arXiv subjects

Jared Kaplan

Publications and source records attributed to Jared Kaplan.

At least 55 records · Page 3Linked to original sources

arXiv2020

Scaling Laws for Neural Language Models

We study empirical scaling laws for language model performance on the cross-entropy loss. The loss scales as a power-law with model size, dataset size, and the amount of compute used for training, with some trends spanning more than seven orders of magnitude. Other architectural details such as network width or depth have minimal effects within a wide range. Simple equations govern the dependence of overfitting on model/dataset size and the dependence of training speed on model size. These relationships allow us to determine the optimal allocation of a fixed compute budget. Larger models are significantly more sample-efficient, such that optimally compute-efficient training involves training very large models on a relatively modest amount of data and stopping significantly before convergence.

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arXiv2019

The Bulk-to-Boundary Propagator in Black Hole Microstate Backgrounds

First-quantized propagation in quantum gravitational AdS$_3$ backgrounds can be exactly reconstructed using CFT$_2$ data and Virasoro symmetry. We develop methods to compute the bulk-to-boundary propagator in a black hole microstate, $\langle ϕ_L \mathcal{O}_L \mathcal{O}_H \mathcal{O}_H\rangle$, at finite central charge. As a first application, we show that the semiclassical theory on the Euclidean BTZ solution sharply disagrees with the exact description, as expected based on the resolution of forbidden thermal singularities, though this effect may appear exponentially small for physical observers.

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arXiv2019

AdS$_3$ Reconstruction with General Gravitational Dressings

The gauge redundancy of quantum gravity makes the definition of local operators ambiguous, as they depend on the choice of gauge or on a `gravitational dressing' analogous to a choice of Wilson line attachments. Recent work identified exact AdS$_3$ proto-fields by fixing to a Fefferman-Graham gauge. Here we extend that work and define proto-fields with general gravitational dressing. We first study bulk fields charged under a $U(1)$ Chern-Simons gauge theory as an illustrative warm-up, and then generalize the results to gravity. As an application, we compute a gravitational loop correction to the bulk-boundary correlator in the background of a black hole microstate, and then verify this calculation using a newly adapted recursion relation. Branch points at the Euclidean horizon are present in the $1/c$ corrections to semiclassical correlators.

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arXiv2019

A Species or Weak-Gravity Bound for Large $N$ Gauge Theories Coupled to Gravity

Causality constrains the gravitational interactions of massive higher spin particles in both AdS and flat spacetime. We explore the extent to which these constraints apply to composite particles, explaining why they do not rule out macroscopic objects or hydrogen atoms. However, we find that they do apply to glueballs and mesons in confining large $N$ gauge theories. Assuming such theories contain massive bound states of general spin, we find parametric bounds in $(3+1)$ spacetime dimensions of the form $N\lesssim \frac{M_{Pl}}{Λ_{\text{QCD}}}$ relating $N$, the QCD scale, and the Planck scale. We also argue that a stronger bound replacing $Λ_{\text{QCD}}$ with the UV cut-off scale may be derived from eikonal scattering in flat spacetime.

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arXiv2018

An Empirical Model of Large-Batch Training

In an increasing number of domains it has been demonstrated that deep learning models can be trained using relatively large batch sizes without sacrificing data efficiency. However the limits of this massive data parallelism seem to differ from domain to domain, ranging from batches of tens of thousands in ImageNet to batches of millions in RL agents that play the game Dota 2. To our knowledge there is limited conceptual understanding of why these limits to batch size differ or how we might choose the correct batch size in a new domain. In this paper, we demonstrate that a simple and easy-to-measure statistic called the gradient noise scale predicts the largest useful batch size across many domains and applications, including a number of supervised learning datasets (MNIST, SVHN, CIFAR-10, ImageNet, Billion Word), reinforcement learning domains (Atari and Dota), and even generative model training (autoencoders on SVHN). We find that the noise scale increases as the loss decreases over a training run and depends on the model size primarily through improved model performance. Our empirically-motivated theory also describes the tradeoff between compute-efficiency and time-efficiency, and provides a rough model of the benefits of adaptive batch-size training.

