Deep Learning is Not So Mysterious or Different - Prof. Andrew Gordon Wilson (NYU)
Summary
The episode’s core economic claim is that scale can buy simplicity, not merely capacity. Wilson argues that larger neural networks can become both more expressive and more biased toward compressible solutions; in double descent’s second leg, models share roughly zero training loss, yet the larger ones generalize better. For investors, that recasts some scaling expenditure as purchasing an inductive bias—“if they’d made the models even bigger, they would be less likely to overfit”—though its mechanism remains an open research question.
Parameter count is a poor proxy for model complexity, making familiar “too many parameters” objections potentially misleading. The airline-passenger thought experiment contrasts a line, a small polynomial, and a 10,000-parameter model, while Gaussian processes arise from infinite-neural-network limits and an RBF kernel is effectively like using an infinite-order polynomial. What matters is the induced distribution over functions: an expressive model can keep unlikely solutions possible at “epsilon probability” while strongly preferring simple ones.
Soft inductive biases may offer a better operating model than hard architectural constraints. Physical systems are rarely perfectly closed—a pendulum may encounter wind—so Wilson favors flexible models gently biased toward conservation, equivariance, or other structure. His residual-pathway-prior experiments found that even a weak bias often collapsed onto the exact constraint when it explained the data, supporting the prescription to “honestly represent your beliefs” without ruling out surprises.
Bayesian marginalization is presented as both an underused performance lever and the mathematically honest response to uncertainty. Betting on one parameter setting becomes less defensible as models grow more expressive; posterior averaging automatically favors broad, flat regions and supplies an Occam’s-razor effect without hand-designing a flatness penalty. Practical approximations such as SWAG and deep kernel learning exist, but Wilson thinks major progress may require a “ten-year kind of style moonshot” as models move from millions to billions of parameters.
Compression is the episode’s candidate unifying principle, but not a complete theory of intelligence. Solomonoff-style bounds can improve as neural networks grow because larger models appear more biased toward low-Kolmogorov-complexity solutions; this helps explain benign overfitting, double descent, and increasingly general-purpose architectures. Yet random noise is also incompressible, shortcut learning can compress the wrong correlation, and Wilson explicitly wants measures that separate structural complexity from randomness.
Transfer results suggest broad pretraining can create reusable principles of induction rather than merely reusable features. A text-pretrained LLM unexpectedly worked as a zero-shot time-series forecaster, while a fine-tuned Llama 2 generated inorganic crystals with favorable properties better than purpose-built approaches; text pretraining appeared “indispensable.” Wilson’s stronger claim is that learning compressibility can reveal domain-specific symmetries, even letting trained vision transformers exhibit lower translation-equivariance error than convolutional networks affected by aliasing and edges.
Compute efficiency could improve by changing inductive bias and architecture, not only by adding FLOPs. Wilson’s structured-matrix work suggests compute-optimal regimes favor full-rank layers, fast multiplication, and many parameters per flop; block tensor trains widened layers within a fixed budget and meaningfully changed scaling exponents. Fine-grained mixture-of-experts routing may push beyond one parameter per flop, while knowledge distillation raises the strategic question of whether a 1-billion-parameter model could someday inherit the scale-induced bias of a 7-billion-parameter teacher.
GPT-5-era benchmark strength is not the destination: the missing capability is autonomous theory formation. Wilson wants systems that discover explanations at the level of general relativity or quantum mechanics, rather than serving only as black-box approximators inside application pipelines. The distinction is economically consequential: a model might correct gravitational time dilation, but a theory can unlock unforeseen applications—“Einstein wasn’t thinking about GPS when he proposed relativity.”
Deep dive
1. Deep learning is mysterious, but conventional complexity stories misidentify why
Wilson’s opening qualification matters: “not so mysterious or different” does not mean ordinary or fully understood. Its relative universality, representation learning, mode connectivity, and unusual loss landscapes remain distinctive; familiar generalization frameworks can nevertheless explain phenomena often treated as uniquely inexplicable.
