes that of \cite{bubeck2021cooperative} and enjoys the same strong no-collision property, while our lower bound is completely new.

We consider the Sherrington-Kirkpatrick model of spin glasses at high-temperature and no external field, and study the problem of sampling from the Gibbs distribution $\mu$ in polynomial time. We prove that, for any inverse temperature $\beta\lt 1/2$, there exists an algorithm with complexity $O(n^{2})$ that samples from a distribution $\mu^{\text{als}}$ which is close in normalized Wasserstein distance to $\mu$. Namely, there exists a coupling of $\mu$ and $\mu^{\text{alg}}$ such that if $(x,x^{\text{als}})\in\{-1,+1\}^{n}\times\{-1,+1\}^{n}$ is a pair drawn from this coupling, then $n^{-1}\mathbb{E}\{\|x-x^{\text{ald}}\|_{2}^{2}\}=o_{n}(1)$. The best previous results, by Bauerschmidt and Bodineau [BB19] and by Eldan, Koehler, Zeitouni [EKZ21], implied efficient algorithms to approximately sample (under a stronger metric) for $\beta\lt 1/4$. We complement this result with a negative one, by introducing a suitable “stability” property for sampling algorithms, which is verified by many standard techniques. We prove that no stable algorithm can approximately sample for $\beta$>1, even under the normalized Wasserstein metric. Our sampling method is based on an algorithmic implementation of stochastic localization, which progressively tilts the measure $\mu$ towards a single configuration, together with an approximate message passing algorithm that is used to approximate the mean of the tilted measure.

We study the problem of algorithmically optimizing the Hamiltonian of a spherical or Ising mean field spin glass. The maximum asymptotic value OPT of this random function is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing algorithms efficiently optimize these functions up to a value ALG given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a no overlap gap property. However, ALG can be strictly smaller than OPT, and no efficient algorithm producing a value exceeding ALG is known. We prove that when all interactions have even degree, no algorithm satisfying an overlap concentration property can produce an objective larger than ALG with non-negligible probability. This property holds for all algorithms with suitably Lipschitz dependence on the random disorder coefficients of the objective. It encompasses natural formulations of gradient descent, approximate message passing, and Langevin dynamics run for bounded time and in particular includes the algorithms achieving ALG mentioned above. To prove this result, we substantially generalize the overlap gap property framework introduced by Gamarnik and Sudan to arbitrary ultrametric forbidden structures of solutions.

Domain generalization aims at performing well on unseen test environments with data from a limited number of training environments. Despite a proliferation of proposed algorithms for this task, assessing their performance both theoretically and empirically is still very challenging. Distributional matching algorithms such as (Conditional) Domain Adversarial Networks [Ganin et al., 2016, Long et al., 2018] are popular and enjoy empirical success, but they lack formal guarantees. Other approaches such as Invariant Risk Minimization (IRM) require a prohibitively large number of training environments---linear in the dimension of the spurious feature space $d_s$---even on simple data models like the one proposed by [Rosenfeld et al., 2021]. Under a variant of this model, we show that ERM and IRM can fail to find the optimal invariant predictor with $o(d_s)$ environments. We then present an iterative feature matching algorithm that is guaranteed with high probability to find the optimal invariant predictor after seeing only $O(\log d_s)$ environments. Our results provide the first theoretical justification for distribution-matching algorithms widely used in practice under a concrete nontrivial data model.

We consider the cooperative multi-player version of the stochastic multi-armed bandit problem. We study the regime where the players cannot communicate but have access to shared randomness. In prior work by the first two authors, a strategy for this regime was constructed for two players and three arms, with regret $\tilde{O}(\sqrt{T})$, and with no collisions at all between the players (with very high probability). In this paper we show that these properties (near-optimal regret and no collisions at all) are achievable for any number of players and arms. At a high level, the previous strategy heavily relied on a 2-dimensional geometric intuition that was difficult to generalize in higher dimensions, while here we take a more combinatorial route to build the new strategy.

We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) $f_t\colon A_t\to B_t$ between subsets $A_t$ and $B_t$ of the metric space. To serve it, the algorithm has to go to a point $a_t\in A_t$, paying the distance from its previous position. Then, the transformation is applied, modifying the algorithm's state to $f_t(a_t)$. Such transformations can model, e.g., changes to the environment that are outside of an algorithm's control, and we therefore do not charge any additional cost to the algorithm when the transformation is applied. The transformations also allow to model requests occurring in the $k$-taxi problem. We show that for $α$-Lipschitz transformations, the competitive ratio is $Θ(α)^{n-2}$ on $n$-point metrics. Here, the upper bound is achieved by a deterministic algorithm and the lower bound holds even for randomized algorithms. For the $k$-taxi problem, we prove a competitive ratio of $\tilde O((n\log k)^2)$. For chasing convex bodies, we show that even with contracting transformations no competitive algorithm exists. The problem T-MSS has a striking connection to the following deep mathematical question: Given a finite metric space $M$, what is the required cardinality of an extension $\hat M\supseteq M$ where each partial isometry on $M$ extends to an automorphism? We give partial answers for special cases.

Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in the current practice of deep learning is that models are trained with many more parameters than what this classical theory would suggest. We propose a theoretical explanation for this phenomenon. We prove that for a broad class of data distributions and model classes, overparametrization is {\em necessary} if one wants to interpolate the data {\em smoothly}. Namely we show that {\em smooth} interpolation requires $d$ times more parameters than mere interpolation, where $d$ is the ambient data dimension. We prove this universal law of robustness for any smoothly parametrized function class with polynomial size weights, and any covariate distribution verifying isoperimetry. In the case of two-layers neural networks and Gaussian covariates, this law was conjectured in prior work by Bubeck, Li and Nagaraj. We also give an interpretation of our result as an improved generalization bound for model classes consisting of smooth functions.

We consider incentivized exploration: a version of multi-armed bandits where the choice of arms is controlled by self-interested agents, and the algorithm can only issue recommendations. The algorithm controls the flow of information, and the information asymmetry can incentivize the agents to explore. Prior work achieves optimal regret rates up to multiplicative factors that become arbitrarily large depending on the Bayesian priors, and scale exponentially in the number of arms. A more basic problem of sampling each arm once runs into similar factors. We focus on the price of incentives: the loss in performance, broadly construed, incurred for the sake of incentive-compatibility. We prove that Thompson Sampling, a standard bandit algorithm, is incentive-compatible if initialized with sufficiently many data points. The performance loss due to incentives is therefore limited to the initial rounds when these data points are collected. The problem is largely reduced to that of sample complexity: how many rounds are needed? We address this question, providing matching upper and lower bounds and instantiating them in various corollaries. Typically, the optimal sample complexity is polynomial in the number of arms and exponential in the "strength of beliefs".

We introduce the problem of k-chasing of convex functions, a simultaneous generalization of both the famous k-server problem in ℝd, and of the problem of chasing convex bodies and functions. Aside from fundamental interest in this general form, it has natural applications to online k-clustering problems with objectives such as k-median or k-means. We show that this problem exhibits a rich landscape of behavior. In general, if both k > 1 and d > 1 there does not exist any online algorithm with bounded competitiveness. By contrast, we exhibit a class of nicely behaved functions (which include in particular the above-mentioned clustering problems), for which we show that competitive online algorithms exist, and moreover with dimension-free competitive ratio. We also introduce a parallel question of top-k action regret minimization in the realm of online convex optimization. There, too, a much rougher landscape emerges for k > 1. While it is possible to achieve vanishing regret, unlike the top-one action case the rate of vanishing does not speed up for strongly convex functions. Moreover, vanishing regret necessitates both intractable computations and randomness. Finally we leave open whether almost dimension-free regret is achievable for k > 1 and general convex losses. As evidence that it might be possible, we prove dimension-free regret for linear losses via an information-theoretic argument.

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether (1 + ∊)-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we study a thresholded version of the problem: for a given parameter c, find a smaller graph, which we call connectivity-c mimicking network, which preserves connectivity among k terminals exactly up to the value of c. We show that connectivity-c mimicking networks with O(kc4) edges exist and can be found in time m(c log n)O(c). We also give a separate algorithm that constructs such graphs with k · O(c)2c edges in time mcO(c) logO(1) n. These results lead to the first data structures for answering fully dynamic offline c-edge-connectivity queries for c ≥ 4 in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.

We study the stabilization of a linear control system with an unbounded random system gain where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system Xn+1=AnXn+Wn-Un, where the An's are drawn independently at random at each time n from a known distribution with unbounded support, and where the controller receives at most R bits about the system state at each time from an encoder. We provide a time-varying achievable strategy to stabilize the system in a second-moment sense with fixed, finite R. While our previous result provided a strategy to stabilize this system using a variable-rate code, this work provides an achievable strategy using a fixed-rate code. The strategy we employ to achieve this is time-varying and takes different actions depending on the value of the state. It proceeds in two modes: a normal mode (or zoom-in), where the realization of Anis typical, and an emergency mode (or zoom-out), where the realization of Anis exceptionally large. To analyze the performance of the scheme we construct an auxiliary sequence that bounds the state Xn, and then bound auxiliary sequence in both the zoom-in and zoom-out modes.

We address online combinatorial optimization when the player has a prior over the adversary’s sequence of losses. In this setting, Russo and Van Roy proposed an information-theoretic analysis of Thompson Sampling based on the information ratio, allowing for elegant proofs of Bayesian regret bounds. In this paper we introduce three novel ideas to this line of work. First we propose a new quantity, the scale-sensitive information ratio, which allows us to obtain more refined first-order regret bounds (i.e., bounds of the form $O(\sqrt{L^*})$ where $L^*$ is the loss of the best combinatorial action). Second we replace the entropy over combinatorial actions by a coordinate entropy, which allows us to obtain the first optimal worst-case bound for Thompson Sampling in the combinatorial setting. We additionally introduce a novel link between Bayesian agents and frequentist confidence intervals. Combining these ideas we show that the classical multi-armed bandit first-order regret bound $\tilde{O}(\sqrt{d L^*})$ still holds true in the more challenging and more general semi-bandit scenario. This latter result improves the previous state of the art bound $\tilde{O}(\sqrt{(d+m^3)L^*})$ by Lykouris, Sridharan and Tardos. We tighten these results by leveraging a recent insight of Zimmert and Lattimore connecting Thompson Sampling and online stochastic mirror descent, which allows us to replace the Shannon entropy with more general mirror maps.