cs.LG↗

arXiv2018

Lightcone Effective Hamiltonians and RG Flows

We present a prescription for an effective lightcone (LC) Hamiltonian that includes the effects of zero modes, focusing on the case of Conformal Field Theories (CFTs) deformed by relevant operators. We show how the prescription resolves a number of issues with LC quantization, including i) the apparent non-renormalization of the vacuum, ii) discrepancies in critical values of bare parameters in equal-time vs LC quantization, and iii) an inconsistency at large N in CFTs with simple AdS duals. We describe how LC quantization can drastically simplify Hamiltonian truncation methods applied to some large N CFTs, and discuss how the prescription identifies theories where these simplifications occur. We demonstrate and check our prescription in a number of examples.

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arXiv2017

The AdS$_3$ Propagator and the Fate of Locality

We recently used Virasoro symmetry considerations to propose an exact formula for a bulk proto-field $ϕ$ in AdS$_3$. In this paper we study the propagator $\langle ϕϕ\rangle$. We show that many techniques from the study of conformal blocks can be generalized to compute it, including the semiclassical monodromy method and both forms of the Zamolodchikov recursion relations. When the results from recursion are expanded at large central charge, they match gravitational perturbation theory for a free scalar field coupled to gravity in our chosen gauge. We find that although the propagator is finite and well-defined at long distances, its perturbative expansion in $G_N = \frac{3}{2c}$ exhibits UV/IR mixing effects. If we nevertheless interpret $\langle ϕϕ\rangle$ as a probe of bulk locality, then when $G_N m_ϕ\ll 1$ locality breaks down at the new short-distance scale $σ_* \sim \sqrt[4]{G_N R_{AdS}^3}$. For $ϕ$ with very large bulk mass, or at small central charge, bulk locality fails at the AdS length scale. In all cases, locality `breakdown' manifests as singularities or branch cuts at spacelike separation arising from non-perturbative quantum gravitational effects.

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arXiv2017

An Exact Operator That Knows Its Location

We use conformal symmetry to define an AdS$_3$ proto-field $ϕ$ as an exact linear combination of Virasoro descendants of a CFT$_2$ primary operator $\mathcal{O}$. We find that both symmetry considerations and a gravitational Wilson line formalism lead to the same results. The operator $ϕ$ has many desirable properties; in particular it has correlators that agree with gravitational perturbation theory when expanded at large $c$, and that automatically take the correct form in all vacuum AdS$_3$ geometries, including BTZ black hole backgrounds. In the future it should be possible to use $ϕ$ to probe bulk locality and black hole horizons at a non-perturbative level.

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arXiv2017

Exact Virasoro Blocks from Wilson Lines and Background-Independent Operators

Aspects of black hole thermodynamics and information loss can be derived as a consequence of Virasoro symmetry. To bolster the connection between Virasoro conformal blocks and AdS$_3$ quantum gravity, we study sl$(2)$ Chern-Simons Wilson line networks and revisit the idea that they compute a variety of CFT$_2$ observables, including Virasoro OPE blocks, exactly. We verify this in the semiclassical large central charge limit and to low orders in a perturbative $1/c$ expansion. Wilson lines connecting the boundary to points in the bulk play a natural role in bulk reconstruction. Because quantum gravity in AdS$_3$ is rigidly fixed by Virasoro symmetry, we argue that sl$(2)$ Wilson lines provide building blocks for background independent bulk reconstruction. In particular, we show explicitly that they automatically compute the uniformizing coordinates appropriate to any background state.

