arXiv subjects
Publications and source records attributed to Mark Sellke.
We study Brownian paths perturbed by semibounded pair potentials and prove upper bounds on the mean square displacement. As a technical tool we derive infinite dimensional versions of key inequalities that were first used in [Sellke; arXiv:2212.14023] in order to study the effective mass of the Fröhlich polaron.
We consider Ising mixed $p$-spin glasses at high-temperature and without external field, and study the problem of sampling from the Gibbs distribution $μ$ in polynomial time. We develop a new sampling algorithm with complexity of the same order as evaluating the gradient of the Hamiltonian and, in particular, at most linear in the input size. We prove that, at sufficiently high-temperature, it produces samples from a distribution $μ^{alg}$ which is close in normalized Wasserstein distance to $μ$. Namely, there exists a coupling of $μ$ and $μ^{alg}$ such that if $({\boldsymbol x},{\boldsymbol x}^{alg})\in\{-1,+1\}^n\times \{-1,+1\}^n$ is a pair drawn from this coupling, then $n^{-1}{\mathbb E}\{\|{\boldsymbol x}-{\boldsymbol x}^{alg}\|_2^2\}=o_n(1)$. For the case of the Sherrington-Kirkpatrick model, our algorithm succeeds in the full replica-symmetric phase. We complement this result with a negative one for sampling algorithms satisfying a certain `stability' property, which is verified by many standard techniques. No stable algorithm can approximately sample at temperatures below the onset of shattering, even under the normalized Wasserstein metric. Further, no algorithm can sample at temperatures below the onset of replica symmetry breaking. Our sampling method implements a discretized version of a diffusion process that has become recently popular in machine learning under the name of `denoising diffusion.' We derive the same process from the general construction of stochastic localization. Implementing the diffusion process requires to efficiently approximate the mean of the tilted measure. To this end, we use an approximate message passing algorithm that, as we prove, achieves sufficiently accurate mean estimation.
We study efficient optimization of the Hamiltonians of multi-species spherical spin glasses. Our results characterize the maximum value attained by algorithms that are suitably Lipschitz with respect to the disorder through a variational principle that we study in detail. We rely on the branching overlap gap property introduced in our previous work and develop a new method to establish it that does not require the interpolation method. Consequently our results apply even for models with non-convex covariance, where the Parisi formula for the true ground state remains open. As a special case, we obtain the algorithmic threshold for all single-species spherical spin glasses, which was previously known only for even models. We also obtain closed-form formulas for pure models which coincide with the $E_{\infty}$ value previously determined by the Kac-Rice formula.
We prove the Fröhlich polaron has effective mass at least $\frac{α^4}{(\log α)^6}$ when the coupling strength $α$ is large. This nearly matches the quartic growth rate $C_*α^4$ predicted by Landau and Pekar in 1948 and complements a recent sharp upper bound of Brooks and Seiringer. Our proof works with the path integral formulation of the problem and systematically applies the Gaussian correlation inequality to exploit quasi-concavity of the interaction terms.
We study pure exploration with infinitely many bandit arms generated i.i.d. from an unknown distribution. Our goal is to efficiently select a single high quality arm whose average reward is, with probability $1-δ$, within $\varepsilon$ of being among the top $η$-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 minimal expected and fixed sample complexity. For fixed confidence, we give an algorithm with expected sample complexity $O\left(\frac{\log (1/η)\log (1/δ)}{η\varepsilon^2}\right)$. This is optimal except for the $\log (1/η)$ factor, and the $δ$-dependence closes a quadratic gap in the literature. For fixed budget, we show the asymptotically optimal sample complexity as $δ\to 0$ is $c^{-1}\log(1/δ)\big(\log\log(1/δ)\big)^2$ to leading order. Equivalently, the optimal failure probability given exactly $N$ samples decays as $\exp\big(-cN/\log^2 N\big)$, up to a factor $1\pm o_N(1)$ inside the exponent. The constant $c$ depends explicitly on the problem parameters (including the unknown arm distribution) through a certain Fisher information distance. Even the strictly super-linear dependence on $\log(1/δ)$ was not known and resolves a question of Grossman and Moshkovitz (FOCS 2016, SIAM Journal on Computing 2020).