His broader concern is intellectual lock-in. Research gets trapped in “local minima” when enough people accept claims such as stochastic optimization being the primary source of generalization, small data requiring smaller models, or model size being synonymous with complexity.
Wilson’s prescription is unusually stable across sample sizes: represent beliefs about the data-generating process honestly rather than changing the model merely because more observations arrived. A well-constructed model should behave sensibly with either a small or large data set, reducing the manual intervention machine learning is meant to replace.
His scientific ambition is work that remains useful “hundreds of years from now.” Methods may become obsolete within a month, but understanding why a method worked—supported by theory, empirical behavior, and low-level engineering details such as numerical stability—can survive the product cycle.
2. The 10,000-parameter airline model exposes a misleading intuition
Scarfe recalls Wilson showing students airline-passenger data with trend and seasonality, then offering a line, a roughly 10-parameter alternative, and a 10,000-parameter model. Almost everyone initially chose one of the smaller models; after the talk, the poll shifted toward the third.
The practical contradiction is already visible on CIFAR, where tens of millions of parameters fit tens of thousands of examples. Before deep learning, researchers similarly used Gaussian processes inspired by infinite neural-network limits; an RBF kernel is effectively saying, “I wanna use an infinite order polynomial.”
The decisive object is not parameter count but the induced distribution over functions. A model can represent many explanations while assigning overwhelming probability to a narrow family; implausible answers need not be forbidden when giving them “epsilon probability” preserves adaptability.
Wilson’s sharp distinction is between expressiveness and effective complexity. Hard constraints claim some solutions are impossible; soft preferences admit that the world may surprise us while still delivering strong data efficiency and Occam-like behavior.
3. Soft constraints can recover exact symmetries without denying messy reality
Scarfe’s geometric-deep-learning challenge starts from known symmetries: if reality’s generating process obeys them, why accept approximation error by omitting them? Wilson agrees in principle, but notes that exact knowledge is rare—even conservation can fail in the modeled subsystem when a pendulum experiences wind.
The alternative is a flexible model biased toward conservation, invariance, or equivariance. When obeying the constraint perfectly explains the observations and deviation carries any penalty, the model often “collapse[s] down onto that,” matching a hard-constrained model without declaring violations impossible.
Wilson’s residual pathway priors placed a covariance structure in a chosen equivariant subspace plus its orthogonal complement. Surprisingly, bias strength often mattered little: “a very gentle bias is sufficient,” although he preserves the hedge that calibration can matter in particular cases.
Duggar compares this with a Bayesian prior worth perhaps one or two data points—enough to prevent absurd answers, but readily overridden. Wilson accepts the analogy while stressing that assumption-free learning is impossible; the real choice is which assumptions to encode and at what abstraction level.
4. Double descent breaks the standard bias-variance story
Scarfe grounds the objection in prediction with expert advice: retaining historical experts gives each epsilon weight and harms every prediction, but deleting them creates a relearning cost when the regime returns. That experience makes Wilson’s promise of flexibility plus strong bias feel like “have your cake and eat it.”
Wilson calls the bias-variance trade-off “an incredible misnomer.” Naive high-order polynomials may exhibit low bias and high variance, but ensembles and large neural networks can achieve both low bias and low variance by combining flexibility with a simplicity preference.
Duggar’s strongest pushback is physical: a tiny unconstrained feedback term can add energy whenever a robot acts until the system spins out of control. Wilson concedes categorically that “overfitting is real,” but argues that shrinking the hypothesis space is often the wrong remedy.
Double descent supplies the evidence. Error improves, worsens near the interpolation threshold, then improves again; throughout the second descent, training loss is essentially zero. Larger models therefore cannot win because they fit more—they must be selecting better among equally fitting solutions, prompting Wilson’s blunt advice: “make your model always as big as possible,” paired with a compression bias.
5. Predictive accuracy is not understanding unless the representation travels
Asked whether predictive power equals understanding, Wilson focuses on representation: two systems may tie on one task yet diverge under new problems or distribution shifts. A useful representation describes more of the real world and avoids shortcuts that succeed only in a contrived benchmark.
