Colin Sandon

Emmanuel Abbe, Samy Bengio, Aryo Lotfi et al.

Can Transformers predict new syllogisms by composing established ones? More generally, what type of targets can be learned by such models from scratch? Recent works show that Transformers can be Turing-complete in terms of expressivity, but this does not address the learnability objective. This paper puts forward the notion of 'globality degree' of a target distribution to capture when weak learning is efficiently achievable by regular Transformers. This measure shows a contrast with the expressivity results of Transformers captured by $TC^0/TC^1$ classes (further studied here), since the globality relates to correlations with the more limited $NC^0$ class. We show here experimentally and theoretically under additional assumptions that distributions with high globality cannot be learned efficiently. In particular, syllogisms cannot be composed on long chains. Further, we develop scratchpad techniques and show that: (i) agnostic scratchpads cannot break the globality barrier, (ii) educated scratchpads can break the globality with intermediate steps, although not all such scratchpads can generalize out-of-distribution (OOD), (iii) a notion of 'inductive scratchpad', that composes the prior information more efficiently, can both break the globality barrier and improve the OOD generalization. In particular, some of our inductive scratchpads can achieve length generalizations of up to $6\times$ for some arithmetic tasks depending on the input formatting.

Emmanuel Abbe, Pritish Kamath, Eran Malach et al.

We study the power of learning via mini-batch stochastic gradient descent (SGD) on the population loss, and batch Gradient Descent (GD) on the empirical loss, of a differentiable model or neural network, and ask what learning problems can be learnt using these paradigms. We show that SGD and GD can always simulate learning with statistical queries (SQ), but their ability to go beyond that depends on the precision $ρ$ of the gradient calculations relative to the minibatch size $b$ (for SGD) and sample size $m$ (for GD). With fine enough precision relative to minibatch size, namely when $b ρ$ is small enough, SGD can go beyond SQ learning and simulate any sample-based learning algorithm and thus its learning power is equivalent to that of PAC learning; this extends prior work that achieved this result for $b=1$. Similarly, with fine enough precision relative to the sample size $m$, GD can also simulate any sample-based learning algorithm based on $m$ samples. In particular, with polynomially many bits of precision (i.e. when $ρ$ is exponentially small), SGD and GD can both simulate PAC learning regardless of the mini-batch size. On the other hand, when $b ρ^2$ is large enough, the power of SGD is equivalent to that of SQ learning.

Ankur Moitra, Elchanan Mossel, Colin Sandon

In this work, we study the computational complexity of determining whether a machine learning model that perfectly fits the training data will generalizes to unseen data. In particular, we study the power of a malicious agent whose goal is to construct a model g that fits its training data and nothing else, but is indistinguishable from an accurate model f. We say that g strongly spoofs f if no polynomial-time algorithm can tell them apart. If instead we restrict to algorithms that run in $n^c$ time for some fixed $c$, we say that g c-weakly spoofs f. Our main results are 1. Under cryptographic assumptions, strong spoofing is possible and 2. For any c> 0, c-weak spoofing is possible unconditionally While the assumption of a malicious agent is an extreme scenario (hopefully companies training large models are not malicious), we believe that it sheds light on the inherent difficulties of blindly trusting large proprietary models or data.

Ankur Moitra, Elchanan Mossel, Colin Sandon

We study learning Censor Markov Random Fields (abbreviated CMRFs). These are Markov Random Fields where some of the nodes are censored (not observed). We present an algorithm for learning high-temperature CMRFs within o(n) transportation distance. Crucially our algorithm makes no assumption about the structure of the graph or the number or location of the observed nodes. We obtain stronger results for high girth high-temperature CMRFs as well as computational lower bounds indicating that our results can not be qualitatively improved.

Emmanuel Abbe, Colin Sandon

The goal of this paper is to characterize function distributions that deep learning can or cannot learn in poly-time. A universality result is proved for SGD-based deep learning and a non-universality result is proved for GD-based deep learning; this also gives a separation between SGD-based deep learning and statistical query algorithms: (1) {\it Deep learning with SGD is efficiently universal.} Any function distribution that can be learned from samples in poly-time can also be learned by a poly-size neural net trained with SGD on a poly-time initialization with poly-steps, poly-rate and possibly poly-noise. Therefore deep learning provides a universal learning paradigm: it was known that the approximation and estimation errors could be controlled with poly-size neural nets, using ERM that is NP-hard; this new result shows that the optimization error can also be controlled with SGD in poly-time. The picture changes for GD with large enough batches: (2) {\it Result (1) does not hold for GD:} Neural nets of poly-size trained with GD (full gradients or large enough batches) on any initialization with poly-steps, poly-range and at least poly-noise cannot learn any function distribution that has super-polynomial {\it cross-predictability,} where the cross-predictability gives a measure of ``average'' function correlation -- relations and distinctions to the statistical dimension are discussed. In particular, GD with these constraints can learn efficiently monomials of degree $k$ if and only if $k$ is constant. Thus (1) and (2) point to an interesting contrast: SGD is universal even with some poly-noise while full GD or SQ algorithms are not (e.g., parities).