We consider the classic question of state tomography: given copies of an unknown quantum state $ρ\in\mathbb{C}^{d\times d}$, output $\widehatρ$ which is close to $ρ$ in some sense, e.g. trace distance or fidelity. When one is allowed to make coherent measurements entangled across all copies, $Θ(d^2/ε^2)$ copies are necessary and sufficient to get trace distance $ε$. 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 $O(d^3/ε^2)$ copies, and multiple papers have posed it as an open question to understand whether or not this rate is tight. In this work, we fully resolve this question, by showing that any protocol using incoherent measurements, even if they are chosen adaptively, requires $Ω(d^3/ε^2)$ copies, matching the best known upper bound. 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 $γ$-close in infidelity to $ρ$ using only $\tilde{O}(d^3/γ)$ copies, which is optimal for incoherent measurements. In contrast, it is known that any nonadaptive algorithm requires $Ω(d^3/γ^2)$ copies. While it is folklore that in $2$ dimensions, one can achieve a scaling of $O(1/γ)$, to the best of our knowledge, our algorithm is the first to achieve the optimal rate in all dimensions.
We consider a random Hamiltonian $H:Σ\to\mathbb R$ defined on a compact space $Σ$ that admits a transitive action by a compact group $\mathcal G$. When the law of $H$ is $\mathcal G$-invariant, we show its expected free energy relative to the unique $\mathcal G$-invariant probability measure on $Σ$ obeys a subadditivity property in the law of $H$ itself. The bound is often tight for weak disorder and relates free energies at different temperatures when $H$ is a Gaussian process. Many examples are discussed including branching random walk, several spin glasses, random constraint satisfaction problems, and the random field Ising model. We also provide a generalization to quantum Hamiltonians with applications to the quantum SK and SYK models.
Given a graph $G$ of degree $k$ over $n$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth $2L$, we develop a local message passing algorithm whose complexity is $O(nkL)$, and that achieves near optimal cut values among all $L$-local algorithms. Focusing on max-cut, the algorithm constructs a cut of value $nk/4+ n\mathsf{P}_\star\sqrt{k/4}+\mathsf{err}(n,k,L)$, where $\mathsf{P}_\star\approx 0.763166$ is the value of the Parisi formula from spin glass theory, and $\mathsf{err}(n,k,L)=o_n(n)+no_k(\sqrt{k})+n \sqrt{k} o_L(1)$ (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, i.e., graphs whose girth becomes $2L$ after removing a small fraction of vertices. Earlier work established that, for random $k$-regular graphs, the typical max-cut value is $nk/4+ n\mathsf{P}_\star\sqrt{k/4}+o_n(n)+no_k(\sqrt{k})$. Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max-cut, and nearly maximum min-bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near-Ramanujan property of random regular graphs.
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. 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 study when a tensor product of irreducible representations of the symmetric group $S_n$ contains all irreducibles as subrepresentations; we say such a tensor product covers $\mathsf{Irrep}(S_n)$. Our results show that this behavior is typical. We first give a general sufficient criterion for tensor products to have this property, which holds asymptotically almost surely for constant-sized collections of (Plancherel or uniformly) random irreducibles. We also consider the minimal tensor power of a single fixed irreducible representation needed to cover $\mathsf{Irrep}(S_n)$. Here a simple lower bound comes from considering dimensions, and we show it is always tight up to a universal constant factor as was recently conjectured by Liebeck, Shalev, and Tiep.
We study the problem of algorithmically optimizing the Hamiltonian $H_N$ of a spherical or Ising mixed $p$-spin glass. The maximum asymptotic value $\mathsf{OPT}$ of $H_N/N$ 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 $H_N/N$ up to a value $\mathsf{ALG}$ 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, $\mathsf{ALG} < \mathsf{OPT}$ can also occur, and no efficient algorithm producing an objective value exceeding $\mathsf{ALG}$ is known. We prove that for mixed even $p$-spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than $\mathsf{ALG}$ with non-negligible probability. This property holds for all algorithms with suitably Lipschitz dependence on the disorder coefficients of $H_N$. It encompasses natural formulations of gradient descent, approximate message passing, and Langevin dynamics run for bounded time and in particular includes the algorithms achieving $\mathsf{ALG}$ 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.