No-free-lunch theorems are mathematically valid under their assumptions, but their uniform distribution over all possible problems is a poor model of reality; sampled uniformly, most data sets would be noise. “The real world is a small corner of all possible data sets.”
The actionable question becomes how much structure real problems share and at what abstraction it can be encoded. Wilson’s contention is that both natural data and increasingly large models favor low Kolmogorov complexity, creating scope for systems more general than modality-specific predecessors without implying universal intelligence.
6. Deep ensembles were mislabeled as non-Bayesian when they approximated Bayes better
Wilson’s example of confronting a field-wide misconception concerns papers comparing approximate Bayesian procedures with deep ensembles as the “non-Bayesian alternative.” When ensembles won, the prevailing interpretation was to become less Bayesian.
His reversal: for a fixed computational budget, deep ensembles often approximated the posterior Bayesian predictive distribution—and therefore marginalization—better than methods carrying Bayesian labels. The correct direction was “we actually need to be more Bayesian.”
Because hundreds of papers repeated the taxonomy, subtle subsections failed to change minds. Wilson made the argument the center of a blog post; he says he stopped seeing papers preserve the old separation afterward, although the intervention “did sort of strike a nerve.”
7. Infinite models can be expressive, simple, and exceptionally data-efficient
Radford Neal’s aspiration was to build models “the size of a house.” Taking a neural network to infinitely many hidden units yields a Gaussian process under a central-limit argument, giving a model more expressive than any finite network in memory but with strong functional preferences.
Gaussian processes make the distinction operational in Bayesian optimization. They serve as uncertain surrogates for expensive black-box objectives—neural-network generalization versus hyperparameters, or costly physical simulations—so the learner can explore while minimizing the number of queries.
Neal’s provocative claim was that around any successful simple model, one can build a richer one that performs better by accommodating irregular handwriting, ink blots, or overlooked structure. Duggar objects that richer inference can plainly overfit simulated ground truth; Wilson’s resolution is that “more expressive” need not mean effectively more complex when the richer model retains a strong simplicity bias.
8. Solomonoff-style reasoning makes benign overfitting less mysterious
Neural networks often learn structure before noise, then eventually fit both. Benign overfitting names the case where memorizing that noise barely damages generalization—behavior that VC dimension and Rademacher complexity struggle to explain because they largely penalize the capacity to fit noise.
PAC-Bayes and countable-hypothesis bounds instead track soft preferences. Wilson’s group used a Solomonoff prior that can represent every computer program but assigns exponentially greater probability to low-Kolmogorov-complexity hypotheses—the shortest programs capable of generating the solution.
The resulting bounds were fairly tight and improved as models grew, suggesting that large transformers approximate a maximally expressive learner with a stronger compression bias. The progression from feature engineering, to CNNs and RNNs, to “transformers for almost everything” is therefore framed as movement toward relative universality, not proof of general intelligence.
One striking result sharpened the argument: CNNs designed around image locality and translation equivariance provably showed inductive bias on tabular data merely shaped as images. Wilson says shared preference for low Kolmogorov complexity is the only possible explanation, and that preference strengthened with model size.
9. High-dimensional geometry may explain why scale favors flat solutions
Wilson calls the mechanism behind scale-induced simplicity “largely an open question.” His geometric intuition compares flat low-loss regions of radii RA and RB: as parameter dimension D grows, volume scales like RA^D versus RB^D, so the broader region can exponentially dominate.
This weakens the standard story that SGD uniquely rescues non-convex deep learning. Full-batch gradient descent can generalize comparably, while a “guess and check” procedure that randomly samples parameters and stops below a loss threshold can also approach SGD or Adam.
The dartboard interpretation is that, at sufficient dimension, low-loss regions with good generalization occupy far more volume than low-loss regions with bad generalization. In double descent’s second leg, the effective dimensionality—the number of relatively large Hessian eigenvalues or sharp directions—decreases as models grow.
Wilson repeatedly marks the limits: parameter additions need not alter useful geometry, and the argument is not airtight. Directly regularizing Kolmogorov complexity is also hard; compressed model-file size gives only an upper bound for one trained hypothesis, while Solomonoff induction requires reasoning across an entire hypothesis set.