Emmanuel Abbe, Colin Sandon

As the success of deep learning reaches more grounds, one would like to also envision the potential limits of deep learning. This paper gives a first set of results proving that certain deep learning algorithms fail at learning certain efficiently learnable functions. The results put forward a notion of cross-predictability that characterizes when such failures take place. Parity functions provide an extreme example with a cross-predictability that decays exponentially, while a mere super-polynomial decay of the cross-predictability is shown to be sufficient to obtain failures. Examples in community detection and arithmetic learning are also discussed. Recall that it is known that the class of neural networks (NNs) with polynomial network size can express any function that can be implemented in polynomial time, and that their sample complexity scales polynomially with the network size. The challenge is with the optimization error (the ERM is NP-hard), and the success behind deep learning is to train deep NNs with descent algorithms. The failures shown in this paper apply to training poly-size NNs on function distributions of low cross-predictability with a descent algorithm that is either run with limited memory per sample or that is initialized and run with enough randomness. We further claim that such types of constraints are necessary to obtain failures, in that exact SGD with careful non-random initialization can be shown to learn parities. The cross-predictability in our results plays a similar role the statistical dimension in statistical query (SQ) algorithms, with distinctions explained in the paper. The proof techniques are based on exhibiting algorithmic constraints that imply a statistical indistinguishability between the algorithm's output on the test model v.s.\ a null model, using information measures to bound the total variation distance.

Emmanuel Abbe, Colin Sandon

In a paper that initiated the modern study of the stochastic block model, Decelle et al., backed by Mossel et al., made the following conjecture: Denote by $k$ the number of balanced communities, $a/n$ the probability of connecting inside communities and $b/n$ across, and set $\mathrm{SNR}=(a-b)^2/(k(a+(k-1)b)$; for any $k \geq 2$, it is possible to detect communities efficiently whenever $\mathrm{SNR}>1$ (the KS threshold), whereas for $k\geq 4$, it is possible to detect communities information-theoretically for some $\mathrm{SNR}<1$. Massoulié, Mossel et al.\ and Bordenave et al.\ succeeded in proving that the KS threshold is efficiently achievable for $k=2$, while Mossel et al.\ proved that it cannot be crossed information-theoretically for $k=2$. The above conjecture remained open for $k \geq 3$. This paper proves this conjecture, further extending the efficient detection to non-symmetrical SBMs with a generalized notion of detection and KS threshold. For the efficient part, a linearized acyclic belief propagation (ABP) algorithm is developed and proved to detect communities for any $k$ down to the KS threshold in time $O(n \log n)$. Achieving this requires showing optimality of ABP in the presence of cycles, a challenge for message passing algorithms. The paper further connects ABP to a power iteration method with a nonbacktracking operator of generalized order, formalizing the interplay between message passing and spectral methods. For the information-theoretic (IT) part, a non-efficient algorithm sampling a typical clustering is shown to break down the KS threshold at $k=4$. The emerging gap is shown to be large in some cases; if $a=0$, the KS threshold reads $b \gtrsim k^2$ whereas the IT bound reads $b \gtrsim k \ln(k)$, making the SBM a good study-case for information-computation gaps.

Emmanuel Abbe, Colin Sandon

Most recent developments on the stochastic block model (SBM) rely on the knowledge of the model parameters, or at least on the number of communities. This paper introduces efficient algorithms that do not require such knowledge and yet achieve the optimal information-theoretic tradeoffs identified in [AS15] for linear size communities. The results are three-fold: (i) in the constant degree regime, an algorithm is developed that requires only a lower-bound on the relative sizes of the communities and detects communities with an optimal accuracy scaling for large degrees; (ii) in the regime where degrees are scaled by $ω(1)$ (diverging degrees), this is enhanced into a fully agnostic algorithm that only takes the graph in question and simultaneously learns the model parameters (including the number of communities) and detects communities with accuracy $1-o(1)$, with an overall quasi-linear complexity; (iii) in the logarithmic degree regime, an agnostic algorithm is developed that learns the parameters and achieves the optimal CH-limit for exact recovery, in quasi-linear time. These provide the first algorithms affording efficiency, universality and information-theoretic optimality for strong and weak consistency in the general SBM with linear size communities.