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arXiv2017

A Numerical Approach to Virasoro Blocks and the Information Paradox

We chart the breakdown of semiclassical gravity by analyzing the Virasoro conformal blocks to high numerical precision, focusing on the heavy-light limit corresponding to a light probe propagating in a BTZ black hole background. In the Lorentzian regime, we find empirically that the initial exponential time-dependence of the blocks transitions to a universal $t^{-\frac{3}{2}}$ power-law decay. For the vacuum block the transition occurs at $t \approx \frac{πc}{6 h_L}$, confirming analytic predictions. In the Euclidean regime, due to Stokes phenomena the naive semiclassical approximation fails completely in a finite region enclosing the `forbidden singularities'. We emphasize that limitations on the reconstruction of a local bulk should ultimately stem from distinctions between semiclassical and exact correlators.

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arXiv2016

On the Late-Time Behavior of Virasoro Blocks and a Classification of Semiclassical Saddles

Recent work has demonstrated that black hole thermodynamics and information loss/restoration in AdS$_3$/CFT$_2$ can be derived almost entirely from the behavior of the Virasoro conformal blocks at large central charge, with relatively little dependence on the precise details of the CFT spectrum or OPE coefficients. Here, we elaborate on the non-perturbative behavior of Virasoro blocks by classifying all `saddles' that can contribute for arbitrary values of external and internal operator dimensions in the semiclassical large central charge limit. The leading saddles, which determine the naive semiclassical behavior of the Virasoro blocks, all decay exponentially at late times, and at a rate that is independent of internal operator dimensions. Consequently, the semiclassical contribution of high-energy states does not resolve a well-known version of the information loss problem in AdS$_3$. However, we identify two infinite classes of sub-leading saddles, and one of these classes does not decay at late times.

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arXiv2016

Degenerate Operators and the $1/c$ Expansion: Lorentzian Resummations, High Order Computations, and Super-Virasoro Blocks

One can obtain exact information about Virasoro conformal blocks by analytically continuing the correlators of degenerate operators. We argued in recent work that this technique can be used to explicitly resolve information loss problems in AdS$_3$/CFT$_2$. In this paper we use the technique to perform calculations in the small $1/c \propto G_N$ expansion: (1) we prove the all-orders resummation of logarithmic factors $\propto \frac{1}{c} \log z$ in the Lorentzian regime, demonstrating that $1/c$ corrections directly shift Lyapunov exponents associated with chaos, as claimed in prior work, (2) we perform another all-orders resummation in the limit of large $c$ with fixed $cz$, interpolating between the early onset of chaos and late time behavior, (3) we explicitly compute the Virasoro vacuum block to order $1/c^2$ and $1/c^3$, corresponding to $2$ and $3$ loop calculations in AdS$_3$, and (4) we derive the heavy-light vacuum blocks in theories with $\mathcal{N}=1,2$ superconformal symmetry.

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arXiv2016

On Information Loss in AdS$_3$/CFT$_2$

We discuss information loss from black hole physics in AdS$_3$, focusing on two sharp signatures infecting CFT$_2$ correlators at large central charge $c$: 'forbidden singularities' arising from Euclidean-time periodicity due to the effective Hawking temperature, and late-time exponential decay in the Lorentzian region. We study an infinite class of examples where forbidden singularities can be resolved by non-perturbative effects at finite $c$, and we show that the resolution has certain universal features that also apply in the general case. Analytically continuing to the Lorentzian regime, we find that the non-perturbative effects that resolve forbidden singularities qualitatively change the behavior of correlators at times $t \sim S_{BH}$, the black hole entropy. This may resolve the exponential decay of correlators at late times in black hole backgrounds. By Borel resumming the $1/c$ expansion of exact examples, we explicitly identify 'information-restoring' effects from heavy states that should correspond to classical solutions in AdS$_3$. Our results suggest a line of inquiry towards a more precise formulation of the gravitational path integral in AdS$_3$.