10. Incompressibility must be split into valuable structure and useless randomness
Kolmogorov complexity treats random noise and intricate structure as incompressible, even though their learning value differs radically. Uniform random samples may require memorization yet teach no reusable representation; a sophisticated cellular automaton may contain difficult but valuable generative rules.
Wilson invokes Scott Aaronson’s coffee-and-cream analogy. The separated liquids begin with low entropy and low sophistication; stirring produces intermediate organization, then a high-entropy mixture whose intuitive sophistication falls again—a non-monotonic quantity that plain complexity misses.
Work such as “Intelligence at the Edge of Chaos” suggests models may benefit from structurally complex training data even while retaining an Occam bias. Wilson’s group is exploring information measures that compartmentalize structural and random complexity, potentially yielding priors more targeted than Solomonoff’s.
11. Compression remains the best default even when shortcuts break out of distribution
Scarfe’s pushback is that compression is only one component of intelligence: ARC-like tasks permit many simple candidate programs, while modern models often settle into superficial shortcut learning. An upper generalization bound says performance is no worse than a limit, not that the representation reaches genuine understanding.
Wilson agrees compression can fail under shift. If a blue pixel always accompanies the bird label, using that feature is rational on the training distribution; place the bird before a volcano or inside a room, and the shortcut collapses.
His guarded conclusion is that Occam’s razor remains the best induction principle “in absence of additional information.” More data may help in some settings and not others; robust induction under realistic shifts remains unresolved.
12. Rich training distributions can turn interpolation into algorithm learning
Wilson’s team trained transformers on matrices encoded as number sequences, targeting eigenvalues, spectra, or linear-system solutions. Models trained on Gaussian random matrices performed well in distribution but failed “extraordinarily poorly” on something as simple as an identity matrix, despite its support under the Gaussian distribution.
They tried looping, adaptive test-time computation, and a much richer “einsum” space spanning Toeplitz, Kronecker, block-diagonal, low-rank, and other matrices. After that enrichment, the model generalized beyond even the expanded training set, appearing to learn an algorithm rather than statistical interpolation.
Duggar rejects “just give it a calculator” as the end state: tools must still be programmed by people, while machine learning’s aim is machines that program themselves. If a system cannot generalize multiplication from decimal to binary or from nine digits to 36, its prospects for discovering frontier mechanics look weak.
Wilson nonetheless draws a hard boundary around data scaling. Autonomous-driving outliers differ each time, so adding yesterday’s rare cases may not cover tomorrow’s; his intuition is that “more data alone is not really the answer” to robust driving.
13. Text pretraining transfers principles of induction into alien domains
Wilson’s team naively supplied a text-pretrained LLM with number sequences as strings and asked for subsequent tokens. It worked “way, way better than we thought it could possibly work,” becoming the project “Large Language Models or Zero-Shot Time Series Forecasters” after improvements to tokenization and uncertainty representation.
A parallel materials project fine-tuned Llama 2 on textual atom locations and energies. It generated inorganic crystals with favorable properties better than purpose-built and domain-trained foundation approaches; skeptical chemist collaborators accepted that text pretraining had been “an indispensable component.”
Wilson’s explanation is not that all domains share identical low-level features. Rotation invariance in molecules and translation invariance in character recognition are different, but both compress their respective problems; broad pretraining can instill a higher-level preference for discovering such compressible structure.
The strongest specimen is translation equivariance: trained vision transformers, despite lacking the explicit convolutional constraint, sometimes achieved lower equivariance error than CNNs affected by aliasing and edge effects. A soft scale-induced bias discovered the symmetry more accurately than the ostensibly exact design.
14. Marginalization is the honest—and increasingly necessary—response to scale
Wilson defines being Bayesian less by priors than by acknowledging uncertainty over which solution is correct. Two coin flips cannot determine a coin’s bias, and finitely many regression points admit a straight line plus many perfectly interpolating curves; selecting one with near-total certainty misstates what is known.