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".
In the Gilbert-Shannon-Reeds shuffle, a deck of $N$ cards is cut into two approximately equal parts which are then riffled uniformly at random. Bayer and Diaconis famously showed that this Markov chain undergoes cutoff in total variation after $\frac{3\log(N)}{2 \log(2)}$ shuffles. We establish cutoff for the more general asymmetric riffle shuffles in which one cuts the deck into differently sized parts before riffling. The value of the cutoff point confirms a conjecture of Lalley from 2000. Some appealing consequences are that asymmetry always slows mixing and that total variation mixing is strictly faster than separation and $L^{\infty}$ mixing.
We study the stochastic multi-player multi-armed bandit problem. In this problem, $m$ players cooperate to maximize their total reward from $K > m$ arms. 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 $\tilde{O}(1/Δ)$ where $Δ$ is the gap between the $m$-th and $m+1$-st best arms. Such guarantees were recently achieved in a model allowing the players to implicitly communicate through intentional collisions. Surprisingly, we show that with no communication at all, such guarantees are not achievable. In fact, obtaining the optimal $\tilde{O}(1/Δ)$ regret for some values of $Δ$ 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 Bubeck, Budzinski, and the second author. As there, our algorithm succeeds even when feedback upon collision can be corrupted by an adaptive adversary, thanks to a strong no-collision property. Our lower bound is based on topological obstructions at multiple scales and is completely new.
We address online combinatorial optimization when the player has a prior over the adversary's sequence of losses. In this framework, Russo and Van Roy proposed an information-theoretic analysis of Thompson Sampling based on the information ratio, resulting in optimal worst-case 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 $\sqrt{L^*}$ 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. Finally, we 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. 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 \emph{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 $L^*$ is almost surely upper-bounded, which we show does not hold for ordinary Thompson sampling.
We consider an unstable scalar linear stochastic system, $X_{n+1}=a X_n + Z_n - U_n$, where $a \geq 1$ is the system gain, $Z_n$'s are independent random variables with bounded $α$-th moments, and $U_n$'s are the control actions that are chosen by a controller who receives a single element of a finite set $\{1, \ldots, M\}$ as its only information about system state $X_i$. We show new proofs that $M > a$ is necessary and sufficient for $β$-moment stability, for any $β< α$. Our achievable scheme is a uniform quantizer of the zoom-in / zoom-out type that codes over multiple time instants for data rate efficiency; the controller uses its memory of the past to correctly interpret the received bits. We analyze its performance using probabilistic arguments. We show a simple proof of a matching converse using information-theoretic techniques. Our results generalize to vector systems, to systems with dependent Gaussian noise, and to the scenario in which a small fraction of transmitted messages is lost.
In the chasing convex bodies problem, an online player receives a request sequence of $N$ convex sets $K_1,\dots, K_N$ contained in a normed space $\mathbb R^d$. The player starts at $x_0\in \mathbb R^d$, and after observing each $K_n$ picks a new point $x_n\in K_n$. At each step the player pays a movement cost of $||x_n-x_{n-1}||$. 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. This conjecture was recently resolved with an exponential $2^{O(d)}$ upper bound on the competitive ratio. We give an improved algorithm achieving competitive ratio $d$ in any normed space, which is exactly tight for $\ell^{\infty}$. In Euclidean space, our algorithm also achieves competitive ratio $O(\sqrt{d\log N})$, nearly matching a $\sqrt{d}$ lower bound when $N$ is subexponential in $d$. The approach extends our prior work for nested convex bodies, which is based on the classical Steiner point of a convex body. We define the functional Steiner point of a convex function and apply it to the associated work function.
Domain generalization aims at performing well on unseen test environments with data from a limited number of training environments. Despite a proliferation of proposal 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 both ERM and IRM cannot generalize with $o(d_s)$ environments. We then present an iterative feature matching algorithm that is guaranteed with high probability to yield a predictor that generalizes after seeing only $O(\log d_s)$ environments. Our results provide the first theoretical justification for a family of distribution-matching algorithms widely used in practice under a concrete nontrivial data model.