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arXiv2016

Conformal Blocks Beyond the Semi-Classical Limit

Black hole microstates and their approximate thermodynamic properties can be studied using heavy-light correlation functions in AdS/CFT. Universal features of these correlators can be extracted from the Virasoro conformal blocks in CFT2, which encapsulate quantum gravitational effects in AdS3. At infinite central charge c, the Virasoro vacuum block provides an avatar of the black hole information paradox in the form of periodic Euclidean-time singularities that must be resolved at finite c. We compute Virasoro blocks in the heavy-light, large c limit, extending our previous results by determining perturbative 1/c corrections. We obtain explicit closed-form expressions for both the `semi-classical' $h_L^2 / c^2$ and `quantum' $h_L / c^2$ corrections to the vacuum block, and we provide integral formulas for general Virasoro blocks. We comment on the interpretation of our results for thermodynamics, discussing how monodromies in Euclidean time can arise from AdS calculations using `geodesic Witten diagrams'. We expect that only non-perturbative corrections in 1/c can resolve the singularities associated with the information paradox.

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arXiv2016

A Quantum Correction To Chaos

We use results on Virasoro conformal blocks to study chaotic dynamics in CFT$_2$ at large central charge c. The Lyapunov exponent $λ_L$, which is a diagnostic for the early onset of chaos, receives $1/c$ corrections that may be interpreted as $λ_L = \frac{2 π}β \left( 1 + \frac{12}{c} \right)$. However, out of time order correlators receive other equally important $1/c$ suppressed contributions that do not have such a simple interpretation. We revisit the proof of a bound on $λ_L$ that emerges at large $c$, focusing on CFT$_2$ and explaining why our results do not conflict with the analysis leading to the bound. We also comment on relationships between chaos, scattering, causality, and bulk locality.

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arXiv2015

Virasoro Conformal Blocks and Thermality from Classical Background Fields

We show that in 2d CFTs at large central charge, the coupling of the stress tensor to heavy operators can be re-absorbed by placing the CFT in a non-trivial background metric. This leads to a more precise computation of the Virasoro conformal blocks between heavy and light operators, which are shown to be equivalent to global conformal blocks evaluated in the new background. We also generalize to the case where the operators carry U(1) charges. The refined Virasoro blocks can be used as the seed for a new Virasoro block recursion relation expanded in the heavy-light limit. We comment on the implications of our results for the universality of black hole thermality in $AdS_3$, or equivalently, the eigenstate thermalization hypothesis for $CFT_2$ at large central charge.

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arXiv2015

Hawking from Catalan

The Virasoro algebra determines all `graviton' matrix elements in AdS$_3$/CFT$_2$. We study the explicit exchange of any number of Virasoro gravitons between heavy and light CFT$_2$ operators at large central charge. These graviton exchanges can be written in terms of new on-shell tree diagrams, organized in a perturbative expansion in $h_H/c$, the heavy operator dimension divided by the central charge. The Virasoro vacuum conformal block, which is the sum of all the tree diagrams, obeys a differential recursion relation generalizing that of the Catalan numbers. We use this recursion relation to sum the on-shell diagrams to all orders, computing the Virasoro vacuum block. Extrapolating to large $h_H/c$ determines the Hawking temperature of a BTZ black hole in dual AdS$_3$ theories.

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arXiv2015

Eikonalization of Conformal Blocks

Classical field configurations such as the Coulomb potential and Schwarzschild solution are built from the t-channel exchange of many light degrees of freedom. We study the CFT analog of this phenomenon, which we term the `eikonalization' of conformal blocks. We show that when an operator $T$ appears in the OPE $\mathcal{O}(x) \mathcal{O}(0)$, then the large spin $\ell$ Fock space states $[TT \cdots T]_{\ell}$ also appear in this OPE with a computable coefficient. The sum over the exchange of these Fock space states in an $\langle \mathcal{O} \mathcal{O} \mathcal{O} \mathcal{O} \rangle$ correlator build the classical `$T$ field' in the dual AdS description. In some limits the sum of all Fock space exchanges can be represented as the exponential of a single $T$ exchange in the 4-pt correlator of $\mathcal{O}$. Our results should be useful for systematizing $1/\ell$ perturbation theory in general CFTs and simplifying the computation of large spin OPE coefficients. As examples we obtain the leading $\log \ell$ dependence of Fock space conformal block coefficients, and we directly compute the OPE coefficients of the simplest `triple-trace' operators.

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