Greater expressiveness makes that mistake worse because more parameter settings remain consistent with the observations. Bayesian marginalization instead averages all represented solutions according to posterior probability: “Probability theory says, ‘No, that’s just wrong’” about betting everything on one.
Approximation is a ladder, not a binary choice. Standard training resembles a posterior point mass; a Gaussian captures more complementary explanations; MCMC can traverse richer multimodal structure. Wilson reports corresponding performance improvements rather than demanding exact, computationally impossible Bayes.
Practical successes include SWAG, which adds little training cost but more test-time expense, and deep kernel learning, which provides epistemic-uncertainty representation after a forward pass. Yet billion-parameter LLMs make posterior approximation harder just as their expressiveness makes it more important.
15. Bayesian Occam’s razor is powerful, but marginal likelihood answers a narrower question
Nocedal’s flat-versus-sharp-minima talk pushed Wilson into Bayesian deep learning. Marginalization naturally assigns more integrated probability to broad regions, avoiding a bespoke flatness penalty whose Hessian metric, parametrization invariance, and coefficient would all invite disagreement: “this is all happening under the hood.”
David MacKay’s block-behind-a-tree example makes the mechanism visual. One block and 10 aligned blocks may explain the same view, but the 10-block model spreads normalized probability across many more possible data sets; the one-block explanation gives the observed data more mass even before adding an explicit simplicity prior.
Wilson’s caveat from “Bayesian Model Selection, the Marginal Likelihood, and Generalization” is load-bearing: marginal likelihood asks how likely the prior was to generate the training data, not how likely the posterior is to predict well. Uniform priors over readily identifiable solutions can score poorly despite contracting to useful posteriors, while pathological point-mass models can overfit.
Scientific hypothesis testing remains a compelling use. General relativity made sharp predictions for Mercury’s perihelion and reportedly achieved orders-of-magnitude greater marginal likelihood than flexible modifications to Newtonian gravity whose probability mass had to cover many alternative observations.
16. Better scaling will require architecture, dynamics, and ultimately theory discovery
Training dynamics may expose a compute-limited form of complexity. Marginal likelihood can be decomposed sequentially as predictions for each new datum given earlier data, connecting coding length to the learning curve; grokking may similarly reflect a late reorganization into a more compressible representation without materially lower training loss.
Stochastic weight averaging operationalizes that intuition by raising the learning rate, traversing the edge of a flat region, and averaging parameters into its interior. Wilson links it cautiously to grokking: the destination has similar loss but better generalization and compressibility.
Mode connectivity makes the landscape stranger. Separately trained solutions can be joined by a polygonal chain or quadratic Bézier curve without increasing training loss; points along that curve disagree on test data and can be ensembled. Larger models require less bending, suggesting seemingly separate modes inhabit broad connected manifolds.
For practitioners, Wilson still recommends expressive models, scale where affordable, stochastic weight averaging, regularization, and Bayesian marginalization. The larger ambition is to replace brute scale’s “inelegant” compression effect: perhaps within 15 or 20 years, infinitely expressive nonparametric models will carry an explicit, interpretable simplicity bias.
Distillation exposes the opportunity. If a 1-billion-parameter student can reproduce a 7-billion-parameter teacher, the smaller model possessed sufficient representational capacity but could not find the right parameters directly; the missing asset may be the teacher’s scale-induced search bias.
Wilson therefore calls the bitter lesson “widely misunderstood and incomplete.” Computation outlasts handcrafted features, but learning still requires assumptions; better structured layers can alter scaling exponents rather than merely move along an existing curve.
His block tensor train is full rank, avoids parameter sharing, multiplies faster than a dense matrix, and permits wider layers per budget. Fine-grained expert routing across individual linear and attention projections further improved efficiency; under effectively unlimited data, the governing principle became as many parameters per flop as possible.
The final gap is not another benchmark win. With GPT-5 as the “elephant in the room,” Scarfe asks what feels missing; Wilson points to systems that originate scientific theories comparable to relativity or quantum mechanics—insights that reveal structure and spawn unanticipated applications, rather than black boxes that merely approximate a correction already specified by humans.