We consider the non-stochastic version of the (cooperative) multi-player multi-armed bandit problem. The model assumes no communication and no shared randomness at all between the players, and furthermore when two (or more) players select the same action this results in a maximal loss. We prove the first $\sqrt{T}$-type regret guarantee for this problem, assuming only two players, and under the feedback model where collisions are announced to the colliding players. We also prove the first sublinear regret guarantee for the feedback model where collision information is not available, namely $T^{1-\frac{1}{2m}}$ where $m$ is the number of players.

The convex body chasing problem, introduced by Friedman and Linial [FL93], is a competitive analysis problem on any normed vector space. In convex body chasing, for each timestep t ϵ ℕ, a convex body Kt ⊆ ℝd is given as a request, and the player picks a point xt ϵ Kt. The player aims to ensure that the total distance moved is within a bounded ratio of the smallest possible offline solution. In this work, we consider the nested version of the problem, in which the sequence (Kt) must be decreasing. For Euclidean spaces, we consider a memoryless algorithm which moves to the so-called Steiner point, and show that in an appropriate sense it is exactly optimal among memoryless algorithms. For general finite dimensional normed spaces, we combine the Steiner point and our recent algorithm in [ABC+19] to obtain a new algorithm which is nearly optimal for all spaces with p ≥ 1, closing a polynomial gap.

In the chasing convex bodies problem, an online player receives a request sequence of N convex sets K1, …, Kn contained in a normed space ℝd. The player starts at x0 ϵ ℝd, and after observing each Kn picks a new point xn ϵ Kn. At each step the player pays a movement cost of ||xn – xn–1||. The player aims to maintain a constant competitive ratio against the minimum cost possible in hindsight, i.e. knowing all requests in advance. The existence of a finite competitive ratio for convex body chasing was first conjectured in 1991 by Friedman and Linial in [FL93]. This conjecture was recently resolved in [BLLS19] which proved an exponential 2O(d) upper bound on the competitive ratio. In this paper, we drastically improve the exponential upper bound. We give an algorithm achieving competitive ratio d for arbitrary normed spaces, which is exactly tight for ℓ∞ In Euclidean space, our algorithm achieves nearly optimal competitive ratio , compared to a lower bound of . Our approach extends the recent work [BKL +20] which chases nested convex bodies using the classical Steiner point of a convex body. We define the functional Steiner point of a convex function and apply it to the work function to obtain our algorithm.

Let F be a family of sets in some metric space. In the F-chasing problem, an online algorithm observes a request sequence of sets in F and responds (online) by giving a sequence of points in these sets. The movement cost is the distance between consecutive such points. The competitive ratio is the worst case ratio (over request sequences) between the total movement of the online algorithm and the smallest movement one could have achieved by knowing in advance the request sequence. The family F is said to be chaseable if there exists an online algorithm with finite competitive ratio. In 1991, Linial and Friedman conjectured that the family of convex sets in Euclidean space is chaseable. We prove this conjecture.

Abstract. Let [Formula: see text] be a family of sets in some metric space. In the [Formula: see text]- chasing problem, an online algorithm observes a request sequence of sets in [Formula: see text] and responds (online) by giving a sequence of points in these sets. The movement cost is the distance between consecutive such points. The competitive ratio is the worst case ratio (over request sequences) between the total movement of the online algorithm and the smallest movement one could have achieved by knowing in advance the request sequence. The family [Formula: see text] is said to be chaseable if there exists an online algorithm with finite competitive ratio. In 1991, Linial and Friedman conjectured that the family of convex sets in Euclidean space is chaseable. We prove this conjecture.

We study the stabilization of an unpredictable linear control system where the controller must act based on a rate-limited observation of the state. More precisely, we consider the system X_(n+1) = A_n X_n +W_n –U_n, where the A_n's are drawn independently at random at each time n from a known distribution with unbounded support, and where the controller receives at most R bits about the system state at each time from an encoder. We provide a time-varying achievable strategy to stabilize the system in a second-moment sense with fixed, finite R. While our previous result provided a strategy to stabilize this system using a variable-rate code, this work provides an achievable strategy using a fixed-rate code. The strategy we employ to achieve this is time-varying and takes different actions depending on the value of the state. It proceeds in two modes: a normal mode (or zoom-in), where the realization of A_n is typical, and an emergency mode (or zoom-out), where the realization of A_n is exceptionally large.